Shape registration has a wide range of applications in geometric modeling, medical imaging, and computer vision. This paper focuses on the registration of the genus-3 vestibular systems and studies the geometric differences between the normal and Adolescent Idiopathic Scoliosis (AIS) groups. The non-trivial topology of the VS poses great technical challenges to the geometric analysis. To tackle these challenges, we present an effective and practical solution to register the vestibular systems. We first extract six geodesic landmarks for the VS, which are stable, intrinsic, and insensitive to the VS’s resolution and tessellation. Moreover, they are highly consistent regardless of the AIS and normal groups. The detected geodesic landmarks partition the VS into three patches, a topological annulus and two topological disks. For each pair of patches of the AIS subject and the control, we compute a bijective map using the holomorphic 1-form and harmonic map techniques. With a carefully designed boundary condition, the three individual maps can be glued in a seamless manner so that the resulting registration is a homeomorphism with exact landmark matching. Our method is robust, automatic and efficient. It takes only a few seconds on a low-end PC, which significantly outperforms the non-rigid ICP algorithm. We conducted a student’s t-test on the test data. Computational results show that using the mean curvature measure E_H, our method can distinguish the AIS subjects and the normal subjects.

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The notion of central stability was first formulated for sequences of representations of the symmetric groups by Putman. A categorical reformulation was subsequently given by Church, Ellenberg, Farb, and Nagpal using the notion of FI-modules, where FI is the category of finite sets and injective maps. We extend the notion of central stability from FI to a wide class of categories, and prove that a module is presented in finite degrees if and only if it is centrally stable. We also introduce the notion of d-step central stability, and prove that if the ideal of relations of a category is generated in degrees at most d, then every module presented in finite degrees is d-step centrally stable.

In this paper, the Lp(p\geq1) mean zonoid of a convex body K is given, and we show that it is the Lp centroid body of radial (n+p)th mean body of K up to a dilation. We also establish some affine inequalities of these bodies by proving that the volume of the new bodies is decreasing under Steiner symmetrization.

Let w be an Abelian differential on a compact Riemann surface of genus g ≥ 1. Then |w|2 defines a flat metric with conical singularities and trivial holonomy on the Riemann surface. We obtain an explicit holomorphic factorization formula for the ζ-regularized determinant of the Laplacian in the metric |w|2, generalizing the classical Ray-Singer result in g = 1.