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 to generalize curvature for non smooth 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 [26, 9]. We derive the computational setting for discrete curves in R 3 , using the Large Deformation Diffeomorphic Metric Mapping framework as model for deformations. We present synthetic and real data experiments and compare with the currents and varifolds approaches. Introduction. Many applications in medical image analysis require a coherent alignment of images as a pre-processing step, using efficient rigid or non-rigid registration algorithms. Moreover, in the field of computational anatomy, the estimation of optimal deformations between images, or geometric structures segmented from the images, is a building block for any statistical analysis of the anatomical variability of organs. Non-rigid registration is classically tackled down by minimizing a functional composed of two terms, one enforcing regularity of the mapping, and the data-attachment term which evaluates dissimilarity between shapes. Defining good data-attachment terms is important, as it may improve the minimization process, and focus the registration on the important features of the shapes to be matched. In [26, 16] a new framework for dissimilarity measures between sub-manifolds was proposed using kernel metrics defined on spaces of currents. This setting is now commonly used in computational anatomy ; its advantages lie in its simple implementation and the fact that it provides a common framework for continuous and discrete shapes (see [11] for a computational analysis of currents and their numerical implementation). However, currents are oriented objects and thus a consistent orientation of shapes is needed for a coherent matching. Moreover, due to this orientation property , artificial cancellation can occur with shapes with high local variations. To deal with this problem, a more advanced model based on varifolds has been introduced recently [8]. Varifolds are measures over fields of non-oriented linear subspaces. See [8], chap. 3 for an exhaustive analysis. In this work, we propose to use a second-order model called normal cycle for defining shape dissimilarities. The normal cycle of a submanifold X is the current associated with its normal bundle N X. The normal cycle encodes second order, i.e. curvature information of X; more precisely one can compute integrals of curvatures by evaluating the normal cycle over simple differential forms. Moreover, it has a canonical orientation which is independent of the orientation of X (in fact X does not need to be oriented)
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SIAM Journal on Imaging Sciences, Society for Industrial and Applied Mathematics, 2016, 9 (4), pp.1991-2038. <http://epubs.siam.org/doi/10.1137/16M1070529>
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Pierre Roussillon, Joan Alexis Glaunès. Kernel Metrics on Normal Cycles and Application to Curve Matching. SIAM Journal on Imaging Sciences, Society for Industrial and Applied Mathematics, 2016, 9 (4), pp.1991-2038. <http://epubs.siam.org/doi/10.1137/16M1070529>. <hal-01305806v2>

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