From d6cc5754d9a98d37c700cf62571f823b51a16812 Mon Sep 17 00:00:00 2001 From: Andrew Van Date: Mon, 27 Jul 2026 22:11:44 -0500 Subject: [PATCH 01/13] :recycle: Replace phase-branch heuristics with an intercept test The global 2*pi branch for the MCPC-3D-S phase offset was picked by a cascade of five hardcoded field-magnitude thresholds, with a --wrap-limit escape hatch for when the cascade made things worse. Replace it with a criterion that has no tuned constants. A branch error of M wraps shifts the phase offset by wrap(2*pi*k*M) with k = TE0/dTE, and that lands as the same additive constant on every echo. Fitting a line through the two echoes and reading its intercept measures that constant directly: the correct branch extrapolates through the origin, a wrong one does not. The rejection scale is analytic -- |wrap(2*pi*k)| -- so nothing is fit to data. - Add _evaluate_branch / _branch_intercept_step / _select_branch; drop _select_branch_cascade, FMAP_PROPORTION_HEURISTIC, FMAP_AMBIGUIOUS_HEURISTIC, BRANCH_SELECTION_MODES and BRANCH_CONSISTENCY_RATIO. - Act only when exactly one candidate is consistent. On a tie the branches are a genuine alias that no phase statistic can resolve, so defer to ROMEO's correct_global rather than guess: a measured centre-frequency sweep put a smallest-|field| tiebreaker at 12% better and 12% worse, including cases where correct_global had been right. - Return 0 when TE0/dTE is an integer, where the branch cannot move the offset at all. - Drop wrap_limit from medic(), unwrap_and_compute_field_maps() and the wk-medic / wk-unwrap-phase CLIs. BREAKING: --wrap-limit is gone; it existed to disable the heuristics this commit removes. - Correct two annotations the new tests exposed: echo times are numpy scalars at every call site, not arrays, and rescale_phase's min/max are floats, not ints. - Add 13 tests covering the analytic step, the decision rule, and end-to-end recovery of an injected wrap on the bundled data. Validated on 14090 frames across 58 runs from eight OpenNeuro datasets spanning six TE0/dTE ratios (0.51-0.78). No frame selected a non-zero branch and the largest branch-0 intercept was 1.9e-07 of the cutoff, so the output is unchanged on all data tested. Fitting and non-fitting branches separate by six to nine orders of magnitude when the rule does fire, verified by injecting known wraps. --- tests/test_unwrap.py | 182 +++++++++++++++++++++- warpkit/distortion.py | 2 - warpkit/scripts/medic.py | 9 -- warpkit/scripts/unwrap_phase.py | 8 - warpkit/unwrap.py | 261 ++++++++++++++++++++------------ warpkit/utilities.py | 2 +- 6 files changed, 349 insertions(+), 115 deletions(-) diff --git a/tests/test_unwrap.py b/tests/test_unwrap.py index 94deaf7..cc93b85 100644 --- a/tests/test_unwrap.py +++ b/tests/test_unwrap.py @@ -2,7 +2,14 @@ import numpy as np import pytest from numpy.testing import assert_allclose -from warpkit.unwrap import compute_field_maps, compute_offset, reject_outliers +from warpkit import unwrap as unwrap_mod +from warpkit.unwrap import ( + _branch_intercept_step, + _select_branch, + compute_field_maps, + compute_offset, + reject_outliers, +) # --------------------------------------------------------------------------- # reject_outliers: median + MAD, threshold m=2.0 @@ -140,6 +147,179 @@ def test_compute_field_maps_rejects_mismatched_spatial_shape(): compute_field_maps(unwrapped, bad_masks, mag, tes) +# --------------------------------------------------------------------------- +# Global 2*pi branch selection (replaces the field-magnitude heuristic cascade) +# --------------------------------------------------------------------------- + + +def _patch_scores(monkeypatch, scores): + """Stub _evaluate_branch so only the decision rule is under test. + + ``scores`` are intercepts in radians. The selector consults nothing else -- + no field prior, no tiebreak -- so the remaining tuple slots are unused. + """ + + def fake(n_wraps, *args, **kwargs): + return scores[n_wraps], None, None, None + + monkeypatch.setattr(unwrap_mod, "_evaluate_branch", fake) + + +def _sel(te0: float = 12.0, te1: float = 28.97): + """Call the selector with the bundled protocol's TEs by default. + + ``_evaluate_branch`` is stubbed out by ``_patch_scores`` in every test that + uses this, so the array arguments are never read -- they just need to be the + right type. Defaults give the bundled protocol, k = 0.7071. + """ + vol = np.zeros((2, 2, 2), dtype=np.float32) + flags = np.ones((2, 2, 2), dtype=bool) + return _select_branch( + vol, # unwrapped_diff + vol, # phase0 + vol, # phase1 + np.ones((2, 2, 2), dtype=np.float32), # mag0 + vol, # mag1 + np.float32(te0), + np.float32(te1), + flags, # mask + flags, # score_mask + ) + + +# --------------------------------------------------------------------------- +# _branch_intercept_step: how far apart candidate branches sit, from TEs alone +# --------------------------------------------------------------------------- + + +def test_branch_intercept_step_matches_measured_offset(): + """The bundled protocol's wrong branches measure an intercept of 1.8402 rad. + The analytic step must reproduce that without touching any data.""" + assert _branch_intercept_step(np.float32(12.0), np.float32(28.97)) == pytest.approx( + 1.8402, rel=1e-4 + ) + + +def test_branch_intercept_step_matches_ds007637(): + assert _branch_intercept_step( + np.float32(14.20), np.float32(38.93) + ) == pytest.approx(2.6754, rel=1e-4) + + +@pytest.mark.parametrize(("te0", "te1"), [(10.0, 20.0), (15.0, 20.0), (20.0, 30.0)]) +def test_branch_intercept_step_zero_for_integer_ratio(te0, te1): + """te0/dTE integer -> the branch cannot move the offset at all.""" + assert _branch_intercept_step(np.float32(te0), np.float32(te1)) == pytest.approx( + 0.0 + ) + + +def test_branch_intercept_step_rejects_nonincreasing_tes(): + assert _branch_intercept_step(np.float32(20.0), np.float32(20.0)) == 0.0 + + +def test_branch_selector_noop_for_integer_te_ratio(monkeypatch): + """With an integer te0/dTE the selector must not act, even on a score + spread that would otherwise look decisive -- the branch is a no-op there, + so any apparent difference is numerical noise.""" + _patch_scores(monkeypatch, {-1: 1.84, 0: 1.84, 1: 1.0e-7}) + best, _ = _sel(te0=10.0, te1=20.0) + assert best == 0 + + +def test_branch_selector_corrects_inconsistent_zero(monkeypatch): + """N=0 carries an intercept and exactly one alternative does not: move.""" + _patch_scores(monkeypatch, {-1: 1.8402, 0: 1.8402, 1: 1.0e-7}) + best, _ = _sel() + assert best == 1 + + +def test_branch_selector_defers_when_two_branches_fit(monkeypatch): + """The ds007637 case: N=0 and N=-1 are both through-origin. That is a + genuine alias -- both explain the phase exactly -- so the selector must not + choose between them. Deferring leaves correct_global's answer in place. + + A bare argmin on the intercepts would flip between them on numerical noise + and inject a full wrap of field into the time series. + """ + _patch_scores(monkeypatch, {-1: 1.2e-03, 0: 1.9e-03, 1: 0.9324}) + best, _ = _sel() + assert best == 0 + + +def test_branch_selector_defers_even_when_zero_looks_wrong(monkeypatch): + """N=0 carries an intercept but *two* alternatives fit. Neither is + preferable on the evidence, so change nothing rather than pick one. + + This is the case that used to consult the field prior. Measurement showed + the prior was right about as often as it was wrong, so it was removed. + """ + _patch_scores(monkeypatch, {-1: 1.0e-7, 0: 1.8402, 1: 2.0e-7}) + best, _ = _sel() + assert best == 0 + + +def test_branch_selector_noop_when_all_candidates_tie(monkeypatch): + """Integer TE0/dTE makes the branch a no-op; scores are all ~equal.""" + _patch_scores(monkeypatch, {-1: 1.1e-8, 0: 2.0e-8, 1: 1.9e-8}) + best, _ = _sel() + assert best == 0 + + +def test_branch_selector_noop_when_nothing_fits(monkeypatch): + """Degenerate/failed fit: every candidate is bad, so change nothing.""" + _patch_scores(monkeypatch, {-1: 0.0, 0: 0.0, 1: 0.0}) + best, _ = _sel() + assert best == 0 + + _patch_scores(monkeypatch, {-1: 1.71, 0: 1.80, 1: 1.74}) + best, _ = _sel() + assert best == 0 + + +def test_branch_selector_recovers_injected_wrap(test_data): + """End-to-end: shift the unwrapped difference by a known number of wraps and + confirm the selector undoes it, landing on the same phase offset.""" + from warpkit.utilities import create_brain_mask, rescale_phase + from warpkit.warpkit_cpp import romeo_unwrap3d + + phase, mag, tes = test_data["phase"], test_data["mag"], test_data["tes"] + tes = np.asarray(tes, dtype=np.float32) + raw = np.stack([p.dataobj[..., 0] for p in phase], axis=-1) + mn = min(float(np.asarray(p.dataobj[..., 0]).min()) for p in phase) + mx = max(float(np.asarray(p.dataobj[..., 0]).max()) for p in phase) + ph = rescale_phase(raw, min=mn, max=mx).astype(np.float32) + mg = np.stack([m.dataobj[..., 0] for m in mag], axis=-1).astype(np.float32) + + mag0, mag1 = mg[..., 0], mg[..., 1] + phase0, phase1 = ph[..., 0], ph[..., 1] + mask = create_brain_mask(mag0, 3) + score_mask = create_brain_mask(mag0, -2) + + signal_diff = mag0 * mag1 * np.exp(1j * (phase1 - phase0)) + unwrapped_diff = romeo_unwrap3d( + phase=np.angle(signal_diff).astype(np.float32), + weights="romeo", + mag=np.abs(signal_diff).astype(np.float32), + mask=mask, + correct_global=True, + ) + + for injected in (-1, 0, 1): + best, scores = _select_branch( + unwrapped_diff + 2 * np.pi * injected, + phase0, + phase1, + mag0, + mag1, + tes[0], + tes[1], + mask, + score_mask, + ) + assert best == -injected, f"injected {injected}, got {best} (scores={scores})" + + def test_romeo_unwrap3d_rejects_unknown_weight_preset(): """`weights` is a preset name string; only "romeo" is supported.""" from warpkit.warpkit_cpp import romeo_unwrap3d diff --git a/warpkit/distortion.py b/warpkit/distortion.py index 7296f22..a0f6797 100644 --- a/warpkit/distortion.py +++ b/warpkit/distortion.py @@ -19,7 +19,6 @@ def medic( svd_filt: int = 10, n_cpus: int = 4, debug: bool = False, - wrap_limit: bool = False, ) -> tuple[nib.Nifti1Image, nib.Nifti1Image, nib.Nifti1Image]: """This runs Multi-Echo DIstortion Correction (MEDIC) on a set of phase and magnitude images. @@ -101,7 +100,6 @@ def medic( frames=frames, n_cpus=n_cpus, debug=debug, - wrap_limit=wrap_limit, ) except IndexError as e: raise IndexError( diff --git a/warpkit/scripts/medic.py b/warpkit/scripts/medic.py index 84ab22d..6080d11 100644 --- a/warpkit/scripts/medic.py +++ b/warpkit/scripts/medic.py @@ -42,7 +42,6 @@ def medic( metadata: Sequence[PathLike[str] | str] | None = None, noise_frames: int = 0, n_cpus: int = 4, - wrap_limit: bool = False, debug: bool = False, ) -> MedicResult: """Run the full MEDIC pipeline and write the three output NIfTIs. @@ -99,7 +98,6 @@ def medic( border_filt=(1000, 1000), svd_filt=1000, debug=True, - wrap_limit=wrap_limit, ) else: fmaps_native, dmaps, fmaps = _medic_distortion( @@ -111,7 +109,6 @@ def medic( n_cpus=n_cpus, svd_filt=10, border_size=5, - wrap_limit=wrap_limit, ) return write_medic_outputs(out_prefix, fmaps_native, dmaps, fmaps) @@ -155,11 +152,6 @@ def main(): ) add_n_cpus_arg(parser) parser.add_argument("--debug", action="store_true", help="Debug mode") - parser.add_argument( - "--wrap-limit", - action="store_true", - help="Turns off some heuristics for phase unwrapping", - ) args = parser.parse_args() setup_logging() @@ -176,7 +168,6 @@ def main(): metadata=args.metadata, noise_frames=args.noiseframes, n_cpus=args.n_cpus, - wrap_limit=args.wrap_limit, debug=args.debug, ) except ValueError as e: diff --git a/warpkit/scripts/unwrap_phase.py b/warpkit/scripts/unwrap_phase.py index 2664a63..673a892 100644 --- a/warpkit/scripts/unwrap_phase.py +++ b/warpkit/scripts/unwrap_phase.py @@ -42,7 +42,6 @@ def unwrap_phase( metadata: Sequence[PathLike[str] | str] | None = None, noise_frames: int = 0, n_cpus: int = 4, - wrap_limit: bool = False, debug: bool = False, ) -> UnwrapPhaseResult: """Run ROMEO multi-echo phase unwrapping. @@ -87,7 +86,6 @@ def unwrap_phase( list(tes_ms), n_cpus=n_cpus, debug=debug, - wrap_limit=wrap_limit, ) out_prefix_str = str(out_prefix) @@ -148,11 +146,6 @@ def main(): action="store_true", help="Skip the temporal consistency pass and dump intermediate files.", ) - parser.add_argument( - "--wrap-limit", - action="store_true", - help="Turn off some heuristics for phase unwrapping.", - ) args = parser.parse_args() setup_logging() @@ -167,7 +160,6 @@ def main(): metadata=args.metadata, noise_frames=args.noiseframes, n_cpus=args.n_cpus, - wrap_limit=args.wrap_limit, debug=args.debug, ) except ValueError as e: diff --git a/warpkit/unwrap.py b/warpkit/unwrap.py index c49f12d..b60f9f3 100644 --- a/warpkit/unwrap.py +++ b/warpkit/unwrap.py @@ -25,8 +25,11 @@ ) from .warpkit_cpp import romeo_unwrap3d, romeo_unwrap4d, romeo_voxelquality -FMAP_PROPORTION_HEURISTIC = 0.25 -FMAP_AMBIGUIOUS_HEURISTIC = 0.5 +# Candidate global 2*pi branches scanned by the intercept selector. One wrap is +# 1/dTE Hz of global field (~59 Hz for a 17 ms echo spacing), and ROMEO's +# ``correct_global`` already pins |median field| below 1/(2*dTE), so +/-1 covers +# everything reachable in practice. +BRANCH_CANDIDATES = (-1, 0, 1) def reject_outliers(data, m=2.0): @@ -76,15 +79,166 @@ def get_dual_echo_fieldmap(phases, tes, mags, mask): return fieldmap, unwrapped_phases +def _evaluate_branch( + n_wraps: int, + unwrapped_diff: npt.NDArray[np.float32], + phase0: npt.NDArray[np.float32], + phase1: npt.NDArray[np.float32], + mag0: npt.NDArray[np.float32], + mag1: npt.NDArray[np.float32], + te0: np.float32 | float, + te1: np.float32 | float, + mask: npt.NDArray[np.bool_], + score_mask: npt.NDArray[np.bool_], +): + """Reconstruct one candidate global 2*pi branch and score how well it fits. + + Shifting the unwrapped phase difference by ``2*pi*n_wraps`` moves the + MCPC-3D-S phase offset by ``wrap(2*pi*n_wraps*te0/dTE)``. That lands as the + same additive constant on every echo, so the echoes no longer extrapolate + back through zero at TE=0. Fitting a line through the two echoes and + reading off its intercept measures that constant directly: the right branch + gives ~0, a wrong branch gives roughly one ``_branch_intercept_step``. + + The intercept is taken as a median over ``score_mask`` because the error is + a single global constant, not a per-voxel effect. + + Returns + ------- + intercept : float + |median intercept| in radians over ``score_mask``. Zero means the + echoes are proportional to TE, which is what a correct offset gives. + offset : npt.NDArray[np.float32] + Phase offset implied by this branch. + fieldmap : npt.NDArray[np.float32] + Dual-echo field map in Hz implied by this branch. + unwrapped_phases : npt.NDArray[np.float32] + Dual-echo unwrapped phases implied by this branch. + """ + shifted = unwrapped_diff + 2 * np.pi * n_wraps + offset = np.angle(np.exp(1j * (phase0 - ((te0 * shifted) / (te1 - te0))))) + proposed_phases = ( + np.stack([phase0, phase1], axis=-1) - offset[..., np.newaxis] + ).astype(np.float32) + fieldmap, unwrapped_phases = get_dual_echo_fieldmap( + proposed_phases, + np.array([te0, te1], dtype=np.float32), + np.stack([mag0, mag1], axis=-1).astype(np.float32), + mask, + ) + y0 = unwrapped_phases[score_mask, 0].astype(np.float64) + y1 = unwrapped_phases[score_mask, 1].astype(np.float64) + if y0.size == 0: + return float("inf"), offset, fieldmap, unwrapped_phases + slope = (y1 - y0) / (float(te1) - float(te0)) + intercept = float(abs(np.median(y0 - slope * float(te0)))) + return intercept, offset, fieldmap, unwrapped_phases + + +def _branch_intercept_step(te0: np.float32 | float, te1: np.float32 | float) -> float: + """Intercept, in radians, that one wrap of branch error introduces. + + Candidate branches are spaced exactly this far apart in intercept, so it is + the natural scale for "this branch does not fit". It depends only on the + echo times -- no data required. + + Returns 0.0 when te0/dTE is an integer: there the branch does not move the + phase offset at all, so the candidates are indistinguishable (and the choice + cannot affect the output either). + """ + dte = float(te1) - float(te0) + if dte <= 0: + return 0.0 + return float(abs(np.angle(np.exp(1j * 2 * np.pi * float(te0) / dte)))) + + +def _select_branch( + unwrapped_diff: npt.NDArray[np.float32], + phase0: npt.NDArray[np.float32], + phase1: npt.NDArray[np.float32], + mag0: npt.NDArray[np.float32], + mag1: npt.NDArray[np.float32], + te0: np.float32 | float, + te1: np.float32 | float, + mask: npt.NDArray[np.bool_], + score_mask: npt.NDArray[np.bool_], +) -> tuple[int, dict[int, float]]: + """Pick the global 2*pi branch. Act only when the data is unambiguous. + + A branch carrying a leftover intercept is not a valid explanation of the + data at all, so discard those. If **exactly one** candidate survives it is + the answer, whatever its field looks like. In every other case -- nothing + fits, or several fit -- return 0 and leave ROMEO's result alone. + + Deferring on a tie is deliberate. When several branches are exactly + through-origin the alias is genuinely reachable: they are equally valid + explanations of the phase, and no statistic computed from the phase can + separate them. The only tiebreaker available is the smallest-|field| prior, + which is what ROMEO's ``correct_global`` has already applied. Re-applying it + here with a slightly stricter statistic adds no information and measurably + does harm -- on a centre-frequency sweep of real data it changed the answer + for the better on 12% of points and for the worse on 12%, including cases + where ``correct_global`` had been right. A coin flip is worse than a no-op, + because a no-op at least stays consistent with the unwrapper. + + So this only ever fires on a call the data decides on its own; measured + separation between fitting and non-fitting branches is 6-9 orders of + magnitude. Across every frame of 58 runs from eight OpenNeuro datasets -- + 14090 frames, six distinct TE0/dTE ratios from 0.51 to 0.78 -- it never + fired, and the largest branch-0 intercept seen was 1.9e-07 of the cutoff. + + That is expected rather than lucky. ``unwrap_4d`` global-corrects only the + template echo, whose unwrapped phase is ``k * unwrapped_diff``; for k < 1 + the scaling pulls values toward zero, so the rounded median cannot be + pushed off it. Every ME-EPI protocol has k < 1, since TE0 is shorter than + the echo spacing. The rule is therefore a safety net, not a hot path. + """ + # Only the intercept is needed; drop each candidate's field map and phases + # rather than holding three of them per frame across the thread pool. + scores = { + n: _evaluate_branch( + n, unwrapped_diff, phase0, phase1, mag0, mag1, te0, te1, mask, score_mask + )[0] + for n in BRANCH_CANDIDATES + } + # Calibrate "how big is a wrong branch" two ways and take the stricter. + # The analytic scale depends only on the TEs, so it still calibrates when + # every candidate happens to fit; the observed scale stays honest when + # ROMEO absorbs part of the intercept into its own 2*pi steps and the real + # penalty comes out smaller than theory predicts. Taking the minimum makes + # the consistency test harder to pass, which biases toward doing nothing. + step = _branch_intercept_step(te0, te1) + if step <= 0: + # te0/dTE is an integer: the branch does not move the phase offset, so + # every candidate returns the same result and there is nothing to pick. + return 0, scores + scale = min(max(scores.values()), step) + if not np.isfinite(scale) or scale <= 0: + return 0, scores + # Nearest-neighbour on the lattice: candidate intercepts sit at 0 or at one + # ``scale``, so the boundary is the midpoint. Not a tuned threshold -- the + # two clusters are separated by 6-9 orders of magnitude on measured data, so + # any boundary strictly inside (0, scale) gives the same answer. The + # midpoint is simply the one that needs no justifying, and it keeps a branch + # that is most of a wrap out from ever counting as a fit. + cutoff = scale / 2 + consistent = [n for n, s in scores.items() if s < cutoff] + if len(consistent) != 1: + # Nothing fits (a failed fit is not evidence for any branch), or several + # fit equally well (a genuine alias the phase cannot resolve). Either + # way there is no call to make here -- defer to correct_global. + return 0, scores + return consistent[0], scores + + def mcpc_3d_s( mag0: npt.NDArray[np.float32], mag1: npt.NDArray[np.float32], phase0: npt.NDArray[np.float32], phase1: npt.NDArray[np.float32], - te0: npt.NDArray[np.float32], - te1: npt.NDArray[np.float32], + te0: np.float32 | float, + te1: np.float32 | float, mask: npt.NDArray[np.bool_], - wrap_limit: bool = False, ): """Apply the MCPC-3D-S algorithm to compute the phase offset. @@ -104,8 +258,6 @@ def mcpc_3d_s( Echo time for the second echo mask : npt.NDArray[np.bool_] Mask of voxels to use for unwrapping - wrap_limit : bool, optional - Limit the phase wrapping, by default False Returns ------- @@ -125,88 +277,17 @@ def mcpc_3d_s( correct_global=True, ) voxel_mask = create_brain_mask(mag0, -2) - phases = np.stack([phase0, phase1], axis=-1) - mags = np.stack([mag0, mag1], axis=-1) - tes = np.array([te0, te1]) - all_tes = np.array([0.0, te0, te1]) - proposed_offset = np.angle( - np.exp(1j * (phase0 - ((te0 * unwrapped_diff) / (te1 - te0)))) - ) - # get the new phases - proposed_phases = phases - proposed_offset[..., np.newaxis] - - # compute the fieldmap - proposed_fieldmap, proposed_unwrapped_phases = get_dual_echo_fieldmap( - proposed_phases, tes, mags, mask + n_wraps, scores = _select_branch( + unwrapped_diff, phase0, phase1, mag0, mag1, te0, te1, mask, voxel_mask + ) + logging.info( + "branch selection: n=%+d (intercepts=%s)", + n_wraps, + {n: f"{s_:.4e}" for n, s_ in scores.items()}, ) - # check if the proposed fieldmap is below 10 - logging.info(f"proposed_fieldmap: {proposed_fieldmap[voxel_mask].mean()}") - if proposed_fieldmap[voxel_mask].mean() < -10: - unwrapped_diff += 2 * np.pi - # check if the propossed fieldmap is between -10 and 0 - elif proposed_fieldmap[voxel_mask].mean() < 0 and not wrap_limit: - # look at proportion of voxels that are positive - voxel_prop = ( - np.count_nonzero(proposed_fieldmap[voxel_mask] > 0) - / proposed_fieldmap[voxel_mask].shape[0] - ) - - # if the proportion of positive voxels is less than 0.25, then add 2pi - if voxel_prop < FMAP_PROPORTION_HEURISTIC: - unwrapped_diff += 2 * np.pi - elif voxel_prop < FMAP_AMBIGUIOUS_HEURISTIC: - # compute mean of phase offset - mean_phase_offset = proposed_offset[voxel_mask].mean() - # print(f"mean_phase_offset: {mean_phase_offset}") - # if less than -1 then - if mean_phase_offset < -1: - phase_fits = np.concatenate( - ( - np.zeros((*proposed_unwrapped_phases.shape[:-1], 1)), - proposed_unwrapped_phases, - ), - axis=-1, - ) - _, residuals_1, _, _, _ = np.polyfit( - all_tes, phase_fits[voxel_mask, :].T, 1, full=True - ) - - # check if adding 2pi makes it better - new_proposed_offset = np.angle( - np.exp( - 1j - * ( - phase0 - - ((te0 * (unwrapped_diff + 2 * np.pi)) / (te1 - te0)) - ) - ) - ) - new_proposed_phases = phases - new_proposed_offset[..., np.newaxis] - new_proposed_fieldmap, new_proposed_unwrapped_phases = ( - get_dual_echo_fieldmap(new_proposed_phases, tes, mags, mask) - ) - # fit linear model to the proposed phases - new_phase_fits = np.concatenate( - ( - np.zeros((*new_proposed_unwrapped_phases.shape[:-1], 1)), - new_proposed_unwrapped_phases, - ), - axis=-1, - ) - _, residuals_2, _, _, _ = np.polyfit( - all_tes, new_phase_fits[voxel_mask, :].T, 1, full=True - ) - if ( - np.isclose( - residuals_1.mean(), residuals_2.mean(), atol=1e-3, rtol=1e-3 - ) - and new_proposed_fieldmap[voxel_mask].mean() > 0 - ): - unwrapped_diff += 2 * np.pi - else: - unwrapped_diff -= 2 * np.pi + unwrapped_diff = unwrapped_diff + 2 * np.pi * n_wraps # compute the phase offset return np.angle( @@ -222,7 +303,6 @@ def unwrap_phase( automask: bool = True, automask_dilation: int = 3, idx: int | None = None, - wrap_limit: bool = False, debug: bool = False, ) -> tuple[npt.NDArray[np.float32], npt.NDArray[np.int8]]: """Unwraps the phase for a single frame of ME-EPI data. @@ -308,7 +388,6 @@ def unwrap_phase( tes[0], tes[1], mask_data, - wrap_limit=wrap_limit, ) if debug: global affine @@ -644,7 +723,6 @@ def unwrap_phases( frames: list[int] | None = None, n_cpus: int = 4, debug: bool = False, - wrap_limit: bool = False, ) -> tuple[list[nib.Nifti1Image], nib.Nifti1Image]: """Unwrap multi-echo phase per frame and enforce temporal consistency. @@ -674,8 +752,6 @@ def unwrap_phases( debug : bool, optional Skip the temporal consistency pass and dump intermediate files, by default False. - wrap_limit : bool, optional - Disable some MCPC-3D-S heuristics, by default False. Returns ------- @@ -810,7 +886,6 @@ def phase_iterator(phase, mag, tes, mask, frames, automask, automask_dilation): automask, automask_dilation, idx, - wrap_limit, debug, ) @@ -1001,7 +1076,6 @@ def unwrap_and_compute_field_maps( frames: list[int] | None = None, n_cpus: int = 4, debug: bool = False, - wrap_limit: bool = False, ) -> nib.Nifti1Image: """Unwrap phase and compute native-space field maps in a single call. @@ -1021,7 +1095,6 @@ def unwrap_and_compute_field_maps( frames=frames, n_cpus=n_cpus, debug=debug, - wrap_limit=wrap_limit, ) return compute_field_maps( unwrapped_imgs, diff --git a/warpkit/utilities.py b/warpkit/utilities.py index 6925002..d3cbddc 100644 --- a/warpkit/utilities.py +++ b/warpkit/utilities.py @@ -141,7 +141,7 @@ def corr2_coeff(a: npt.NDArray, b: npt.NDArray) -> npt.NDArray: def rescale_phase( - data: npt.NDArray[Any], min: int = -4096, max: int = 4096 + data: npt.NDArray[Any], min: float = -4096, max: float = 4096 ) -> npt.NDArray[Any]: """Rescale phase data to [-pi, pi] From a6cbe6e42a05b5d3349e8c7732a85cc998c379c3 Mon Sep 17 00:00:00 2001 From: Andrew Van Date: Mon, 27 Jul 2026 22:31:18 -0500 Subject: [PATCH 02/13] :memo: Correct scalar-TE docstrings and split the tie/no-fit selector test Addresses review feedback on #32 and unblocks the lint job. - `mcpc_3d_s` documents `te0`/`te1` as `np.float32 | float` to match the scalar signature; `rescale_phase` documents `min`/`max` as floats. - Split `test_branch_selector_noop_when_nothing_fits` in two. The all-zero intercepts case is an all-way perfect fit that collapses the observed scale, not a failed fit, so it gets its own test and an accurate docstring. - Reformat the README's Python block: ruff 0.16 formats code in markdown, which fails `ruff format --check` on main as well as here. --- README.md | 6 +++--- tests/test_unwrap.py | 12 ++++++++++-- warpkit/unwrap.py | 4 ++-- warpkit/utilities.py | 4 ++-- 4 files changed, 17 insertions(+), 9 deletions(-) diff --git a/README.md b/README.md index 72a9787..733568f 100644 --- a/README.md +++ b/README.md @@ -66,9 +66,9 @@ from warpkit.utilities import displacement_map_to_field # each list entry is a different echo phases = [nib.load(p) for p in phases_paths] magnitudes = [nib.load(p) for p in magnitude_paths] -TEs = [TE1, TE2, ...] # milliseconds -total_readout_time = ... # seconds -phase_encoding_direction = ... # one of i, j, k, i-, j-, k-, x, y, z, x-, y-, z- +TEs = [TE1, TE2, ...] # milliseconds +total_readout_time = ... # seconds +phase_encoding_direction = ... # one of i, j, k, i-, j-, k-, x, y, z, x-, y-, z- field_maps_native, displacement_maps, field_maps = medic( phases, magnitudes, TEs, total_readout_time, phase_encoding_direction diff --git a/tests/test_unwrap.py b/tests/test_unwrap.py index cc93b85..1b4124b 100644 --- a/tests/test_unwrap.py +++ b/tests/test_unwrap.py @@ -266,12 +266,20 @@ def test_branch_selector_noop_when_all_candidates_tie(monkeypatch): assert best == 0 -def test_branch_selector_noop_when_nothing_fits(monkeypatch): - """Degenerate/failed fit: every candidate is bad, so change nothing.""" +def test_branch_selector_noop_when_every_candidate_fits_perfectly(monkeypatch): + """All intercepts are exactly 0, so every candidate is through-origin. The + observed scale collapses to 0 and there is no wrong-branch magnitude left to + calibrate a cutoff against, so the selector must not act -- an all-way tie is + an alias the phase cannot resolve, same as the two-branch case above.""" _patch_scores(monkeypatch, {-1: 0.0, 0: 0.0, 1: 0.0}) best, _ = _sel() assert best == 0 + +def test_branch_selector_noop_when_nothing_fits(monkeypatch): + """Every candidate carries roughly a full step of intercept, so none of them + explains the phase. A failed fit is not evidence for any branch, so change + nothing rather than take the least-bad one.""" _patch_scores(monkeypatch, {-1: 1.71, 0: 1.80, 1: 1.74}) best, _ = _sel() assert best == 0 diff --git a/warpkit/unwrap.py b/warpkit/unwrap.py index b60f9f3..0c2f008 100644 --- a/warpkit/unwrap.py +++ b/warpkit/unwrap.py @@ -252,9 +252,9 @@ def mcpc_3d_s( Phase image for the first echo phase1 : npt.NDArray[np.float32] Phase image for the second echo - te0 : npt.NDArray[np.float32] + te0 : np.float32 | float Echo time for the first echo - te1 : npt.NDArray[np.float32] + te1 : np.float32 | float Echo time for the second echo mask : npt.NDArray[np.bool_] Mask of voxels to use for unwrapping diff --git a/warpkit/utilities.py b/warpkit/utilities.py index d3cbddc..3f97ff5 100644 --- a/warpkit/utilities.py +++ b/warpkit/utilities.py @@ -151,9 +151,9 @@ def rescale_phase( ---------- data : npt.NDArray[Any] phase data to be rescaled - min : int, optional + min : float, optional min value that should be mapped to -pi, by default -4096 - max : int, optional + max : float, optional max value that should be mapped to pi, by default 4096 Returns From 06209ffdc5392bfe2b7450f43b562ba1c0c8492f Mon Sep 17 00:00:00 2001 From: Andrew Van Date: Mon, 27 Jul 2026 22:45:08 -0500 Subject: [PATCH 03/13] :bug: Keep the field prior for branch ties; it fixes a real 40 Hz flip Restores the smallest-|field| tiebreaker removed earlier in this branch. The evidence for removing it came from an induced centre-frequency sweep, which only exercises the regime where the prior is meant to lose -- a genuine global field offset. It never covered the opposite regime, an artifactual one-wrap flip, and that is the one that actually occurs. Found on ds006131 sub-20828 (TEs 14.2/38.93/63.66 ms, k=0.5742, wrap 40.44 Hz). That subject's global field sits at 17.7 Hz against a 20.22 Hz half-wrap boundary, so correct_global's median of rounded wrap counts is on a knife edge and tips over on individual frames: its ballot reads 0, +1, 0, -1 frame to frame, and each tip flips the dual-echo field a full wrap for one frame. The intercept test detects the flip -- the fitting set moves from {-1, 0} to {0, +1} -- but both survivors score ~1e-7, so it cannot rank them. The prior can, and correctly: branch +1 restores +17.7 Hz against branch 0's -22.8 Hz. It is the same rule correct_global applies, on a better conditioned statistic (magnitude-weighted, eroded mask, on the field itself rather than an unweighted median of rounded counts over a dilated mask), which is what makes it stable where correct_global is not. Full pipeline on that run, before and after: before field +16.46..+53.36 Hz max step 35.939 Hz 34 wrap-sized steps after field +16.38..+19.62 Hz max step 1.462 Hz 0 wrap-sized steps 19 of 243 frames were single-frame ~40 Hz excursions; all are gone, and the 224 healthy frames are untouched. - Restore _weighted_median and the tie branch of _select_branch. - Rewrite the docstring around this measurement instead of the sweep. - Replace the two deferral tests with ones built from the real frame-44 intercepts and fields. --- tests/test_unwrap.py | 61 ++++++++++++++++++----------- warpkit/unwrap.py | 93 ++++++++++++++++++++++++++------------------ 2 files changed, 94 insertions(+), 60 deletions(-) diff --git a/tests/test_unwrap.py b/tests/test_unwrap.py index 1b4124b..cbc7a02 100644 --- a/tests/test_unwrap.py +++ b/tests/test_unwrap.py @@ -152,15 +152,17 @@ def test_compute_field_maps_rejects_mismatched_spatial_shape(): # --------------------------------------------------------------------------- -def _patch_scores(monkeypatch, scores): +def _patch_scores(monkeypatch, scores, fields=None): """Stub _evaluate_branch so only the decision rule is under test. - ``scores`` are intercepts in radians. The selector consults nothing else -- - no field prior, no tiebreak -- so the remaining tuple slots are unused. + ``scores`` are intercepts in radians; ``fields`` are per-branch field maps + (constant-valued), consulted only when the consistency test ties. """ + fields = fields or dict.fromkeys(scores, 0.0) def fake(n_wraps, *args, **kwargs): - return scores[n_wraps], None, None, None + fmap = np.full((2, 2, 2), fields[n_wraps], dtype=np.float32) + return scores[n_wraps], None, fmap, None monkeypatch.setattr(unwrap_mod, "_evaluate_branch", fake) @@ -234,43 +236,58 @@ def test_branch_selector_corrects_inconsistent_zero(monkeypatch): assert best == 1 -def test_branch_selector_defers_when_two_branches_fit(monkeypatch): - """The ds007637 case: N=0 and N=-1 are both through-origin. That is a - genuine alias -- both explain the phase exactly -- so the selector must not - choose between them. Deferring leaves correct_global's answer in place. +def test_branch_selector_tie_keeps_zero_when_its_field_is_smaller(monkeypatch): + """A healthy ds006131 frame: N=0 and N=-1 are both through-origin, so the + field prior decides. N=0 has the smaller |field|, so nothing moves. A bare argmin on the intercepts would flip between them on numerical noise and inject a full wrap of field into the time series. """ - _patch_scores(monkeypatch, {-1: 1.2e-03, 0: 1.9e-03, 1: 0.9324}) - best, _ = _sel() + _patch_scores( + monkeypatch, + {-1: 9.12e-08, 0: 3.86e-08, 1: 0.9324}, + fields={-1: -24.29, 0: 16.15, 1: 16.15}, + ) + best, _ = _sel(te0=14.2, te1=38.93) assert best == 0 -def test_branch_selector_defers_even_when_zero_looks_wrong(monkeypatch): - """N=0 carries an intercept but *two* alternatives fit. Neither is - preferable on the evidence, so change nothing rather than pick one. +def test_branch_selector_tie_recovers_correct_global_flip(monkeypatch): + """The ds006131 sub-20828 failure, frame 44. - This is the case that used to consult the field prior. Measurement showed - the prior was right about as often as it was wrong, so it was removed. + correct_global's ballot tipped over the half-wrap boundary and the + dual-echo field flipped a full wrap, from +16.2 Hz to -22.8 Hz. Branch 0 + and branch +1 are both through-origin so the intercept cannot rank them, + but +1 restores +17.7 Hz against 0's -22.8 Hz and the prior picks it. + + This is the case that motivated keeping the tiebreaker: 19 of 243 frames in + that run, each a single-frame ~40 Hz excursion in the field time series. """ - _patch_scores(monkeypatch, {-1: 1.0e-7, 0: 1.8402, 1: 2.0e-7}) - best, _ = _sel() - assert best == 0 + _patch_scores( + monkeypatch, + {-1: 0.9324, 0: 8.64e-08, 1: 6.43e-08}, + fields={-1: -22.78, 0: -22.78, 1: 17.66}, + ) + best, _ = _sel(te0=14.2, te1=38.93) + assert best == 1 def test_branch_selector_noop_when_all_candidates_tie(monkeypatch): """Integer TE0/dTE makes the branch a no-op; scores are all ~equal.""" - _patch_scores(monkeypatch, {-1: 1.1e-8, 0: 2.0e-8, 1: 1.9e-8}) + _patch_scores( + monkeypatch, + {-1: 1.1e-8, 0: 2.0e-8, 1: 1.9e-8}, + fields={-1: -58.8, 0: 0.1, 1: 58.9}, + ) best, _ = _sel() assert best == 0 def test_branch_selector_noop_when_every_candidate_fits_perfectly(monkeypatch): """All intercepts are exactly 0, so every candidate is through-origin. The - observed scale collapses to 0 and there is no wrong-branch magnitude left to - calibrate a cutoff against, so the selector must not act -- an all-way tie is - an alias the phase cannot resolve, same as the two-branch case above.""" + observed scale collapses to 0, leaving no wrong-branch magnitude to calibrate + a cutoff against, so the selector bails before the consistency test runs at + all -- the field prior never gets to break this tie.""" _patch_scores(monkeypatch, {-1: 0.0, 0: 0.0, 1: 0.0}) best, _ = _sel() assert best == 0 diff --git a/warpkit/unwrap.py b/warpkit/unwrap.py index 0c2f008..895c4ed 100644 --- a/warpkit/unwrap.py +++ b/warpkit/unwrap.py @@ -135,6 +135,17 @@ def _evaluate_branch( return intercept, offset, fieldmap, unwrapped_phases +def _weighted_median(values: npt.NDArray, weights: npt.NDArray) -> float: + """Weighted median of ``values``. Used with magnitude-squared weights so + low-SNR voxels do not sway the global field estimate.""" + order = np.argsort(values) + values, weights = values[order], weights[order] + cumulative = np.cumsum(weights) + if cumulative[-1] <= 0: + return float(np.median(values)) + return float(values[np.searchsorted(cumulative, 0.5 * cumulative[-1])]) + + def _branch_intercept_step(te0: np.float32 | float, te1: np.float32 | float) -> float: """Intercept, in radians, that one wrap of branch error introduces. @@ -163,44 +174,43 @@ def _select_branch( mask: npt.NDArray[np.bool_], score_mask: npt.NDArray[np.bool_], ) -> tuple[int, dict[int, float]]: - """Pick the global 2*pi branch. Act only when the data is unambiguous. - - A branch carrying a leftover intercept is not a valid explanation of the - data at all, so discard those. If **exactly one** candidate survives it is - the answer, whatever its field looks like. In every other case -- nothing - fits, or several fit -- return 0 and leave ROMEO's result alone. - - Deferring on a tie is deliberate. When several branches are exactly - through-origin the alias is genuinely reachable: they are equally valid - explanations of the phase, and no statistic computed from the phase can - separate them. The only tiebreaker available is the smallest-|field| prior, - which is what ROMEO's ``correct_global`` has already applied. Re-applying it - here with a slightly stricter statistic adds no information and measurably - does harm -- on a centre-frequency sweep of real data it changed the answer - for the better on 12% of points and for the worse on 12%, including cases - where ``correct_global`` had been right. A coin flip is worse than a no-op, - because a no-op at least stays consistent with the unwrapper. - - So this only ever fires on a call the data decides on its own; measured - separation between fitting and non-fitting branches is 6-9 orders of - magnitude. Across every frame of 58 runs from eight OpenNeuro datasets -- - 14090 frames, six distinct TE0/dTE ratios from 0.51 to 0.78 -- it never - fired, and the largest branch-0 intercept seen was 1.9e-07 of the cutoff. - - That is expected rather than lucky. ``unwrap_4d`` global-corrects only the - template echo, whose unwrapped phase is ``k * unwrapped_diff``; for k < 1 - the scaling pulls values toward zero, so the rounded median cannot be - pushed off it. Every ME-EPI protocol has k < 1, since TE0 is shorter than - the echo spacing. The rule is therefore a safety net, not a hot path. + """Pick the global 2*pi branch, in two stages. + + 1. **Consistency.** A branch carrying a leftover intercept is not a valid + explanation of the data at all, so discard those. If exactly one + candidate survives it is the answer, whatever its field looks like. If + none survives, return 0 -- a failed fit is not evidence for any branch. + 2. **Prior, only to break a tie.** Depending on TE0/dTE several branches can + be exactly through-origin; the alias is genuinely reachable and no + statistic computed from the phase can separate them. Among the survivors, + take the smallest magnitude-weighted median field. + + Stage 2 is the same prior ``correct_global`` already applies, but on a much + better conditioned statistic: magnitude-weighted, over an eroded brain mask, + on the field itself rather than an unweighted median of rounded wrap counts + over a dilated mask. That difference is the whole point. ``correct_global``'s + ballot becomes unstable when the global field sits near 1/(2*dTE), and then + it flips frame to frame. + + Measured on `ds006131` sub-20828 (k = 0.5742, wrap 40.44 Hz, half-wrap + 20.22 Hz), whose field sits at 17.7 Hz -- 2.5 Hz under the boundary. On 19 + of 243 frames ``correct_global``'s median jumps to +/-1 and the dual-echo + field flips from +16.2 Hz to -22.8 Hz, a full wrap, for a single frame at a + time. The intercept test detects that something moved (the fitting set goes + from {-1, 0} to {0, +1}) but cannot rank the two survivors. The prior can: + branch +1 restores +17.7 Hz against branch 0's -22.8 Hz, so it fixes all 19 + and leaves the other 224 untouched. + + The prior is consulted *only* between branches that already fit, so it can + never select something the data contradicts. """ - # Only the intercept is needed; drop each candidate's field map and phases - # rather than holding three of them per frame across the thread pool. - scores = { + evaluated = { n: _evaluate_branch( n, unwrapped_diff, phase0, phase1, mag0, mag1, te0, te1, mask, score_mask - )[0] + ) for n in BRANCH_CANDIDATES } + scores = {n: e[0] for n, e in evaluated.items()} # Calibrate "how big is a wrong branch" two ways and take the stricter. # The analytic scale depends only on the TEs, so it still calibrates when # every candidate happens to fit; the observed scale stays honest when @@ -223,12 +233,19 @@ def _select_branch( # that is most of a wrap out from ever counting as a fit. cutoff = scale / 2 consistent = [n for n, s in scores.items() if s < cutoff] - if len(consistent) != 1: - # Nothing fits (a failed fit is not evidence for any branch), or several - # fit equally well (a genuine alias the phase cannot resolve). Either - # way there is no call to make here -- defer to correct_global. + if not consistent: + # nothing fits; a failed fit is not evidence for any branch return 0, scores - return consistent[0], scores + if len(consistent) == 1: + return consistent[0], scores + # Tie: every survivor explains the phase equally well, so fall back to the + # prior and take the one closest to zero global field. + weights = np.square(mag0[score_mask].astype(np.float64)) + fields = { + n: _weighted_median(evaluated[n][2][score_mask].astype(np.float64), weights) + for n in consistent + } + return min(consistent, key=lambda n: abs(fields[n])), scores def mcpc_3d_s( From 973cb00a52b18733dc4214659fd593cbda121e9d Mon Sep 17 00:00:00 2001 From: Andrew Van Date: Mon, 27 Jul 2026 22:51:37 -0500 Subject: [PATCH 04/13] :white_check_mark: Add the ds006131 branch-flip regression fixture Pins the failure that motivated keeping the field prior, using the real data rather than hand-written numbers. Two frames from ds006131 sub-20828 run-01 (TEs 14.2/38.93/63.66 ms, k=0.5742, wrap 40.44 Hz): the original frame 43, which is healthy, and frame 44, one of 19 frames out of 243 where correct_global's ballot tips and the dual-echo field jumps a full wrap. The selector must leave the first alone and return +1 on the second. - tests/data/branch_flip/ (9.7 MB, excluded from the sdist by the existing tool.scikit-build exclude of tests/data). - Cropped to the brain bounding box, which reproduces the behaviour exactly. Deliberately NOT decimated: correct_global's median of rounded wrap counts is on a knife edge here, so resampling tips it the other way -- 2x decimation makes the flip vanish on frame 44 and appear on the healthy frame 43. That would silently invert what the test checks. - Also pins that the broken frame is a genuine {0, +1} tie on the intercept, so a classify-only rule provably cannot fix it and the prior is load-bearing. --- tests/conftest.py | 32 ++++++++ .../sub-20828_echo-1_part-mag_bold.json | 5 ++ .../sub-20828_echo-1_part-mag_bold.nii.gz | Bin 0 -> 1736907 bytes .../sub-20828_echo-1_part-phase_bold.nii.gz | Bin 0 -> 1748663 bytes .../sub-20828_echo-2_part-mag_bold.json | 5 ++ .../sub-20828_echo-2_part-mag_bold.nii.gz | Bin 0 -> 1608137 bytes .../sub-20828_echo-2_part-phase_bold.nii.gz | Bin 0 -> 1762238 bytes .../sub-20828_echo-3_part-mag_bold.json | 5 ++ .../sub-20828_echo-3_part-mag_bold.nii.gz | Bin 0 -> 1513963 bytes .../sub-20828_echo-3_part-phase_bold.nii.gz | Bin 0 -> 1764575 bytes tests/test_unwrap.py | 74 ++++++++++++++++++ 11 files changed, 121 insertions(+) create mode 100644 tests/data/branch_flip/sub-20828_echo-1_part-mag_bold.json create mode 100644 tests/data/branch_flip/sub-20828_echo-1_part-mag_bold.nii.gz create mode 100644 tests/data/branch_flip/sub-20828_echo-1_part-phase_bold.nii.gz create mode 100644 tests/data/branch_flip/sub-20828_echo-2_part-mag_bold.json create mode 100644 tests/data/branch_flip/sub-20828_echo-2_part-mag_bold.nii.gz create mode 100644 tests/data/branch_flip/sub-20828_echo-2_part-phase_bold.nii.gz create mode 100644 tests/data/branch_flip/sub-20828_echo-3_part-mag_bold.json create mode 100644 tests/data/branch_flip/sub-20828_echo-3_part-mag_bold.nii.gz create mode 100644 tests/data/branch_flip/sub-20828_echo-3_part-phase_bold.nii.gz diff --git a/tests/conftest.py b/tests/conftest.py index b3bf045..7779913 100644 --- a/tests/conftest.py +++ b/tests/conftest.py @@ -9,6 +9,7 @@ # get this directory THISDIR = Path(__file__).parent TEST_DATA_DIR = THISDIR / "data" / "test_data" +BRANCH_FLIP_DIR = THISDIR / "data" / "branch_flip" # fixture for test data @@ -31,6 +32,37 @@ def test_data(): } +@fixture(scope="session") +def branch_flip_data(): + """Two frames from `ds006131` sub-20828 run-01 that expose a real + ``correct_global`` branch flip. + + That subject's global field sits at 17.7 Hz against a 20.22 Hz half-wrap + (TEs 14.2/38.93/63.66 ms), so ROMEO's median of rounded wrap counts is on a + knife edge and tips over on individual frames. Frame 0 here is the original + frame 43 and is healthy; frame 1 is the original frame 44, where the + dual-echo field flips a full 40.44 Hz wrap. + + Cropped to the brain bounding box, which preserves the behaviour exactly. + It is *not* decimated: resampling changes which voxels vote in the ballot + and tips it the other way, which would destroy the very thing under test. + """ + mag = sorted(BRANCH_FLIP_DIR.glob("*part-mag*.nii.gz")) + phase = sorted(BRANCH_FLIP_DIR.glob("*part-phase*.nii.gz")) + sidecar = sorted(BRANCH_FLIP_DIR.glob("*part-mag*.json")) + metadata = [] + for s in sidecar: + with s.open() as f: + metadata.append(load(f)) + return { + "phase": [cast(Nifti1Image, nib.load(str(p))) for p in phase], + "mag": [cast(Nifti1Image, nib.load(str(m))) for m in mag], + "tes": [m["EchoTime"] * 1000 for m in metadata], + # frame index -> branch the selector must choose + "expected_branch": {0: 0, 1: 1}, + } + + @fixture(scope="session") def test_data_paths(): """File paths for the bundled BIDS-style MEDIC test data, suitable for diff --git a/tests/data/branch_flip/sub-20828_echo-1_part-mag_bold.json b/tests/data/branch_flip/sub-20828_echo-1_part-mag_bold.json new file mode 100644 index 0000000..f133cc6 --- /dev/null +++ b/tests/data/branch_flip/sub-20828_echo-1_part-mag_bold.json @@ -0,0 +1,5 @@ +{ + "EchoTime": 0.0142, + "RepetitionTime": 1.761, + "_comment": "ds006131 sub-20828 run-01, frames 43 (healthy) and 44 (correct_global flip)" +} \ No newline at end of file diff --git a/tests/data/branch_flip/sub-20828_echo-1_part-mag_bold.nii.gz b/tests/data/branch_flip/sub-20828_echo-1_part-mag_bold.nii.gz new file mode 100644 index 0000000000000000000000000000000000000000..4987a1738bd98601880615d2d097e502dcc72c3c GIT binary patch literal 1736907 zcmV(qK<~dFiwFP!00004|IEB)m{dp9E?QOF+ueKGr)AoE+02KlqRQ4-G$PL}_oO{P6&M z-T&hn{2TP+`j2;b760-6;~sebKmUsV&;9@B`+x79QeywTi%}TugvUyt7QmwlFz9B~ z7J(_}i=rH;N zHHY5~1zG%rHlaM!5A8x2AA_fhkPdBt5n96MB`{-8WQ4n>p(|(Qh3Df}9N14cus-T+4 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zitdr!l}o|*>NdYJai58hAUF}}Ar*dw@XQ#ZE8Q@9ZY&b^+lAByVo$fW(GqWq&P2v2 zt%$ku9*T^LV5!tdt~61adTwp@TCjbME)jzV(LzflJ@9$2Bd&WT@waGujn($TjO};m zigb`#GgqLcwlAmokXPfLZHTU`|{@%h>5wD-!7R`ao)kl0SMVy_EA% zUm>sJyAtrpm{~z(CyC<3p=ud>x!hX12~^f?#waTg4gjC6I?+IROILJe0c-5Ge_7s# zB=IfyU&1WlHuYxnScARpZ0R7u9vy8(a|s=h#aIIE#lu~A=~^ky6sa2IH*GT$sSG5E zWn5kFKRjLgHLMBgTDEm|Lq7qE^D#=)Us0aj)?W z;6h9?7I?{sCspS>4ioE`KX?ncVR9<9L3t>tYJFx!)KYMSZ21^jBIqmDvOAh7MyKWlW*jp@SGQ9>i zicPhb1RtEsF)5|<^^kJPb7wR6&ImNoc@||L@7?8Ez$6hH#3u{^lc~eh1f#HGX`kc| z%IUZg_FGgl`c9jq-GVo}trlTM`E86E^mW4~mzc@y5C88kc$>K`(2j0JFEY2gVBW-D zA+7NvKv-`cs`vw>6#rKJJWO&Iiw*JQZ~~a026BU=a&aa6FQXIu)%@S@^EbQEZYj-| zapSrpLEEmpdJ5lx-J`eDzB%8&sZH~yYtMiY+90eeq>B Date: Mon, 27 Jul 2026 23:21:40 -0500 Subject: [PATCH 05/13] :zap: Use the half-sample mode, not a weighted median, for the field prior The tie-breaking prior asks which candidate branch puts the global field closest to zero. It was estimating that level with a magnitude-weighted median over an eroded brain mask. A median is the wrong statistic for this quantity. The field distribution over a brain is a sharp peak at bulk-tissue value with long tails from sinuses, dropout and edges. The scanner's frequency adjustment centres that peak near zero -- not the median-including-tails -- so a median estimates something the shim never set, and is pulled toward whichever tail is heavier. That is the same defect that makes correct_global's own estimate tip over: its dilated mask simply has more tail. Margins to the +/-1/(2*dTE) flip boundary, ds006131 sub-20828, worst of the healthy/broken frame pair: correct_global (median, dilated) -0.26 Hz flips median, eroded 1.25 Hz magnitude-weighted median, eroded 2.57 Hz (previous) half-sample mode, eroded 10.27 Hz (this commit) Across all 243 frames of that run the two estimators pick identically -- same 18 frames corrected, zero disagreements -- and the full pipeline output is unchanged, but the worst-case margin goes from 1.51 Hz to 9.34 Hz. The half-sample mode takes the shortest interval containing half the remaining points, recursively. It has no bin width and no bandwidth, so it introduces no tuned constant; a binned mode agrees with it to within 9.8-11.2 Hz of margin across a 64x range of bin counts, confirming the choice is not load-bearing. It is also cheaper than the weighted median it replaces (2.8 ms vs 6.9 ms at 120k voxels) because it never sorts a weight array. --- warpkit/unwrap.py | 79 +++++++++++++++++++++++++++++++++-------------- 1 file changed, 56 insertions(+), 23 deletions(-) diff --git a/warpkit/unwrap.py b/warpkit/unwrap.py index 895c4ed..df92544 100644 --- a/warpkit/unwrap.py +++ b/warpkit/unwrap.py @@ -135,15 +135,40 @@ def _evaluate_branch( return intercept, offset, fieldmap, unwrapped_phases -def _weighted_median(values: npt.NDArray, weights: npt.NDArray) -> float: - """Weighted median of ``values``. Used with magnitude-squared weights so - low-SNR voxels do not sway the global field estimate.""" - order = np.argsort(values) - values, weights = values[order], weights[order] - cumulative = np.cumsum(weights) - if cumulative[-1] <= 0: - return float(np.median(values)) - return float(values[np.searchsorted(cumulative, 0.5 * cumulative[-1])]) +def _half_sample_mode(values: npt.NDArray) -> float: + """Mode of ``values`` by the half-sample method: repeatedly keep the + shortest interval containing half the remaining points. + + Used instead of a median to locate the global field level. The field + distribution over a brain is a sharp peak at bulk-tissue value with long + tails from sinuses, dropout and edges. The scanner's frequency adjustment + centres that *peak* near zero, not the median-including-tails, so a median + estimates the wrong quantity and is dragged toward whichever tail is + heavier. On the `ds006131` sub-20828 frame where ROMEO's ballot tips, the + mode sits 10.3 Hz from the half-wrap boundary against the weighted median's + 2.6 Hz -- four times the margin, from the same single frame. + + Parameter-free by construction: no bin width, no bandwidth, nothing to + tune. A binned mode gives the same answer (9.8-11.2 Hz of margin across a + 64x range of bin counts), so the choice is not load-bearing; this form just + avoids having to justify one. It is also cheaper than the weighted median + it replaced, since it never sorts a weight array. + """ + x = np.sort(np.asarray(values, dtype=np.float64)) + if x.size == 0: + return 0.0 + while x.size > 3: + half = (x.size + 1) // 2 + widths = x[half - 1 :] - x[: x.size - half + 1] + start = int(np.argmin(widths)) + x = x[start : start + half] + if x.size == 3: + left, right = x[1] - x[0], x[2] - x[1] + if left < right: + return float(np.mean(x[:2])) + if right < left: + return float(np.mean(x[1:])) + return float(np.mean(x)) def _branch_intercept_step(te0: np.float32 | float, te1: np.float32 | float) -> float: @@ -183,14 +208,26 @@ def _select_branch( 2. **Prior, only to break a tie.** Depending on TE0/dTE several branches can be exactly through-origin; the alias is genuinely reachable and no statistic computed from the phase can separate them. Among the survivors, - take the smallest magnitude-weighted median field. - - Stage 2 is the same prior ``correct_global`` already applies, but on a much - better conditioned statistic: magnitude-weighted, over an eroded brain mask, - on the field itself rather than an unweighted median of rounded wrap counts - over a dilated mask. That difference is the whole point. ``correct_global``'s - ballot becomes unstable when the global field sits near 1/(2*dTE), and then - it flips frame to frame. + take the one whose bulk-tissue field level -- the half-sample mode over + an eroded brain mask -- is closest to zero. + + Stage 2 is the same prior ``correct_global`` already applies, but on a far + better conditioned statistic. ``correct_global`` takes an unweighted median + of rounded wrap counts over a *dilated* mask, which is pulled toward the + tails contributed by sinuses, dropout and edge voxels; that mask carries + roughly twice the voxels of the brain core. Its estimate therefore sits + close to the +/-1/(2*dTE) decision boundary and tips frame to frame. + + Measured margins to that boundary on the frame below, worst of the two: + + ========================================== ============= + estimator margin + ========================================== ============= + ``correct_global`` (median, dilated) -0.26 Hz (flips) + median, eroded 1.25 Hz + magnitude-weighted median, eroded 2.57 Hz + **half-sample mode, eroded** **10.27 Hz** + ========================================== ============= Measured on `ds006131` sub-20828 (k = 0.5742, wrap 40.44 Hz, half-wrap 20.22 Hz), whose field sits at 17.7 Hz -- 2.5 Hz under the boundary. On 19 @@ -239,12 +276,8 @@ def _select_branch( if len(consistent) == 1: return consistent[0], scores # Tie: every survivor explains the phase equally well, so fall back to the - # prior and take the one closest to zero global field. - weights = np.square(mag0[score_mask].astype(np.float64)) - fields = { - n: _weighted_median(evaluated[n][2][score_mask].astype(np.float64), weights) - for n in consistent - } + # prior and take the one whose bulk-tissue field level is closest to zero. + fields = {n: _half_sample_mode(evaluated[n][2][score_mask]) for n in consistent} return min(consistent, key=lambda n: abs(fields[n])), scores From 96242ad09a6bceafb38a12a522932007a37409eb Mon Sep 17 00:00:00 2001 From: Andrew Van Date: Mon, 27 Jul 2026 23:34:17 -0500 Subject: [PATCH 06/13] :memo: Add a theory note for global 2*pi branch selection Derivation only -- no dataset names, measurements or validation numbers. Every quantity is a function of the echo times or the signal model, so it stays valid independent of what we happen to have tested against. Covers: - The signal model and the two derived scales, wrap spacing 1/dTE and the half-wrap 1/(2*dTE) that any smallest-|f| prior decides against. - The two ambiguities, each with a proof. The exact alias (theta - W(2*pi*k*N), f + N/dTE) reproduces the wrapped data identically and is therefore undetectable; an offset-branch mismatch instead lands a constant W(2*pi*k*M) on every echo and is. - Why extra echoes buy nothing: with even spacing every pair aliases at the same period, so five echoes carry as much branch information as two. - The intercept criterion, and why its rejection scale |W(2*pi*k)| is analytic rather than tuned. - Why the intercept and the field prior are both required -- the intercept is blind to a sign-preserving shift of the field, the prior to a sign-reversing one, so neither is sufficient alone. - Why a median is the wrong estimator for the prior. Writing the field distribution as (1-eps)*p_bulk + eps*p_tail, the mode is consistent for the bulk centre while the median is not, and mask dilation raises eps. The half-sample mode estimates the mode without a bin width or bandwidth. - Structural limits: ambiguity A is unaddressable by construction, dTE is bounded below by the EPI readout and above by T2*, the two ultra-high-field effects act in opposite directions, and a whole-turn shift of a multi-echo unwrapper's template echo is a pure slope change that no intercept test can see. Source is committed alongside the PDF so it stays regenerable with `typst compile notes/branch-selection-theory.typ`. --- notes/branch-selection-theory.pdf | Bin 0 -> 150612 bytes notes/branch-selection-theory.typ | 379 ++++++++++++++++++++++++++++++ 2 files changed, 379 insertions(+) create mode 100644 notes/branch-selection-theory.pdf create mode 100644 notes/branch-selection-theory.typ diff --git a/notes/branch-selection-theory.pdf b/notes/branch-selection-theory.pdf new file mode 100644 index 0000000000000000000000000000000000000000..435a453e4d4f912bb6811a57370d8117b0e63a23 GIT binary patch literal 150612 zcmeFa1wa&C7cgw=BPw>snjp&7Y#JzK2O_qpD=eXuu%Ot2-Q9)VVxVGUqu4%xf!*C5 z|G6=%V9Yz*;5 zK$Y;F0{1Y48v&#O-zsZo9AuKH(A_exKvR&>ULo-cFiOJ0L;D69gX_xNj1lm=3O(;> z2FF-|c3j3`5q&!2LD<$nhO$Jk7Xo$oR;wy;=^bauvNrFRtjX@k97*jJTh@+YDAbn%TU^Y|O3{d49 z+&9oaGSCz)ac&?n0iR6akw%FjD9{kW!N(#p!$%;pf{$s#_Y(Mxnz=V?CYLjeT&{G7 zRtc?KE>|h!a*Z4O2VXRL_)jaBYt`g$tqT5zYxq9+KgOB-tpPY14f#(C@HBA0R;83H 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false): it => box( + fill: luma(240), inset: (x: 3pt, y: 0pt), outset: (y: 3pt), radius: 2pt, it, +) + +#let dte = $Delta "TE"$ +#let phiuw = $Delta phi_"uw"$ + +#let key(body) = block( + width: 100%, fill: luma(245), stroke: (left: 2pt + luma(140)), + inset: 9pt, radius: 2pt, body, +) +#let defn(body) = block( + width: 100%, stroke: 0.5pt + luma(160), inset: 9pt, radius: 2pt, body, +) +#let proof(body) = block(inset: (left: 12pt), text(size: 10pt, body)) + +#align(center)[ + #text(size: 17pt, weight: "bold")[ + Global $2 pi$ branch selection in MCPC-3D-S + ] + #v(0.3em) + #text(size: 11pt)[Theory and derivation] + #v(0.2em) + #text(size: 9.5pt, style: "italic")[warpkit — phase offset estimation for MEDIC] +] + +#v(1em) + +#outline(indent: auto, depth: 2) + +#v(1em) + += The problem + +Multi-echo distortion correction needs the receive-coil phase offset $theta$ — +the phase each voxel carries at $"TE" = 0$, independent of the field. MCPC-3D-S +recovers it from the first two echoes. The recovered offset is pinned only up to +a whole number of turns of the unwrapped phase difference, and choosing the +wrong turn corrupts $theta$ by a fixed constant. This note derives what that +constant is, when it is detectable, and what decides the choice when it is not. + +Nothing here is specific to a dataset; every quantity is a function of the echo +times or of the signal model. + += Notation and signal model + +Let $W(x) = arg(e^(i x))$ denote wrapping to $(-pi, pi]$. Echo times are +evenly spaced, +$ + t_e = "TE"_0 + e dot #dte, quad e = 0, 1, ..., E-1, +$ +and we write the dimensionless echo-time ratio +$ + k = "TE"_0 / (#dte). +$ +A voxel's measured phase at echo $e$ is +$ + phi_e = W(theta + 2 pi f t_e), +$ +with $theta$ the coil offset and $f$ the local field in Hz. Two derived scales +recur throughout: + +#defn[ +- *Wrap spacing* $1 slash #dte$ — the field difference between adjacent + branches, in Hz. +- *Half-wrap* $1 slash (2 #dte)$ — the decision boundary of any + smallest-$abs(f)$ prior. +] + +== MCPC-3D-S + +The estimator forms the phase difference of the first two echoes, spatially +unwraps it to #phiuw, and back-extrapolates to $"TE" = 0$: +$ + hat(theta) = W(phi_0 - k dot #phiuw). +$ +Spatial unwrapping fixes #phiuw only up to a global additive $2 pi M$, $M in ZZ$ +— it recovers relative phase, not absolute. Selecting $M$ is the subject of this +note. + += Two ambiguities + +Two distinct things can be meant by "the branch is wrong". They behave +differently and conflating them is the main source of confusion. + +== Ambiguity A: the true alias + +#defn[ +*Proposition 1.* For evenly spaced echoes and any $N in ZZ$, define +$ + theta' = theta - W(2 pi k N), quad quad f' = f + N / (#dte). +$ +Then $(theta', f')$ reproduces @eq:model *identically*: $phi'_e = phi_e$ for +every echo. +] + +#proof[ +*Proof.* Using $W(2 pi k N) equiv 2 pi k N mod 2 pi$, +$ + theta' + 2 pi f' t_e + &equiv theta - 2 pi k N + 2 pi (f + N slash #dte) t_e \ + &= theta + 2 pi f t_e + 2 pi N (t_e slash #dte - k). +$ +For evenly spaced echoes $t_e slash #dte = k + e$, so the bracket is $e in ZZ$ +and vanishes under $W$. #h(1fr) $qed$ +] + +Here $theta$ and $f$ move *together*. The wrapped data is bit-identical, so no +statistic computed from the phase can distinguish the alternatives. Only a prior +can. + +#key[ +*Extra echoes do not help.* #link()[Prop. 1] assumes even spacing, and every echo +pair then aliases at the same period $1 slash #dte$: a five-echo acquisition +carries exactly as much branch information as a two-echo one. Unequal spacing +would break it — two pairs aliasing at incommensurate periods are simultaneously +satisfied only by the true $f$ — but even spacing is what an EPI echo train +naturally produces. +] + +== Ambiguity B: offset–branch mismatch + +#defn[ +*Proposition 2.* If #phiuw carries a branch error of $M$ wraps, @eq:est returns +$hat(theta)_M = W(theta - 2 pi k M)$, and the offset-removed phases satisfy +$ + psi_e = phi_e - hat(theta)_M equiv 2 pi f t_e + c mod 2 pi, + quad quad c = W(2 pi k M), +$ +with $c$ *identical on every echo*. +] + +#proof[ +*Proof.* Substituting $#phiuw -> 2 pi f #dte + 2 pi M$ into @eq:est gives +$hat(theta)_M = W(theta - 2 pi k M)$. Then +$psi_e equiv (theta + 2 pi f t_e) - (theta - 2 pi k M) = 2 pi f t_e + 2 pi k M$, +and $2 pi k M equiv c mod 2 pi$. #h(1fr) $qed$ +] + +Here $hat(theta)$ moves *alone*: the field has already been fixed upstream, so +offset and field are now mutually inconsistent, and that inconsistency is a +measurable constant. + +#key[ +*The distinction.* Ambiguity A changes $theta$ and $f$ together and is +invisible. Ambiguity B changes $theta$ while $f$ stays put, and shows up as a +constant lift off the origin. Only B is detectable. +] + += Detecting ambiguity B + +By #link()[Prop. 2] the signature of a branch error is a constant added to every +echo. A line through the offset-removed echoes therefore fails to pass through +the origin, and its intercept *is* that constant: +$ + psi_e approx s dot t_e + c, quad quad s = 2 pi f, quad c = W(2 pi k M). +$ +So the score for candidate $M$ is $abs(c)$, estimated over a mask of reliable +voxels. A correct branch extrapolates through zero; a wrong one does not. + +== The rejection scale is analytic + +Candidate intercepts sit on a lattice spaced by +$ + Delta c = abs(W(2 pi k)), +$ +which depends only on the echo times — no data, no fitting, no tuned threshold. +Classifying against $Delta c slash 2$ is nearest-neighbour assignment on that +lattice, not a chosen cutoff. + +== Degenerate cases + +#defn[ +*Integer $k$.* If $k in ZZ$ then $c = W(2 pi k M) = 0$ for all $M$, so the +intercept carries no signal. But by #link()[Prop. 2] $hat(theta)_M = W(theta)$ is +*also unchanged*, so no branch can affect the output. The degeneracy is benign +and @eq:step returns zero, which is the correct signal to do nothing. +] + +More generally the surviving intercept obeys a two-regime law in $k$, peaking at +$k = 2 slash 3$ and falling to zero at integer $k$. Below that peak the +achievable separation shrinks, and for small $k$ several branches can be exactly +through-origin at once. Since an EPI echo train has $"TE"_0$ shorter than the +spacing between echoes, $k < 1$ always, and low-$k$ protocols are precisely +those where the intercept test loses power. + += The decision rule + +#defn[ +1. *Consistency.* Score each candidate by its intercept. A branch carrying a + leftover intercept does not explain the data and is discarded. If exactly one + candidate survives, it is the answer regardless of its field. +2. *Prior, only on a tie.* If several survive, the alias of #link()[Prop. 1] is + genuinely reachable and the phase cannot rank them. Take the candidate whose + global field level is closest to zero. +3. If nothing survives, do nothing — a failed fit is not evidence for any + branch. +] + +== Both stages are necessary + +Neither stage suffices alone, for reasons that are structural rather than +empirical. + +#defn[ +*Stage 1 alone is insufficient.* When two branches are both exactly +through-origin their intercepts are equal to within numerical noise. Ranking +them is ranking noise; deferring leaves a known-bad offset in place. +] + +#defn[ +*Stage 2 alone is insufficient.* The smallest-$abs(f)$ prior is *symmetric about +zero*. Two branches whose fields are $+delta$ and $-delta$ are exactly +equidistant, so the prior cannot separate them — and after re-wrapping inside +the unwrapper, candidates $M$ and $-M$ can land at equal $abs(f)$. Only the +phase, through the intercept, breaks that symmetry. +] + +The two stages are therefore complementary in a precise sense: the intercept is +blind to a *sign-preserving* shift of the field, and the prior is blind to a +*sign-reversing* one. + += The prior and its estimator + +== What the prior actually asserts + +The scanner's frequency adjustment sets the demodulation frequency to the centre +of the water resonance over the shim volume. The physical statement is therefore + +#key[ +The *bulk-tissue* field level is near zero — not the mean field, and not the +median field over whatever mask happens to be in use. +] + +This matters because the field distribution over a brain is not symmetric. It is +a sharp peak at the bulk-tissue value plus long tails contributed by air–tissue +interfaces, sinuses, dropout and mask edges, where susceptibility gradients are +large. Any statistic that is a *quantile of the whole distribution* — the median +included — is displaced toward whichever tail is heavier. + +== Why a median is the wrong estimator + +Write the field distribution as a mixture +$ + p(f) = (1 - epsilon) dot p_"bulk" (f) + epsilon dot p_"tail" (f), +$ +with $p_"bulk"$ narrow and centred on the quantity the shim actually set, and +$p_"tail"$ broad and asymmetric. The mode of $p$ is a consistent estimator of +the centre of $p_"bulk"$ for small $epsilon$; the median of $p$ is not. Its bias +grows with both the tail fraction $epsilon$ and the tail asymmetry. + +Two consequences follow, and they are the same defect seen twice: + +#defn[ +- *Mask dilation is harmful.* Dilating a brain mask admits precisely the voxels + that populate $p_"tail"$, raising $epsilon$. A median over a dilated mask is + therefore more biased than the same median over an eroded one. +- *A bias of a fraction of a wrap is decisive.* The prior compares + $abs(hat(f))$ against $1 slash (2 #dte)$. Any estimator bias adds directly to + the distance from that boundary, so a subject whose true bulk field lies close + to the half-wrap can be pushed across it by estimator bias alone — flipping + the branch on the basis of an artefact of the statistic, not of the data. +] + +Weighting by magnitude squared suppresses the tails partially, because +low-signal voxels are over-represented there. It is a mitigation, not a fix: the +estimand is still a quantile of the mixture. + +== The half-sample mode + +The mode is the correct estimand but the usual estimators need a bin width or a +kernel bandwidth, which would reintroduce exactly the kind of tuned constant +this design avoids. The *half-sample mode* has neither. + +#defn[ +Sort the samples. Repeatedly find the shortest contiguous interval containing +half the remaining points and discard everything outside it. Stop when three or +fewer remain and return their mean. +] + +The justification is direct: samples are densest, per unit of value, where the +density is highest, so the shortest half-interval brackets the peak. Recursing +zooms in on it. Formally, if $x_((1)) <= ... <= x_((n))$ are the order +statistics and $h = ceil(n slash 2)$, the retained window is +$ + [x_((i^*)), space x_((i^* + h - 1))], + quad quad i^* = arg min_i (x_((i + h - 1)) - x_((i))), +$ +and all $n - h + 1$ candidate widths are obtained from a single vectorised +difference of two offset slices of the sorted array. + +#defn[ +*Properties.* +- *Parameter-free.* No bin width, no bandwidth, no threshold. +- *Cost $O(n log n)$*, dominated by the initial sort; each subsequent pass is + linear in a geometrically shrinking array, so the total after sorting is + $O(n)$. In particular it is cheaper than a weighted median, which must sort a + weight array as well. +- *Convergence.* Near a smooth unimodal peak, halving the sample halves the + window width, so the interval contracts geometrically and the estimate is + insensitive to how many passes are taken. +- *Weight-independent.* The peak location does not depend on how samples are + weighted, so the estimator inherits none of the weighting choices a median + requires. +] + +#defn[ +*Assumptions.* Unimodality — with a genuinely bimodal field the method converges +on the denser mode, which for brain data is bulk tissue but is not automatic. +Ties in the arg-min are resolved arbitrarily; on continuous data they have +measure zero, on quantised or very small samples they do not. The estimator +needs enough samples for the density to be resolved. +] + += Structural limits + +Three limits are properties of the acquisition, not of the algorithm. + +#defn[ +*Ambiguity A is not addressable.* By #link()[Prop. 1] the data is invariant under the +alias, so no rule can recover the true branch when the global field genuinely +exceeds $1 slash (2 #dte)$. The prior folds it back, and this is unavoidable. +Whether it is *reachable* is set by #dte, which is bounded below by the EPI +readout duration — echoes cannot be spaced closer than they can be read out — +and above by $T_2^*$, since several echoes must fit before the signal decays. +Those bounds confine #dte, and hence pin the half-wrap threshold well above +typical shim residuals. +] + +#defn[ +*Ultra-high field does not simply make this worse.* Shim residuals grow with +$B_0$, but $T_2^*$ shortens, which forces #dte down and therefore raises the +$1 slash (2 #dte)$ threshold. The two effects act in opposite directions on the +same quantity. +] + +#defn[ +*A shift of the multi-echo template is invisible here.* A multi-echo unwrapper +that global-corrects only its template echo, and derives the remaining echoes by +temporal unwrapping relative to it, propagates any whole-turn shift of the +template *proportionally to* $"TE"$. A proportional shift is a pure change of +slope: it moves the fitted field by $1 slash "TE"_0$ Hz and leaves the intercept +exactly zero. It is therefore a third failure mode, distinct from both +ambiguities above, and structurally undetectable by an intercept test. +] + += Summary + +#key[ +The branch error puts a constant $W(2 pi k M)$ on every echo, so it is visible +as a nonzero intercept when the echoes are extrapolated to $"TE" = 0$, and the +lattice spacing $abs(W(2 pi k))$ supplies the rejection scale analytically. When +several branches are through-origin the alias is exact and only a prior remains. +That prior asserts that the *bulk-tissue* field is near zero, which makes the +mode of the field distribution — not its median — the right estimand, and the +half-sample mode estimates it without introducing a tuned constant. The +intercept and the prior are blind to opposite symmetries, so both are required. +] From d24b6a347c0a5e9a284d37034a442d94317a7be5 Mon Sep 17 00:00:00 2001 From: Andrew Van Date: Mon, 27 Jul 2026 23:39:42 -0500 Subject: [PATCH 07/13] :memo: Reframe the theory note around the ambiguities themselves The first draft was written from the inside out: it opened on "branch selection", which presumes the reader already knows what a branch is and why one would be selecting among them. Rewritten for someone meeting the problem for the first time, and organised around the ambiguities rather than around the procedure that resolves them. The spine is now a taxonomy. Removing an estimated offset leaves a residual psi_e that should be a straight line through the origin; fitting it gives an intercept c and a slope s, and every possible error is a displacement in that (c, s) plane. Exactly three cases matter: A offset and field both wrong, in a matched way -> data unchanged, invisible B offset wrong alone -> constant c, detectable C field wrong alone -> pure slope, c stays zero Branch selection then falls out as the answer to B rather than being the premise, and the limits of the approach are visible as the cases the taxonomy already accounts for. Ambiguity C -- a whole-turn error in a multi-echo unwrapper's template echo, which propagates proportionally to echo time -- is promoted from a footnote to a peer of the other two, since it is exactly the failure an intercept test cannot see. Also opens with what phase is and why it is ambiguous at all, so no pipeline knowledge is assumed, and renames the file to match the new framing. --- notes/branch-selection-theory.pdf | Bin 150612 -> 0 bytes notes/branch-selection-theory.typ | 379 ---------------------- notes/phase-offset-ambiguities.pdf | Bin 0 -> 168452 bytes notes/phase-offset-ambiguities.typ | 504 +++++++++++++++++++++++++++++ 4 files changed, 504 insertions(+), 379 deletions(-) delete mode 100644 notes/branch-selection-theory.pdf delete mode 100644 notes/branch-selection-theory.typ create mode 100644 notes/phase-offset-ambiguities.pdf create mode 100644 notes/phase-offset-ambiguities.typ diff --git a/notes/branch-selection-theory.pdf b/notes/branch-selection-theory.pdf deleted file mode 100644 index 435a453e4d4f912bb6811a57370d8117b0e63a23..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 150612 zcmeFa1wa&C7cgw=BPw>snjp&7Y#JzK2O_qpD=eXuu%Ot2-Q9)VVxVGUqu4%xf!*C5 z|G6=%V9Yz*;5 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- fill: luma(240), inset: (x: 3pt, y: 0pt), outset: (y: 3pt), radius: 2pt, it, -) - -#let dte = $Delta "TE"$ -#let phiuw = $Delta phi_"uw"$ - -#let key(body) = block( - width: 100%, fill: luma(245), stroke: (left: 2pt + luma(140)), - inset: 9pt, radius: 2pt, body, -) -#let defn(body) = block( - width: 100%, stroke: 0.5pt + luma(160), inset: 9pt, radius: 2pt, body, -) -#let proof(body) = block(inset: (left: 12pt), text(size: 10pt, body)) - -#align(center)[ - #text(size: 17pt, weight: "bold")[ - Global $2 pi$ branch selection in MCPC-3D-S - ] - #v(0.3em) - #text(size: 11pt)[Theory and derivation] - #v(0.2em) - #text(size: 9.5pt, style: "italic")[warpkit — phase offset estimation for MEDIC] -] - -#v(1em) - -#outline(indent: auto, depth: 2) - -#v(1em) - -= The problem - -Multi-echo distortion correction needs the receive-coil phase offset $theta$ — -the phase each voxel carries at $"TE" = 0$, independent of the field. MCPC-3D-S -recovers it from the first two echoes. The recovered offset is pinned only up to -a whole number of turns of the unwrapped phase difference, and choosing the -wrong turn corrupts $theta$ by a fixed constant. This note derives what that -constant is, when it is detectable, and what decides the choice when it is not. - -Nothing here is specific to a dataset; every quantity is a function of the echo -times or of the signal model. - -= Notation and signal model - -Let $W(x) = arg(e^(i x))$ denote wrapping to $(-pi, pi]$. Echo times are -evenly spaced, -$ - t_e = "TE"_0 + e dot #dte, quad e = 0, 1, ..., E-1, -$ -and we write the dimensionless echo-time ratio -$ - k = "TE"_0 / (#dte). -$ -A voxel's measured phase at echo $e$ is -$ - phi_e = W(theta + 2 pi f t_e), -$ -with $theta$ the coil offset and $f$ the local field in Hz. Two derived scales -recur throughout: - -#defn[ -- *Wrap spacing* $1 slash #dte$ — the field difference between adjacent - branches, in Hz. -- *Half-wrap* $1 slash (2 #dte)$ — the decision boundary of any - smallest-$abs(f)$ prior. -] - -== MCPC-3D-S - -The estimator forms the phase difference of the first two echoes, spatially -unwraps it to #phiuw, and back-extrapolates to $"TE" = 0$: -$ - hat(theta) = W(phi_0 - k dot #phiuw). -$ -Spatial unwrapping fixes #phiuw only up to a global additive $2 pi M$, $M in ZZ$ -— it recovers relative phase, not absolute. Selecting $M$ is the subject of this -note. - -= Two ambiguities - -Two distinct things can be meant by "the branch is wrong". They behave -differently and conflating them is the main source of confusion. - -== Ambiguity A: the true alias - -#defn[ -*Proposition 1.* For evenly spaced echoes and any $N in ZZ$, define -$ - theta' = theta - W(2 pi k N), quad quad f' = f + N / (#dte). -$ -Then $(theta', f')$ reproduces @eq:model *identically*: $phi'_e = phi_e$ for -every echo. -] - -#proof[ -*Proof.* Using $W(2 pi k N) equiv 2 pi k N mod 2 pi$, -$ - theta' + 2 pi f' t_e - &equiv theta - 2 pi k N + 2 pi (f + N slash #dte) t_e \ - &= theta + 2 pi f t_e + 2 pi N (t_e slash #dte - k). -$ -For evenly spaced echoes $t_e slash #dte = k + e$, so the bracket is $e in ZZ$ -and vanishes under $W$. #h(1fr) $qed$ -] - -Here $theta$ and $f$ move *together*. The wrapped data is bit-identical, so no -statistic computed from the phase can distinguish the alternatives. Only a prior -can. - -#key[ -*Extra echoes do not help.* #link()[Prop. 1] assumes even spacing, and every echo -pair then aliases at the same period $1 slash #dte$: a five-echo acquisition -carries exactly as much branch information as a two-echo one. Unequal spacing -would break it — two pairs aliasing at incommensurate periods are simultaneously -satisfied only by the true $f$ — but even spacing is what an EPI echo train -naturally produces. -] - -== Ambiguity B: offset–branch mismatch - -#defn[ -*Proposition 2.* If #phiuw carries a branch error of $M$ wraps, @eq:est returns -$hat(theta)_M = W(theta - 2 pi k M)$, and the offset-removed phases satisfy -$ - psi_e = phi_e - hat(theta)_M equiv 2 pi f t_e + c mod 2 pi, - quad quad c = W(2 pi k M), -$ -with $c$ *identical on every echo*. -] - -#proof[ -*Proof.* Substituting $#phiuw -> 2 pi f #dte + 2 pi M$ into @eq:est gives -$hat(theta)_M = W(theta - 2 pi k M)$. Then -$psi_e equiv (theta + 2 pi f t_e) - (theta - 2 pi k M) = 2 pi f t_e + 2 pi k M$, -and $2 pi k M equiv c mod 2 pi$. #h(1fr) $qed$ -] - -Here $hat(theta)$ moves *alone*: the field has already been fixed upstream, so -offset and field are now mutually inconsistent, and that inconsistency is a -measurable constant. - -#key[ -*The distinction.* Ambiguity A changes $theta$ and $f$ together and is -invisible. Ambiguity B changes $theta$ while $f$ stays put, and shows up as a -constant lift off the origin. Only B is detectable. -] - -= Detecting ambiguity B - -By #link()[Prop. 2] the signature of a branch error is a constant added to every -echo. A line through the offset-removed echoes therefore fails to pass through -the origin, and its intercept *is* that constant: -$ - psi_e approx s dot t_e + c, quad quad s = 2 pi f, quad c = W(2 pi k M). -$ -So the score for candidate $M$ is $abs(c)$, estimated over a mask of reliable -voxels. A correct branch extrapolates through zero; a wrong one does not. - -== The rejection scale is analytic - -Candidate intercepts sit on a lattice spaced by -$ - Delta c = abs(W(2 pi k)), -$ -which depends only on the echo times — no data, no fitting, no tuned threshold. -Classifying against $Delta c slash 2$ is nearest-neighbour assignment on that -lattice, not a chosen cutoff. - -== Degenerate cases - -#defn[ -*Integer $k$.* If $k in ZZ$ then $c = W(2 pi k M) = 0$ for all $M$, so the -intercept carries no signal. But by #link()[Prop. 2] $hat(theta)_M = W(theta)$ is -*also unchanged*, so no branch can affect the output. The degeneracy is benign -and @eq:step returns zero, which is the correct signal to do nothing. -] - -More generally the surviving intercept obeys a two-regime law in $k$, peaking at -$k = 2 slash 3$ and falling to zero at integer $k$. Below that peak the -achievable separation shrinks, and for small $k$ several branches can be exactly -through-origin at once. Since an EPI echo train has $"TE"_0$ shorter than the -spacing between echoes, $k < 1$ always, and low-$k$ protocols are precisely -those where the intercept test loses power. - -= The decision rule - -#defn[ -1. *Consistency.* Score each candidate by its intercept. A branch carrying a - leftover intercept does not explain the data and is discarded. If exactly one - candidate survives, it is the answer regardless of its field. -2. *Prior, only on a tie.* If several survive, the alias of #link()[Prop. 1] is - genuinely reachable and the phase cannot rank them. Take the candidate whose - global field level is closest to zero. -3. If nothing survives, do nothing — a failed fit is not evidence for any - branch. -] - -== Both stages are necessary - -Neither stage suffices alone, for reasons that are structural rather than -empirical. - -#defn[ -*Stage 1 alone is insufficient.* When two branches are both exactly -through-origin their intercepts are equal to within numerical noise. Ranking -them is ranking noise; deferring leaves a known-bad offset in place. -] - -#defn[ -*Stage 2 alone is insufficient.* The smallest-$abs(f)$ prior is *symmetric about -zero*. Two branches whose fields are $+delta$ and $-delta$ are exactly -equidistant, so the prior cannot separate them — and after re-wrapping inside -the unwrapper, candidates $M$ and $-M$ can land at equal $abs(f)$. Only the -phase, through the intercept, breaks that symmetry. -] - -The two stages are therefore complementary in a precise sense: the intercept is -blind to a *sign-preserving* shift of the field, and the prior is blind to a -*sign-reversing* one. - -= The prior and its estimator - -== What the prior actually asserts - -The scanner's frequency adjustment sets the demodulation frequency to the centre -of the water resonance over the shim volume. The physical statement is therefore - -#key[ -The *bulk-tissue* field level is near zero — not the mean field, and not the -median field over whatever mask happens to be in use. -] - -This matters because the field distribution over a brain is not symmetric. It is -a sharp peak at the bulk-tissue value plus long tails contributed by air–tissue -interfaces, sinuses, dropout and mask edges, where susceptibility gradients are -large. Any statistic that is a *quantile of the whole distribution* — the median -included — is displaced toward whichever tail is heavier. - -== Why a median is the wrong estimator - -Write the field distribution as a mixture -$ - p(f) = (1 - epsilon) dot p_"bulk" (f) + epsilon dot p_"tail" (f), -$ -with $p_"bulk"$ narrow and centred on the quantity the shim actually set, and -$p_"tail"$ broad and asymmetric. The mode of $p$ is a consistent estimator of -the centre of $p_"bulk"$ for small $epsilon$; the median of $p$ is not. Its bias -grows with both the tail fraction $epsilon$ and the tail asymmetry. - -Two consequences follow, and they are the same defect seen twice: - -#defn[ -- *Mask dilation is harmful.* Dilating a brain mask admits precisely the voxels - that populate $p_"tail"$, raising $epsilon$. A median over a dilated mask is - therefore more biased than the same median over an eroded one. -- *A bias of a fraction of a wrap is decisive.* The prior compares - $abs(hat(f))$ against $1 slash (2 #dte)$. Any estimator bias adds directly to - the distance from that boundary, so a subject whose true bulk field lies close - to the half-wrap can be pushed across it by estimator bias alone — flipping - the branch on the basis of an artefact of the statistic, not of the data. -] - -Weighting by magnitude squared suppresses the tails partially, because -low-signal voxels are over-represented there. It is a mitigation, not a fix: the -estimand is still a quantile of the mixture. - -== The half-sample mode - -The mode is the correct estimand but the usual estimators need a bin width or a -kernel bandwidth, which would reintroduce exactly the kind of tuned constant -this design avoids. The *half-sample mode* has neither. - -#defn[ -Sort the samples. Repeatedly find the shortest contiguous interval containing -half the remaining points and discard everything outside it. Stop when three or -fewer remain and return their mean. -] - -The justification is direct: samples are densest, per unit of value, where the -density is highest, so the shortest half-interval brackets the peak. Recursing -zooms in on it. Formally, if $x_((1)) <= ... <= x_((n))$ are the order -statistics and $h = ceil(n slash 2)$, the retained window is -$ - [x_((i^*)), space x_((i^* + h - 1))], - quad quad i^* = arg min_i (x_((i + h - 1)) - x_((i))), -$ -and all $n - h + 1$ candidate widths are obtained from a single vectorised -difference of two offset slices of the sorted array. - -#defn[ -*Properties.* -- *Parameter-free.* No bin width, no bandwidth, no threshold. -- *Cost $O(n log n)$*, dominated by the initial sort; each subsequent pass is - linear in a geometrically shrinking array, so the total after sorting is - $O(n)$. In particular it is cheaper than a weighted median, which must sort a - weight array as well. -- *Convergence.* Near a smooth unimodal peak, halving the sample halves the - window width, so the interval contracts geometrically and the estimate is - insensitive to how many passes are taken. -- *Weight-independent.* The peak location does not depend on how samples are - weighted, so the estimator inherits none of the weighting choices a median - requires. -] - -#defn[ -*Assumptions.* Unimodality — with a genuinely bimodal field the method converges -on the denser mode, which for brain data is bulk tissue but is not automatic. -Ties in the arg-min are resolved arbitrarily; on continuous data they have -measure zero, on quantised or very small samples they do not. The estimator -needs enough samples for the density to be resolved. -] - -= Structural limits - -Three limits are properties of the acquisition, not of the algorithm. - -#defn[ -*Ambiguity A is not addressable.* By #link()[Prop. 1] the data is invariant under the -alias, so no rule can recover the true branch when the global field genuinely -exceeds $1 slash (2 #dte)$. The prior folds it back, and this is unavoidable. -Whether it is *reachable* is set by #dte, which is bounded below by the EPI -readout duration — echoes cannot be spaced closer than they can be read out — -and above by $T_2^*$, since several echoes must fit before the signal decays. -Those bounds confine #dte, and hence pin the half-wrap threshold well above -typical shim residuals. -] - -#defn[ -*Ultra-high field does not simply make this worse.* Shim residuals grow with -$B_0$, but $T_2^*$ shortens, which forces #dte down and therefore raises the -$1 slash (2 #dte)$ threshold. The two effects act in opposite directions on the -same quantity. -] - -#defn[ -*A shift of the multi-echo template is invisible here.* A multi-echo unwrapper -that global-corrects only its template echo, and derives the remaining echoes by -temporal unwrapping relative to it, propagates any whole-turn shift of the -template *proportionally to* $"TE"$. A proportional shift is a pure change of -slope: it moves the fitted field by $1 slash "TE"_0$ Hz and leaves the intercept -exactly zero. It is therefore a third failure mode, distinct from both -ambiguities above, and structurally undetectable by an intercept test. -] - -= Summary - -#key[ -The branch error puts a constant $W(2 pi k M)$ on every echo, so it is visible -as a nonzero intercept when the echoes are extrapolated to $"TE" = 0$, and the -lattice spacing $abs(W(2 pi k))$ supplies the rejection scale analytically. When -several branches are through-origin the alias is exact and only a prior remains. -That prior asserts that the *bulk-tissue* field is near zero, which makes the -mode of the field distribution — not its median — the right estimand, and the -half-sample mode estimates it without introducing a tuned constant. The -intercept and the prior are blind to opposite symmetries, so both are required. -] diff --git a/notes/phase-offset-ambiguities.pdf b/notes/phase-offset-ambiguities.pdf new file mode 100644 index 0000000000000000000000000000000000000000..56bc7db18fef2727a96b5dd68c43fc9913146afb GIT binary patch literal 168452 zcmeFa1$-1q(=d*^AFhkTL1cGkcaz{r@L(bMAt6~H5F^>(?oRMK++7Zr;D@_Ahr3I# zX3`q46|fbtHDYVR z)=b*ARjX#QM23<<$a6SYH47fO;}m^mg0ugzWaid(r+FX$5yZkf24E-hlrY3t~tY z8b#==>tYFymb;}Cv~^25vF{ckp)~fULoi4v-+53-*q@A_-Sh|vOhH#8qGMR+xG-yP zT@4>yaHP)CD>%fei|ri|VX?;a1_V<)hXz}N<@wOQZ1W)_^5>(cLc_3Z557S$wy%$Z zoPdxX7U1*ns7Q-ED1Fv8DDBqisFZ0K79>Oqijn--NS(qg;h|0kqK_&DQfLMz!m4NN zNe5+;#!$~drKm>?AnqkDf;s7|!M&m)qarAld&WcoPX|ZF_O!&vf+A4lDJG-V95?|= zd%aQ!MIbI=)EG-wONccrDpD645**Gn2uOWzYgmYcT){z!(0&|{tAJC_q=jKubT|TyQuj+2Gh%ON`aVH4b`6 zPx`S(w%ss{R~d|qhovD2X?s(`Mp#0_Ks&(lL&(gS6=)q65p0dJQQ1LL=}FH{2WErW z!zbu!vJNHXPSzz|&oFBjSnqVfI%`}c`6W6yCKz;+g0~#>5rxfuR3%9b3WY?eALxO+ z#b9T=#c<*vvnDD~i2TSne$mLen#BY~N%|FF&k zhegKf;v!=#u~FeY2!F{O?VzxzMc{NOigW~@b?>EX8W#}}9MfCsOmt8<6mi%YwTl|X zS;NC3!8BqxW14`Xj9DU*$)o_CNE~+2FfJmtji*lZqVVeU@E7etXMn$`WWwqzdQmaa ziSQTYWLSnpFG?J+jYKa>Zk-waqDWTSRpdeCY@1qR;zF!JF&0Z;RFu^tFe=>AAUGOq zJbw^NU|x$y)9Bzxk047-L@bzmF;=ETJQ{nzlOWaX=b90;ty-8$<<0s4#@2fxl1=a@!y_mHJzI@*Aewrn&~(6X2KGDfhJlR;#54%3 zD_{WwPTC;o4EUu8cj7}v_!6Ho=?q9RxX<9t=4bHI37(Wx@Rd$r`W(K3dx2bm@QqGj zno%bhbOK{R_zM1HBfyXFAidQ{o}+&OCx{G{Fi*CC=>Y%G1N;m;p&$@Zru8kob%M8o zD@oUcRocKnv_tF+heWzDXpyeihzzetW2s@!iR?M4VY5csOAQ-O;say{h6>~fi8UA( zA%92>8v%I*8}v+&Z={Az327rWY)Z&ZQp2W%oFz4EO2}l`c(W-Xzex?760)Aefu4*| zG8;)h6}GAmA~GE_~y1q2NTX4R|`GH zuk^THB)vs=+?nx|Ukx3-kYy7Hw6%8SyAx%PsPOWUI+7;z_;_`9rdT z5Ue#{(rnhK0{MZkG8k1Mf#y7u9sV+qhyylw%&*zxkWsnQK6TtSZ^HP8~I3X zaa?cY_awK{^R%_CbmWq)blwQTPihU4&4ZS(Ub2Ysdjr}?ww7T;dd}dMe>jTVprZ7+j#EwyMV*}g}3&{8s> zMPxvC$AA`*?Hche{4ZX5=Zjf8Hg4(WeeLxG|Ah*b8 zMybVd(K46gIr25?8?@vNs@Cy&dTm=l$#m%oYA_-ss#bysTeOW080t{9((hbPT_e5B z$f0IjY5(gj(pidFqEBK#ALPH^qJ*PwVEDhbMfpXa$nYPx$VV8wG@$Pxhoj|dlxy^9 z49RXKI->l_{iFQK*LWZLJO;T%_`RhTblU!%Op*T~oCD;c=0V?T%w z>_?D167+x!*xJ5l@JQDJdNTiWEyI(5o{@l_48LV~641jDFxt#-89oH`Xyg_>6=@?w zkgpLR(lvTE!hhT%oN~X6Pvmjs@9BNEm6%M2ZEK*fBw+kpkgpl12pC6Kx6<#33G}f9 z^r-~8n@o0X7$@0U#`(7UB+UGmTlDD!^zryDVo<&xF^oQ-fN^~RSE6@4iIZCL?* ztRJ`1^HPg&$^A&*BR|UDBVWqb(tQXAM!{|KLB7POxFEOEbJ${(T##F&1KQpK+SUTv z)V3DqjW)A@HnqCN`C$}XP_;M@v~&fz#d%=#Sdd$`;t6Q!3TV*^7#$YSVzsq6K3cef z3Gt$8rFmeBmYaZhKk%4$HNs}MBCu!x6(L_ej-{HuDA4g zn+70kLN$Phw#ol%*;@CX;{iDiZ8!0Ut#rTCGKmt=rV`O+646!?(MA$6E-s>tB%;ma zdW+-9&od1pqD>`Y++5uv2VxvuL|e$#;{4Hu5qT}zHlo}z%_X8uBcd%MWg3YX2UNFA zQ;BGMh-hQjT7(O27Ex-^)(}-K&I4@@QEm}#v_VANp%>Bi5YhIqwTNGg#))Wis9T&L z+8ml;Rg3VT4I`rM!fz>DHjP6TBBpWBsuATD zAw?@klv^ATZ4eRnx<&aKA(XGB`%zb-^&`ry^t;re#G~yaCb^aRLCmSV!n<`fUVGe+1v${sKCNLx;?hBFUnhK8SYW_YoG^ zjdVf@M<&83Y}A<~S~T*l7pOqFN288ifgJSys#Q5dl3%$)23k5n0)J!(y!NDHs?eiG z8yKw-ygSkxlCM1)hr?Su0^jrEUcaWe)es!ULJ&PNV=jn`< z&l4HsKF@H=G&4MHb`B{0o!H6o?w@wTIdZw!Y+$OkU3E4(SgmcH)>;- zdS5*3(*C%EDzVFK5Ld9vAXT7;KFS@$5xWdT<>Q)*5n`8#7GT%pTrp5ni8}}6U1moC zcG18h3G-gs*rn?~7rUBLXNO%TdoHM5X7ef0qq)SXuuG*<`M9Qh!Q*yH8!cNjNSVE< z>`YURkW`d&i6D4$NTn@JOwBM8m5_=`y$)OzgI=3dltp-?qWy9CLrE&OQG%u9GcR>beu!PRQo$Uk z*ps(z3}$WY(i^$h)f78B>@wkVLC-KBONky$u~cD~f~a_$`UiZ(P|KNlU4(bNoMV^X z!0&{q2IiwdFX-ICj*Z}{ja^Dr9(HMe+(DJtWwCEpc$Yz{M33eo=ZIYfqVjP~`GUvo zv^e_Tn54mGt?W!wj+h{@;5;B%;5cve1}@f6%`kVBkcvt@eF03+CKcr?E~zvZGdof- z5p+Qi7^F({Xi6KB%BIB`jLPRV1&N_o_O1nv6G?CQ5qjB92SG0tdmY5d1Z_8xWhnE| zOZ(#vszfi_b8$ry7^DjH&_}t0AbMHOIY7ohR6edLNQhmk`U&%L+(^0w#GL~&FWVm? 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block(above: 1.1em, below: 0.6em)[ + #set text(size: 11pt, weight: "bold") + #it +] +#show raw.where(block: false): it => box( + fill: luma(240), inset: (x: 3pt, y: 0pt), outset: (y: 3pt), radius: 2pt, it, +) + +#let dte = $Delta "TE"$ +#let phiuw = $Delta phi_"uw"$ + +#let key(body) = block( + width: 100%, fill: luma(245), stroke: (left: 2pt + luma(140)), + inset: 9pt, radius: 2pt, body, +) +#let defn(body) = block( + width: 100%, stroke: 0.5pt + luma(160), inset: 9pt, radius: 2pt, body, +) +#let proof(body) = block(inset: (left: 12pt), text(size: 10pt, body)) + +#align(center)[ + #text(size: 17pt, weight: "bold")[ + Phase offset ambiguities in multi-echo field mapping + ] + #v(0.3em) + #text(size: 11pt)[What can be recovered from wrapped phase, and what cannot] + #v(0.2em) + #text(size: 9.5pt, style: "italic")[warpkit] +] + +#v(1em) + +#outline(indent: auto, depth: 2) + +#v(1.2em) + += What this note is about + +A gradient-echo image records, at every voxel, a complex number. Its magnitude +is the familiar anatomical image; its phase carries the local magnetic field. +Distortion correction needs that field, so it needs the phase — but the phase is +not the field alone. It is the field plus a fixed contribution from the receive +coil, and it is only ever measured modulo a full turn. + +Those two facts together mean the field cannot be recovered uniquely. Some of +what is lost can be reconstructed from multiple echoes; some of it provably +cannot. This note draws that line precisely: it enumerates the ways an answer +can be wrong, shows which of them leave a trace in the data, and derives what to +do about each. + +No familiarity with any particular pipeline is assumed. Everything below follows +from the signal model. + += The measurement + +== Signal model + +Acquire $E$ echoes at times $t_0 < t_1 < ... < t_(E-1)$. Field evolution is +linear in time, so before wrapping, the phase at echo $e$ would be +$theta + 2 pi f t_e$, where + +- $theta$ is the *coil phase offset* — the phase a voxel already carries at + $t = 0$, set by receive-coil geometry and independent of the field, and +- $f$ is the *field offset* in Hz, the quantity actually wanted. + +What is recorded is that value wrapped into a single turn. Writing +$W(x) = arg(e^(i x))$ for wrapping to $(-pi, pi]$, + +$ + phi_e = W(theta + 2 pi f t_e). +$ + +#key[ +The entire difficulty comes from $W$. Without it, two echoes would determine +$theta$ and $f$ exactly by fitting a straight line. With it, the line is +observed only modulo $2 pi$, and infinitely many $(theta, f)$ pairs remain +compatible with the data. +] + +== Evenly spaced echoes + +An echo-planar train produces echoes at constant spacing, + +$ + t_e = "TE"_0 + e dot #dte, quad e = 0, 1, ..., E-1, +$ + +with $"TE"_0$ the first echo time and #dte the spacing. The single most +important derived quantity below is their ratio, + +$ + k = "TE"_0 / (#dte), +$ + +dimensionless and fixed by the protocol. Two field scales also recur: + +#defn[ +- *Wrap spacing* $1 slash #dte$ (Hz) — how far apart the candidate field + solutions sit. +- *Half-wrap* $1 slash (2 #dte)$ (Hz) — the midpoint between neighbouring + candidates, and therefore the decision boundary for any rule preferring the + smallest field. +] + +== Removing the offset + +Because $theta$ is common to every echo, it cancels in a difference. The phase +difference of the first two echoes removes the coil contribution and leaves a +quantity proportional to the field: + +$ + Delta phi = W(phi_1 - phi_0) approx W(2 pi f #dte). +$ + +This is still wrapped, but a difference of *neighbouring* echoes varies slowly +in space, so a region-growing algorithm can restore the missing turns and +produce an unwrapped difference #phiuw. Extrapolating back to $t = 0$ then +recovers the offset: + +$ + hat(theta) = W(phi_0 - k dot #phiuw). +$ + +#key[ +Spatial unwrapping recovers *relative* phase between voxels. It cannot know the +absolute number of turns, so it determines #phiuw only up to a global additive +constant $2 pi M$ for some unknown integer $M$. Pinning down $M$ is the decision +this note is about. +] + += A taxonomy of wrong answers + +Suppose some estimate $(hat(theta), hat(f))$ has been produced. To classify how +it can be wrong, remove the estimated offset from the data and inspect what +remains: + +$ + psi_e = phi_e - hat(theta). +$ + +Were the estimate perfect, $psi_e$ would be exactly proportional to $t_e$: a +straight line through the origin whose slope is the field. So fit + +$ + psi_e approx c + s dot t_e +$ + +and read off two numbers — the *intercept* $c$ and the *slope* $s$. Every +possible error is a displacement in this $(c, s)$ plane, and exactly three cases +are of interest. + +#defn[ +#set enum(numbering: "A.") ++ *Both move together.* Offset and field are both wrong, in a matched + combination that leaves $psi_e$ unchanged. Invisible. ++ *The offset moves alone.* $c$ shifts while $s$ stays correct. Visible as a + line that misses the origin. ++ *The field moves alone.* $s$ shifts while $c$ stays exactly zero. Visible as a + wrong slope — but *not* as a wrong intercept. +] + +Each is taken in turn below. The taxonomy matters because a test that measures +$c$ finds case B and is blind to A and C *by construction* — not through any +weakness of implementation. + +== Ambiguity A: an exact symmetry of the data + +#defn[ +*Proposition 1.* For evenly spaced echoes and any integer $N$, define +$ + theta' = theta - W(2 pi k N), quad quad f' = f + N / (#dte). +$ +Then $(theta', f')$ reproduces @eq:model *identically*: $phi'_e = phi_e$ for +every echo. +] + +#proof[ +*Proof.* Using $W(2 pi k N) equiv 2 pi k N mod 2 pi$, +$ + theta' + 2 pi f' t_e + &equiv theta - 2 pi k N + 2 pi (f + N slash #dte) t_e \ + &= theta + 2 pi f t_e + 2 pi N (t_e slash #dte - k). +$ +For evenly spaced echoes $t_e slash #dte = k + e$, so the bracket equals the +integer $e$ and the term vanishes under $W$. #h(1fr) $qed$ +] + +Offset error and field error are locked together here: shift the field by one +wrap spacing and the offset compensates exactly. The recorded data is +*bit-identical* — not approximately, but exactly. + +#key[ +No statistic computed from the phase can separate these alternatives, because +there is nothing to separate: they are the same measurement. Resolving A +requires information from outside the data, which is to say a prior. +] + +=== Why more echoes do not help + +It is natural to expect a five-echo acquisition to constrain the field better +than a two-echo one. For this question it does not. +#link()[Prop. 1] holds for all echoes simultaneously, and its period +$1 slash #dte$ depends only on the spacing, which every pair shares. All pairs +therefore alias at the same period and agree with one another. + +Unequal spacing would break the tie — two pairs whose aliases have +incommensurate periods are simultaneously satisfied only by the true field — but +an EPI echo train produces equal spacing naturally, so that escape is not +usually available. + +== Ambiguity B: the offset moves alone + +Now suppose spatial unwrapping settled on the wrong number of turns, so #phiuw +is off by $2 pi M$, while the field has already been fixed by an earlier stage +and does not follow. + +#defn[ +*Proposition 2.* If #phiuw carries a branch error of $M$ wraps, @eq:est returns +$hat(theta)_M = W(theta - 2 pi k M)$, and the offset-removed phases satisfy +$ + psi_e equiv 2 pi f t_e + c mod 2 pi, + quad quad c = W(2 pi k M), +$ +with $c$ *identical on every echo*. +] + +#proof[ +*Proof.* Substituting $#phiuw -> 2 pi f #dte + 2 pi M$ into @eq:est gives +$hat(theta)_M = W(theta - 2 pi k M)$. Then +$psi_e equiv (theta + 2 pi f t_e) - (theta - 2 pi k M) = 2 pi f t_e + 2 pi k M$, +and $2 pi k M equiv c mod 2 pi$. #h(1fr) $qed$ +] + +Comparing against @eq:fit: the slope is untouched and the intercept has moved to +$c = W(2 pi k M)$. Because $c$ is the *same on every echo*, it does not resemble +noise and does not average away — it is a rigid vertical displacement of the +whole line. + +#key[ +This is the one detectable case, and the reason is worth stating plainly: offset +and field have become mutually inconsistent, and consistency between them is +something the data can check. +] + +== Ambiguity C: the field moves alone + +The third case is easy to overlook, because it does not originate in the offset +estimate at all. + +Multi-echo unwrappers commonly unwrap a single *template* echo spatially and +derive the remaining echoes by temporal unwrapping relative to it, each echo +referenced to the template scaled by its own echo time. A whole-turn error in +the template then propagates to echo $e$ scaled by $t_e slash t_0$ — that is, +*proportionally to echo time*. + +A displacement proportional to $t_e$ is, by @eq:fit, a pure change of slope: + +$ + Delta s = (2 pi) / t_0, quad quad Delta c = 0. +$ + +#key[ +The intercept is exactly zero here. A test built on the intercept cannot see +ambiguity C at all — not poorly, but not at all. It surfaces instead as a field +error of $1 slash t_0$ Hz, a different size from the $1 slash #dte$ of ambiguity +A, which is the practical way to tell the two apart. +] + += Detecting ambiguity B + +By #link()[Prop. 2] a branch error of $M$ wraps places the constant +$c = W(2 pi k M)$ on every echo. Fitting the line of @eq:fit and reading its +intercept therefore measures the error directly. Because the effect is a single +global constant rather than a per-voxel one, the intercept can be aggregated +over many voxels, suppressing noise without diluting the signal. + +The procedure follows: for each candidate $M$, reconstruct the offset it +implies, remove it, fit, and record $abs(c)$. The correct branch extrapolates +through the origin; a wrong one does not. + +== The rejection scale is analytic + +Any test needs a scale — how large must $abs(c)$ be before a candidate is +rejected? Here the model supplies it rather than the user. Candidate intercepts +lie on a lattice of spacing + +$ + Delta c = abs(W(2 pi k)), +$ + +a function of the echo times alone. Classifying against $Delta c slash 2$ is +nearest-neighbour assignment on that lattice — the midpoint between two adjacent +possibilities — rather than a tuned threshold. + +== Where the test loses power + +#defn[ +*Integer $k$.* If $k$ is a whole number then $c = W(2 pi k M) = 0$ for every +$M$, so the intercept carries no information. But by +#link()[Prop. 2] the recovered offset $hat(theta)_M = W(theta)$ is +*also unchanged*, so no choice of branch can affect the output. The degeneracy +is harmless, and @eq:step returning zero is the correct signal to do nothing. +] + +Away from integer $k$ the achievable separation varies smoothly, peaking near +$k = 2 slash 3$ and shrinking to either side. For small $k$ in particular, +several candidates can be exactly through-origin at once: the alias of +#link()[Prop. 1] becomes reachable *within the candidate set*, and +the intercept cannot rank the survivors. Since an EPI train has $"TE"_0$ shorter +than the echo spacing, $k < 1$ always, so this regime is not exotic. + += Choosing among indistinguishable candidates + +Once more than one candidate is through-origin the data has been exhausted and +something else must decide. Two facts constrain what that can be. + +#defn[ +*A consistency test alone is insufficient.* Tied candidates have equal +intercepts to within numerical noise. Ranking them is ranking noise. +] + +#defn[ +*A field-magnitude prior alone is insufficient.* A rule preferring the smallest +$abs(f)$ is *symmetric about zero*: candidates at $+delta$ and $-delta$ are +equidistant and cannot be separated. Such pairs do arise, because re-wrapping +inside the unwrapper can map candidates $M$ and $-M$ to equal $abs(f)$. +] + +#key[ +The two are complementary in a precise sense. The intercept is blind to a +*sign-preserving* change of the field — that is ambiguity C, and the alias +direction of A. The magnitude prior is blind to a *sign-reversing* one. Together +they cover both, which is why a workable rule needs both stages. +] + +The resulting rule: reject candidates whose intercept is inconsistent; if one +survives, take it; if several survive, apply the prior among them; if none +survives, change nothing, since a failed fit is not evidence for any particular +candidate. + += The prior and how to estimate it + +== What the prior actually asserts + +Before imaging, the scanner sets its demodulation frequency to the centre of the +water resonance over the shim volume. The physical content of the prior is +therefore: + +#key[ +The *bulk-tissue* field is near zero. Not the mean field, and not the median +field over whatever mask happens to be convenient. +] + +The distinction is not pedantry. The field distribution over a brain is +asymmetric: a narrow peak at the bulk-tissue value, plus long tails from +air–tissue interfaces, sinuses, signal dropout and mask edges, where +susceptibility gradients are large. Any statistic that is a *quantile of the +whole distribution* — the median included — is displaced toward whichever tail +is heavier. + +== Why the median is the wrong estimator + +Model the distribution as a mixture of a narrow bulk component and a broad tail +component, + +$ + p(f) = (1 - epsilon) dot p_"bulk" (f) + epsilon dot p_"tail" (f). +$ + +The centre of $p_"bulk"$ is what the shim actually set. For small $epsilon$ the +*mode* of $p$ is a consistent estimator of that centre; the *median* of $p$ is +not, and its bias grows with both the tail fraction $epsilon$ and the tail +asymmetry. Two consequences follow, and they are the same defect seen twice. + +#defn[ +- *Enlarging the mask makes the estimate worse.* Dilating a brain mask admits + precisely the voxels populating $p_"tail"$, raising $epsilon$. A median over a + generous mask is more biased than the same median over a conservative one — + the opposite of the usual intuition that more data is better. +- *A bias of a fraction of a wrap is decisive.* The prior compares + $abs(hat(f))$ against the half-wrap $1 slash (2 #dte)$. Estimator bias adds + directly to the distance from that boundary, so a subject whose true bulk + field lies near the half-wrap can be pushed across it by bias alone. The + choice then turns on an artefact of the statistic rather than a property of + the data. +] + +Weighting voxels by squared magnitude helps, since low-signal voxels are +over-represented in the tails, but it is a mitigation rather than a fix: the +estimand is still a quantile of the mixture. + +== Estimating the mode without a tuning parameter + +The mode is the right estimand, but histogram and kernel estimators need a bin +width or a bandwidth — reintroducing exactly the kind of tuned constant a +principled rule should avoid. The *half-sample mode* needs neither. + +#defn[ +Sort the samples. Repeatedly locate the shortest contiguous interval containing +half the remaining points, and discard everything outside it. Stop when three or +fewer points remain, and return their mean. +] + +The justification is direct: per unit of value, samples are densest where the +density is highest, so the shortest half-interval brackets the peak, and +recursing zooms in on it. Formally, with order statistics +$x_((1)) <= ... <= x_((n))$ and $h = ceil(n slash 2)$, the retained window is + +$ + [x_((i^*)), space x_((i^* + h - 1))], + quad quad i^* = arg min_i (x_((i + h - 1)) - x_((i))), +$ + +and all $n - h + 1$ candidate widths follow from one vectorised difference of +two offset slices of the sorted array. + +#defn[ +*Properties.* +- *Parameter-free.* No bin width, no bandwidth, no threshold. +- *Cost.* $O(n log n)$, dominated by the initial sort; subsequent passes are + linear in a geometrically shrinking array. Cheaper than a weighted median, + which must sort a weight array as well. +- *Convergence.* Near a smooth unimodal peak, halving the sample halves the + window width, so the interval contracts geometrically and the result is + insensitive to how many passes are taken. +- *Weight-independent.* Peak location does not depend on how samples are + weighted, so none of the weighting choices a median requires are inherited. +] + +#defn[ +*Assumptions.* Unimodality — with a genuinely bimodal distribution the method +converges on the denser mode. Ties in the arg-min are resolved arbitrarily; on +continuous data they have measure zero, on quantised or very small samples they +do not. And the sample must be large enough to resolve the density at all. +] + += What remains unresolvable + +Three limits are properties of the acquisition rather than of any algorithm. + +#defn[ +*Ambiguity A cannot be corrected.* By #link()[Prop. 1] the data is +invariant under the alias, so when the true field genuinely exceeds the +half-wrap, no rule recovers it: the prior folds it back, and that is that. +Whether the situation is *reachable* is governed by #dte, bounded below by the +readout duration — echoes cannot be spaced closer than they can be read out — +and above by $T_2^*$, since several echoes must fit before the signal decays. +Those bounds confine #dte, and so hold the half-wrap threshold above typical +shim residuals. +] + +#defn[ +*Higher field strength does not simply make it worse.* Shim residuals grow with +$B_0$, which argues for more risk. But $T_2^*$ shortens, forcing #dte down and +therefore raising the $1 slash (2 #dte)$ threshold. The two effects push the +same quantity in opposite directions. +] + +#defn[ +*Ambiguity C needs a different detector.* A whole-turn error in a template echo +produces a pure slope change with exactly zero intercept, so no refinement of an +intercept test will find it. Recognising it means looking at the field itself, +and its characteristic size is $1 slash t_0$ Hz rather than the $1 slash #dte$ +of the other cases. +] + += Summary + +#key[ +Wrapped phase determines the coil offset and the field only up to a family of +alternatives. Sorting them out means asking, of each candidate, whether the +offset and the field it implies are mutually consistent. + +Three cases exhaust the possibilities. When offset and field are both wrong in a +matched way (*A*) the data is unchanged, and only a prior can help. When the +offset is wrong alone (*B*) every echo carries the same constant, so +extrapolating the echoes back to $t = 0$ exposes it, against a rejection scale +$abs(W(2 pi k))$ fixed by the echo times. When the field is wrong alone (*C*) +the displacement is proportional to echo time — a pure slope change no intercept +test can see. + +Where several candidates remain consistent, the prior that decides is that +*bulk-tissue* field is near zero. That makes the mode of the field distribution, +not its median, the correct estimand, and the half-sample mode estimates it +without introducing a tuned constant. Consistency and prior are blind to +opposite symmetries, so a workable rule needs both. +] From 6586db0d681985c753efcc2ea335295f14af2d3d Mon Sep 17 00:00:00 2001 From: Andrew Van Date: Mon, 27 Jul 2026 23:43:17 -0500 Subject: [PATCH 08/13] :memo: Tighten the note's opening and add a byline - Cut the gradient-echo primer. The reader is assumed to know what MR phase is; what needs motivating is why the coil offset has to be recovered at all. The opening now makes that case directly: field mapping wants the rate of phase accumulation, theta is the constant it accumulates from, and theta cannot be ignored because it varies rapidly in space and so breaks the smoothness that spatial unwrapping depends on. - Drop the now-redundant restatement of theta in the signal model. - Byline and date; author metadata instead of the project name. --- notes/phase-offset-ambiguities.pdf | Bin 168452 -> 166708 bytes notes/phase-offset-ambiguities.typ | 51 ++++++++++++++++------------- 2 files changed, 29 insertions(+), 22 deletions(-) diff --git a/notes/phase-offset-ambiguities.pdf b/notes/phase-offset-ambiguities.pdf index 56bc7db18fef2727a96b5dd68c43fc9913146afb..b8f9b5676ec8318c859b706ed2fe74313d31beb4 100644 GIT binary patch literal 166708 zcmeFa1$-1q(=d)5dbs=Io@94sCBQ*Qa1FtOhGZduBn!zV5ZoOOIXF1nT@O2)G zcRAb%{;%$?nVsF)9yWP+-rxWIzVBi0>CEm-b(eN^b#+atz?!uy>AlPuOAUhmWz4A8 z`RHt&yJf6e)w_8@536@zNLOpDcYrM}+OD&BH;4#}ZEG}{;ie#+(PY8am$haeYzYwf z-E6>Cz*Zz}`)buPTBE~AC**hREHEU-3SAiJBi@azz3n;^x!b#Wggw$)R)h*6tMH41znIe$`&6H-IbDt5k($5z1IPkpzaJ;No+kOoj2`-@t}4@4A7Ke z(lCUOm*U4np^_kBr)v&OiL!>o#>H5pfQ54R^ma%0L}r}3r_zI91TQ_XfbclV7Gp&U z>B4NfXq#OZZjJ0AgQXmFfF(ej!6GJ<$5b%A875R{y(uQ_PsBv;7=o@wROg7UaS`?e zU5(1RkZ7H?cSxvR7n=|jWwpm7fYebshlSWfB_WLWBwu$Ik1QP!{sunr(Rgw0G?f!7f*9oZaQcCl1?0@-cHtTSi) z1YIo#P%`dhUefi7u!q~??79%0JuaI3(jz1$1Zgw~RB zBZ!$xI`SZN0%Xf?q6&q?j}Fs?+j{BjHXYIdR7;);1Q&BfJJlGIVk$z`Fcz4B0l;&Z zXF?*PV|8)SG1gdHWIT~CS)yGG7WD|+c14Mf=riyZU9-5TsF0Wh37F_&a46vr7@jhn_U18GYmm)m_YSf}S{sG* z07tJTs3i!m)w@}bkZAAb)|jYR@cUxy%!qggdc%{T)$He5QM9dA&AUNc(LyQIp4<&n zKLHKk5HEqUf}q840j$_y1)Y&$Eb5FBN+t>jFfuW8%{p{0EjsifeRXE`ILsgt#q>I) z30Oj31_7+KF9QJ<%9qka2ZF8lVWb7I*84E(fg#W{bE*RY*E8wYfxsEqpg=YZ>`q{& zL13MO2pI4$0d8bH8AP}fA2P!4_>@U!z~R9o246-Fg9W|?#stAPI>Cqi2{W63LxI^x zV3tuQ7<2*?LHG*(WIe!-@FW2MmW@q`Isqg>WMhGGvI$HFS_crA!>wQtF{TZy2|B@7 zA(aF)5tX*k5e1FFuuFyzl7MM5JkRcAjmXH0td<(~9I_o7BjXc_fYh-5BynKf8X0Iw zG*Cj=08vb&hOq=i1{*UIG87-FVM9Vu!iGt)BvmL}sCURgsbNDxS(6$zBosTTVM9U* zlmyWSRb7%sxn&s3qA7_e^OhWPD#B=ou1h_XxUKJEsjh6 z9kqo`74V#N?fN|9n;sQ~-y(kUebV?CmGmgljusUcRYlKjB{;NgFRv>Jm#IAKQA5-% zVvNe7XR4x;TU0zfDuiCXMvPHM_20Xd`jJ}ZzUUnNGXd6v$`eKPxRv1{#pP?Hv~(@C zlJ%CZQKj`F((SvqC|PnpC~;_5^xtfy=aD~X4)y9P(CD=Qf{UDkl*N#>;Gq4$BahU$7VEiO!SiLi;Drh z+#;uOX`q)|T z&$n9Xe#U!L0V2p*HVi?o zwT>B&41?_+;2KGgTP6ttu5r{YqnvIrEj^8p03Ao0QTU@)y`zi7@Qb@YSwT|!~ zw@g9=87`9?$F=M4h!@LAfE^m?T56FF7}po%R{9CY!w{#q`rKRG{EG_zU z0{U_S`gZCTH4A+^L2i*!=;H};iyX$dv4Fmrqs5VORjD9fBc0`I%YNCkoQGSafz&Tw+pW*wrQa_BNBKjJhx6<>>0*L6N z{J-`;r3XqJ`flPkTj_qOWf~=-PbH$yB%-e*qK_nETwFvSNkpH?^A`J)pGV0@pGw5I zxw=Jp!#KK#zL2BE@uLqT@>=w5M7d>_OGKYWL|;ZqbP_QRsBW315~Z{w^f5%aMZC}_ z5v3M=4N=wNIMC-1v-Q zC|@In^0jn7S~m1?M7foIms-?x^nJu6w-OwrlWH6izNiuCGl}k72Zsn+ghSGq9G1vg zE7A(7n2?fdhij#X}GR2LwTk z3!<)GL5${Q18mWDNK1uOa7b0p7=*2S7!lFk$*3d_;nL2e zGfV7j;$J^V!8PwDox1@?sjO71=Dte`joW3AwF?|Uhz^C*TROH<^sdnsdTRpd+zF6A z?j0Bj=a9%@9dG|gTWELEOM|uyO+^AReU#ft^CkiH0KT4uA_#>khRWw9%OWA_B)5#V z%&5cD77n7CvfmJCOZA%!fjPInkmD=$>6W-G@W&-C0msGLiMZ5YctTu;r;<3D>ZB$v zMN#>@rY<3IsTBaa=s7w~r${cQ5_PGyBGhGO4g6QVpiN$Czq#bqR5>T|QXTM!ycAO< zc{Ftr$;)P7f*D0o@i_GX_=KTmfyE*Sbvd?$-oOXKWS4mp00rF+@VfOTZR%1X=TVo! z<94c~F7sPFp)T%lI;Io)D7O<=>M|6S)YVilCcDgwBGfgzw+!?K?if&ZnGZ&&OZA5) z`ul2Am--M~>S}786Lp#Hd0=-Lrb_Z?&T%U0Qms@zuBl(}xUmieYiV48Zw zSj8MvAhLnueb5`e#VWRWOjyN61j`(QL7P>Sr#x0sc-&5vtYVI|CyKx@Rgi~1%IySM z<(MKFj>_k?r#EV6m{(57%khcm4ctMXCNQ5Kph%nWaE!#LO

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zI8;Zd)&Mk)d|V879cT|40)jxN@U#ro&SDz(sR8Yx{uPwEsFsBGGwLIW0X7NGFO@!1askjRhTPC^%EAs`GAWK)Hj1?s_?K%upSGgm5@Jp87hq9 z@VJ;2_62|jUL;%(K*+y%Q7NVJNSRUxp5Xh)6J_dm^`dWY`Z={$;8DN&K|fGI9qRu$ pjW_?_emBCx{D1qE3yXknzegois=htmhk6R0r0DePH_T)B{{Yob_-p_G diff --git a/notes/phase-offset-ambiguities.typ b/notes/phase-offset-ambiguities.typ index 05ae299..b271d4a 100644 --- a/notes/phase-offset-ambiguities.typ +++ b/notes/phase-offset-ambiguities.typ @@ -1,6 +1,6 @@ #set document( title: "Phase offset ambiguities in multi-echo field mapping", - author: "warpkit", + author: "Andrew Van", ) #set page( paper: "us-letter", @@ -39,10 +39,12 @@ #text(size: 17pt, weight: "bold")[ Phase offset ambiguities in multi-echo field mapping ] - #v(0.3em) + #v(0.35em) #text(size: 11pt)[What can be recovered from wrapped phase, and what cannot] - #v(0.2em) - #text(size: 9.5pt, style: "italic")[warpkit] + #v(0.9em) + #text(size: 10.5pt)[Andrew Van] + #v(0.15em) + #text(size: 9.5pt)[July 27, 2026] ] #v(1em) @@ -51,22 +53,30 @@ #v(1.2em) -= What this note is about += Why the phase offset matters -A gradient-echo image records, at every voxel, a complex number. Its magnitude -is the familiar anatomical image; its phase carries the local magnetic field. -Distortion correction needs that field, so it needs the phase — but the phase is -not the field alone. It is the field plus a fixed contribution from the receive -coil, and it is only ever measured modulo a full turn. +Multi-echo field mapping reads the field off the *rate* at which phase +accumulates across echoes. That rate is the quantity of interest; the constant +it accumulates from is a nuisance. Every voxel carries a receive-coil phase +offset $theta$ — present already at $t = 0$, identical on every echo, and set by +coil geometry rather than by the field. -Those two facts together mean the field cannot be recovered uniquely. Some of -what is lost can be reconstructed from multiple echoes; some of it provably -cannot. This note draws that line precisely: it enumerates the ways an answer -can be wrong, shows which of them leave a trace in the data, and derives what to -do about each. +$theta$ cannot simply be ignored. It varies rapidly in space, steeply near coil +elements and discontinuously between channels, so the measured phase is not +spatially smooth even where the field is. Spatial unwrapping assumes +neighbouring voxels differ by less than half a turn, and on raw phase that +assumption fails. Removing $theta$ first restores the smoothness that everything +downstream depends on. -No familiarity with any particular pipeline is assumed. Everything below follows -from the signal model. +The difficulty is that $theta$ is never measured. It has to be inferred from the +same wrapped phase it corrupts, and that inference is not unique: several +distinct $(theta, f)$ pairs explain the same data. Some of the resulting +ambiguity can be resolved from additional echoes, and some of it provably +cannot. + +This note draws that line. It enumerates the ways an estimate can be wrong, +shows which of them leave a trace in the data, and derives what to do about +each. Everything below follows from the signal model. = The measurement @@ -74,11 +84,8 @@ from the signal model. Acquire $E$ echoes at times $t_0 < t_1 < ... < t_(E-1)$. Field evolution is linear in time, so before wrapping, the phase at echo $e$ would be -$theta + 2 pi f t_e$, where - -- $theta$ is the *coil phase offset* — the phase a voxel already carries at - $t = 0$, set by receive-coil geometry and independent of the field, and -- $f$ is the *field offset* in Hz, the quantity actually wanted. +$theta + 2 pi f t_e$, with $f$ the *field offset* in Hz — the quantity wanted — +and $theta$ the coil offset described above. What is recorded is that value wrapped into a single turn. Writing $W(x) = arg(e^(i x))$ for wrapping to $(-pi, pi]$, From 81703f25c4ae5578a058bd43c7eb09341b5cdbd6 Mon Sep 17 00:00:00 2001 From: Andrew Van Date: Mon, 27 Jul 2026 23:43:54 -0500 Subject: [PATCH 09/13] :memo: Drop the note's subtitle --- notes/phase-offset-ambiguities.pdf | Bin 166708 -> 166696 bytes notes/phase-offset-ambiguities.typ | 2 -- 2 files changed, 2 deletions(-) diff --git a/notes/phase-offset-ambiguities.pdf b/notes/phase-offset-ambiguities.pdf index b8f9b5676ec8318c859b706ed2fe74313d31beb4..65f7b1288cf263e9323dbbd969d029c660f11153 100644 GIT binary patch delta 96076 zcmc$G2SAfcw(t!|M>>Q~DAEa}CyFS7iVegrh!t!oSg@dIu%Ot15hE5<>|GSB*s-8u z!GeN_Vy}P|1@S-gee)&p-n(!2z5l)4UA&V_J7>_l)@)SCGDc%G!cdfkF*>6$hb2ZGG?Q2obVPGB zI;?4e{9B}G@MUOki;=7VWT7<-rkJXgj;3gtAS3PZFiDE`NHkO11bx&V!xEuEI#W=z zjtP3AGan`ysXG?M>6)O&x`7%38FJJc**Sp1iN_I2dDJI9eBx7|1n@~feG)XktR0o)no(E>6I|?`=8k&$JVx|+x$ZTR7l!POe zP&HDHghoilk3!W4{3Xy?6a=3pso-jWJ&YmGm@xF^5mkfPQ=#y==ae zBM2bFWE=s3BM5-L6hr}!BM{I41bp~NV=oZ2;0T05w8y{^w(zP!A|axXMxIcTDQNhF zs!TyoL@3M@G$4lBgy=LPg!0S=(0fI8K`fBd+d>nCYzZkOH0j9pFa?bh*(#=>aU#3M z6f{nRa!f(vMD`L2Io=%^jTG5erl4^mJIri!0NGDwx0NNmX64p1ED%7rhT#KcNn^+J z1}35yL|M`p@_=KgdSDc~U4~4IoSX@Grj}2bPE|rLHu5*pl@T5UMxy8Wb4lPT+ZlLb zlwrhnCVWj8llRvW_d=(QTB4;!y6BowEA-vSL{1h93`fBvOvfX9_xvN+{; zvUtINw`3MXB!hB3LLTLJx}TgUAXH=;$VuV}$-e!smxR(Z9KRh;CJ}(c>y>IU%U%KU5C?p8^(>jTMoNXG&!~;UJL=T`}{p zVRXb_N>pLygiIdk$cg;ti3!Vz3CsS&lIcef5EFnzUh;?({a4F2s*VIl7n{BM?gM$HmqhsPK_%6hUpGP-2?D@$a{HkH#1%qJ@PzhBZU%nzV9 zlFwLM0kl+8eWxj>`c9LdFINs=CZb8o5BO&#)1TfsKCvPAJhC8VNwXlI*c7UE8bLXq z*c5y)M6o{RgPBT~%y)u_vYn|XROJ%@V3$(kh?T)tekYVsekYWnzk@N$XC@*OE59@K zWQ9z^or#VtVjNDQ7t9+ z_zMD3)MOhmjIf-L2nZqbohb>+D(eZ$68DeKltRLa%y$uySRx)Wt!AU~4!UTogGm@U z-y#ByxQ+aOSW^8|N(&=JDJ^u!L8c)NKpz~eP?Lim1S*Uit&qK=9uhfPGmf^HaN}P} zvL@n+@`=HvbikG036{hF{xPO3ILxl<)(I=kfo)OJ+WDewFhRUeJ(A(LEOlbNoqchc6@D|56f$RMrzC zgV@3Re^~x`J|ZTVc{F4)#C2seN}n-ciNPu$hV1`&iM~~vqAFKiQ(6iNh%pN#Vm)S~ zL^oX|cQauLk?CV6LNWo_dnUZ7`c6|`^_`aX0%D*3k4pw#DD!~{X42nPf*-@91iumm z%Elv0gB3 zqYH>?77(LO^_^@yu?hu$Eztx|HUxv=zovwW%DF^TEzRiyCb}+^qMn{?G{jREP4kq( zrGn`b54Zui33ctCU<%j4r=CxAkd>Dy8u~;R`FnBD^d}~0h?gglds(B6PfS2nZ+hYc zV>fzPYe)pB#>)fUgswK;9Q5jmacd%Z1w`!pwIr)1Ubm{9kG?-~l9P3ips(QHUCRHr z021LPATm=~D(lJCD8JJ(5aOU@*NJdcepl9$qsx4k5qc`W(}vq$#*x^#fJj9F@#|G3 z#lv!rfP@QF4Gg9vW3KX|HmbL#Dzqz(Ir~uYls&4no$2X`C7x} zmU9hzG!05f#FeU~#hUUvkv_t|FO|{=E?9^(5)wJXlqlHG1$FbY#{cwaLP8T7=hu#S zwnE~V|JzIXe+n>x8~uN`Vlgu0`V$Q%giAOq_=O~}A>W%700vJNL zYNNl?qF6{SNB((9#*I}@KzNF11|i(bQ4{~Aq|d34ggAsG@}MdS&ypB`P*tLdHC9Mo z!vod5sm4Q(^aI)aFhGJH*XSZw9tU;fnV{Z0Ph|3xjhvp^5N#ZKp?{f?>=SVugd{5bHzisn2r#47l7F}8 zfEfRtqc|e57~L1}RMk9XwVhC{ny;)LEcA5?^yWk;oTiNL!+_?YwkoI5B9C$P-m})Y0M5M+fPM`RL{;9ZPCQ0G5Oqjps#? zVlRw}I~`TPSkjy&<;ea96|`Jr1EM%nOoNT==A}bhBQZ!%@;lgA2?f@Q+CG?nWsGj^Wb#3k=TV{~hGFVhp)fLP2Ip(b>(OIQY+AU0h~uHi?&fwh&J@XI6gkvL-b02lFw_w7DHLcJG3cV~ zHx!&(6BH;r2BRSUjqGDvQz&$x11QkWT&zLDf%I>dui^n(zpEj%yEQ>bj1e-ULohKx zLCg(-2xUe&V2)tOBX1+J^Y01S3`)||M@U4US#-9IKPI0uecz99m#O2fni-uCy%)UIaZ6$Y?(Azg`3-2GkbOFOXUU?C=wjmktd%2Hgn!ou|CpQdD)J13mzduRNES zKq}0{C_mf*qvpZTpNSnoriSouP(%5Fn7s zXrO@h^bwONPd=@}K%)r){tkFTI}Chcdm1>{^q2

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z4R>7e@kK+39qDj8PsJIfMI~)tgP-Jj@O{c@57h0GwXIYD{OZd4GPdh)9QHtVV z(p~wqNu3=u}gSDca$nve8xrGi0Bix9qNkM3r=Y+4uBHg9lE-4sPPF05v#*> zKyi0DOaKTx{QN+%MfGF2~7k91g|+eiiAQ=nFC!T;#cfks;af-w~Q z@fbRW(eAJWeN~2wCaTv2dLD7l4_9VJrjc6(3f>lgz=QccTFjq9qv;SEsPP?F6l2JG zx;o6eg>_(#s^_}=lge1=;$lvDlWkmG8PO5O&+O@0)@kx2)j4zg)aw@Y;6H5k=hccac(q#*lEW2^Z!bJlyp9#sp)A2(IQ~nRgfgPa$ diff --git a/notes/phase-offset-ambiguities.typ b/notes/phase-offset-ambiguities.typ index b271d4a..f028feb 100644 --- a/notes/phase-offset-ambiguities.typ +++ b/notes/phase-offset-ambiguities.typ @@ -39,8 +39,6 @@ #text(size: 17pt, weight: "bold")[ Phase offset ambiguities in multi-echo field mapping ] - #v(0.35em) - #text(size: 11pt)[What can be recovered from wrapped phase, and what cannot] #v(0.9em) #text(size: 10.5pt)[Andrew Van] #v(0.15em) From a3c26b258be0ad48487f51f3fb8569944881d3a8 Mon Sep 17 00:00:00 2001 From: Andrew Van Date: Tue, 28 Jul 2026 00:00:00 -0500 Subject: [PATCH 10/13] :rewind: Revert the half-sample mode; keep the weighted median Reverts the field-prior estimator to a magnitude-weighted median. The mode regressed a run that the median handled correctly, caught by the corpus scan before this left the branch. On ds005354 sub-02 run-00 (k=0.5089, wrap 36.35 Hz) the mode made 108 of 223 frames select a non-zero branch, producing a 32.7 Hz step in the field time series. The median selects branch 0 on all 223 and the run is flat. That is the only run in the corpus scan with a wrap-sized step under either estimator, and the median baseline has none. The cause is the assumption the mode carries. That subject's field distribution is bimodal -- peaks 21 Hz apart at 95% and 100% of maximum -- so the argmax flips on noise between adjacent frames. Magnitude weighting does not separate peaks that close (they stay within 5%), and restricting to high-SNR voxels made it worse, swinging between the two peaks on consecutive frames. The general lesson, now recorded on the function: continuity matters more here than freedom from bias. The estimate feeds a threshold comparison against 1/(2*dTE) that must be stable frame to frame, and a median is a continuous functional of the data where any argmax is not. The earlier claim of a 4x margin improvement came from a single subject and did not generalise -- the mode-minus-median offset is subject-specific in sign as well as magnitude. A continuous alternative that avoids both defects, scoring candidates by field mass at zero, was also tested. It reproduces the median's decisions on both stress cases but does not improve on them, so it was not adopted either. Also drops the mode material from notes/phase-offset-ambiguities.typ and replaces it with the estimator actually used, plus the continuity requirement that rules the alternatives out. --- .gitignore | 3 + notes/phase-offset-ambiguities.pdf | Bin 166696 -> 156017 bytes notes/phase-offset-ambiguities.typ | 89 +++++++++-------------------- warpkit/unwrap.py | 85 ++++++++++----------------- 4 files changed, 59 insertions(+), 118 deletions(-) diff --git a/.gitignore b/.gitignore index eaf10a2..bfeff3c 100644 --- a/.gitignore +++ b/.gitignore @@ -151,3 +151,6 @@ warpkit/build-tmp*/ warpkit/share/ # ignore .secrets + +# macOS +.DS_Store diff --git a/notes/phase-offset-ambiguities.pdf b/notes/phase-offset-ambiguities.pdf index 65f7b1288cf263e9323dbbd969d029c660f11153..dd7c1d25062c936b2c7c8b53076b9929aee69ce1 100644 GIT binary patch delta 78218 zcmc${30zIv+c@r&M$OTn=`>MkI`cWEMk)!(5UC_lhN3dZ$($jo6^Ia zU*zN)H8n0sz!kEc*`wJ4t_YW6TuP{t=Zs4ZF1e}E7A#I`>HvAZn44NL0ty~tv%;X{ 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method: repeatedly keep the - shortest interval containing half the remaining points. - - Used instead of a median to locate the global field level. The field - distribution over a brain is a sharp peak at bulk-tissue value with long - tails from sinuses, dropout and edges. The scanner's frequency adjustment - centres that *peak* near zero, not the median-including-tails, so a median - estimates the wrong quantity and is dragged toward whichever tail is - heavier. On the `ds006131` sub-20828 frame where ROMEO's ballot tips, the - mode sits 10.3 Hz from the half-wrap boundary against the weighted median's - 2.6 Hz -- four times the margin, from the same single frame. - - Parameter-free by construction: no bin width, no bandwidth, nothing to - tune. A binned mode gives the same answer (9.8-11.2 Hz of margin across a - 64x range of bin counts), so the choice is not load-bearing; this form just - avoids having to justify one. It is also cheaper than the weighted median - it replaced, since it never sorts a weight array. +def _weighted_median(values: npt.NDArray, weights: npt.NDArray) -> float: + """Weighted median of ``values``. Used with magnitude-squared weights so + low-SNR voxels do not sway the global field estimate. + + Continuity is why this is a median and not a mode. The estimate feeds a + threshold comparison against 1/(2*dTE) that has to give the same answer on + consecutive frames, and a median moves smoothly with the data. Estimators + defined by an argmax do not: field distributions are often bimodal, and a + near-tie between peaks makes them flip frame to frame. A half-sample mode + was tried here and reverted for exactly that reason. """ - x = np.sort(np.asarray(values, dtype=np.float64)) - if x.size == 0: - return 0.0 - while x.size > 3: - half = (x.size + 1) // 2 - widths = x[half - 1 :] - x[: x.size - half + 1] - start = int(np.argmin(widths)) - x = x[start : start + half] - if x.size == 3: - left, right = x[1] - x[0], x[2] - x[1] - if left < right: - return float(np.mean(x[:2])) - if right < left: - return float(np.mean(x[1:])) - return float(np.mean(x)) + order = np.argsort(values) + values, weights = values[order], weights[order] + cumulative = np.cumsum(weights) + if cumulative[-1] <= 0: + return float(np.median(values)) + return float(values[np.searchsorted(cumulative, 0.5 * cumulative[-1])]) def _branch_intercept_step(te0: np.float32 | float, te1: np.float32 | float) -> float: @@ -208,26 +191,14 @@ def _select_branch( 2. **Prior, only to break a tie.** Depending on TE0/dTE several branches can be exactly through-origin; the alias is genuinely reachable and no statistic computed from the phase can separate them. Among the survivors, - take the one whose bulk-tissue field level -- the half-sample mode over - an eroded brain mask -- is closest to zero. - - Stage 2 is the same prior ``correct_global`` already applies, but on a far - better conditioned statistic. ``correct_global`` takes an unweighted median - of rounded wrap counts over a *dilated* mask, which is pulled toward the - tails contributed by sinuses, dropout and edge voxels; that mask carries - roughly twice the voxels of the brain core. Its estimate therefore sits - close to the +/-1/(2*dTE) decision boundary and tips frame to frame. - - Measured margins to that boundary on the frame below, worst of the two: - - ========================================== ============= - estimator margin - ========================================== ============= - ``correct_global`` (median, dilated) -0.26 Hz (flips) - median, eroded 1.25 Hz - magnitude-weighted median, eroded 2.57 Hz - **half-sample mode, eroded** **10.27 Hz** - ========================================== ============= + take the smallest magnitude-weighted median field. + + Stage 2 is the same prior ``correct_global`` already applies, but on a much + better conditioned statistic: magnitude-weighted, over an eroded brain mask, + on the field itself rather than an unweighted median of rounded wrap counts + over a dilated mask. That difference is the whole point. ``correct_global``'s + ballot becomes unstable when the global field sits near 1/(2*dTE), and then + it flips frame to frame. Measured on `ds006131` sub-20828 (k = 0.5742, wrap 40.44 Hz, half-wrap 20.22 Hz), whose field sits at 17.7 Hz -- 2.5 Hz under the boundary. On 19 @@ -276,8 +247,12 @@ def _select_branch( if len(consistent) == 1: return consistent[0], scores # Tie: every survivor explains the phase equally well, so fall back to the - # prior and take the one whose bulk-tissue field level is closest to zero. - fields = {n: _half_sample_mode(evaluated[n][2][score_mask]) for n in consistent} + # prior and take the one closest to zero global field. + weights = np.square(mag0[score_mask].astype(np.float64)) + fields = { + n: _weighted_median(evaluated[n][2][score_mask].astype(np.float64), weights) + for n in consistent + } return min(consistent, key=lambda n: abs(fields[n])), scores From 30979fa601e3fc77d5e0cefb306e96cd0d92fd97 Mon Sep 17 00:00:00 2001 From: Andrew Van Date: Tue, 28 Jul 2026 00:04:31 -0500 Subject: [PATCH 11/13] :fire: Remove ambiguity C from the theory note It was never observed, and the derivation behind it was wrong. The claim was that a whole-turn error in a multi-echo unwrapper's template echo propagates to echo e scaled by t_e/t_0, i.e. proportionally to echo time, and that a proportional displacement is a pure slope change with exactly zero intercept -- so no intercept test could ever see it. The proportional step does not survive the implementation. Non-template echoes are unwrapped by unwrap_voxel (include/romeo/algorithm.h), which returns new - 2*pi*round((new - old)/2*pi): the correction is a whole number of turns. Shifting the reference therefore shifts echo e by 2*pi*round(t_e/t_0) turns, not 2*pi*t_e/t_0. For TEs 14.2/38.93/63.66 those ratios are 2.742 and 4.483, which round to 3 and 4 -- not proportional to echo time, so the intercept is not zero and an intercept test would in fact partially see it. Nor was it ever seen in data. Whole-run scans counted frame-to-frame steps at 1000/TE0 Hz, its predicted signature, and found none on any run of any protocol. The one supporting measurement I had -- the template echo's pre-correction wrap count going non-zero on some frames -- shows correct_global doing its job rather than failing at it, and is not evidence of the failure at all. The taxonomy is now the two cases the propositions actually establish: offset and field wrong together (invisible), and offset wrong alone (a constant on every echo). Both are proved, and B is the one the selector acts on. --- notes/phase-offset-ambiguities.pdf | Bin 156017 -> 147474 bytes notes/phase-offset-ambiguities.typ | 67 +++++++---------------------- 2 files changed, 16 insertions(+), 51 deletions(-) diff --git a/notes/phase-offset-ambiguities.pdf b/notes/phase-offset-ambiguities.pdf index dd7c1d25062c936b2c7c8b53076b9929aee69ce1..d219183adffe0f2b82737b6059638988e031a66f 100644 GIT binary patch delta 86066 zcmc$`2UJwcvM}s`5=~k!7Vp z0C67>n~}!mcpy1B7$!#Hazo_!9%!eW70Ux%l1r51b5IX?D-Y;VW_{N0E5@Buq&KiixGW_ZyA>jiFNtTsgh~B`H{&$WkYinaxF* zVlO$q5PenvvdA8C@d8w=2xN&+nxeIa>=ZsyRdSZ&i_twrD>=*qbyGCgr_l2W$Rv6J zTB77FC-6W*rQgw$Y$1skB1L64IROXNDOvX%a)kd_$hAYzB7PntA(%5+C} z|5#WVPeAOfx3_9+(zxU?6I6i+r0gii5u@YEMx${a;7>h0-3O#kN{#ej^E^lq;jnoe z0tp*~zeq9R!e5l2z+a@O@Zm3l#cTomMJy^x1{>NJYl!-(^q2Dxp+Xf?2{C}pVRK0= z0Gk8@pb5UQd2Bw3jL+r^a25ev0ZE9EEg&(8*aC98m@Ooy1F1rCK9DLRX#`S5Bm^Lp z5_k#FDkgydtzv4a0uWycj*yK}e~H)_HBijPIOO>O!NR!Y5-<)MqyEBRB#8*-VtkSg 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zh(!Pzq?rXE0@8H`h>AiIBMcfypEy$kl{-`=e>lkotqMrW0~$6E+y++={^_RtgD_|m z;Xi(LGI0sP)3KU?!OO*f?0fP9Q2xvG+H2{P&6JIfk4p+$yN!Vmj!_qaI6bCOToaPpo;+pOsqQ4zJXMn z-~tc?=fE-XuLR(CWXFJ8AVeVur1s$KiAby%I4Eo(G~nG3+TT}e{!O94De-@fBmW=S z0TI%Y05nK}4$5gr90oLqzhl5{8;E4UX&jxXce3x5aZmh8v)Q~ zL5rSPtKfpozs&rtLC^t*NEbAsp%3FB84~7K$C7vv(CC9OjYR$(xBfu~3<0#qpbrBJ zg-nQrN(OY>Ale75MB)?zos>5R;jdJX-B!)8avu;{yJJOsSJ4EUVV?)le+PcR`+6a_D@lBV+i0e7h0G5`Po diff --git a/notes/phase-offset-ambiguities.typ b/notes/phase-offset-ambiguities.typ index a5aea36..1a47219 100644 --- a/notes/phase-offset-ambiguities.typ +++ b/notes/phase-offset-ambiguities.typ @@ -168,8 +168,8 @@ $ $ and read off two numbers — the *intercept* $c$ and the *slope* $s$. Every -possible error is a displacement in this $(c, s)$ plane, and exactly three cases -are of interest. +possible error is a displacement in this $(c, s)$ plane, and the branch +ambiguity produces two cases. #defn[ #set enum(numbering: "A.") @@ -177,13 +177,11 @@ are of interest. combination that leaves $psi_e$ unchanged. Invisible. + *The offset moves alone.* $c$ shifts while $s$ stays correct. Visible as a line that misses the origin. -+ *The field moves alone.* $s$ shifts while $c$ stays exactly zero. Visible as a - wrong slope — but *not* as a wrong intercept. ] -Each is taken in turn below. The taxonomy matters because a test that measures -$c$ finds case B and is blind to A and C *by construction* — not through any -weakness of implementation. +Both are taken in turn below. The distinction matters because a test that +measures $c$ finds case B and is blind to case A *by construction* — not through +any weakness of implementation. == Ambiguity A: an exact symmetry of the data @@ -264,30 +262,6 @@ and field have become mutually inconsistent, and consistency between them is something the data can check. ] -== Ambiguity C: the field moves alone - -The third case is easy to overlook, because it does not originate in the offset -estimate at all. - -Multi-echo unwrappers commonly unwrap a single *template* echo spatially and -derive the remaining echoes by temporal unwrapping relative to it, each echo -referenced to the template scaled by its own echo time. A whole-turn error in -the template then propagates to echo $e$ scaled by $t_e slash t_0$ — that is, -*proportionally to echo time*. - -A displacement proportional to $t_e$ is, by @eq:fit, a pure change of slope: - -$ - Delta s = (2 pi) / t_0, quad quad Delta c = 0. -$ - -#key[ -The intercept is exactly zero here. A test built on the intercept cannot see -ambiguity C at all — not poorly, but not at all. It surfaces instead as a field -error of $1 slash t_0$ Hz, a different size from the $1 slash #dte$ of ambiguity -A, which is the practical way to tell the two apart. -] - = Detecting ambiguity B By #link()[Prop. 2] a branch error of $M$ wraps places the constant @@ -349,10 +323,11 @@ inside the unwrapper can map candidates $M$ and $-M$ to equal $abs(f)$. ] #key[ -The two are complementary in a precise sense. The intercept is blind to a -*sign-preserving* change of the field — that is ambiguity C, and the alias -direction of A. The magnitude prior is blind to a *sign-reversing* one. Together -they cover both, which is why a workable rule needs both stages. +The two are complementary in a precise sense. The intercept is blind to the +alias direction of A, along which offset and field move together; the magnitude +prior is blind to a sign reversal, since $+delta$ and $-delta$ are equidistant +from zero. Together they cover both, which is why a workable rule needs both +stages. ] The resulting rule: reject candidates whose intercept is inconsistent; if one @@ -422,7 +397,7 @@ For this decision, stability across frames outweighs freedom from bias. = What remains unresolvable -Three limits are properties of the acquisition rather than of any algorithm. +Two limits are properties of the acquisition rather than of any algorithm. #defn[ *Ambiguity A cannot be corrected.* By #link()[Prop. 1] the data is @@ -442,14 +417,6 @@ therefore raising the $1 slash (2 #dte)$ threshold. The two effects push the same quantity in opposite directions. ] -#defn[ -*Ambiguity C needs a different detector.* A whole-turn error in a template echo -produces a pure slope change with exactly zero intercept, so no refinement of an -intercept test will find it. Recognising it means looking at the field itself, -and its characteristic size is $1 slash t_0$ Hz rather than the $1 slash #dte$ -of the other cases. -] - = Summary #key[ @@ -457,13 +424,11 @@ Wrapped phase determines the coil offset and the field only up to a family of alternatives. Sorting them out means asking, of each candidate, whether the offset and the field it implies are mutually consistent. -Three cases exhaust the possibilities. When offset and field are both wrong in a -matched way (*A*) the data is unchanged, and only a prior can help. When the -offset is wrong alone (*B*) every echo carries the same constant, so -extrapolating the echoes back to $t = 0$ exposes it, against a rejection scale -$abs(W(2 pi k))$ fixed by the echo times. When the field is wrong alone (*C*) -the displacement is proportional to echo time — a pure slope change no intercept -test can see. +Two cases arise. When offset and field are both wrong in a matched way (*A*) the +data is unchanged, and only a prior can help. When the offset is wrong alone +(*B*) every echo carries the same constant, so extrapolating the echoes back to +$t = 0$ exposes it, against a rejection scale $abs(W(2 pi k))$ fixed by the echo +times. Where several candidates remain consistent, the prior that decides is that *bulk-tissue* field is near zero, estimated by a weighted median over a From afe462599db917a8cbd0a537872ac9c8882bedd2 Mon Sep 17 00:00:00 2001 From: Andrew Van Date: Tue, 28 Jul 2026 11:18:09 -0500 Subject: [PATCH 12/13] :zap: Weight the field prior by the second echo, not the first The dual-echo field is read from phase1 - phase0, whose noise is dominated by the weaker echo. Weighting on mag0 gave full weight to fast-T2* voxels, a healthy echo 0 alongside a collapsed echo 1, which are exactly the air-tissue interfaces the prior should discount. Measured over 100 runs and 24k frames from ten OpenNeuro ME-EPI datasets, 71 of them held out from the comparison that chose the weighting. The decision boundary is fixed at 1/(2*dTE) and does not move, and the worst-case margin barely changes (0.884 -> 1.052 Hz on the tightest subject). What improves is stability: the estimator's offset varies by 0.67 Hz across subjects under mag1**2 against 2.04 Hz under mag0**2, and that between-subject spread is what predicts a wrap flip. Zero errors under either weighting. Also record in the docstrings why the estimator is a median rather than a mean or M-estimator: the field distribution is broad and right-skewed and the half-wrap boundary falls inside its body, so anything pulled toward the mean reads closer to the boundary without being more accurate. --- warpkit/unwrap.py | 44 +++++++++++++++++++++++++++++++------------- 1 file changed, 31 insertions(+), 13 deletions(-) diff --git a/warpkit/unwrap.py b/warpkit/unwrap.py index 6644834..10aebf2 100644 --- a/warpkit/unwrap.py +++ b/warpkit/unwrap.py @@ -139,12 +139,20 @@ def _weighted_median(values: npt.NDArray, weights: npt.NDArray) -> float: """Weighted median of ``values``. Used with magnitude-squared weights so low-SNR voxels do not sway the global field estimate. - Continuity is why this is a median and not a mode. The estimate feeds a - threshold comparison against 1/(2*dTE) that has to give the same answer on - consecutive frames, and a median moves smoothly with the data. Estimators - defined by an argmax do not: field distributions are often bimodal, and a - near-tie between peaks makes them flip frame to frame. A half-sample mode - was tried here and reverted for exactly that reason. + A median, not a mean or an M-estimator. The field distribution over the + brain is wide (~2 wraps) and right-skewed, and the 1/(2*dTE) threshold this + feeds sits *inside* its bulk, between the 50th and 75th percentiles. So the + question the threshold asks -- is most of the brain within half a wrap of + zero -- is a question about the 50% point, and the median is the statistic + that answers it. Anything pulled toward the mean reads ~2 Hz higher and + moves the estimate toward the boundary for no gain in accuracy: a Huber + M-estimator tried here cut the worst-case margin from 1.5 Hz to 0.05 Hz. + + Nor an argmax. The threshold comparison has to give the same answer on + consecutive frames, and a median moves smoothly with the data; field + distributions are often bimodal and a near-tie between peaks makes an + argmax flip frame to frame. A half-sample mode was tried and reverted for + exactly that reason. """ order = np.argsort(values) values, weights = values[order], weights[order] @@ -191,14 +199,24 @@ def _select_branch( 2. **Prior, only to break a tie.** Depending on TE0/dTE several branches can be exactly through-origin; the alias is genuinely reachable and no statistic computed from the phase can separate them. Among the survivors, - take the smallest magnitude-weighted median field. + take the smallest weighted-median field. Stage 2 is the same prior ``correct_global`` already applies, but on a much - better conditioned statistic: magnitude-weighted, over an eroded brain mask, - on the field itself rather than an unweighted median of rounded wrap counts - over a dilated mask. That difference is the whole point. ``correct_global``'s - ballot becomes unstable when the global field sits near 1/(2*dTE), and then - it flips frame to frame. + better conditioned statistic: weighted by ``mag1**2``, over an eroded brain + mask, on the field itself rather than an unweighted median of rounded wrap + counts over a dilated mask. That difference is the whole point. + ``correct_global``'s ballot becomes unstable when the global field sits near + 1/(2*dTE), and then it flips frame to frame. + + The weight is the *second* echo's magnitude because the field comes from + ``phase1 - phase0``, whose noise is dominated by the weaker echo. Weighting + on ``mag0`` instead lets voxels with fast T2* decay -- a healthy echo 0 and a + collapsed echo 1, i.e. exactly the air-tissue interfaces -- carry full + weight. Measured over 29 runs, this does not move the decision boundary + (that is fixed at 1/(2*dTE)) and barely moves the margin; what it buys is + stability. The estimator's offset varies by 0.43 Hz across subjects under + ``mag1**2`` against 0.99 Hz under ``mag0**2``, and that between-subject + spread, not the per-subject margin, is what predicts a wrap flip. Measured on `ds006131` sub-20828 (k = 0.5742, wrap 40.44 Hz, half-wrap 20.22 Hz), whose field sits at 17.7 Hz -- 2.5 Hz under the boundary. On 19 @@ -248,7 +266,7 @@ def _select_branch( return consistent[0], scores # Tie: every survivor explains the phase equally well, so fall back to the # prior and take the one closest to zero global field. - weights = np.square(mag0[score_mask].astype(np.float64)) + weights = np.square(mag1[score_mask].astype(np.float64)) fields = { n: _weighted_median(evaluated[n][2][score_mask].astype(np.float64), weights) for n in consistent From d066cf7f85c78338bfa70d43658cf1ca3b15b247 Mon Sep 17 00:00:00 2001 From: Andrew Van Date: Tue, 28 Jul 2026 11:19:32 -0500 Subject: [PATCH 13/13] :fire: Move the theory note out of this branch The note is independent of the selector change and is going up separately on docs/phase-offset-note. Keeps this PR to the code change. --- notes/phase-offset-ambiguities.pdf | Bin 147474 -> 0 bytes notes/phase-offset-ambiguities.typ | 437 ----------------------------- 2 files changed, 437 deletions(-) delete mode 100644 notes/phase-offset-ambiguities.pdf delete mode 100644 notes/phase-offset-ambiguities.typ diff --git a/notes/phase-offset-ambiguities.pdf b/notes/phase-offset-ambiguities.pdf deleted file mode 100644 index d219183adffe0f2b82737b6059638988e031a66f..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 147474 zcmeFa2S60d(lBbyIVW^gR1C0THle5>Dh5!FjkOs#h&t+Rr*=wZ0;tfDPP zTQzEXHfmJB5bBSBqrcNZynMq909u7X$lDsCOfoI%E%ymB1sfccvd)3gGE<;IHZagP z+#oad>lbb?$wGWhCPP?wZFy5exJjl~YtSQR_-I^9_yi^fk0(RHtDKaw4nYB-zQM8( zqrV|o0EqjW0itJ^0P)~*uP{TT3?nSSR>_)08bX7kW#I#T{S5vxgI}Q0V4k$ja?%J) zdShU~mXqdW-i8okWKd`TCJiBqFn7!y^V|_CmT<+yRuaPFK``6o&4a>#rUVmDU-*^a zM?}DpH?zTpvJh7e$(Fgz9t^Y92u%i;0-C>YL32`oT74l#xqD1~JHMp>xQ 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That rate is the quantity of interest; the constant -it accumulates from is a nuisance. Every voxel carries a receive-coil phase -offset $theta$ — present already at $t = 0$, identical on every echo, and set by -coil geometry rather than by the field. - -$theta$ cannot simply be ignored. It varies rapidly in space, steeply near coil -elements and discontinuously between channels, so the measured phase is not -spatially smooth even where the field is. Spatial unwrapping assumes -neighbouring voxels differ by less than half a turn, and on raw phase that -assumption fails. Removing $theta$ first restores the smoothness that everything -downstream depends on. - -The difficulty is that $theta$ is never measured. It has to be inferred from the -same wrapped phase it corrupts, and that inference is not unique: several -distinct $(theta, f)$ pairs explain the same data. Some of the resulting -ambiguity can be resolved from additional echoes, and some of it provably -cannot. - -This note draws that line. It enumerates the ways an estimate can be wrong, -shows which of them leave a trace in the data, and derives what to do about -each. Everything below follows from the signal model. - -= The measurement - -== Signal model - -Acquire $E$ echoes at times $t_0 < t_1 < ... < t_(E-1)$. Field evolution is -linear in time, so before wrapping, the phase at echo $e$ would be -$theta + 2 pi f t_e$, with $f$ the *field offset* in Hz — the quantity wanted — -and $theta$ the coil offset described above. - -What is recorded is that value wrapped into a single turn. Writing -$W(x) = arg(e^(i x))$ for wrapping to $(-pi, pi]$, - -$ - phi_e = W(theta + 2 pi f t_e). -$ - -#key[ -The entire difficulty comes from $W$. Without it, two echoes would determine -$theta$ and $f$ exactly by fitting a straight line. With it, the line is -observed only modulo $2 pi$, and infinitely many $(theta, f)$ pairs remain -compatible with the data. -] - -== Evenly spaced echoes - -An echo-planar train produces echoes at constant spacing, - -$ - t_e = "TE"_0 + e dot #dte, quad e = 0, 1, ..., E-1, -$ - -with $"TE"_0$ the first echo time and #dte the spacing. The single most -important derived quantity below is their ratio, - -$ - k = "TE"_0 / (#dte), -$ - -dimensionless and fixed by the protocol. Two field scales also recur: - -#defn[ -- *Wrap spacing* $1 slash #dte$ (Hz) — how far apart the candidate field - solutions sit. -- *Half-wrap* $1 slash (2 #dte)$ (Hz) — the midpoint between neighbouring - candidates, and therefore the decision boundary for any rule preferring the - smallest field. -] - -== Removing the offset - -Because $theta$ is common to every echo, it cancels in a difference. The phase -difference of the first two echoes removes the coil contribution and leaves a -quantity proportional to the field: - -$ - Delta phi = W(phi_1 - phi_0) approx W(2 pi f #dte). -$ - -This is still wrapped, but a difference of *neighbouring* echoes varies slowly -in space, so a region-growing algorithm can restore the missing turns and -produce an unwrapped difference #phiuw. Extrapolating back to $t = 0$ then -recovers the offset: - -$ - hat(theta) = W(phi_0 - k dot #phiuw). -$ - -#key[ -Spatial unwrapping recovers *relative* phase between voxels. It cannot know the -absolute number of turns, so it determines #phiuw only up to a global additive -constant $2 pi M$ for some unknown integer $M$. Pinning down $M$ is the decision -this note is about. -] - -= A taxonomy of wrong answers - -Suppose some estimate $(hat(theta), hat(f))$ has been produced. To classify how -it can be wrong, remove the estimated offset from the data and inspect what -remains: - -$ - psi_e = phi_e - hat(theta). -$ - -Were the estimate perfect, $psi_e$ would be exactly proportional to $t_e$: a -straight line through the origin whose slope is the field. So fit - -$ - psi_e approx c + s dot t_e -$ - -and read off two numbers — the *intercept* $c$ and the *slope* $s$. Every -possible error is a displacement in this $(c, s)$ plane, and the branch -ambiguity produces two cases. - -#defn[ -#set enum(numbering: "A.") -+ *Both move together.* Offset and field are both wrong, in a matched - combination that leaves $psi_e$ unchanged. Invisible. -+ *The offset moves alone.* $c$ shifts while $s$ stays correct. Visible as a - line that misses the origin. -] - -Both are taken in turn below. The distinction matters because a test that -measures $c$ finds case B and is blind to case A *by construction* — not through -any weakness of implementation. - -== Ambiguity A: an exact symmetry of the data - -#defn[ -*Proposition 1.* For evenly spaced echoes and any integer $N$, define -$ - theta' = theta - W(2 pi k N), quad quad f' = f + N / (#dte). -$ -Then $(theta', f')$ reproduces @eq:model *identically*: $phi'_e = phi_e$ for -every echo. -] - -#proof[ -*Proof.* Using $W(2 pi k N) equiv 2 pi k N mod 2 pi$, -$ - theta' + 2 pi f' t_e - &equiv theta - 2 pi k N + 2 pi (f + N slash #dte) t_e \ - &= theta + 2 pi f t_e + 2 pi N (t_e slash #dte - k). -$ -For evenly spaced echoes $t_e slash #dte = k + e$, so the bracket equals the -integer $e$ and the term vanishes under $W$. #h(1fr) $qed$ -] - -Offset error and field error are locked together here: shift the field by one -wrap spacing and the offset compensates exactly. The recorded data is -*bit-identical* — not approximately, but exactly. - -#key[ -No statistic computed from the phase can separate these alternatives, because -there is nothing to separate: they are the same measurement. Resolving A -requires information from outside the data, which is to say a prior. -] - -=== Why more echoes do not help - -It is natural to expect a five-echo acquisition to constrain the field better -than a two-echo one. For this question it does not. -#link()[Prop. 1] holds for all echoes simultaneously, and its period -$1 slash #dte$ depends only on the spacing, which every pair shares. All pairs -therefore alias at the same period and agree with one another. - -Unequal spacing would break the tie — two pairs whose aliases have -incommensurate periods are simultaneously satisfied only by the true field — but -an EPI echo train produces equal spacing naturally, so that escape is not -usually available. - -== Ambiguity B: the offset moves alone - -Now suppose spatial unwrapping settled on the wrong number of turns, so #phiuw -is off by $2 pi M$, while the field has already been fixed by an earlier stage -and does not follow. - -#defn[ -*Proposition 2.* If #phiuw carries a branch error of $M$ wraps, @eq:est returns -$hat(theta)_M = W(theta - 2 pi k M)$, and the offset-removed phases satisfy -$ - psi_e equiv 2 pi f t_e + c mod 2 pi, - quad quad c = W(2 pi k M), -$ -with $c$ *identical on every echo*. -] - -#proof[ -*Proof.* Substituting $#phiuw -> 2 pi f #dte + 2 pi M$ into @eq:est gives -$hat(theta)_M = W(theta - 2 pi k M)$. Then -$psi_e equiv (theta + 2 pi f t_e) - (theta - 2 pi k M) = 2 pi f t_e + 2 pi k M$, -and $2 pi k M equiv c mod 2 pi$. #h(1fr) $qed$ -] - -Comparing against @eq:fit: the slope is untouched and the intercept has moved to -$c = W(2 pi k M)$. Because $c$ is the *same on every echo*, it does not resemble -noise and does not average away — it is a rigid vertical displacement of the -whole line. - -#key[ -This is the one detectable case, and the reason is worth stating plainly: offset -and field have become mutually inconsistent, and consistency between them is -something the data can check. -] - -= Detecting ambiguity B - -By #link()[Prop. 2] a branch error of $M$ wraps places the constant -$c = W(2 pi k M)$ on every echo. Fitting the line of @eq:fit and reading its -intercept therefore measures the error directly. Because the effect is a single -global constant rather than a per-voxel one, the intercept can be aggregated -over many voxels, suppressing noise without diluting the signal. - -The procedure follows: for each candidate $M$, reconstruct the offset it -implies, remove it, fit, and record $abs(c)$. The correct branch extrapolates -through the origin; a wrong one does not. - -== The rejection scale is analytic - -Any test needs a scale — how large must $abs(c)$ be before a candidate is -rejected? Here the model supplies it rather than the user. Candidate intercepts -lie on a lattice of spacing - -$ - Delta c = abs(W(2 pi k)), -$ - -a function of the echo times alone. Classifying against $Delta c slash 2$ is -nearest-neighbour assignment on that lattice — the midpoint between two adjacent -possibilities — rather than a tuned threshold. - -== Where the test loses power - -#defn[ -*Integer $k$.* If $k$ is a whole number then $c = W(2 pi k M) = 0$ for every -$M$, so the intercept carries no information. But by -#link()[Prop. 2] the recovered offset $hat(theta)_M = W(theta)$ is -*also unchanged*, so no choice of branch can affect the output. The degeneracy -is harmless, and @eq:step returning zero is the correct signal to do nothing. -] - -Away from integer $k$ the achievable separation varies smoothly, peaking near -$k = 2 slash 3$ and shrinking to either side. For small $k$ in particular, -several candidates can be exactly through-origin at once: the alias of -#link()[Prop. 1] becomes reachable *within the candidate set*, and -the intercept cannot rank the survivors. Since an EPI train has $"TE"_0$ shorter -than the echo spacing, $k < 1$ always, so this regime is not exotic. - -= Choosing among indistinguishable candidates - -Once more than one candidate is through-origin the data has been exhausted and -something else must decide. Two facts constrain what that can be. - -#defn[ -*A consistency test alone is insufficient.* Tied candidates have equal -intercepts to within numerical noise. Ranking them is ranking noise. -] - -#defn[ -*A field-magnitude prior alone is insufficient.* A rule preferring the smallest -$abs(f)$ is *symmetric about zero*: candidates at $+delta$ and $-delta$ are -equidistant and cannot be separated. Such pairs do arise, because re-wrapping -inside the unwrapper can map candidates $M$ and $-M$ to equal $abs(f)$. -] - -#key[ -The two are complementary in a precise sense. The intercept is blind to the -alias direction of A, along which offset and field move together; the magnitude -prior is blind to a sign reversal, since $+delta$ and $-delta$ are equidistant -from zero. Together they cover both, which is why a workable rule needs both -stages. -] - -The resulting rule: reject candidates whose intercept is inconsistent; if one -survives, take it; if several survive, apply the prior among them; if none -survives, change nothing, since a failed fit is not evidence for any particular -candidate. - -= The prior and how to estimate it - -== What the prior actually asserts - -Before imaging, the scanner sets its demodulation frequency to the centre of the -water resonance over the shim volume. The physical content of the prior is -therefore: - -#key[ -The *bulk-tissue* field is near zero. Not the mean field, and not the median -field over whatever mask happens to be convenient. -] - -The distinction is not pedantry. The field distribution over a brain is -asymmetric: a narrow peak at the bulk-tissue value, plus long tails from -air–tissue interfaces, sinuses, signal dropout and mask edges, where -susceptibility gradients are large. Any statistic that is a *quantile of the -whole distribution* — the median included — is displaced toward whichever tail -is heavier. - -== Estimating the level - -Model the distribution as a mixture of a narrow bulk component and a broad tail -component, - -$ - p(f) = (1 - epsilon) dot p_"bulk" (f) + epsilon dot p_"tail" (f), -$ - -with the centre of $p_"bulk"$ the quantity the shim actually set. Two practical -consequences follow. - -#defn[ -- *Enlarging the mask makes the estimate worse.* Dilating a brain mask admits - precisely the voxels populating $p_"tail"$, raising $epsilon$ and displacing - any quantile of $p$ further from the bulk centre. A generous mask is worse - than a conservative one — the opposite of the usual intuition that more data - helps. -- *A bias of a fraction of a wrap is decisive.* The prior compares - $abs(hat(f))$ against the half-wrap $1 slash (2 #dte)$. Estimator bias adds - directly to the distance from that boundary, so a subject whose bulk field - lies near the half-wrap can be pushed across it by bias alone, and the choice - then turns on an artefact of the statistic rather than a property of the data. -] - -The estimator used is a magnitude-squared weighted median over an eroded brain -mask. The erosion and the weighting both reduce $epsilon$: erosion by excluding -edge voxels geometrically, weighting by suppressing the low-signal voxels that -are over-represented in the tails. - -#key[ -*Continuity is the binding requirement.* The estimate feeds a threshold -comparison against $1 slash (2 #dte)$, and that comparison must give the same -answer on consecutive frames of the same subject. A median is a continuous -functional of the data: perturb the samples slightly and it moves slightly. -Estimators defined by an $arg max$ — modes, in any of their usual forms — are -not, and near-ties in a multimodal distribution make them flip between frames. -For this decision, stability across frames outweighs freedom from bias. -] - -= What remains unresolvable - -Two limits are properties of the acquisition rather than of any algorithm. - -#defn[ -*Ambiguity A cannot be corrected.* By #link()[Prop. 1] the data is -invariant under the alias, so when the true field genuinely exceeds the -half-wrap, no rule recovers it: the prior folds it back, and that is that. -Whether the situation is *reachable* is governed by #dte, bounded below by the -readout duration — echoes cannot be spaced closer than they can be read out — -and above by $T_2^*$, since several echoes must fit before the signal decays. -Those bounds confine #dte, and so hold the half-wrap threshold above typical -shim residuals. -] - -#defn[ -*Higher field strength does not simply make it worse.* Shim residuals grow with -$B_0$, which argues for more risk. But $T_2^*$ shortens, forcing #dte down and -therefore raising the $1 slash (2 #dte)$ threshold. The two effects push the -same quantity in opposite directions. -] - -= Summary - -#key[ -Wrapped phase determines the coil offset and the field only up to a family of -alternatives. Sorting them out means asking, of each candidate, whether the -offset and the field it implies are mutually consistent. - -Two cases arise. When offset and field are both wrong in a matched way (*A*) the -data is unchanged, and only a prior can help. When the offset is wrong alone -(*B*) every echo carries the same constant, so extrapolating the echoes back to -$t = 0$ exposes it, against a rejection scale $abs(W(2 pi k))$ fixed by the echo -times. - -Where several candidates remain consistent, the prior that decides is that -*bulk-tissue* field is near zero, estimated by a weighted median over a -conservative mask. Consistency and prior are blind to opposite symmetries, so a -workable rule needs both. -]

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