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Persistent Homology Transform for Modeling Shapes and Surfaces
Katharine Turner, Sayan Mukherjee, Doug M Boyer
TL;DR
The paper seeks a statistically usable representation of shapes and surfaces without discarding relevant information. It introduces the persistent homology transform, proves its sufficiency through injectivity, and demonstrates distance-based applications on simulated and real data, while noting computational and theoretical scope limits.
Problem
Statistical shape modeling needs representations of shapes and surfaces suitable for inference, while conventional approaches may rely on user-specified landmarks.
Method
The paper represents shapes and surfaces with the persistent homology transform, a collection of persistence diagrams, and develops algorithms for computing and comparing these representations.
Results
The PHT is proved sufficient for shape and surface models, and is applied to compute distances between aligned and unaligned objects in simulated and real data.
Takeaways & Limitations
The PHT can represent shapes without user-specified landmarks and can measure distances between shapes that are not isomorphic.
Takeaways & Limitations
The rotation-accounting method is computationally heavy for surfaces in R3, though reasonable for shapes in R2.
Abstract
from arXiv · showhide
In this paper we introduce a statistic, the persistent homology transform (PHT), to model surfaces in $\mathbb{R}^3$ and shapes in $\mathbb{R}^2$. This statistic is a collection of persistence diagrams - multiscale topological summaries used extensively in topological data analysis. We use the PHT to represent shapes and execute operations such as computing distances between shapes or classifying shapes. We prove the map from the space of simplicial complexes in $\mathbb{R}^3$ into the space spanned by this statistic is injective. This implies that the statistic is a sufficient statistic for probability densities on the space of piecewise linear shapes. We also show that a variant of this statistic, the Euler Characteristic Transform (ECT), admits a simple exponential family formulation which is of use in providing likelihood based inference for shapes and surfaces. We illustrate the utility of this statistic on simulated and real data.
1 Introduction
The paper addresses the challenge of representing shapes and surfaces for statistical modeling by introducing the persistent homology transform (PHT). It connects multiscale topology with shape statistics, geometric analysis, and applications such as morphology.
- Contribution: The PHT is introduced as a sufficient statistic for modeling objects and surfaces in R3 and shapes in R2.It is a collection of persistence diagrams, which are multiscale topological summaries used in topological data analysis.
- Motivation: The central challenge is obtaining a representation of shapes and surfaces that can be used in statistical models.Applications include comparing heel bones across primates and relating morphological distances to genetic distances.
- Contribution: The paper proves that persistence diagrams can be sufficient statistics for shape and surface models without loss of information from persistent homology.The result is presented as a formal advance for both topological data analysis and shape statistics.
- Related work: The approach is positioned alongside landmark-based shape spaces, infinite-dimensional manifold models, and integral-geometric transforms.The paper notes that manifold-based approaches can be computationally intensive and require parameterizing the shape manifold.
- Applications: The PHT is used to compute distances between aligned objects and compare unaligned objects in simulated and real data.Applications include distances between bone surfaces and meshes, including primate bones and teeth.
- Applications: Surface-distance methods based on conformal geometry can encounter problems when compared objects are not isomorphic, such as when a tooth is broken.The PHT is presented as applicable to distances between such topologically different objects.
2 Persistence diagrams and height functions
The paper constructs persistence diagrams from filtrations of simplicial complexes and uses height functions to form the PHT. These diagrams support metrics, continuity results, and the paper’s injectivity and sufficiency claims for shapes and surfaces.
- Topological preliminaries: A simplicial complex is a collection of simplices closed under faces, with simplex intersections required to be empty or common faces.The paper works with chains, boundaries, cycles, and homology over the field Z2.
- Persistence diagrams: Persistent homology tracks how homology classes are born and die across an inclusion-based filtration of simplicial complexes.Persistence diagrams encode these birth and death times, including essential classes and zero-persistence classes represented by diagonal copies.
- Height filtrations: Height filtrations include simplices according to their position relative to a direction, allowing changes in 0-dimensional homology to be summarized by persistence diagrams.For the illustrated construction, the subcomplex at time t contains simplices entirely at or below height t; each simplex enters at its maximal height.
- Metrics and continuity: The map v 7→Xk(M, v) is Lipschitz for finite simplicial complexes, and in the p = ∞ case its Lipschitz constant is bounded by the farthest point’s distance from the origin.This provides continuity of the persistence-diagram representation as the height direction changes.
- PHT properties: The PHT is injective on the paper’s spaces of simplicial complexes in R^2 and R^3, making it a sufficient statistic and theoretically invertible through reconstruction.The paper also uses the PHT to define distances between shapes or surfaces; for certain sphere-like objects, higher-dimensional diagrams can be constructed from two 0-dimensional transforms.
3 Injectivity of the transform
The paper proves that the persistent homology transform is injective for piecewise-linear simplicial complexes in R^3 and R^2, making it theoretically invertible and sufficient for shape models. The proof reconstructs vertices and links from directional persistence information, while related results establish sufficiency, distance construction, and injectivity of the Euler Characteristic Transform.
- Injectivity in R^3: The PHT is injective on simplicial complexes in R^3, and the proof gives a constructive reconstruction procedure.The procedure finds vertices and then determines their links; for piecewise-linear complexes, these suffice to reconstruct the complex.
- Reconstructing links: For each essential edge, changes in relative Euler characteristic around its perpendicular great circle determine the link of that edge.If the relevant semicircle contains k components, the link changes the relative Euler characteristic by k−1; the resulting function is equivalent to the bird’s-eye link description.
- Reconstruction: Vertices are recovered by scanning them in height order along a direction with distinct vertex heights and identifying directional homology changes.At each stage, the method uses the known sublevel set and partitions directions around the current vertex according to relative homology.
- Reconstructing links: The reconstruction algorithm resolves opposing-edge cancellations by subtracting the contribution of a known edge and attributing the residual function to the opposite edge when nonzero.When the residual is zero, no opposite edge is present; otherwise its link can be determined from the residual.
- Consequences: The PHT is injective in R^2, and its image can represent piecewise-linear shapes for defining distances and sufficient-statistic models.The R^2 result follows by embedding the shapes in R^3; the third-dimensional persistence diagram is empty, and equality of planar PHTs implies equality of the shapes.
- Consequences: The injectivity and continuity results imply that the inverse reconstruction map is Borel measurable.The paper states that the PHT is Borel continuous and uses injectivity to conclude Borel continuity of the inverse.
- Consequences: The ECT is also injective and sufficient, and admits a simple inner-product structure for exponential-family modeling.The paper additionally states injectivity of the 0-th dimensional PHT for surfaces homeomorphic to specified spheres and circles.
- Consequences: For unaligned objects, the induced PHT distance gives a metric on unaligned objects in R^d.The paper describes alignment-related distance construction and states that the resulting unaligned distance is a metric.
4 Results real and simulated data
The paper evaluates PHT-based distances on planar silhouettes and primate calcanei, using sampled height directions, persistence-diagram distances, alignment, and multidimensional scaling. Results cover both shape-class structure and agreement with manual and automated analyses.
- Distance computation: The distance algorithm approximates PHT distances by averaging persistence-diagram distances across finitely sampled directions.It uses height functions and sublevel-set persistence diagrams for each object-direction pair.
- Distance computation: For planar shapes, the study used 64 evenly spaced directions; for simplicial complexes in R3, it used 162 directions from an icosahedron-based grid.The 0-dimensional persistence computations used a union-find algorithm, while diagram distances used the Hungarian algorithm.
- Planar silhouette data: The silhouette experiment contained 1,400 shapes across seven classes, after alignment for scaling and translation and comparison under rotations.The classes were Bone, Heart, Glass, Fountain, Key, Fork, and Axe.
- Planar silhouette data: Multidimensional scaling clustered most silhouette classes, whereas Axe and Fork did not form tight clusters.The projections were examined in two and three dimensions.
- Primate calcanei: The calcaneal analysis compared manual landmarks, an automated pseudolandmark protocol, and PHT distances for 106 extant and extinct primates.The PHT used the same alignment procedure as the automated protocol before computing pairwise distances.
- Primate calcanei: 0.014, 0.015, and 0.016 were the distances between Automated protocol–PHT, Manual–Automated protocol, and Manual–PHT, respectively.A qualitative analysis suggested that PHT distances may outperform the other two methods, while noting alignment choices affect the distances.
5 Discussion
The discussion presents PHT as an information-preserving shape statistic with practical advantages, while identifying open questions about scope and computational alignment.
- Contributions: The paper’s main result is that the PHT is sufficient for capturing information in a shape.The authors emphasize that sufficiency follows from the statistic’s injectivity.
- Open questions: The authors suspect sufficiency extends beyond dimensions 2 and 3 and to more general compact Euclidean subsets, but have no proof.The stated extension includes higher-dimensional simplicial complexes and manifolds.
- Open questions: The rotation-accounting method is computationally heavy for surfaces in R3, although it is considered reasonable for planar shapes.The discussion identifies more robust alignment methods as a remaining priority.
- Open questions: The paper also poses whether other geometric and topological summaries could clarify classic shape-space models.
A Calcaneal data set
The appendix documents the 106-specimen calcaneal dataset and its visualization through multidimensional scaling and phenetic clustering. It highlights separation between several primate groups and weaker clustering for some classes in the silhouette comparison.
- Dataset: The calcaneal dataset contains 106 extant and extinct primates, with specimen information indexed by bone number.The appendix provides taxon, specimen ID, and bone-number records.
- Silhouette visualization: Figure 12 uses multidimensional scaling projections to visualize seven silhouette classes in two and three dimensions.The two-dimensional panels include a full projection and a projection excluding Fork and Axe.
- Silhouette visualization: Axe and Fork do not form tight clusters, while the remaining classes cluster when those two classes are excluded.
- Specimen illustration: Figure 13 shows a calcaneus from two different viewing angles.
- Calcaneal clustering: Figure 14 presents phenetic clustering of 106 primate calcanei representing 67 genera, with more primitive prosimians clustering separately from simians.The caption identifies groups including strepsirrhines, platyrrhines, cercopithecoids, omomyiforms, adapiforms, parapithecids, and hominoids.
B Examples of PHT of families of surfaces
The appendix studies PHTs for parameterized families of quadric ellipsoids and restricted-z hyperboloids to make the transform and its induced distances more intuitive.
- Parameterized surfaces: The appendix examines PHTs and PHT distances for quadric ellipsoids and hyperboloids with restricted z-values.It explains the persistence diagrams in each direction and considers normalization for ellipsoids.
- Parameterized surfaces: The examples use algebraic structure to compute and describe the transforms and analyze resulting distance matrices with multidimensional scaling.The appendix also gives a geometrical interpretation of the multidimensional-scaling coordinates.
B.1 PHT of ellipsoids
The ellipsoid PHT can be computed from directional height-function critical values, and its distances reveal low-dimensional geometric structure. For ellipsoids with fixed size and location, multidimensional scaling identifies coordinates associated with size and axis ratios.
- PHT computation: The PHT of an ellipsoid has no off-diagonal H1 points, while H0 and H2 each have one essential off-diagonal point.These classes correspond to first contact with and completion of the ellipsoid, respectively.
- PHT computation: Directional persistence diagrams can be computed from the minimum and maximum height values, using algebraic ellipsoid descriptions and Lagrange multipliers.The relevant geometric condition occurs where the surface normal is ±v.
- Distance calculation: The ellipsoid distance calculation uses numerical integration because the spherical integral lacks a convenient closed form, with optional size normalization.The normalization rescales each ellipsoid according to the size functional I(a, b, c).
- Distance calculation: For ellipsoids E(a, b, c) with a, b, c in {1, 2, 3, 4, 5}, multidimensional scaling of pairwise PHT distances yields an effectively three-dimensional configuration.The ellipsoids have fixed size and location in this analysis.
- MDS interpretation: The first MDS coordinate is effectively linear in ellipsoid size, while the second and third coordinates reflect symmetry in the ratios of a, b, and c.For spheres, where a = b = c, both of the latter coordinates are zero.
B.2 PHT of hperboloids with restricted z-values
For cut-off hyperboloids, the PHT is determined by boundary-ellipse events and interior points whose normals align with the filtration direction. A one-parameter family with fixed boundary geometry exhibits a simple distance relationship and one-dimensional MDS structure.
- Definition and filtration: A cut-off hyperboloid is a surface with restricted z-values and boundary consisting of a pair of ellipses.Its directional filtration is analyzed for unit vectors v, using symmetry to reduce the directions considered.
- Definition and filtration: Homological changes occur when the filtration first contacts or completes a boundary ellipse or encounters an interior point whose normal is ±v.The corresponding event heights are denoted by boundary and interior critical-value formulas.
- Persistence diagrams: The H0 diagram always contains one essential class, and may contain a second component when the upper boundary is reached before a connecting path forms.The components merge when a path between the boundary ellipses is completed.
- Persistence diagrams: The H1 diagram always contains exactly one essential class, born when the loop around the hyperboloid is completed.Its birth height depends on whether relevant interior critical points exist.
- Distance analysis: For the family Hyp(a) with fixed boundary ellipses, PHT distances are exactly twice PH0T distances, up to finite-approximation error.Because PH0Ts are significantly faster to compute, the analysis uses them instead.
- Distance analysis: MDS of Hyp(a) for a from 0.125 to 1.875 in increments of 0.125 has only one non-zero eigenvalue.The resulting scores are plotted against the family parameter a.