Towards a general sampling theory for shape preservation

Computerized image analysis makes statements about the continuous world by looking at a discrete representation. Therefore, it is important to know precisely which information is preserved during digitization. We analyze this question in the context of shape recognition. Existing results in this area are based on very restricted models and thus not applicable to real imaging situations. We present generalizations in several directions: first, we introduce a new shape similarity measure that approximates human perception better. Second, we prove a geometric sampling theorem for arbitrary dimensional spaces. Third, we extend our sampling theorem to two-dimensional images that are subjected to blurring by a disk point spread function. Our findings are steps towards a general sampling theory for shapes that shall ultimately describe the behavior of real optical systems.