Integral Invariant Signatures

For shapes represented as closed planar contours, we introduce a class of functionals that are invariant with respect to the Euclidean and similarity group, 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 (in the limit), they are not as sensitive to noise in the data. We exploit the integral invariants to define a unique signature, from which the original shape can be reconstructed uniquely up to the symmetry group, and a notion of scale-space that allows analysis at multiple levels of resolution. The invariant signature can be used as a basis to define various notions of distance between shapes, and we illustrate the potential of the integral invariant representation for shape matching on real and synthetic data.