QCMC: Quasi-conformal Parameterizations for Multiply-connected domains

This paper presents a method to compute the {\it quasi-conformal parameterization} (QCMC) for a multiply-connected 2D domain or surface. QCMC computes a quasi-conformal map from a multiply-connected domain $S$ onto a punctured disk $D_S$ associated with a given Beltrami differential. The Beltrami differential, which measures the conformality distortion, is a complex-valued function $\mu:S\to\mathbb{C}$ with supremum norm strictly less than 1. Every Beltrami differential gives a conformal structure of $S$. Hence, the conformal module of $D_S$, which are the radii and centers of the inner circles, can be fully determined by $\mu$, up to a M\"obius transformation. In this paper, we propose an iterative algorithm to simultaneously search for the conformal module and the optimal quasi-conformal parameterization. The key idea is to minimize the Beltrami energy subject to the boundary constraints. The optimal solution is our desired quasi-conformal parameterization onto a punctured disk. The parameterization of the multiply-connected domain simplifies numerical computations and has important applications in various fields, such as in computer graphics and vision. Experiments have been carried out on synthetic data together with real multiply-connected Riemann surfaces. Results show that our proposed method can efficiently compute quasi-conformal parameterizations of multiply-connected domains and outperforms other state-of-the-art algorithms. Applications of the proposed parameterization technique have also been explored.Comment: 26 pages, 23 figures, submitted. arXiv admin note: text overlap with arXiv:1402.6908, arXiv:1307.2679 by other author

Parallelizable global conformal parameterization of simply-connected surfaces via partial welding

Conformal surface parameterization is useful in graphics, imaging and visualization, with applications to texture mapping, atlas construction, registration, remeshing and so on. With the increasing capability in scanning and storing data, dense 3D surface meshes are common nowadays. While meshes with higher resolution better resemble smooth surfaces, they pose computational difficulties for the existing parameterization algorithms. In this work, we propose a novel parallelizable algorithm for computing the global conformal parameterization of simply-connected surfaces via partial welding maps. A given simply-connected surface is first partitioned into smaller subdomains. The local conformal parameterizations of all subdomains are then computed in parallel. The boundaries of the parameterized subdomains are subsequently integrated consistently using a novel technique called partial welding, which is developed based on conformal welding theory. Finally, by solving the Laplace equation for each subdomain using the updated boundary conditions, we obtain a global conformal parameterization of the given surface, with bijectivity guaranteed by quasi-conformal theory. By including additional shape constraints, our method can be easily extended to achieve disk conformal parameterization for simply-connected open surfaces and spherical conformal parameterization for genus-0 closed surfaces. Experimental results are presented to demonstrate the effectiveness of our proposed algorithm. When compared to the state-of-the-art conformal parameterization methods, our method achieves a significant improvement in both computational time and accuracy

A Linear Formulation for Disk Conformal Parameterization of Simply-Connected Open Surfaces

Surface parameterization is widely used in computer graphics and geometry processing. It simplifies challenging tasks such as surface registrations, morphing, remeshing and texture mapping. In this paper, we present an efficient algorithm for computing the disk conformal parameterization of simply-connected open surfaces. A double covering technique is used to turn a simply-connected open surface into a genus-0 closed surface, and then a fast algorithm for parameterization of genus-0 closed surfaces can be applied. The symmetry of the double covered surface preserves the efficiency of the computation. A planar parameterization can then be obtained with the aid of a M\"obius transformation and the stereographic projection. After that, a normalization step is applied to guarantee the circular boundary. Finally, we achieve a bijective disk conformal parameterization by a composition of quasi-conformal mappings. Experimental results demonstrate a significant improvement in the computational time by over 60%. At the same time, our proposed method retains comparable accuracy, bijectivity and robustness when compared with the state-of-the-art approaches. Applications to texture mapping are presented for illustrating the effectiveness of our proposed algorithm

The Theory of Computational Quasi-conformal Geometry on Point Clouds

Quasi-conformal (QC) theory is an important topic in complex analysis, which studies geometric patterns of deformations between shapes. Recently, computational QC geometry has been developed and has made significant contributions to medical imaging, computer graphics and computer vision. Existing computational QC theories and algorithms have been built on triangulation structures. In practical situations, many 3D acquisition techniques often produce 3D point cloud (PC) data of the object, which does not contain connectivity information. It calls for a need to develop computational QC theories on PCs. In this paper, we introduce the concept of computational QC geometry on PCs. We define PC quasi-conformal (PCQC) maps and their associated PC Beltrami coefficients (PCBCs). The PCBC is analogous to the Beltrami differential in the continuous setting. Theoretically, we show that the PCBC converges to its continuous counterpart as the density of the PC tends to zero. We also theoretically and numerically validate the ability of PCBCs to measure local geometric distortions of PC deformations. With these concepts, many existing QC based algorithms for geometry processing and shape analysis can be easily extended to PC data

Fast Disk Conformal Parameterization of Simply-connected Open Surfaces

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 significantly 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

TEMPO: Feature-Endowed Teichm\"uller Extremal Mappings of Point Clouds

In recent decades, the use of 3D point clouds has been widespread in computer industry. The development of techniques in analyzing point clouds is increasingly important. In particular, mapping of point clouds has been a challenging problem. In this paper, we develop a discrete analogue of the Teichm\"{u}ller extremal mappings, which guarantee uniform conformality distortions, on point cloud surfaces. Based on the discrete analogue, we propose a novel method called TEMPO for computing Teichm\"{u}ller extremal mappings between feature-endowed point clouds. Using our proposed method, the Teichm\"{u}ller metric is introduced for evaluating the dissimilarity of point clouds. Consequently, our algorithm enables accurate recognition and classification of point clouds. Experimental results demonstrate the effectiveness of our proposed method

Spherical Conformal Parameterization of Genus-0 Point Clouds for Meshing

Point cloud is the most fundamental representation of 3D geometric objects. 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 simplified 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

Computing Quasiconformal Maps on Riemann surfaces using Discrete Curvature Flow

Surface mapping plays an important role in geometric processing. They induce both area and angular distortions. If the angular distortion is bounded, the mapping is called a {\it quasi-conformal} map. Many surface maps in our physical world are quasi-conformal. The angular distortion of a quasi-conformal map can be represented by Beltrami differentials. According to quasi-conformal Teichm\"uller theory, there is an 1-1 correspondence between the set of Beltrami differentials and the set of quasi-conformal surface maps. Therefore, every quasi-conformal surface map can be fully determined by the Beltrami differential and can be reconstructed by solving the so-called Beltrami equation. In this work, we propose an effective method to solve the Beltrami equation on general Riemann surfaces. The solution is a quasi-conformal map associated with the prescribed Beltrami differential. We firstly formulate a discrete analog of quasi-conformal maps on triangular meshes. Then, we propose an algorithm to compute discrete quasi-conformal maps. The main strategy is to define a discrete auxiliary metric of the source surface, such that the original quasi-conformal map becomes conformal under the newly defined discrete metric. The associated map can then be obtained by using the discrete Yamabe flow method. Numerically, the discrete quasi-conformal map converges to the continuous real solution as the mesh size approaches to 0. We tested our algorithm on surfaces scanned from real life with different topologies. Experimental results demonstrate the generality and accuracy of our auxiliary metric method

Optimization of Surface Registrations using Beltrami Holomorphic Flow

In shape analysis, finding an optimal 1-1 correspondence between surfaces within a large class of admissible bijective mappings is of great importance. Such process is called surface registration. The difficulty lies in the fact that the space of all surface diffeomorphisms is a complicated functional space, making exhaustive search for the best mapping challenging. To tackle this problem, we propose a simple representation of bijective surface maps using Beltrami coefficients (BCs), which are complex-valued functions defined on surfaces with supreme 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 can be represented by a unique BC. Conversely, given a BC, we can reconstruct the unique surface map associated to 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

Beltrami Representation and its applications to texture map and video compression

Surface parameterizations and registrations are important in computer graphics and imaging, where 1-1 correspondences between meshes are computed. In practice, surface maps are usually represented and stored as 3D coordinates each vertex is mapped to, which often requires lots of storage memory. This causes inconvenience in data transmission and data storage. To tackle this problem, we propose an effective algorithm for compressing surface homeomorphisms using Fourier approximation of the Beltrami representation. The Beltrami representation is a complex-valued function defined on triangular faces of the surface mesh with supreme norm strictly less than 1. Under suitable normalization, there is a 1-1 correspondence between the set of surface homeomorphisms and the set of Beltrami representations. Hence, every bijective surface map is associated with a unique Beltrami representation. Conversely, given a Beltrami representation, the corresponding bijective surface map can be exactly reconstructed using the Linear Beltrami Solver introduced in this paper. Using the Beltrami representation, the surface homeomorphism can be easily compressed by Fourier approximation, without distorting the bijectivity of the map. The storage memory can be effectively reduced, which is useful for many practical problems in computer graphics and imaging. In this paper, we proposed to apply the algorithm to texture map compression and video compression. With our proposed algorithm, the storage requirement for the texture properties of a textured surface can be significantly reduced. Our algorithm can further be applied to compressing motion vector fields for video compression, which effectively improve the compression ratio.Comment: 30 pages, 23 figure