Equi-affine Invariant Geometries of Articulated Objects Dan Raviv, Alexander M. Bronstein, Michael M. Bronstein, Ron Kimmel, and Nir Sochen Technion, Computer Science Department, Israel Tel Aviv University, School of Electrical Engineering, Israel Universit`a della Svizzera Italiana, Faculty of Informatics, Switzerland Technion, Computer Science Department, Israel Tel Aviv University, Department of Applied Mathematics Abstract. We introduce an (equi-)affine invariant geometric structure by which surfaces that go through squeeze and shear transformations can still be properly analyzed. The definition of an affine invariant metric enables us to evaluate a new form of geodesic distances and to construct an invariant Laplacian from which local and global diffusion geometry is constructed. Applications of the proposed framework demonstrate its power in generalizing and enriching the existing set of tools for shape analysis. 1 Introduction Shape analysis has been one of the principal research fields in computer vision for many years. Numerous methods are based on modeling shapes as Riemnnian manifolds, from which it is possible to derive many geometric invariances. Dif- ferential geometry and diffusion geometry have been bold players in this growing field. Schwartz et al. [22] proposed to embed a non-rigid shape in an Euclidean domain both conformal and isometric, followed by Elad et al. [14] that discussed embeddings in higher dimensions, and presented a practical representation of shapes referred to as canonical forms. Later on Elad et al. [13] and Bronstein et al. [5] showed that for some surfaces, such as faces, a spherical domain better captures intrinsic properties. In 2005 Memoli et al. [17] pointed the importance of Gromov-Hausdorff distance for shape analysis, followed by Bronstein et al. [6] who introduced a variational framework that minimizes the Gromov-Hausdorff distance by a direct embedding between two non-rigid shapes which does not suffer from an unbounded distortion of an intermediate ambient space. Diffusion geometry, referred to as spectral geometry, based on heat diffusion on manifolds and the properties of the Laplace Bertrami operator have become growingly popular in shape analysis in the past years. Driving inspiration from Berard et. al. 1994 work [2], Lafon et al. [10] proposed in 2006 a probabilistic analysis of algorithms using graph Laplacians. In 2007, Rustamov [21] showed how shapes can be analyzed using the eigen-functions of the Laplace Beltrami operator, and later on Gebal et. al. [15] discussed auto diffusion functions. Sun et al. [24] used F. Dellaert et al. (Eds.): Real-World Scene Analysis 2011, LNCS 7474, pp. 177–190, 2012. c  Springer-Verlag Berlin Heidelberg 2012