Full and Partial Symmetries of Non-rigid Shapes

Abstract

Symmetry and self-similarity are the cornerstone of Nature, exhibiting themselves through the shapes of natural creations and ubiquitous laws of physics. Since many natural objects are symmetric, the absence of symmetry can often be an indication of some anomaly or abnormal behavior. Therefore, detection of asymmetries is important in numerous practical applications, including crystallography, medical imaging, and face recognition, to mention a few. Conversely, the assumption of underlying shape symmetry can facilitate solutions to many problems in shape reconstruction and analysis. Traditionally, symmetries are described as extrinsic geometric properties of the shape. While being adequate for rigid shapes, such a description is inappropriate for non-rigid ones: extrinsic symmetry can be broken as a result of shape deformations, while its intrinsic symmetry is preserved. In this paper, we present a generalization of symmetries for non-rigid shapes and a numerical framework for their analysis, addressing the problems of full and partial exact and approximate symmetry detection and classification.