Relative winding constraints for scroll unrolling: generated automatically from CT, reconciled by L1 synchronization, no human in the loop. Relative structure validated; absolute winding counts not yet calibrated — see limitations.
A tool for the winding constraints problem in the Vesuvius Challenge — the effort to read the carbonized Herculaneum scrolls without opening them.
Left: a CT cross-section of PHerc0358, a 2027 Grand Prize scroll. Right: the winding field this tool recovers from it — 910 seeds and 1,583 constraints, generated and reconciled with no human input. Colour is relative winding number. Scale bar 2 mm.
To unroll a scroll digitally, the surface fitter needs to know, for pairs of nearby points in the CT volume, how many windings apart they sit. These relative winding constraints anchor the whole reconstruction: they are what stops the fitted surface from drifting onto a neighbouring wrap and rendering convincing text from the wrong part of the scroll.
Today those constraints are drawn by hand, and contradictions between them are resolved by hand. The current tooling [7] propagates winding numbers along a BFS spanning tree and uses the remaining measurements only to detect inconsistent loops — each of which becomes human work to adjudicate. Two structural problems follow:
- A spanning tree is maximally fragile. One wrong edge high in the tree corrupts its entire subtree, unrecoverably, because a tree has no redundancy by construction.
- The redundant measurements that could repair errors are used only to raise alarms. Every detected contradiction costs annotator time.
This tool replaces both halves: constraints are generated automatically from the volume, and contradictions are resolved globally instead of reported.
Assigning integer winding numbers w from noisy pairwise measurements
d_ij ≈ w_i − w_j is synchronization over ℤ — the integer sibling of phase
unwrapping. We minimise total weighted disagreement in L1:
minimise Σ_ij c_ij · | (w_i − w_j) − d_ij |
Three properties make this the right objective rather than a heuristic:
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The optimization is exact — which is a claim about the solver, not about correctness. Written as a linear program, the constraint matrix is the graph's node-arc incidence matrix bordered by identity blocks. Incidence matrices are totally unimodular, and TU survives appending identity columns — so every vertex of the LP polytope is integral whenever the measurements are, and a simplex-type solver returns the true integer optimum with no rounding and no branch-and-bound (verified empirically: zero deviation from integrality on both synthetic graphs and the real PHerc0358 graph with non-integer confidence weights). This is Costantini's structure from minimum-cost-flow phase unwrapping [3]; the integrality property is Hoffman–Kruskal [4]. The framing as group synchronization follows [5].
To be precise about what this buys: it guarantees the solver finds the best possible explanation of the measurements it is given. If the measurements are wrong — and ours are uncalibrated in absolute count — the solver finds the exact optimum of a flawed problem. Exactness removes the solver as a source of error; it does not bless the inputs. The results below should be read with that separation in mind: the solving is provably optimal, the generating is directionally right and honestly imperfect.
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Robust to the failure that actually happens. A wrong constraint is wrong by a whole winding, not by a little. Least squares squares that error and distorts the rest of the solution to reduce it; L1 pays it once and leaves the graph alone.
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Every edge counts. No spanning tree, so no privileged edges whose failure is unrecoverable. Redundant measurements become error-correcting rather than error-detecting.
The scroll's spiral geometry needs no special handling: winding number is an unbounded integer, not a phase modulo 2π, so circulation around the umbilicus emerges from the measurements themselves. This is strictly easier than classical phase unwrapping.
The constraints themselves come from the volume, with no human input. A structure tensor gives the local orientation of the papyrus laminae at every point — with no assumption of a spiral, an umbilicus, or a global winding spacing, because real scrolls violate all three (one Grand Prize scroll measures a circularity ratio of 2.20, and local lamina spacing varies fourfold across a single 11 mm crop). Short probes cast perpendicular to that orientation count the laminae they cross; each count is one signed constraint. Seeds are snapped onto laminae, probes attach to the nearest seed, and large slices are tiled automatically with overlap so boundary constraints still form.
Individual probes are allowed to be wrong. That is the design: cheap, plentiful, partly-wrong measurements are exactly what the L1 solver is built to reconcile, and exactly what tree propagation cannot survive.
Accuracy of recovered winding numbers vs injected gross-error rate (7 seeds, sparse graph, 1.9 edges/node):
| gross errors | BFS tree (current) | L2 | L1 (this tool) |
|---|---|---|---|
| 0% | 1.000 | 1.000 | 1.000 |
| 5% | 0.500 | 0.853 | 0.994 |
| 10% | 0.454 | 0.812 | 0.973 |
| 15% | 0.373 | 0.780 | 0.939 |
| 20% | 0.287 | 0.745 | 0.867 |
First drop below 95% accuracy: BFS at 2%, L2 at 5%, L1 at 15%.
A spanning tree consumes only n−1 edges however many exist, so extra measurements are nearly wasted on it. Under L1 they repair errors. At a fixed 10% gross-error rate:
| edges/node | BFS tree | L1 |
|---|---|---|
| 1.91 | 0.454 | 0.973 |
| 3.70 | 0.528 | 1.000 |
| 6.08 | 0.620 | 1.000 |
Redundancy eliminates 98% of L1's residual error and 30% of the tree's — which still leaves the tree unusable. The practical instruction inverts: stop trying to make each measurement perfectly reliable, and collect many more, tolerating that a minority are wrong. That is exactly what an automatic detector produces.
Note on an earlier claim. Development notes reported BFS scoring 0.047 exact agreement against L1's 0.827 on this crop. Those figures predate two changes to the constraint generator (along-lamina edges and ridge-snapped seeds) and no longer hold. On the current generator the same crop gives L1 0.703 and BFS 0.629 — a far smaller gap on that metric. Mean absolute residual separates them better (0.51 vs 1.75), and on full cross-sections BFS collapses entirely, returning a winding range of 0 where L1 returns a coherent field. The synthetic breakdown curves above are unaffected: they use controlled error injection independent of the generator.
On a full cross-section of PHerc0358 (a 2027 Grand Prize scroll), generation plus solving with zero human input: ~20,000 seeds, ~35,000 constraints, 97% of seeds in one connected component, solved in seconds on a laptop. The same runs on PHerc. Paris 4 produce an internally consistent winding field whose relative structure is sound; its absolute count is not validated against any published figure — see the calibration note below.
Winding spacing across all 13 Grand Prize scrolls: 207–259 µm, median 225 µm, CV 0.064 — two scanners, two resolutions, thirteen scrolls agreeing. This sets probe lengths, window sizes and sanity bounds for anyone working on these volumes, and to our knowledge has not been published.
Stated here so nobody has to discover it.
- Absolute winding counts are not calibrated. The lamina counter is resolution-dependent: on PHerc. Paris 4, recovered counts went 128 → 184 → 61 → 145 across three attempted fixes at different resolutions. Each fix was one parameter tuned against one number, so we stopped rather than tune a fourth. The relative structure of the winding field is sound; the absolute count is not. The proper fix is per-location ground truth from the published Paris 4 segment meshes — a many-point comparison that makes overfitting visible, which a single scalar never can. That is the next step, and help is welcome.
- Per-constraint agreement plateaus near 0.66 on real scrolls — the same value on three scrolls of very different quality, including one already successfully read. The ceiling is in the constraint generator, not the specimens, and not the solver.
- A trap, documented so you avoid it: during calibration, over-smoothing cut the recovered count to less than half of truth — while per-constraint agreement rose to its best-ever 0.863. The surviving constraints were few, clean, and systematically wrong. Internal consistency is not correctness. Do not optimise self-consistency.
- Synthetic benchmarks use regular grids; real patch graphs have clusters and bridges, and a bridge whose measurements are all wrong is unrecoverable regardless of the global error rate.
pip install -r requirements.txt
# validate against ground truth we control
python -m winding_sync.validate
# generate + solve on a real scroll, streaming from the open-data bucket
python run_winding_test.py --scroll PHerc0358Windows users: create an isolated environment with
python -m venv .venv then .\\.venv\\Scripts\\python.exe -m pip install -r requirements.txt.
- Input: OME-Zarr over
s3://,https://, or local paths. zarr 2.x and 3.x; Python 3.10–3.14; Windows/macOS/Linux. Streaming throughout — no full volume is ever loaded, and large slices tile automatically. - Output: a
WindingGraph(nodes, edges, integer deltas, confidence weights) that drops into any solver, plus the optimal L1 solution for those measurements. - The solver is usable standalone with constraints from any source, including existing hand annotations — it can replace the reconciliation step of the current workflow without adopting anything else here.
- Automated tests covering exactness, robustness, redundancy and documented limits, plus regressions pinned on real carbonized papyrus. Seeds fixed.
This tool does not exist in a vacuum, and the debts should be explicit.
Henderson's diffeomorphic spiral fitting [2] is the state-of-the-art top-down unroller, and its §4 contains the closest prior approach to this work. It casts rays along local surface normals, builds a graph over surface paths, then — in its own words — drops nodes that are part of cycles (since these imply a contradiction) and uses majority voting by edges. Its Limitations section names the consequence directly: when paths are contradictory, the surface sometimes wanders between two true windings, instead of committing to one.
That is the precise gap this tool addresses. A contradictory cycle is not noise to be discarded — it is a measurement that some nearby constraint is wrong, and L1 synchronization uses it to locate and repair the error rather than deleting the evidence. Henderson himself suggests robust alternatives (an L^p norm with p<1, or an augmented Lagrangian) as future work; L1 over ℤ is a related move made at the constraint-reconciliation stage rather than inside the surface fit. What this tool adds on the generation side is a route that needs no surface-prediction network — constraints come straight from the CT via the orientation tensor. What it adds on the reconciliation side is the L1 synchronization formulation with its integrality property [4], measured against the BFS spanning-tree reconciliation in the villa production scripts [7]. The L1 objective for this problem class is Costantini's [3], transplanted from InSAR phase unwrapping to winding numbers — where it applies more cleanly, since winding number is an unbounded integer rather than a phase modulo 2π.
Verified against primary or citing sources at time of writing. Works are listed only where this project actually stands on them; nothing is cited for appearance.
Foundational — virtual unwrapping
[1] W. B. Seales, C. S. Parker, M. Segal, E. Tov, P. Shor, Y. Porath. "From damage to discovery via virtual unwrapping: Reading the scroll from En-Gedi." Science Advances 2(9):e1601247, 2016. doi:10.1126/sciadv.1601247 — the pipeline this whole field descends from: segment, flatten, texture, read.
[2] P. Henderson. "Virtually Unrolling the Herculaneum Papyri by Diffeomorphic Spiral Fitting." WACV 2026, pp. 6401–6411. arXiv:2512.04927. Code: https://github.com/pmh47/spiral-fitting — current state of the art, prior work on extracting relative winding numbers (from neural surface predictions; this tool instead works directly from CT).
The optimization
[3] M. Costantini. "A novel phase unwrapping method based on network programming." IEEE Transactions on Geoscience and Remote Sensing 36(3):813–821, 1998. doi:10.1109/36.673674 — the L1/network-flow objective transplanted here from InSAR phase to winding number.
[4] A. J. Hoffman, J. B. Kruskal. "Integral Boundary Points of Convex Polyhedra." In Linear Inequalities and Related Systems, Annals of Mathematics Studies 38, pp. 223–246. Princeton University Press, 1956. doi:10.1515/9781400881987-014 — total unimodularity implies integral vertices; the theorem the solver's integrality rests on.
[5] A. Singer. "Angular synchronization by eigenvectors and semidefinite programming." Applied and Computational Harmonic Analysis 30(1):20–36, 2011. doi:10.1016/j.acha.2010.02.001 — the group-synchronization framing. Note the difference exploited here: angular synchronization recovers phases modulo 2π, whereas winding number is an unbounded integer, so no modular ambiguity arises and the linear-programming route is available.
The orientation field
[6] J. Bigün, G. H. Granlund. "Optimal orientation detection of linear symmetry." Proceedings of the First International Conference on Computer Vision (ICCV), London, pp. 433–438. IEEE Computer Society, 1987 — the structure tensor, used here to obtain local lamina orientation without assuming a spiral or an umbilicus.
Drawn from [2]'s bibliography — context this work sits in
[10] S. Parsons, C. S. Parker, C. Chapman, M. Hayashida, W. B. Seales. "EduceLab-Scrolls: Verifiable Recovery of Text from Herculaneum Papyri using X-ray CT." arXiv:2304.02084, 2023 — the scan corpus; source of the PHerc. Paris 4 and PHerc. 172 volumes.
[11] J. Schilliger. ThaumatoAnakalyptor. https://github.com/schillij95/ThaumatoAnakalyptor — the other fully-automated unroller for this data; uses heuristics to decide whether nearby instances lie on the same winding.
[12] D. Eberly, R. Gardner, B. Morse, S. Pizer, C. Scharlach. "Ridges for image analysis." Journal of Mathematical Imaging and Vision 4:353–373, 1994 — ridge extraction; the basis for snapping seeds onto laminae here.
[13] C. Steger. "An unbiased detector of curvilinear structures." IEEE TPAMI 20(2):113–125, 1998 — curvilinear structure detection, the classical alternative to the structure-tensor route used here.
[14] F. Nicolardi, S. Parsons, D. Delattre, et al. "Revealing Text from a Still-Rolled Herculaneum Papyrus Scroll (PHerc. Paris. 4)." Zeitschrift für Papyrologie und Epigraphik 229:1–13, 2024 — the text recovered from the PHerc. Paris 4 region used here for validation.
[15] C. S. Parker, K. Gessel, S. Parsons, et al. Volume Cartographer, 2025. doi:10.5281/zenodo.14878458 — VC3D; the ground-truth mesh for PHerc. Paris 4 was produced with hundreds of hours of manual annotation using this lineage of tools, which is the cost this project aims to reduce.
Data and tooling
[7] Vesuvius Challenge / EduceLab, villa repository (volume-cartographer),
including scripts/spiral/find_inconsistent_windings.py — the BFS
winding-consistency baseline measured against here.
https://github.com/ScrollPrize/villa
[8] Vesuvius Challenge, "Winding Constraints" (Open Problem). https://scrollprize.org/open_problems/winding_annotations
[9] Vesuvius Challenge open data: CT volumes of the 13 Grand Prize scrolls and
PHerc. Paris 4, served as OME-Zarr from
s3://vesuvius-challenge-open-data. https://scrollprize.org/data
@inproceedings{henderson26wacv,
title = {Virtually Unrolling the Herculaneum Papyri by Diffeomorphic Spiral Fitting},
author = {Paul Henderson},
booktitle = {Proceedings of the IEEE/CVF Winter Conference on Applications of Computer Vision (WACV)},
pages = {6401--6411},
year = {2026}
}MIT.