Integral Invariant Signatures

Abstract

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.

Shape Palindromes: Analysis of Intrinsic Symmetries in 2D Articulated Shapes

“…A simple example of such a descriptor is the curvature κ(t) Since the curvature involves second-order derivatives, it is sensitive to noise (more generally, all differential invariants tend to be sensitive to noise). Alternatively, we can use as a S the integral invariant proposed in [14], defined as I(t) = S S(t) − S(t ) q(t, t )dt , where · denotes the Euclidean distance, and q is a local kernel decreasing with the distance used to localize the descriptor. Such a descriptor is also invariant to Euclidean isometries.…”

Principal curvatures from the integral invariant viewpoint

“…Taubin and Cooper (1992) for applications in Computer Vision), local integral invariants are a rather new topic in geometric computing. Manay et al (2004) investigate integral invariants for curves in the plane and show their superior performance on noisy data, especially for the reliable retrieval of shapes from geometric databases. A special case of an integral invariant, defined for 2D curves or 3D surfaces, has been used by Connolly (1986) for molecular shape analysis.…”

“…This is an approach initiated by Manay et al (2004) and Clarenz et al (2004b,a). The present paper serves as a theoretical foundation for the work by Yang et al (2006), which presents numerical and experimental results.…”

Curvature Approximation From Parabolic Sectors

“…However a problem of the curvature invariant is that derivatives have to be computed, amplifying the effect of noise. To overcome this limitation, Manay et al (2004) introduced integral invariant functionals for shapes represented as closed planar contours, as opposed to the traditional differential ones and they proved that such functionals are far less sensitive to noise, while retaining the nice features of differential invariants such as locality; that is, they obtained results which are used to calculate curvature based on integral computing. Pottmann et al (2009) extended the application of these integral invariant functionals and others defined via distance functions, to surfaces in R 3 , where, using Taylor expansions, they showed the relation of these integral invariants to differential invariants, as curvature.…”