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

Quantitative analysis of biological shape is crucial for understanding how form informs function. However, analyzing three-dimensional morphology presents challenges due to the intricate geometries and topologies encountered in biological specimens. Current methods either utilize handcrafted metrics that lack generalizability or employ data-driven neural models that compromise interpretability and robustness.

In this study, researchers introduce Minkowski Tensors (MT), a series of integral-geometric shape descriptors that broaden classical scalar measures of volume, surface area, and curvature to tensor-valued quantities that capture information about direction and anisotropy. MT constitute a provably complete, compact, and interpretable feature set that encapsulates volumetric and curvature-based data in a transformation- and scale-covariant, topology-aware format.

The MT were evaluated on a variety of segmented 3D biomedical images, including vascular and adrenal structures from magnetic resonance angiography (MRA) and computed tomography (CT), as well as dividing cell nuclei imaged by confocal microscopy. The results demonstrated that MT achieved competitive or superior classification performance compared to classical geometric descriptors and neural baselines, while maintaining direct biological interpretability.

These findings establish Minkowski tensors as a concise yet powerful framework for discerning complex 3D biological morphology.