In shape analysis, finding an optimal 1-1 correspondence between 3D surfaces within a large class of admissible bijective mappings is of great importance. Such a process is called surface registration. The difficulty lies in the fact that the space of all surface diffeomorphisms is a complicated functional space, making it challenging to exhaustively search for the best mapping. To tackle this problem, we propose a simple representation of bijective surface maps using Beltrami coefficients (BCs)—complex-valued functions defined on surfaces with supremum norm less than 1. Fixing any 3 points on a pair of surfaces, there is a 1-1 correspondence between the set of surface diffeomorphisms between them and the set of BCs. Hence, every bijective surface map may be represented by a unique BC. Conversely, given a BC, we can reconstruct the unique surface map associated with it using the Beltrami Holomorphic flow (BHF) method. Using BCs to represent surface maps is advantageous because it is a much simpler functional space, which captures many essential features of a surface map. By adjusting BCs, we equivalently adjust surface diffeomorphisms to obtain the optimal map with desired properties. More specifically, BHF gives us the variation of the associated map under the variation of BC. Using this, a variational problem over the space of surface diffeomorphisms can be easily reformulated into a variational problem over the space of BCs. This makes the minimization procedure much easier. More importantly, the diffeomorphic property is always preserved. We test our method on synthetic examples and real medical applications. Experimental results demonstrate the effectiveness of our proposed algorithm for surface registration.

Superpixels are perceptually meaningful atomic regions that can effectively capture image features. Among various methods for computing uniform superpixels, simple linear iterative clustering (SLIC) is popular due to its simplicity and high performance. In this paper, we extend SLIC to compute content-sensitive superpixels, i.e., small superpixels in content-dense regions with high intensity or colour variation and large superpixels in content-sparse regions. Rather than using the conventional SLIC method that clusters pixels in R^5, we map the input image I to a 2-dimensional manifoldMR5, whose area elements are a good measure of the content density in I. We propose a simple method, called intrinsic manifold SLIC (IMSLIC), for computing a geodesic centroidal Voronoi tessellation (GCVT)—a uniform tessellation—onM, which induces the content-sensitive superpixels in I. In contrast to the existing algorithms, IMSLIC characterizes the content sensitivity by measuring areas of Voronoi cells onM. Using a simple and fast approximation to a closed-form solution, the method can compute the GCVT at a very low cost and guarantees that all Voronoi cells are simply connected. We thoroughly evaluate IMSLIC and compare it with eleven representative methods on the BSDS500 dataset and seven representative methods on the NYUV2 dataset. Computational results show that IMSLIC outperforms existing methods in terms of commonly used quality measures pertaining to superpixels such as compactness, adherence to boundaries, and achievable segmentation accuracy. We also evaluate IMSLIC and seven representative methods in an image contour closure application, and the results on two datasets, WHD and WSD, show that IMSLIC achieves the best foreground segmentation performance.

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.

In this paper we show that stability for holomorphic vector bundles are equivalent to the existence of solutions to certain system of Monge Ampere equations parametrized by a parameter k. We solve this fully nonlinear elliptic system by singular perturbation technique and show that the vanishing of obstructions for the perturbation is given precisely by the stability condition. This can be interpreted as an infinite dimensional analog of the equivalencybetween Geometric Invariant Theory and Symplectic Reduction for moduli space of vector bundles.

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.