Intrinsic and extrinsic operators for shape analysis

Abstract Geometric operators are common objects in surface-based shape analysis and geometry processing. While the intrinsic Laplace–Beltrami operator has been a ubiquitous choice thanks to its intuitive and often desirable properties, it fails to capture the spatial embedding of a shape because it discards extrinsic information; furthermore, it is not always sensitive to the geometric features relevant for a given shape analysis task. To address these challenges, several alternative operators for shape analysis have been proposed in recent work, with an emphasis on operators sensitive to extrinsic features. Many operators appearing in previous work on other problems also encode aspects of extrinsic geometry and are potentially suitable for shape analysis. In this survey, we unify discussion of operators for shape analysis, highlighting key theoretical properties as well as their numerical discretizations. Additionally, we provide numerical experiments on model tasks in the operator-based shape analysis pipeline, including computation of descriptors, distances, and segmentations, to demonstrate the effect of using different operators on the qualitative behaviour of algorithms in this space.