Topology-Inspired Morphological Descriptor

• Topology-inspired morphological descriptors are quantitative representations that combine topological invariants with geometric embedding to capture shape complexity.
• They employ methods such as persistent homology, skeleton reduction, and cycle counting to summarize connectivity, branching, and multiscale organization.
• Applications span neuroscience, materials science, computer vision, and soft robotics, offering robust and interpretable metrics for complex form analysis.

A topology-inspired morphological descriptor is a quantitative representation of shape that encodes morphology through topological invariants, topology-preserving reductions, or combinatorial summaries derived from filtrations, skeletons, nerve graphs, or cycle structure. In current arXiv literature, the term spans several related constructions: persistent-homology barcodes and persistence images for rooted trees, point clouds, and cubical complexes; skeleton-to-graph pipelines for biological objects and cracks; ring-frequency vectors for disordered materials; and Morse-theoretic critical-point counts for soft continuum robots. Despite their heterogeneity, these methods share a common objective: converting geometrically complex forms into stable and comparable descriptors that preserve branching, connectivity, loops, cavities, or multiscale organization (Kanari et al., 2016, Minamitani et al., 2021, Ryan et al., 2023, Chung et al., 2021).

1. Scope and principal descriptor families

Topology-inspired descriptors are best understood as a family of representations rather than a single formalism. Some methods are explicitly topological in the homological sense, using filtrations and persistence diagrams; others are topology-preserving graph reductions or cycle-counting schemes that retain connectivity and branching while adding geometric weights. This broader usage is visible across neuroscience, materials science, image morphology, computer vision, and soft robotics.

Framework	Core construction	Representative use
TMD (Kanari et al., 2016)	Rooted tree with spherical filtration mapped to a 0-dimensional persistence barcode	Neuronal morphology discrimination
Ball Mapper (Dłotko, 2019)	Greedy $\varepsilon$-net cover and nerve graph of overlapping balls	Exploratory shape summary for point clouds
ToSkA (Ryan et al., 2023)	Topological skeleton converted to a spatial graph with network metrics	Shape profiling of segmented objects
C2DTD (Hawthorne et al., 2 Apr 2026)	Concatenation $\Delta=[G,R,T]$ of geometry, radial statistics, and primitive ring fractions	Two-dimensional carbon networks
Soft-robot Morse signature (Wu et al., 1 Aug 2025)	Directional projections of a center-line with discrete Morse critical-point counts	Morphology classification and control
CrackMorph-XAI-Net (Ajjarapu et al., 8 May 2026)	Learned topology-preserving skeleton, junction heatmaps, and graph-derived crack metrics	Automated crack morphology

A recurring distinction separates descriptors that summarize filtration dynamics from those that summarize a reduced structural graph. Persistent-homology methods emphasize birth, death, and persistence of features across scale; skeleton and network methods emphasize degrees, branch counts, cycle counts, or weighted path structure. Ring-based descriptors, such as those for amorphous and two-dimensional carbon systems, lie between these categories: they are graph-theoretic rather than homological in the strict sense, yet they explicitly target medium-range topological organization.

2. Filtrations, homology, and barcode-based morphology

The dominant mathematical substrate is the filtration. Given a space or embedded combinatorial object together with a scalar function, one forms a nested family of subobjects and records how connected components, loops, or voids appear and disappear. In persistent homology, the $k$-th Betti number at scale $\alpha$ is

$\beta_k(\alpha)=\mathrm{rank}\;H_k(F(\alpha)),$

and the persistence diagram is

$\mathrm{PD}^k=\{(b_i,d_i)\},$

with persistence $\pi_i=d_i-b_i$ (Minamitani et al., 2021).

In the Topological Morphology Descriptor for neurons, the object is a rooted tree $T=(V,E)$ embedded in $\mathbb{R}^3$ with soma vertex $v_{\mathrm{soma}}$. The filtration function is the radial distance

$\Delta=[G,R,T]$0

Sublevel sets $\Delta=[G,R,T]$1 add vertices and edges with $\Delta=[G,R,T]$2; new leaves generate birth events, and component mergers at branch points generate death events. The descriptor is the resulting 0-dimensional persistence barcode $\Delta=[G,R,T]$3 or persistence diagram $\Delta=[G,R,T]$4, which encodes the branching anatomy of the tree (Kanari et al., 2016).

For point-cloud data, the filtration is often simplicial. In amorphous materials, atomic coordinates $\Delta=[G,R,T]$5 are lifted to a Vietoris–Rips filtration

$\Delta=[G,R,T]$6

or alternatively an $\Delta=[G,R,T]$7-complex filtration. Persistent homology in dimensions $\Delta=[G,R,T]$8 then yields descriptors of connected components and loops, with the one-dimensional diagram $\Delta=[G,R,T]$9 subsequently vectorized for prediction tasks (Minamitani et al., 2021).

For surface and image data, the filtration is cubical. A depth map or grayscale image defines a function on 2-cells, and lower-dimensional cells inherit coface values, producing a sublevel filtration on a 2D cubical complex. This construction supports $k$0 and $k$1 persistence for pits, basins, ridges, and enclosed rims in 3D surface texture analysis (Zeppelzauer et al., 2016, Zeppelzauer et al., 2017).

A more general extension replaces a single scalar parameter by a multi-parameter grid generated by classical morphological operators. For nested structuring elements $k$2, erosion, dilation, opening, and closing induce one-parameter and then $k$3-parameter filtrations. For a multi-index $k$4, the sets

$k$5

satisfy $k$6 whenever $k$7 coordinatewise, yielding persistence modules, Betti functions, and rank invariants that encode image structure across multiple morphological scales (Chung et al., 2021).

Ball Mapper occupies a nearby but distinct position. It does not require homology computation on a full filtration. Instead, a greedy $k$8-net of centers defines a cover by $k$9-balls, and the 1-skeleton of the nerve becomes a graph whose connected components, cycles, and high-degree nodes summarize shape at scale $\alpha$0 (Dłotko, 2019).

3. Skeleton, graph, and cycle formulations

A second major lineage begins not from filtrations but from topology-preserving reduction to a skeleton or graph. In ToSkA, a compact object $\alpha$1 is reduced to a one-voxel-thick topological skeleton by 26-neighbourhood thinning using the Lee 3D algorithm. Terminal points with $\alpha$2 and branch points with $\alpha$3 become graph nodes, maximal voxel chains become edges, and each edge receives Euclidean length

$\alpha$4

Morphology is then described by degree distributions, betweenness centrality, clustering coefficients, branch-length distributions, the longest leaf-to-leaf path $\alpha$5, and the cyclomatic number

$\alpha$6

for graph $\alpha$7 (Ryan et al., 2023).

CrackMorph-XAI-Net uses a related graph view, but the skeleton is learned rather than obtained by classical thinning. From the binarized crack skeleton and crack mask, it computes total crack length

$\alpha$8

average width

$\alpha$9

orientation from PCA on skeleton coordinates, tortuosity $\beta_k(\alpha)=\mathrm{rank}\;H_k(F(\alpha)),$0, junction count $\beta_k(\alpha)=\mathrm{rank}\;H_k(F(\alpha)),$1, and a topology class determined by $\beta_k(\alpha)=\mathrm{rank}\;H_k(F(\alpha)),$2: Linear, Branched, Complex, or Network (Ajjarapu et al., 8 May 2026).

In topology-optimized bio-inspired networks, skeletonization yields a weighted planar graph $\beta_k(\alpha)=\mathrm{rank}\;H_k(F(\alpha)),$3 in which branch nodes become vertices, struts become edges, and weights $\beta_k(\alpha)=\mathrm{rank}\;H_k(F(\alpha)),$4 are mean local thicknesses measured along each strut. The descriptor suite then includes node degree, mean degree, node strength, total edge weight, modularity

$\beta_k(\alpha)=\mathrm{rank}\;H_k(F(\alpha)),$5

and stress-distribution measures such as $\beta_k(\alpha)=\mathrm{rank}\;H_k(F(\alpha)),$6 and $\beta_k(\alpha)=\mathrm{rank}\;H_k(F(\alpha)),$7 for fracture localization (Nguyen et al., 2019).

Cycle-based descriptors for materials replace explicit skeletons by ring topology. In C2DTD, a periodic neighbor graph is built from atomic coordinates, local geometric quantities are summarized by global moments, a compact RDF histogram is computed, and primitive rings are identified as chordless cycles in a supercell. Ring counts $\beta_k(\alpha)=\mathrm{rank}\;H_k(F(\alpha)),$8 are normalized as

$\beta_k(\alpha)=\mathrm{rank}\;H_k(F(\alpha)),$9

and concatenated with geometric and radial statistics into a fixed-length descriptor $\mathrm{PD}^k=\{(b_i,d_i)\},$0 (Hawthorne et al., 2 Apr 2026).

Soft continuum robots introduce a Morse-theoretic variant. For a center-line $\mathrm{PD}^k=\{(b_i,d_i)\},$1 and a direction $\mathrm{PD}^k=\{(b_i,d_i)\},$2, one defines

$\mathrm{PD}^k=\{(b_i,d_i)\},$3

and counts non-degenerate critical points of this projection. In the pseudo-rigid-body discretization, a practical proxy is the sign change

$\mathrm{PD}^k=\{(b_i,d_i)\},$4

leading to the discrete Morse number

$\mathrm{PD}^k=\{(b_i,d_i)\},$5

The resulting signature $\mathrm{PD}^k=\{(b_i,d_i)\},$6 classifies J-, C-, and S-shapes and is also used as a control target (Wu et al., 1 Aug 2025).

4. Vectorization, distances, stability, and computation

Because many topology-inspired objects are multisets or graphs rather than fixed-length vectors, practical deployment typically requires vectorization or task-specific metrics. For persistence diagrams, standard distances are the bottleneck distance

$\mathrm{PD}^k=\{(b_i,d_i)\},$7

and the $\mathrm{PD}^k=\{(b_i,d_i)\},$8-Wasserstein distance

$\mathrm{PD}^k=\{(b_i,d_i)\},$9

The 1-Wasserstein distance is likewise defined by optimal matchings with summed $\pi_i=d_i-b_i$0 costs (Kanari et al., 2016, Beers et al., 2022).

For TMD, the original stability statement is a bottleneck bound. If two rooted trees share the same vertex set but have filtration functions $\pi_i=d_i-b_i$1 and $\pi_i=d_i-b_i$2, then

$\pi_i=d_i-b_i$3

Beers–Harrington–Goriely extend this picture by proving 1-Wasserstein stability under four perturbation classes: vertex perturbation, short leaf-edge addition, subdivision plus attachment, and retraction. Their bounds include

$\pi_i=d_i-b_i$4

for vertex perturbation and

$\pi_i=d_i-b_i$5

for short-edge addition or retraction, with an improved $\pi_i=d_i-b_i$6 bound under a stronger nondegeneracy condition (Beers et al., 2022).

Vectorization of persistence diagrams is dominated by persistence landscapes, persistence images, aggregated statistics, and related surfaces. In the persistence-image formulation, one smooths each off-diagonal point with a Gaussian kernel, weights by persistence, and integrates over a rectangular grid. For TMD, a common coordinate change is

$\pi_i=d_i-b_i$7

followed by a Gaussian persistence surface and pixel integration; Adams et al. are cited for the fact that the map from diagrams to persistence images is Lipschitz with respect to the 1-Wasserstein distance (Beers et al., 2022). Surface-texture studies also compare PD_AGG, persistence landscapes, Betti curves, and persistence images, finding that these vectorizations capture complementary information and remain empirically robust to moderate Gaussian noise, spatial shifts, and parameter variation (Zeppelzauer et al., 2017).

Perturbed Topological Signatures provide another vectorization route. A persistence diagram is converted into a Gaussian persistence surface, randomly perturbed, stacked into a matrix, and reduced by SVD to a low-dimensional subspace on the Grassmann manifold. The resulting representation enjoys a stability bound of the form

$\pi_i=d_i-b_i$8

and was reported to be much faster than classical diagram matchings in 3D shape analysis (Som et al., 2018).

Computational cost varies sharply across constructions. TMD extraction can be performed in linear time $\pi_i=d_i-b_i$9 because each vertex and edge is processed once (Kanari et al., 2016). Cubical and Vietoris–Rips pipelines are heavier; surface studies note worst-case growth roughly as $T=(V,E)$0 for cubical grids, though runtimes for $T=(V,E)$1 patches are described as manageable (Zeppelzauer et al., 2017). C2DTD reports $T=(V,E)$2 neighbor search and approximately $T=(V,E)$3 per 300-atom sheet on a single CPU for ring detection, with trivial parallelization across structures (Hawthorne et al., 2 Apr 2026).

5. Reported applications and empirical performance

In neuroscience, TMD was introduced as a stable descriptor for any tree-like morphology and empirically evaluated on artificial binary trees and reconstructed neuronal arbors. On random-tree benchmarks, groups differing only in tree depth or branch length were separated with $T=(V,E)$4 accuracy; differences in bifurcation angle yielded $T=(V,E)$5, and differences in randomness $T=(V,E)$6. On rat cortical pyramidal-cell subtypes, persistent images combined with a decision-tree classifier recovered expert-labeled classes with accuracy up to $T=(V,E)$7, while random relabeling gave near-chance performance (Kanari et al., 2016).

In amorphous materials, Minamitani et al. used persistent homology of atomic configurations, persistent images with $T=(V,E)$8, $T=(V,E)$9, weighting $\mathbb{R}^3$0, and Gaussian bandwidth $\mathbb{R}^3$1, followed by ridge regression with $\mathbb{R}^3$2. On $\mathbb{R}^3$3 samples, five-fold cross-validation produced mean $\mathbb{R}^3$4 and RMSE $\mathbb{R}^3$5. Their inverse analysis identified a five-atom ring as the primary medium-range-order unit and associated narrow-distribution five-vertex cycles with higher conductivity (Minamitani et al., 2021).

For two-dimensional carbon networks, C2DTD concatenates 56 geometric and radial statistics with 14 ring fractions. In benchmark comparisons against matminer features, reported performance includes $\mathbb{R}^3$6 versus $\mathbb{R}^3$7 at test size $\mathbb{R}^3$8, and $\mathbb{R}^3$9 versus $v_{\mathrm{soma}}$0 at test size $v_{\mathrm{soma}}$1. In feature-ablation studies at test size $v_{\mathrm{soma}}$2, the full descriptor achieved $v_{\mathrm{soma}}$3 and RMSE $v_{\mathrm{soma}}$4, while rings only gave $v_{\mathrm{soma}}$5 and RMSE $v_{\mathrm{soma}}$6, supporting the importance of ring topology. Vacancy-engineered graphene experiments further showed a progression from hexagon-dominated to topologically disordered networks as vacancy fraction increased from $v_{\mathrm{soma}}$7 to $v_{\mathrm{soma}}$8 (Hawthorne et al., 2 Apr 2026).

In 3D surface texture analysis, Zeppelzauer et al. reported Dice Similarity Coefficient values of approximately $v_{\mathrm{soma}}$9 for PD_AGG and $\Delta=[G,R,T]$00 for persistence images on the small dataset, with early fusion of PI and ESDD reaching approximately $\Delta=[G,R,T]$01. A related study on high-resolution archaeological rock surfaces reported best persistence-image performance of $\Delta=[G,R,T]$02 and an ESDD+PI combination of $\Delta=[G,R,T]$03, with statistically significant improvements over baseline or PD_AGG features (Zeppelzauer et al., 2017, Zeppelzauer et al., 2016).

In image-based topological object detection, the Shape of Orientation Histogram descriptor measures symmetry and smoothness of a normalized orientation histogram rather than using raw bins directly. SOH-based proposal generation achieved recall approximately $\Delta=[G,R,T]$04–$\Delta=[G,R,T]$05 on Artcode versus non-Artcode classification, ran in under $\Delta=[G,R,T]$06 per image, and, with Random Forests or RBF-SVM on SMOTE-balanced data, yielded overall accuracy approximately $\Delta=[G,R,T]$07 and ROC-AUC approximately $\Delta=[G,R,T]$08 (Xu et al., 13 Aug 2025).

In automated crack morphology, CrackMorph-XAI-Net reported a mean Dice coefficient of $\Delta=[G,R,T]$09 for learned skeleton extraction with topology preserved in $\Delta=[G,R,T]$10 of test images, junction detection recall $\Delta=[G,R,T]$11 and F1-score $\Delta=[G,R,T]$12, and descriptor-level correlations exceeding $\Delta=[G,R,T]$13 for length, width, orientation, junction count, and tortuosity. Topology classification across four crack classes reached $\Delta=[G,R,T]$14 accuracy (Ajjarapu et al., 8 May 2026).

6. Interpretive issues, misconceptions, and open problems

A common misconception is that topology-inspired descriptors are purely topological and therefore discard geometry. The literature shows the opposite. TMD is driven by Euclidean distance from the soma; ToSkA weights edges by Euclidean length and uses $\Delta=[G,R,T]$15 as a proxy for principal-axis variability; fracture graphs weight edges by mean local thickness; C2DTD explicitly concatenates local geometric moments and an RDF with ring topology; crack descriptors depend on area, path length, PCA orientation, and endpoint distance (Kanari et al., 2016, Ryan et al., 2023, Nguyen et al., 2019, Hawthorne et al., 2 Apr 2026, Ajjarapu et al., 8 May 2026). A more precise statement is that these methods couple topological organization to geometric embedding rather than replacing geometry altogether.

A second misconception is that topological construction automatically guarantees scale invariance or parameter insensitivity. Ball Mapper depends directly on the choice of $\Delta=[G,R,T]$16: if $\Delta=[G,R,T]$17, the graph becomes highly fragmented, whereas if $\Delta=[G,R,T]$18 it may become a clique (Dłotko, 2019). Surface persistence images are empirically robust, but their performance still depends on normalization, grid resolution, and kernel bandwidth (Zeppelzauer et al., 2017). Soft-robot signatures depend on the selected directions $\Delta=[G,R,T]$19 or on a spherical search for the direction maximizing $\Delta=[G,R,T]$20 (Wu et al., 1 Aug 2025).

Topology preservation also has limits. ToSkA notes that very simple convex shapes may reduce to a point or line skeleton and that skeletonization can over-branch on extremely noisy boundaries if not pre-smoothed (Ryan et al., 2023). Beers–Harrington–Goriely emphasize that the TMD vertex-perturbation bound grows linearly in the number of leaves and that larger reorganizations, such as re-attaching a long branch to a different parent, are not handled by the current stability theory (Beers et al., 2022). In multi-parameter persistence for mathematical morphology, computation of the full rank invariant is described as #P-hard in general, which constrains exhaustive use of the framework (Chung et al., 2021).

These limitations suggest that the main design tension is not between topology and geometry, but between descriptive richness, computational tractability, and interpretability. Current work already points to several concrete directions: richer filtrations incorporating thickness or bifurcation angles, controlled stability results for graph-edit operations, multiscale sweeps rather than single-parameter choices, and inverse mappings from descriptor space back to representative cycles or actuation parameters for interpretation and control (Beers et al., 2022, Minamitani et al., 2021, Wu et al., 1 Aug 2025).