Persistent Homology Transform Overview

• Persistent Homology Transform is a topological method that encodes the complete geometric and topological information of 2D/3D shapes into a structured family of persistence diagrams.
• It applies directional height functions to finite simplicial complexes and uses Wasserstein metrics for effective, distance-based shape comparisons.
• By demonstrating injectivity, the PHT acts as a sufficient statistic in probabilistic shape analysis, supporting applications on both simulated and real-world datasets.

The Persistent Homology Transform (PHT) is a topological statistic designed to encode complete geometric and topological information of two- and three-dimensional shapes, modeled as finite simplicial complexes in $\mathbb{R}^2$ or $\mathbb{R}^3$, into a structured family of persistence diagrams. For each direction on the unit sphere $S^{d-1}$, the PHT associates the persistent homology (typically over $\mathbb{Z}_2$ coefficients) of the sublevel sets of the height function. The main result demonstrates that this descriptor is injective, rendering it a sufficient statistic for probabilistic inference on the space of piecewise linear shapes. The PHT supports distance-based shape comparisons and can be explicitly linked to likelihood-based models via the Euler Characteristic Transform (ECT). Applications span both simulated and real datasets, confirming the practical utility and theoretical sufficiency of the transform.

1. Topological and Statistical Framework

The PHT encodes a shape $M\subset\mathbb{R}^d$ by mapping to a collection of persistence diagrams, each summarizing the behavior of homology groups of sublevel sets under height functions

$h_v(x) = x\cdot v,\qquad v\in S^{d-1},\ x\in M.$

The sublevel sets $M_{v, r} = \{x \in M : x \cdot v \leq r\}$ induce a filtration, and the persistent homology of this filtration in each homological degree $k = 0, 1, \dots, d-1$ is recorded as a diagram $X_k(M, v)$. The collection

$$\mathrm{PHT}(M) : S^{d-1} \to D^d,\qquad v \mapsto (X_0(M,v), X_1(M,v),\dots,X_{d-1}(M,v))$$

captures the comprehensive, directional evolution of topological features in $M$.

With appropriate choices of distance, such as the $L_1$ Wasserstein-type metric between persistence diagrams,

$$\mathrm{dist}_p (X, Y) = \left( \inf_\varphi \sum_{x\in X} \|x - \varphi(x)\|_p^p \right)^{1/p}$$

(where the infimum is over all matchings $\varphi$ between diagram points and diagonals), shape comparison and classification are facilitated.

2. Injectivity and Sufficient Statistic Property

A cornerstone result is the injectivity of the map

$M \mapsto \mathrm{PHT}(M),$

from the space $\mathcal{M}_d$ of finite simplicial complexes in $\mathbb{R}^d$ ($d = 2,3$) into $C(S^{d-1}, D^d)$. The proof, based on Theorem 3.1, is constructive:

• Vertex Identification: Each vertex $x\in M$ induces a “critical event” in the persistence diagrams at height $h_v(x)$ in a nontrivial neighborhood of directions $v$.
• Local Linear Dependence: The birth/death times of features vary linearly with the vertex positions in a neighborhood, making the correspondence between changes in diagrams and the combinatorics of $M$ explicit.
• Link Recovery and Reconstruction: By scanning $S^{d-1}$ and piecing together how these critical values shift, one reconstructs vertices and their links, and thereby the entire simplicial complex.

Consequently, the PHT loses no information and is a sufficient statistic for distributions on $\mathcal{M}_d$ (in the sense of the Fisher–Neyman factorization theorem, appropriately formulated for shape spaces).

3. Euler Characteristic Transform (ECT) and Likelihood-Based Models

The Euler Characteristic Transform (ECT) is a scalar-valued variant, mapping each direction to the Euler characteristic curve

$\chi(M, v)(r) = \chi(M_{v, r}),$

where, for polyhedral objects, $\,\chi(M_{v, r}) = V - E + F$ (vertices, edges, faces of $M_{v, r}$). The ECT is injective under appropriate conditions and produces a family of curves in a tractable function space well suited for likelihood-based inference.

In particular, the ECT allows modeling of shape populations with exponential family likelihoods:

$$p_\theta(x) = a(\theta) h(x) \exp(-\langle\theta, T(x)\rangle),$$

with sufficient statistics $T(x)$ derived from collections of Euler characteristic curves over a finite set of directions.

Shapes can then be represented as $K\times T$ matrices recording Euler characteristic curves in $K$ directions over $T$ sampled threshold values, providing a framework for quantitative and statistical shape analysis.

4. Applications to Simulated and Real Shape Data

Simulated Shape Models

• Ellipsoids: Explicit analytic descriptions of persistence diagrams are given for ellipsoids $(x^2/a^2)+(y^2/b^2)+(z^2/c^2)=1$, with birth and completion times in diagrams determined by formulas such as $\sqrt{a^2 v_1^2 + b^2 v_2^2 + c^2 v_3^2}$.
• Hyperboloids: For cut-off hyperboloids, formulas precisely describe the critical heights arising in the persistence diagrams, with multidimensional scaling (MDS) of PHT-based distances revealing the shape space structure.

Real-World Data

• Primate Calcanei: Digital meshes of primate heel bones are transformed using the PHT, leading to pairwise distances whose MDS projections reflect meaningful evolutionary and taxonomic structure, matching or improving upon landmark- or Procrustes-based methods.
• 2D Silhouette Database: For classes of silhouettes (e.g. Bone, Heart, Glass, etc.), after geometric normalization, 0-dimensional persistence diagrams in 64 directions yield clustering by class under MDS, demonstrating capacity to resolve subtle geometric differences; known challenging pairs (e.g. Axe and Fork) remain difficult.

5. Theoretical Extensions and Research Directions

The following future research avenues are identified:

• Higher-Dimensional Generalization: Extending sufficiency and injectivity to simplicial complexes in dimensions $d>3$ and to more general (e.g., smooth manifold) settings.
• Efficient Alignment: The alignment procedures currently required (especially for unaligned 3D surfaces) can be computationally expensive; algorithmic improvement is needed.
• Alternative Summaries: Investigating sufficiency properties of other topological/geometric transforms, and their statistical utility relative to the PHT.
• Full Shape-Analysis Pipelines: Integration of PHT-based statistics into broader applications, particularly in medical imaging, evolutionary biology, and other domains requiring unbiased, landmark-free shape analysis.

6. Comparison of PHT and ECT

Feature	Persistent Homology Transform (PHT)	Euler Characteristic Transform (ECT)
Output	Family of persistence diagrams (all degrees)	Family of Euler characteristic curves
Injectivity	Yes (for PL $\mathbb{R}^2/\mathbb{R}^3$)	Yes (under suitable PL conditions)
Sufficient for	Any distribution on PL shapes	Any distribution on PL shapes
Probabilistic	No explicit exponential family structure	Compatible with exponential families
Computationally	More expensive (many diagrams per direction)	Efficient (scalar curve per direction)

ECT is particularly advantageous for statistical inference due to lower dimensionality of summaries and amenability to explicit likelihood models. Both PHT and ECT are injective and theoretically sufficient, but the choice between full persistence diagrams and scalar summaries (Euler characteristic) is dictated by the balance between statistical efficiency, computational tractability, and sufficiency for discrimination of shapes.

7. Summary and Significance

The Persistent Homology Transform enables the full reconstruction and statistical modeling of shapes via a multiscale, invertible, and complete collection of topological invariants parameterized by direction. Its injectivity guarantees that it functions as a sufficient statistic in statistical shape theory, and its computation enables both metric-based and likelihood-based shape inference. The ECT provides a computationally tractable and statistically convenient scalar summary. Demonstrated efficacy on both simulated and biologically meaningful datasets establishes the PHT as a central object in topological and statistical shape analysis, with ongoing developments targeting scalability, generalization, and integration into comprehensive analysis pipelines.