The Push-Forward Transform for Continuous and Robust Comparison of Dynamic Shapes
2026-08-03 • Computer Vision and Pattern Recognition
Computer Vision and Pattern RecognitionMachine Learning
AI summaryⓘ
The authors developed a new math tool called the Push-Forward Transform to compare shapes in a way that ignores changes like moving, rotating, or resizing the shape, but still notices real geometric differences. Their method works by converting shapes into functions that describe distances, capturing both the outline and inside details. This helps measure how similar shapes are and find patterns like symmetry or structure. It works for 2D and 3D shapes, including shapes that change over time, and can also analyze extra information mapped onto shapes. The paper includes the math behind the method, an algorithm to use it, and tests with different shape data.
Push-Forward TransformShape comparisonSigned Distance FunctionInvariant representationGeometric morphometricsShape analysisTemporal shape evolutionTopological featuresScalar fields on shapesShape parametrization
Authors
Roua Rouatbi, Juan-Esteban Suarez Cardona, Ivo F. Sbalzarini
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.