Same Dice, Different Geometry

The question

Dice measures how much a predicted mask $P$ overlaps a reference $G$ :

D \;=\; \frac{2\,|P \cap G|}{|P| + |G|} \;=\; \frac{2\,\mathrm{TP}}{2\,\mathrm{TP} + \mathrm{FP} + \mathrm{FN}}

It depends on how many pixels are wrong, not on where they are. For thin, branching structures such as retinal vessels, that matters: a mask can score highly while breaking connections, losing terminal branches or growing false ones.

Most method comparisons let overlap and structure change together, which makes it hard to see what the overlap score is missing. This study inverts the setup: hold Dice constant, and vary only the spatial organisation of the error.

Method

Data: 20 expert vessel annotations from the DRIVE training set, restricted to their official field-of-view masks.
Five controlled error families: centerline-preserving width loss, connectivity cuts, terminal-vessel loss, thickness gain and false spurs.
Matched overlap: solving the equation above for the error count, for each target Dice (0.99, 0.97, 0.95) the exact pixel budget was solved analytically from the Dice equation, then spent according to each error rule. 300 surrogate segmentations in total; realised Dice stayed within 1.01×10⁻⁵ of target.
Structural panel: clDice, connected components, Betti-number error, endpoint statistics, a finite-scale Rényi entropy profile (box sizes 2–64 px) and a block-lacunarity curve.
Statistics: paired Friedman tests per outcome and Dice level, with Wilcoxon contrasts under Holm correction. The mask is the experimental unit (n = 20).

Results

At effectively identical Dice, the five error families looked very different structurally.

Median structural outcomes at matched Dice 0.99, across 20 masks. Source: manuscript Table 1.
Error family	clDice	Component change	Betti error	Rényi profile RMSE
Width loss	1.0000	0	0	0.0021
Connectivity cuts	0.9788	47.5	61	0.0221
Terminal-vessel loss	0.9825	2	3	0.0297
Thickness gain	0.9963	0	4.5	0.0053
False spurs	0.9701	0	24	0.0384

Chart loads with JavaScript; the same values are in the table on this page.

Median clDice by matched Dice target (manuscript Table 2). Overlap falls identically for every family; centerline fidelity does not. Marker shape identifies the family.

Same budget, different damage. Width loss and connectivity cuts remove the same number of pixels. One keeps clDice at 1.0000; the other creates a median of 47.5 extra components.
Component count is not enough. Thickness gain and false spurs both leave the component count unchanged, yet false spurs raise Betti error from 4.5 to 24 and produce the largest multiscale deviation, about eighteen times that of width loss.
The gap widens as overlap falls. At Dice 0.95, median clDice ranged from 1.0000 to 0.8649 and Rényi-profile deviation from 0.0096 to 0.1711.
The error families differed on all five pre-specified structural outcomes at every Dice level (largest of 15 Friedman p-values: 4.84×10⁻¹⁴).

What it means

The point is not to abandon Dice — it answers the overlap question well. The problem starts when overlap is treated as if it also answered questions about connectivity, topology or multiscale geometry.

The study supports a panel of metrics matched to the property that matters, rather than one replacement score. Each diagnostic catches different failures: component counts detect fragmentation but miss attached false branches; clDice detects centerline mismatch but not which kind; multiscale profiles react to redistribution across scales but are not topological invariants.

The same lesson applies beyond medical imaging: when an intelligent system’s output is a structured object, evaluation should measure the structure it is expected to preserve.

Limitations

The masks are controlled surrogates, not outputs of trained models. The study establishes a property of evaluation — not model superiority, not how often these failures occur in practice, and not clinical validity.

It uses 20 annotations from a single dataset, one skeletonisation and connectivity convention, and finite-scale descriptors over six box sizes that should not be read as clinical fractal biomarkers or as evidence of multifractality.

Try it yourself: the interactive explainer lets you move the same number of pixels on a small grid and watch connectivity change while Dice stays fixed.

The natural next step is to freeze these metric definitions and apply them to real predictions from several model families and datasets.

Related: the earlier Applied Sciences study evaluated a PDE-refined local fractal-dimension prior with topology-aware refinement. This work isolates a measurement question that arose from it.