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.

The study of 2D shapes is a central problem in the field of computer vision. In 2D shape analysis, classification and recognition of objects from their observed silhouettes are extremely crucial and yet difficult. It usually involves an efficient representation of 2D shape space with natural metric, so that its mathematical structure can be used for further analysis. Although significant progress has been made for the study of 2D simply-connected shapes, very few works have been done on the study of 2D objects with arbitrary topologies. In this work, we propose a representation of general 2D domains with arbitrary topologies using conformal geometry. A natural metric can be defined on the proposed representation space, which gives a metric to measure dissimilarities between objects. The main idea is to map the exterior and interior of the domain conformally to unit disks and circle domains, using holomorphic 1-forms. A set of diffeomorphisms from the unit circle S^1 to itself can be obtained, which together with the conformal modules are used to define the shape signature. We prove mathematically that our proposed signature uniquely represents shapes with arbitrary topologies. We also introduce a reconstruction algorithm to obtain shapes from their signatures. This completes our framework and allows us to move back and forth between shapes and signatures. Experiments show the efficacy of our proposed algorithm as a stable shape representation scheme.

Finding surface mappings with least distortion arises from many applications in various fields. Extremal Teichmüller maps are surface mappings with least conformality distortion. The existence and uniqueness of the extremal Teichmüller map between Riemann surfaces of finite type are theoretically guaranteed (see Fletcher and Markovic, Quasiconformal maps and Teichmüller theory, Oxford Graduate Texts in Math., vol. 11, Oxford University Press, Oxford, 2007). Recently, a simple iterative algorithm for computing the Teichmüller maps between connected Riemann surfaces with given boundary value was proposed by Lui, Lam, Yau, and Gu in Teichmüller extremal mapping and its applications to landmark matching registration, arXiv:1211.2569. Numerical results were reported in the paper to show the effectiveness of the algorithm. The method was successfully applied to landmarkmatching registration. The purpose of this paper is to prove the iterative algorithm proposed in loc. cit., indeed converges.

Geodesic based Voronoi diagrams play an important role in many applications of computer graphics. Constructing such Voronoi diagrams usually resorts to exact geodesics. However, exact geodesic computation always consumes lots of time and memory, which has become the bottleneck of constructing geodesic based Voronoi diagrams. In this paper, we propose the window-VTP algorithm, which can effectively reduce redundant computation and save memory. As a result, constructing Voronoi diagrams using the proposed window-VTP algorithm runs 3-8 times faster than Liu et al.’s method [LCT11], 1.2 times faster than its FWP-MMP variant and more importantly uses 10-70 times less memory than both of them.

Computing discrete geodesic distance over triangle meshes is one of the fundamental problems in computational geometry and computer graphics. In this problem, an effective window pruning strategy can significantly affect the actual running time. Due to its importance, we conduct an in-depth study of window pruning operations in this paper, and produce an exhaustive list of scenarios where one window can make another window partially or completely redundant. To identify a maximal number of redundant windows using such pairwise cross checking, we propose a set of procedures to synchronize local window propagation within the same triangle by simultaneously propagating a collection of windows from one triangle edge to its two opposite edges. On the basis of such synchronized window propagation, we design a new geodesic computation algorithm based on a triangle-oriented region growing scheme. Our geodesic algorithm can remove most of the redundant windows at the earliest possible stage, thus significantly reducing computational cost and memory usage at later stages. In addition, by adopting triangles instead of windows as the primitive in propagation management, our algorithm significantly cuts down the data management overhead. As a result, it runs 4-15 times faster than MMP and ICH algorithms, 2-4 times faster than FWP-MMP and FWP-CH algorithms, and also incurs the least memory usage.