Affine-permutation symmetry: invariance and shape space

Studying similarity of objects by looking at their shapes arises naturally in many applications. However, under different viewpoints one and the same object appears to have different shapes. In addition, the correspondences between their feature points are unknown to the viewer. In this paper, we introduce the concept of intrinsic shape of an object that is invariant to affine-permutation shape distortions. We study geometry of the intrinsic shape space in the framework of differentiable manifolds with the emphasis on the computational aspects. We represent the intrinsic shape space as a folded Grassmann manifold. This allows us to easily analyze and compare different intrinsic shapes under the affine-permutation distortion without explicitly computing and recovering these intrinsic shapes. We present the mathematical equations for connecting two intrinsic shapes by a geodesic, measuring their similarity, and morphing one intrinsic shape onto another.