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Topology-Aware Segmentation Using Discrete Morse Theory

Xiaoling Hu, Yusu Wang, Li Fuxin, Dimitris Samaras, Chao Chen

arXiv:2103.09992v1cs.CVcs.CG

TL;DR

Fine-scale segmentation can preserve pixel accuracy while still missing topological structure that matters for downstream analysis. The paper uses discrete Morse theory to identify global structures and builds a loss around them; it reports better per-pixel and topological performance, especially near challenging locations.

  • Problem

    Pixel-accurate segmentation can still break thin connections or leave holes, damaging downstream analysis, while topology is difficult to learn near blurred membranes or weak connections.

    Source: §1 · p. 1

  • Method

    The DMT-loss uses discrete Morse theory to identify 1D skeletons and 2D patches and constrains cross-entropy training to these topologically critical structures.

    Source: §2 · p. 3

  • Results

    The DMT-loss outperforms other methods in DICE score and topological accuracy on four 2D datasets and in all three topology-aware metrics on three 3D datasets.

    Source: §3 · pp. 7–8

  • Takeaways & Limitations

    The method's networks perform significantly better near topologically challenging locations, including weak connections and membranes.

    Source: §4 · p. 9

    Not all spurious structures can be pruned or cancelled, although most can in practice.

    Source: §2.2 · p. 5

Abstract

from arXiv · show

In the segmentation of fine-scale structures from natural and biomedical images, per-pixel accuracy is not the only metric of concern. Topological correctness, such as vessel connectivity and membrane closure, is crucial for downstream analysis tasks. In this paper, we propose a new approach to train deep image segmentation networks for better topological accuracy. In particular, leveraging the power of discrete Morse theory (DMT), we identify global structures, including 1D skeletons and 2D patches, which are important for topological accuracy. Trained with a novel loss based on these global structures, the network performance is significantly improved especially near topologically challenging locations (such as weak spots of connections and membranes). On diverse datasets, our method achieves superior performance on both the DICE score and topological metrics.

1 INTRODUCTION

Fine-scale segmentation can achieve high per-pixel accuracy yet still break connections or leave holes, damaging downstream analysis. The paper uses discrete Morse structures to guide training toward topologically difficult locations and reports gains in topological and per-pixel accuracy.

1 INTRODUCTION

  • Structural errors such as broken thin connections and membrane holes can damage downstream analysis, even when per-pixel accuracy is high.Topology-challenging spots include blurred membranes and weak vessel connections, and the paper says more unbalanced annotations do not alleviate the issue.Source: §1 · p. 1
  • The method identifies critical structures in likelihood maps using discrete Morse theory, including 1D and 2D manifold pieces that reveal topological information.The approach treats likelihood as a terrain and uses its structures to guide the network toward segmentations with the ground truth Betti number.Source: §1 · p. 1
  • The DMT-loss assigns higher penalties along Morse structures, which encompass potential object skeletons and target difficult locations during training.Because structures are identified from potentially noisy likelihoods, they can represent false negatives or false positives, and the loss is intended to correct both.Source: §1 · pp. 1–2
  • Discrete Morse theory, an approximation for computing 2D structures, and persistent-homology pruning address computational cost and irrelevant structures.The introduction contrasts this approach with sparse critical-point losses and reports a 3-time speedup over TopoLoss; it also notes limitations of pretrained-filter losses and neuron-specific region methods.Source: §1 · pp. 2–3
  • The DMT-loss outperforms state-of-the-art methods on multiple topology-relevant metrics across 2D and 3D benchmarks, with superior Betti number error.The introduction also reports high performance in both topological and per-pixel accuracy.Source: §1 · p. 2

2 METHOD

The method uses discrete Morse theory (DMT) to find topologically critical structures in a likelihood map and trains a network with cross-entropy focused on those structures. It prunes less salient structures and approximates costly 2D manifolds for efficient computation.

2 METHOD

  • DMT identifies 1D skeletons and 2D sheets as critical structures, and the DMT-loss constrains cross-entropy to these structures during end-to-end training.The method targets locations that may be critical for structural accuracy in 2D and 3D images.Source: §2 · p. 3

2.1 MORSE THEORY

  • For a likelihood map viewed as terrain, stable manifolds trace ridges from maxima to saddles, including ridges associated with broken connections.The method uses these structures to guide learning near topologically challenging pixels.Source: §2.1 · pp. 3–4

2.2 SIMPLIFICATION AND COMPUTATION

  • Persistence below threshold ϵ prunes corresponding Morse structures, while the approximation of 2-stable manifolds uses a maximum spanning forest computable in O(n log n).The approximation avoids the costly persistence computation required for the original 2-stable-manifold computation.Source: §2.2 · p. 5

2.3 THE DMT-BASED LOSS FUNCTION AND TRAINING DETAILS

  • The total loss combines cross-entropy with a weighted DMT-loss, which applies cross-entropy on pixels in the retained 1- and approximated 2-stable manifolds.The structures and mask are recomputed each epoch; the mask remains constant in a neighborhood of the current likelihood, so the DMT-loss gradient exists.Source: §2.3 · p. 6

3 EXPERIMENTS ON 2D AND 3D DATASETS

Across 2D and 3D benchmarks, DMT-loss improves topological accuracy and, in 2D, DICE performance. Ablations compare its efficiency and robustness with alternative losses and structure-selection strategies.

Quantitative and qualitative results

  • DMT-loss outperforms other methods in DICE score and topological accuracy across four 2D datasets, including ARI, VOI, and Betti Error.The reported comparison covers ISBI13, CREMI, CrackTree, and Road; TopoLoss uses a heavily engineered backbone, with same-backbone comparisons relegated to the appendix.Source: §3 · p. 7, Table 1 · p. 7
  • On three 3D datasets, DMT-loss outperforms existing methods in all three topology-aware metrics.The datasets are ISBI13, CREMI, and 3Dircadb.Source: §3 · p. 8, Table 2 · p. 8

The benefit of the proposed DMT-loss

  • On CREMI 2D, DMT-loss reaches Betti Error 0.956 in about 1.2 hours, versus TopoLoss at 1.113 in about 3 hours.The comparison uses the same backbone and starts from reasonable pretrained likelihood maps; standard U-Net takes about 0.5 hours.Source: §3 · p. 8

Ablation study for persistence threshold ϵ

  • On CREMI 2D, persistence threshold ϵ = 0.2 gives the best Betti Error, 0.982, versus 1.206 at 0.1 and 2.105 at 0.3.The authors attribute the poorer results to excess irrelevant structures at 0.1 and missed important structures at 0.3; ϵ is selected by cross-validation.Source: §3 · p. 9

Comparison with other simpler choices

  • On CREMI 2D, DMT-loss has Betti Error 0.982, compared with 2.971 for Canny edges, 2.507 for ridge detection, and 2.753 for reweighted cross-entropy.The paper attributes the simpler baselines' weaker topology results to noisy structures or weighting misclassified pixels without distinguishing topology-critical ones.Source: §3 · p. 9

Robustness of the proposed method

  • On CREMI 2D, DMT-loss maintains good performance under high Gaussian noise; its Morse structures are computed from a likelihood map described as robust to noise.The paper also reports that DMT-loss correctly segments fine structures such as membranes, roads, and vessels, with improvements over U-Net across datasets.Source: §3 · pp. 7, 9

4 CONCLUSION

The conclusion presents DMT-loss as a way to improve topological accuracy in deep image segmentation. It identifies 1D skeletons and 2D patches with discrete Morse theory and reports better performance near topologically challenging locations.

4 CONCLUSION

  • DMT-loss trains deep image segmentation networks for better topological accuracy.The approach is based on discrete Morse theory.Source: §4 · p. 9
  • Discrete Morse theory identifies 1D skeletons and 2D patches that matter for topological accuracy.Source: §4 · p. 9
  • Networks trained with this loss perform significantly better near challenging locations such as weak connections and membranes.Source: §4 · p. 9

A APPENDIX

The appendix previews additional material on topological errors, method details, and experiments.

A APPENDIX

A.1.1 ILLUSTRATION OF 3D TOPOLOGICAL ERRORS

The appendix illustrates two kinds of 3D topological errors: an index-1 error in synthetic data and an index-2 error in vessel data. The index-1 example is a hole in a sheet, associated with a saddle point and its stable manifold.

A.1.1 ILLUSTRATION OF 3D TOPOLOGICAL ERRORS

  • Figure 8 depicts an index-1 topological error: a hole in a 3D synthetic sheet after the likelihood is weakened in the middle.Source: §A.1.1 · p. 14, Figure 8 · p. 14
  • The appendix identifies index-2 topological errors in 3D vessel data as a separate illustrated case.Source: §A.1.1 · p. 14
  • In the index-1 example, saddle point S has one negative and two positive Hessian eigenvalues, and its stable manifold is a 2D plane through the saddle.The plane cuts the segmentation into two thin slices; negative-gradient V-paths flow toward S.Source: Figure 8 · p. 14

A.1.2 FALSE NEGATIVE AND FALSE POSITIVE ERRORS

The DMT-loss identifies false-negative and false-positive topological errors, then adjusts likelihoods on the implicated structures to support correct segmentation.

A.1.2 FALSE NEGATIVE AND FALSE POSITIVE ERRORS

  • The DMT-loss captures missing membrane regions as false negatives and mitochondria mistaken for neuron boundaries as false positives.The examples associate the missing membrane with a blurred region and the false positive with mitochondria.Source: §A.1.2 · p. 14
  • After identifying these errors, the DMT-loss raises or lowers likelihood values on the implicated structures to encourage correct topology.Source: §A.1.2 · p. 14

A.2.1 DISCRETE MORSE THEORY

The paper represents images as cubical complexes and uses discrete gradient fields to identify critical cells and their connecting structures. These structures provide a combinatorial description of topological features and errors.

A.2.1 DISCRETE MORSE THEORY

  • A dD image is represented as a cubical complex built from cells corresponding to vertices, edges, squares, and cubes.Here d is 2 or 3.Source: §A.2.1 · p. 14
  • The illustrated 3D vessel error is a broken connection associated with an index-2 saddle, while the 2D illustration shows critical structures extracted from a likelihood map.The 3D figure depicts the saddle's 1-stable manifold; the 2D figure identifies structures that are used in the DMT-loss.Source: Figure 9 · p. 15, Figure 10 · p. 15
  • A discrete gradient field pairs cells into V-pairs, with each cell in at most one pair and all V-paths acyclic; unpaired cells are critical.The paper interprets V-pairs as discrete flow directions and critical cells as locations with vanishing gradient.Source: §A.2.1 · pp. 15–16
  • Critical cells correspond to minima, saddle points, or maxima, with their cell dimensions determining the saddle index in 3D.In 2D, critical 0-, 1-, and 2-cells correspond to minima, saddles, and maxima; in 3D, critical 1- and 2-cells are index-1 and index-2 saddles.Source: §A.2.1 · p. 16
  • Morse cancellation reverses V-pairs along a unique V-path between a cancellable pair of critical cells, removing both from the critical set.The paper describes cancellation as a combinatorial way to reduce noisy stable manifolds.Source: §A.2.1 · p. 16

A.2.2 PERSISTENCE PRUNING

The method uses persistence to simplify a discrete gradient field, then constructs stable manifolds from the remaining important saddles.

A.2.2 PERSISTENCE PRUNING

  • The method extends the vertex-valued function to cells using each cell's maximum vertex value, initializes all cells as critical, and computes persistence pairs.The pairing uses a super-level set filtration and follows prior approaches cited in the passage.Source: §A.2.2 · p. 16
  • Persistence pairs record the birth and death of topological features during a super-level filtration, and their persistence measures feature lifetime and importance.The filtration adds cells in decreasing function values; pair persistence is the difference between the paired cells' function values.Source: §A.2.2 · p. 16
  • It cancels cancellable persistence pairs in increasing order of persistence when their persistence is below threshold ϵ.Source: §A.2.2 · p. 16
  • After pruning low-persistence critical cells, the method constructs 1- and 2-stable manifolds for the remaining high-persistence saddles.These remaining saddles are treated as important structures.Source: §A.2.2 · p. 16

A.2.3 MORE DETAILS ON THE APPROXIMATION OF S2 VIA ˆS2.

The appendix describes an approximation of 2-stable manifolds and reports experiments spanning biomedical and natural-image segmentation, with several metrics and ablations.

A.2.3 MORE DETAILS ON THE APPROXIMATION OF S2 VIA ˆS2.

  • The approximation of S2 uses the boundary of minima's stable manifolds, estimated with spanning trees in a watershed-like basin-growing procedure.Source: §A.2.3 · p. 16

A.3.1 DATASETS; A.3.2 EVALUATION METRICS

  • Experiments span neuron-membrane segmentation, natural-image structures, and retinal vessels, and evaluation uses five metrics including DICE and Betti number error.The datasets include electron microscopy images and natural-image datasets whose structures are described as vital to their functionality.Source: §A.3.1 · p. 17, §A.3.2 · p. 17

A.3.3 QUANTITATIVE RESULTS FOR MORE 2D DATASETS

  • Table 3 reports quantitative results for ISBI12 and DRIVE, but the supplied passage does not state the results themselves.Source: §A.3.3 · p. 17, Table 3 · p. 18

A.3.4 FAIRNESS COMPARISONS WITH SAME BACKBONE NETWORKS

  • The same-backbone comparison applies U-Net and DIVE on CREMI 2D, with Betti Error results listed in Table 4.The authors describe DMT-loss as backbone-agnostic and choose U-Net as a lightweight, generalizable backbone.Source: §A.3.4 · p. 17, Table 4 · p. 18

A.3.5 QUALITATIVE RESULTS FOR 3D DATASETS

A.3.6 THE ABLATION STUDY FOR BALANCED TERM β.

  • On CREMI 2D, β = 3 gives 0.982 Betti Error, compared with 1.074 at β = 2 and 1.181 at β = 4.The passage states that β is dataset-dependent and chosen by cross-validation.Source: §A.3.6 · p. 18

A.3.7 COMPARISON WITH OTHER SIMPLER CHOICES

  • Table 5 compares Canny edge detection and ridge detection with the proposed method, without giving the table values in the supplied passages.Source: §A.3.7 · p. 18, Table 5 · p. 18

A.3.8 RESULTS WITH DIFFERENT NOISE LEVELS

  • Table 6 varies corrupted-pixel percentage and Gaussian-noise standard deviation; the passage gives standard U-Net performance as 3.016 Betti Error for reference.Source: §A.3.8 · p. 18, Table 6 · p. 19

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