Proper scales of shapes: A curved scale space

Abstract

We present an extension of the scale space idea to surfaces, with the aim of extending ideas like Gaussian derivatives to function on curved spaces. This is done by using the fact, also valid for normal images, that among the continuous range of scales at which one can look at an image, or surface, there is a infinite discrete subset which has a natural geometric interpretation. We call them “proper scales", as they are defined by eigenvalues of an elliptic partial differential operator associated with the image, or shape. The computations are performed using the Finite Element technique.