Surface registration, which transforms different sets of surface data into one common reference space, is an important process which allows us to compare or integrate the surface data effectively. If a nonrigid transformation is required, surface registration is commonly done by parameterizing the surfaces onto a simple parameter domain, such as the unit square or sphere. In this work, we are interested in looking for meaningful registrations between surfaces through parameterizations, using prior features in the form of landmark curves on the surfaces. In particular, we generate optimized conformal parameterizations which match landmark curves exactly with shape-based correspondences between them. We propose a variational method to minimize a compound energy functional that measures the harmonic energy of the parameterization maps and the shape dissimilarity between mapped points on the landmark curves. The novelty is that the computed maps are guaranteed to align the landmark features consistently and give a shape-based diffeomorphism between the landmark curves. We achieve this by intrinsically modeling our search space of maps as flows of smooth vector fields that do not flow across the landmark curves. By using the local surface geometry on the curves to define a shape measure, we compute registrations that ensure consistent correspondences between anatomical features. We test our algorithm on synthetic surface data. An application of our model to medical imaging research is shown, using experiments on brain cortical surfaces, with anatomical (sulcal) landmarks delineated, which show that our computed maps give a shape-based alignment of the sulcal curves without significantly impairing conformality. This ensures correct averaging and comparison of data across subjects.

Surface parameterizations have been widely used in computer graphics and geometry processing. In particular, as simply-connected open surfaces are conformally equivalent to the unit disk, it is desirable to compute the disk conformal parameterizations of the surfaces. In this paper, we propose a novel algorithm for the conformal parameterization of a simply-connected open surface onto the unit disk, which significantly speeds up the computation, enhances the conformality and stability, and guarantees the bijectivity. The conformality distortions at the inner region and on the boundary are corrected by two steps, with the aid of an iterative scheme using quasi-conformal theories. Experimental results demonstrate the effectiveness of our proposed method.

This paper proposes a novel algorithm to extract feature landmarks on the vestibular system (VS), for the analysis of Adolescent Idiopathic Scoliosis (AIS) disease. AIS is a 3-D spinal deformity commonly occurred in adolescent girls with unclear etiology. One popular hypothesis was suggested to be the structural changes in the VS that induce the disturbed balance perception, and further cause the spinal deformity. The morphometry of VS to study the geometric differences between the healthy and AIS groups is of utmost importance. However, the VS is a genus-3 structure situated in the inner ear. The high-genus topology of the surface poses great challenge for shape analysis. In this work, we present a new method to compute exact geodesic loops on the VS. The resultant geodesic loops are in Euclidean metric, thus characterizing the intrinsic geometric properties of the VS based on the real background geometry. This leads to more accurate results than existing methods, such as the hyperbolic Ricci flow method. Furthermore, our method is fully automatic and highly efficient, e.g., one order of magnitude faster than. We applied our algorithm to the VS of normal and AIS subjects. The promising experimental results demonstrate the efficacy of our method and reveal more statistically significant shape difference in the VS between right-thoracic AIS and normal subjects.

In this paper, we investigate how to topologically and geometrically characterize the intersection relations between a movable convex polygon and a set  of possibly overlapping polygons fixed in the plane. More specifically, a subset   is called an intersection relation if there exists a placement of that intersects, and only intersects, . The objective of this paper is to design an efficient algorithm that finds a finite and discrete representation of all of the intersection relations between and . Past related research only focuses on the complexity of the free space of the configuration space between and  and how to move or place an object in this free space. However, there are many applications that require the knowledge of not only the free space, but also the intersection relations. Examples are presented to demonstrate the rich applications of the formulated problem on intersection relations.

Point cloud is the most fundamental representation of 3D geometric objects as it provides the most exact information of them. Analyzing and processing point cloud surfaces is important in computer graphics and computer vision. However, most of the existing algorithms for surface analysis require connectivity information. Therefore, it is desirable to develop a mesh structure on point clouds. This task can be simplified with the aid of a parameterization. In particular, conformal parameterizations are advantageous in preserving the geometric information of the point cloud data. In this paper, we extend a state-of-the-art spherical conformal parameterization algorithm for genus-0 closed meshes to the case of point clouds, using an improved approximation of the Laplace-Beltrami operator on data points. Then, we propose an iterative scheme called the North-South reiteration for achieving a spherical conformal parameterization. A balancing scheme is introduced to enhance the distribution of the spherical parameterization. High quality triangulations and quadrangulations can then be built on the point clouds with the aid of the parameterizations. Also, the meshes generated are guaranteed to be genus-0 closed meshes. Moreover, using our proposed spherical conformal parameterization, multilevel representations of point clouds can be easily constructed. Experimental results demonstrate the effectiveness of our proposed framework.