Corruption-Resilient Skeletal Representation

• Corruption-resilient skeletal representation is a robust method that extracts the essential structure of shapes and point clouds despite noise and boundary perturbations.
• Techniques such as diffusion filtering, persistent homology, dictionary-based constraints, and deep learning enforce stability and invariant matching across varied data corruptions.
• This approach provides theoretical guarantees and empirical improvements in recognition, registration, and reconstruction tasks, offering a foundation for robust 2D and 3D perception applications.

A corruption-resilient skeletal representation is a mathematical and algorithmic construct designed to encode the salient structural features of a shape, signal, motion, or point cloud in a way that is explicitly robust to boundary perturbations, noise, outliers, or other corruptions. Unlike traditional skeletons—such as the classical connected medial axis, which are highly sensitive to small boundary irregularities—corruption-resilient approaches suppress or regularize unstable details so that only stable, meaningful structural primitives remain. This stability enables both theoretical guarantees and practical advantages for tasks such as recognition, matching, 3D reconstruction, registration, and motion analysis in environments with challenging or corrupted data.

1. Mathematical Foundations and Forms of Corruption-Resilient Skeletonization

Corruption resilience in skeletonization departs fundamentally from classical medial axis calculations by emphasizing strong regularization and abstraction:

• Over-Smoothing and Coarse Scale Pruning: In the “Disconnected Skeleton: Shape at its Absolute Scale” framework (Aslan et al., 2011), resistance to corruption is achieved via “excessive regularization,” wherein the solution φ to the linear diffusion equation

$$\frac{\partial \phi(x,y,\sigma)}{\partial \sigma} = \Delta\phi(x,y,\sigma)$$

with boundary condition $\phi|_\Gamma = 1$ on the domain boundary Γ, is evolved up to a shape-dependent coarse scale σ*. All minor, unstable features are “washed out,” leaving only isolated, dominant symmetry axes. These axes form a set of disconnected skeletal branches, each characterized by its disconnection point, type, and length, with no reliance on local detail.

• Homologically Persistent Skeletons (HoPeS): “Skeletonisation Algorithms with Theoretical Guarantees for Unorganised Point Clouds with High Levels of Noise” (Kurlin et al., 2019) constructs a one-dimensional skeleton from noisy point clouds by leveraging persistent homology. After building the minimum spanning tree of the sample, critical edges—those corresponding to persistent homology classes with long lifetimes—are augmented to the tree, forming skeletons that are provably topologically correct (homotopy equivalent to the underlying graph) and geometrically close to the original data.
• Dictionary-Based Representations with Coefficient Nullification: The method described in (Ren, 2015) achieves resilience to gross sample-specific corruption in signal data by enforcing a constraint that the coefficient vector for corrupted samples is zero in a dictionary representation. The optimization problem

$$\min_{Z, E} \|Z\|_{2,1} + \lambda F_2(E)\quad \text{subject to } X = DZ + E$$

ensures that only clean data influence the dictionary coefficients, segregating the effect of corrupted samples to the error term E.

• Explicit Regularization with Geometric Deep Learning: Learning-based skeletal methods (Khargonkar et al., 2023, Bahri et al., 26 May 2025) use geometric deep neural architectures (e.g., PointNet++-based encoders, or transformer-based models for pose) and introduce geometric consistency losses (spread, medial, point-to-sphere) or utilize robust skeleton prediction branches to extract representations inherently insensitive to small perturbations or missing data.

2. Algorithmic Strategies for Achieving Corruption Resilience

The core algorithmic mechanisms supporting corruption-resilient skeletal representations include:

• Diffusion-Based Skeleton Extraction: By solving the diffusion/heat equation up to the coarsest meaningful scale (Aslan et al., 2011), the skeletonization process eliminates short branches and unstable points. The absolute scale σ* acts as an adaptive cutoff, retaining only robust axes located in low-curvature, “ribbon-like” regions.
• Persistent Homology Filtration and Optimal Subgraph Selection: HoPeS (Kurlin et al., 2019) builds on a filtration of neighborhood graphs parameterized by scale, extracting cycles (i.e., skeleton branches) with birth-death persistence gaps. Only subgraphs with sufficiently large diagonal or vertical gaps are retained to mitigate spurious structures due to noise.
• Skeletal Representation Learning via Deep Networks: Both point-based (Khargonkar et al., 2023, Bahri et al., 26 May 2025) and sequence-based (transformer) networks (Jiang et al., 2021) incorporate architectural features (global attention, pointwise feature fusion, skeleton-specific branches) and regularization (radius regularization, spread loss, geometric constraint loss) to ensure noise robustness and semantic integrity across corrupted or incomplete data.
• Disconnection Point and Primitive Attrition: Branches are terminated at stable “break” points, typically corresponding to coarse skeleton events rather than artifacts induced by noise. This reduces the risk of spurious correspondences in downstream tasks.
• Null-Space Constraints for Outlier Suppression: For dictionary-based data representations (Ren, 2015), L_{2,1} norm regularization enforces that corrupted samples exert no representational influence, segregating their impact to a dedicated error variable.

3. Effects on Recognition, Matching, and Registration

Corruption-resilient skeletal representations confer direct advantages for recognition, matching, and registration:

• Invariant and Articulation-Tolerant Matching: By parameterizing the skeletal primitives in a global Euclidean frame (via dominant negative-curvature branches and the diffusion field centroid) (Aslan et al., 2011), branch attributes—type, disconnection location, normalized length—can be compared using Mahalanobis or Euclidean-based similarities. The approach is inherently invariant to translation, rotation, and scale and can be configured to yield either articulation-invariant or -sensitive descriptions using semi-local coordinate frames.
• Guaranteed Topological Fidelity in Point Cloud Skeletons: HoPeS provides theoretical guarantees that, up to a noise threshold, the skeleton recovers the correct number of cycles (Betti number) and is geometrically close to the uncorrupted ground truth (Kurlin et al., 2019). Its success rates in synthetic (grids, wheels, hexagons) and real (BSD500) data under various noise models validate its resilience.
• Robust Sample Matching and Error Segregation: In dictionary-based representations (Ren, 2015), the forced zeroing out of coefficients for corrupted samples ensures that matching and clustering tasks are performed exclusively on reliable, clean samples; gross outliers do not interfere with the subspace structure of usable data.
• Accurate Registration in Corrupted 3D Point Clouds: Skeleton-based fusion of alignment transformations—in conjunction with a distribution distance loss (Wang et al., 29 Sep 2025)—mitigates the impact of density distortions, noise, and geometric deformations, enabling robust and accurate point cloud registration critical for autonomous perception and medical imaging.

4. Performance Assessment and Empirical Results

Strong empirical evidence demonstrates the practical value of corruption-resilient skeletal representations:

• High Retrieval and Precision Scores: The disconnected skeleton method (Aslan et al., 2011) scores 98% in the “Bull’s eye test” and maintains precision of ~88% at full recall on a broad 2D shapes database, indicating effective and robust shape retrieval across scales, positions, orientations, and articulations.
• Theoretical and Empirical Noise Thresholds: HoPeS (Kurlin et al., 2019) achieves correct topology reconstruction (Betti-1 success rates) for noise levels up to sharply quantified thresholds for Gaussian and uniform perturbations.
• Downstream Task Improvement: In body-pose estimation, transformer-based skeletal correction (Jiang et al., 2021) exhibits reduced MSE under frames/joints masking and delivers downstream gains (e.g., BLEU/ROUGE improvement in sign language translation).
• Point Cloud Registration Gains: SRRF (Wang et al., 29 Sep 2025) reduces rotation/translation RMSE compared to ICP, DCP, RGM, and others, particularly in heavily corrupted scenarios, owing to its global structural abstraction and distributional consistency enforcement.

5. Design Considerations and Applications

The structural and algorithmic choices underlying corruption-resilient skeletal representations are guided by intended downstream use and real-world data properties:

• Choice of Scale and Regularization: The diffusion scale σ*, persistence diagram thresholds, and loss hyperparameters (e.g., spread, medial, point-to-sphere) control the abstraction level, trading detail for robustness.
• Global vs. Local Coordinate Frames: Global frames increase invariance and resilience, while semi-local enrichment permits articulation sensitivity for tasks such as pose recognition or comparative morphology.
• Sparse vs. Dense Representations: The enforcement of sparsity or structural abstraction (e.g., using only the dominant branches or persistent cycles) reduces overfitting to noise and focuses representational capacity on stable, interpretably meaningful parts.
• Application Domains: These representations are employed in shape retrieval, motion capture, 3D registration, subspace clustering, medical imaging analysis, and robust face recognition, capitalizing on their resilience to occlusions, artifacts, and varied noise models.

6. Limitations and Future Directions

Despite demonstrated success, certain limitations and avenues for further research remain:

• Adaptivity to Novel Classes and Category-Generalization: While methods like the category-agnostic implicit skeleton (Zhang et al., 16 Jan 2024) overcome fixed-template shortcomings, further work is needed to dynamically tune abstraction thresholds or zoning in higher-noise, multi-instance scenarios.
• Scaling to Higher Dimensions and Complex Scenes: The skeletonization of dense or higher-dimensional datasets, as well as the integration with multi-modal data (e.g., RGB, depth, semantic labels), presents additional challenges in balancing robustness and descriptive power.
• Dynamic and Online Adaptation: Extensions to real-time, distribution-shifting environments (as in SMART-PC (Bahri et al., 26 May 2025)) require efficient test-time adaptation protocols that preserve skeletal integrity under continuous input corruption.
• Unified Robustness Certifications: Formal guarantees—extending from persistent homology to distributionally robust training objectives (e.g., HR loss in (Bennouna et al., 2023))—are increasingly sought for neural and geometric representations in adversarial and noisy regimes.

Corruption-resilient skeletal representation, through judicious abstraction, regularization, and topologically meaningful structure extraction, provides a robust foundation for shape analysis, recognition, and registration under imperfect or adversarial data. This paradigm is continuously extended via persistent topology, deep geometric learning, and domain-agnostic regularization, underpinning robust 2D and 3D perception systems for a range of scientific and engineering applications.