Minkowski Tensors as a Lightweight and Interpretable Representation for Three-Dimensional Morphology

Abstract

Quantitative analysis of biological shape is central to understanding how form encodes function. Despite rapid advances in volumetric imaging and segmentation, analysis and of three-dimensional morphology remains challenging due to complex geometries and topologies found in biology. Existing approaches either rely on handcrafted metrics that lack generalizability or employ data-driven neural models that sacrifice interpretability and robustness. Here, we introduce Minkowski Tensors (MT), a family of integral-geometric shape descriptors that extend classical scalar measures of volume, surface area, and curvature into tensor-valued quantities encoding direction and anisotropy. MT form a provably complete yet compact and interpretable feature set that captures volumetric and curvature-based information in a transformation-and scale-covariant, topology-aware manner. We systematically derive rotation-invariant and covariant subgroups of the tensor family, explicitly constructing measures for invariance and covariance, together with Euler-characteristic-based subgroups that capture topological features such as holes and connected components. We bench-mark MT on diverse segmented 3D biomedical images spanning vascular and adrenal structures from magnetic resonance angiography (MRA) and computed tomography (CT), and dividing cell nuclei imaged by confocal microscopy, demonstrating competitive or superior classification performance relative to classical geometric descriptors and neural baselines while retaining direct biological interpretability. Our results establish Minkowski tensors as a concise yet powerful framework for dis-criminative analysis of complex 3D biological morphology.