Intrinsic vs extrinsic symmetry of non-rigid shapes
Examples of extrinsically (left) and intrinsically (right) symmetric objects.
Symmetry, referred to in some contexts as self-similarity or invariance is the cornerstone of Nature, exhibiting itself through the shapes of natural creations we see every day as well as through less evident yet omnipresent laws of physics. The interest in symmetries of shapes dates back to the dawn of the human civilization. Early evidences that our predecessors attributed importance to symmetries can be found in many cultural heritages, ranging from monumental architecture of the Egyptian pyramids to traditional ancient Greek decorations. Johannes Kepler was among the first who attempted to give a geometric formulation to symmetries in as early as 1611. A few centuries later, the study of symmetric shapes became a cornerstone of crystallography. Finally, symmetries of more complicated higher-dimensional objects underlie modern physics theories about the nature of matter, space and time. 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. By considering shapes as metric spaces, symmetries can be described as self-isometries, i.e. automorphisms that preserve some metric defined on the shape. Using the Euclidean metric gives us traditional extrinsic symmetry. Using intrinsic, e.g. geodesic or diffusion metrics, allows to generalize the notion of symmetry to non-rigid shapes. This construction readily extends to approximate or partial symmetries. Generalized multidimensional scaling is used as a numerical framework for intrinsic symmetry analysis, addressing the problems of full and partial exact and approximate symmetry detection and classification.
A. Hooda, M. M. Bronstein, A. M. Bronstein, R. Horaud, "Shape palindromes: analysis of intrinsic symmetries in 2D articulated shapes", Proc. Conf. on Scale Space and Variational Methods in Computer Vision (SSVM), 2011.
N. Mitra, A. M. Bronstein, M. M. Bronstein, "Intrinsic regularity detection in 3D geometry", Proc. European Conf. Computer Vision (ECCV), 2010.
D. Raviv, A. M. Bronstein, M. M. Bronstein, R. Kimmel, G. Sapiro, "Diffusion symmetries of non-rigid shapes", Proc. Intl. Symposium on 3D Data Processing, Visualization and Transmission (3DPVT), 2010.
D. Raviv, A. M. Bronstein, M. M. Bronstein, R. Kimmel, "Full and partial symmetries of non-rigid shapes", Intl. Journal of Computer Vision (IJCV), Vol. 89/1, pp. 18-39, August 2010.
D. Raviv, A. M. Bronstein, M. M. Bronstein, R. Kimmel, "Symmetries of non-rigid shapes", Proc. Workshop on Non-rigid Registration and Tracking through Learning (NRTL), 2007.
Non-rigid shape similarity and correspondence
Partial similarity and correspondence