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

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.

Push-Forward Signed Distance Functions enable interpretable and robust continuous shape quantification

We introduce the Push-Forward Signed Distance Morphometric (PF-SDM), a novel method for shape quantification in biomedical imaging that is continuous, interpretable, and invariant to shape-preserving transformations. PF-SDM effectively captures the geometric properties of shapes, including their topological skeletons and radial symmetries. This results in a robust and interpretable shape descriptor that generalizes to capture temporal shape dynamics. Importantly, PF-SDM avoids certain issues of previous geometric morphometrics, like Elliptical Fourier Analysis and Generalized Procrustes Analysis, such as coefficient correlations and landmark choices. We present the PF-SDM theory, provide a practically computable algorithm, and benchmark it on synthetic data.