The Push-Forward Transform for Continuous and Robust Comparison of Dynamic Shapes

2026-08-03

AI Summary

This work addresses the challenge of continuously, robustly, and interpretably comparing dynamic shapes and their temporal evolution while preserving invariance under transformations such as translation, rotation, reflection, reparameterization, and uniform scaling. To this end, the authors propose a novel framework based on Push-Forward mappings—introduced here for the first time in shape analysis—that transports signed distance functions (SDFs) onto a common reference domain, yielding a continuous invariant representation that simultaneously encodes both boundary and interior geometric information. The approach unifies the treatment of 2D, 3D, and dynamic shapes, supports joint analysis with associated scalar fields, and enables an interpretable shape metric capable of revealing structural features such as skeletal topology and rotational symmetries. Extensive experiments demonstrate the method’s effectiveness and robustness across diverse datasets.

Abstract

We introduce a mathematical framework for shape comparison based on mapping functions from the shape domain to a common reference domain. This Push-Forward Transform enables invariant and robust comparison of shapes, preserving intrinsic geometric information. Quantitatively comparing shapes and their temporal evolution is a fundamental challenge in image analysis. Meaningful shape comparison requires representations that are invariant to transformations that do not alter shape itself, such as translation, rotation, reflection, re-parametrization, and uniform scaling, while remaining sensitive to intrinsic geometric variation. Existing approaches often rely on sensitive parameterizations, landmark correspondence, or learned representations that are difficult to interpret and reproduce. We show that the Push-Forward Transform (PF-T) applied to Signed Distance Functions (SDFs) yields a continuous representation that captures both boundary and interior geometry. We derive an interpretable morphometric that quantifies shape similarity and reveals features such as skeletal topology and rotational symmetries. The push-forward transform applies consistently to two- and three-dimensional shapes, extends to time-evolving geometries, and supports the joint analysis of shape and additional scalar fields defined over shapes, such as intensity or molecular signals. We present the mathematical formulation, describe an efficient algorithm, and benchmark the approach on 2D, 3D, and temporal data sets.

Problem

• shape comparison
• dynamic shapes
• geometric invariance
• temporal evolution
• robust representation

Innovation

• Push-Forward Transform
• Signed Distance Function
• Shape Comparison
• Morphometric Analysis
• Geometric Invariance