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Geometric Mappings With Low Distortion

Posted on:2017-04-24Degree:DoctorType:Dissertation
Country:ChinaCandidate:C X WangFull Text:PDF
GTID:1108330485953655Subject:Computational Mathematics
Abstract/Summary:PDF Full Text Request
With the development of 3D scanning technology, digital geometric data has be-come a new type of multi-media followed sound, image and video, and widely used in the fields of surface modeling, computer animation and vision, geographic information system, physical simulation, virtual reality, scientific computing visualization and so on. The research of this thesis is based on the basic digital geometric data of triangular mesh. Finding geometric mappings with low distortion between surfaces is a basic and important problem in computer graphics, computer vision, medical image processing and so on. Among them, surface parameterization and registration are two important techniques. With the development of digital geometry processing and the wide applica-tion of surface differential geometry in computer science, more and more engineering problems based on the mappings with low distortion are solved. In this thesis, based on triangular meshes, we focus on the problem of finding mappings with low distortion, in-cluding spherical parametrization with low distortion and mappings with low distortion between planar meshes.For closed genus-zero triangular meshes, we present an as-rigid-as-possible spher-ical parametrization method (ARAP method), which is an extension of planar ARAP parametrization approach to spherical domain. Our aim is to find an optimal sphere on which parametrization triangles can be embedded in a rigidity-preserving manner. We analyze the smooth and discrete ARAP energy and formulate our spherical parametriza-tion model from the discrete ARAP energy. The solution is nontrivial as the energy in-volves a large system of non-linear equations with additional spherical constraints. To this end, we propose an efficient two-step iterative algorithm including a local/global iterative scheme to calculate the parametrization coordinates and iterative updating ra-dius. This method optimizes rigid distortion directly, which overcomes the drawbacks of previous work optimizing only angle distortion or area distortion. Experimental re-sults show that ARAP spherical parametrization has the best rigidity-preserving prop-erty compared with previous methods.For closed genus-zero triangular meshes, we present a method to find a bijective spherical parametrization with low distortion (BLD method), including conformal and isometric spherical parametrization. Previous methods for spherical parametrization cannot, in general, control the worst case distortion of all triangles nor guarantee bi- jectivity. To surmount these disadvantages, we formulate our spherical parametrization model based on AMIPS energy. Given an initial bijective spherical parametrization, even with high distortion, we develop a non-linear constrained optimization problem to refine it, with objective penalizing the presence of degenerate triangles and maximal distortion. By using a dynamic adjusting parameter and a constrained, iterative inexact Block Coordinate Descent optimization method, we efficiently and robustly achieve a bijective and low distortion parametrization with an optimal sphere radius. Experimen-tal results show that, our method can achieve both the lowest maximal distortion and average distortion on numerous models undergoing simple to complex shapes. In ad-dition, our method is fast, efficient, robust to initial parametrization and insensitive to parameter choice.For the feature matching problem of images, we find geometrically consistent cor-respondences in two images by constructing the mappings with low distortion between planar meshes. Given a set of candidate matches provided by SIFT descriptors[1], which may include many outliers. Our goal is to select a subset of these matches retaining much more geometric information. To solve this problem, we present a filtering method based on the space of all diffeomorphisms, formulate a constrained optimization in-volving both the Beltrami coefficient term and quasi-conformal map, and solved by an efficient iterative algorithm based on variable splitting method and iterative reweighted least squares method. In each iteration, we solve two subproblems including a lin-ear system and a linearly constrained convex quadratic programming. To measure the matching accuracy, we define a statistic F-measure and test our algorithm on both syn-thetic data and real images. Experimental results show that our method enables pro-ducing more correct correspondences compared with the state-of-the-art approaches, is insensitive to parameter choice and robust to outliers.
Keywords/Search Tags:digital geometric data, geometric mappings with low distortion, triangular meshes, spherical parametrization, feature matching of images, quasi-conformal maps
PDF Full Text Request
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