Persistent Homology Filtrations

• Persistent homology-based filtrations are topological methods that extract multiscale geometric features using spectral descriptors from the Laplace–Beltrami operator.
• They employ energy-based filters such as HKS and WKS to organize data by spectral properties, ensuring spatial localization and invariance.
• Empirical studies show that WKS provides high true-positive rates in point matching and robust performance under geometric noise.

Persistent homology-based filtrations are central to topological data analysis and shape analysis, providing a rigorous framework to extract multiscale topological features from complex geometric data. In spectral shape analysis, filtration techniques have been significantly influenced by the development of spectral descriptors derived from the eigenstructure of the Laplace–Beltrami operator, notably the heat kernel signature (HKS) and the wave kernel signature (WKS). These approaches induce filtrations of the data by using spectral information to organize features across scales, optimizing for stability, invariance, and discriminative power in shape correspondence, registration, and retrieval tasks (Bronstein, 2011).

1. Spectral Foundations of Filtrations

Let $X$ be a compact two-dimensional Riemannian manifold, possibly with boundary $\partial X$. The Laplace–Beltrami operator $\Delta$ is defined via the convention $\Delta \phi = -\text{div} \, \text{grad} \, \phi$. Its discrete spectrum $\{\lambda_k\}_{k=1}^\infty$ ($\lambda_1=0<\lambda_2\le \lambda_3\le \dots$) admits an orthonormal basis of real eigenfunctions $\{\phi_k(x)\}$ satisfying: $$\Delta\,\phi_k(x)=\lambda_k\,\phi_k(x), \qquad \int_X \phi_k(x)\,\phi_l(x)\,da(x) = \delta_{kl}.$$ This spectral decomposition encodes the intrinsic geometry of $X$, providing a foundation for the construction of pointwise invariant descriptors and associated filtrations.

2. Quantum Probability and Descriptor Construction

Persistent homology-based filtrations in the spectral paradigm often involve quantum mechanical interpretations. The time-dependent Schrödinger equation on $X$,

$i\,\partial_t \psi(x,t) = -\Delta\,\psi(x,t),$

admits as solution

$$\psi(x,t) = \sum_k e^{i\lambda_k t} f(\lambda_k)\phi_k(x),$$

where $f(\lambda_k)$ defines the initial energy distribution in spectral space. The time-averaged probability density for observing a quantum particle at $x$ is

$$p_f(x) = \lim_{T \to \infty}\frac{1}{T}\int_0^T |\psi(x,t)|^2 dt = \sum_k |f(\lambda_k)|^2 \phi_k^2(x).$$

This framework enables spectral descriptors to be interpreted as probability densities induced by energy filtrations parameterized over the spectrum.

3. Wave Kernel Signature (WKS) and Energy-Based Filtrations

The WKS defines a canonical example of a persistent homology-inspired filtration using log-normal energy filters. For each energy center $e>0$ and variance $\sigma > 0$, the energy distribution is

$$f_e(\lambda) = C_e \exp\left(-\frac{(\log \lambda - \log e)^2}{2\sigma^2}\right),$$

with normalization $C_e$ such that $\|f_e\|_2=1$. Discrete energy centers $\{e_j\}_{j=1}^n$ are chosen uniformly on the log-scale between $\lambda_\text{min}$ and $\lambda_\text{max}$ (e.g., $\lambda_2$ to the 95th percentile of the spectrum). The $j$th coordinate of the WKS at $x$ is then

$$\operatorname{WKS}(x)_j = p_{f_{e_j}}(x) = \sum_k |f_{e_j}(\lambda_k)|^2 \phi_k^2(x).$$

In matrix terms,

$\operatorname{WKS}(x) = F\Phi(x),$

where $F_{j,k} = |f_{e_j}(\lambda_k)|^2$, and $\Phi(x) = (\phi_1^2(x),\ldots,\phi_K^2(x))^T$.

4. Band-Pass Filters, Localization, and Comparison to Other Filtrations

The band-pass nature of WKS derives from its use of log-Gaussian filters in spectral space, as contrasted with HKS, whose low-pass exponential filters emphasize only low-frequency components:

• HKS at time $t$: $$\operatorname{HKS}_t(x) = \sum_k e^{-\lambda_k t} \phi_k^2(x)$$.
• Each $f_{e_j}(\lambda)$ in WKS forms a “Gaussian bump” in $\log \lambda$.

Key consequences for persistent filtrations include:

• WKS achieves sharp spatial localization, with values decaying rapidly under small geodesic shifts.
• WKS is less influenced by low-frequency modes, being sensitive to mid- and high-frequency geometric variations.
• Empirical studies on the SHREC’10 benchmark show WKS attains higher true-positive rates at low false-positive thresholds in point-matching, while HKS may outperform WKS when extremely high recall is required in retrieval (Bronstein, 2011).

5. Numerical Implementation and Parameter Choices

Spectral filtrations are computed as follows:

• Discretize $\Delta$ using finite-elements or the cotangent-weight mesh Laplacian, applying Neumann boundary conditions.
• Assemble stiffness ($S$) and mass ($M$) matrices; solve $S\phi=\lambda M\phi$ for the first $K\approx 300$ eigenpairs, omitting $\lambda_1=0$.
• Typical parameter choices: $n=12$ energy centers $e_j$, sampled uniformly in $[\log \lambda_2, \log \lambda_\text{max}]$; $$\sigma \approx (\log \lambda_\text{max} - \log \lambda_2)/n$$ to ensure ~50% overlap of adjacent Gaussians.
• Precompute $G_{k,v} = \phi_k^2(v)$ for mesh vertex $v$, form filter bank $F_{j,k}$ as above, and compute $\operatorname{WKS}(v) = F G(:,v)$ in $O(nK)$ time.

6. Empirical Properties and Stability

Spectral persistent filtrations derived from WKS manifest several principled properties:

• Isometry invariance: dependence only on the intrinsic Laplace spectrum and eigenfunctions.
• Spatial localization: band-pass filters yield rapid value changes under local surface variations, enhancing feature detection and correspondence.
• Discriminativity: WKS demonstrates higher hit rates in point-matching at low false-positive rates; HKS may be more effective in high-recall retrieval (Bronstein, 2011).
• Robustness: the global nature of spectral features and the band-pass property confer stability under topological noise, holes, and mesh decimation.

7. Applications and Significance in Shape Analysis

Spectral filtration-based signatures such as WKS are foundational in deformable shape analysis, supporting state-of-the-art performance in correspondence, registration, and retrieval tasks. Their mathematical rigor, invariance, and computational efficiency render them particularly suitable for complex geometric and topological settings. The parametric generalization and learning-based optimization of these descriptors, as explored in Mahalanobis metric learning schemes, further enhance their adaptability to domain-specific tasks (Bronstein, 2011). A plausible implication is that as spectral filtrations are further tuned to the signal and noise statistics of a shape population, their role in persistent homology-based pipelines may expand, bridging developments in shape analysis, topological data analysis, and quantum-inspired geometric methods.