Explainable Binary Classification of Separable Shape Ensembles

Scientists, engineers, biologists, and technology specialists universally leverage image segmentation to extract shape ensembles containing many thousands of curves representing patterns in observations and measurements. These large curve ensembles facilitate statistical inferences about important changes when comparing and contrasting images. We introduce novel pattern recognition formalisms combined with two-sample hypothesis testing over large ensembles of segmented curves. Our formalism involves accurately approximating eigenspaces of composite integral operators to motivate discrete, dual representations of curves collocated at quadrature nodes. Approximations are projected onto underlying matrix manifolds, and the resulting separable shape tensors constitute Euclidean invariant decompositions of curves into generalized (linear) scale variations and complementary (nonlinear) undulations. Hypothesis testing over the separable feature space can be conceived of as an explainable binary classification mapping from pairs of distributions into a binary conclusion—reject or fail to reject—offering principled decisions with logical implications. With thousands of curves segmented from pairs of images, we demonstrate how data-driven features of separable shape tensors inform explainable binary classification utilizing a product maximum mean discrepancy, absent labeled data, building interpretable feature spaces in seconds without high performance computation, and detecting discrepancies below cursory visual inspections.