Kernel Metrics on Normal Cycles and Application to Curve Matching

Abstract

In this work we introduce a new dissimilarity measure for shape registration using the notion of normal cycles, a concept from geometric measure theory which allows us to generalize curvature for nonsmooth subsets of the Euclidean space. Our construction is based on the definition of kernel metrics on the space of normal cycles which take explicit expressions in a discrete setting. This approach is closely similar to previous works based on currents and varifolds [M. Vaillant and J. Glaunès, Surface matching via currents, in Information Processing in Medical Imaging, G. E. Christensen and M. Sonka, eds., Lecture Notes in Comput. Sci. 3565, Springer, Berlin, 2005, pp. 381--392; N. Charon and A. Trouvé, SIAM J. Imaging Sci., 6 (2013), pp. 2547--2580]. We derive the computational setting for discrete curves in $\mathbb{R}^3$, using the large deformation diffeomorphic metric mapping framework as the model for deformations. We present synthetic and real data experiments and compare them with the currents and varifolds approaches.