Integral invariants and shape matching

For shapes represented as closed planar contours, we introduce a class of functionals which are invariant with respect to the Euclidean group, and which are obtained by performing integral operations. While such integral invariants enjoy some of the desirable properties of their differential cousins, such as locality of computation (which allows matching under occlusions) and uniqueness of representation (asymptotically), they do not exhibit the noise sensitivity associated with differential quantities and therefore do not require pre-smoothing of the input shape. Our formulation allows the analysis of shapes at multiple scales. Based on integral invariants, we define a notion of distance between shapes. The proposed distance measure can be computed efficiently, it allows for shrinking and stretching of the boundary, and computes optimal correspondence. Numerical results on shape matching demonstrate that this framework can match shapes despite the deformation of subparts, missing parts, and noise. As a quantitative analysis, we report matching scores for shape retrieval from a database.