A fast and simple stretch-minimizing mesh parameterization

Yoshizawa, Shin; Belyaev, Alexander; Seidel, Hans‐Peter

doi:10.1109/smi.2004.1314507

Cited by 78 publications

(105 citation statements)

References 20 publications

(38 reference statements)

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“…There are many different unwrapping methods to map a 3D mesh to a 2D plane to produce a parameterization [2], [6], [11], [19]- [22]. To a great extent, they all are able to produce an initial parameterization.…”

Section: B Bar-net Based Initial Parameterizationmentioning

confidence: 99%

Constrained Texture Mapping and Foldover-Free Condition

Yu¹,

Yang²,

Zhang³

2013

IJCTE

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Abstract-Texture mapping has been widely used in image processing and graphics to enhance the realism of CG scenes. However to perfectly match the feature points of a 3D model with the corresponding pixels in texture images, the parameterisation which maps a 3D mesh to the texture space must satisfy the positional constraints. Despite numerous research efforts, the construction of a mathematically robust foldover-free parameterisation subject to internal constraints is still a remaining issue. In this paper, we address this challenge by developing a two-step parameterisation method. First, we produce an initial parameterisation with a method traditionally used to solve structural engineering problems, called the bar-network. We then derive a mathematical foldover-free condition, which is incorporated into a Radial Basis Function based scheme. This method is therefore able to guarantee that the resulting parameterization meets the hard constraints without foldovers.Index Terms-Foldover, constrained texture mapping, parameterization. I. INTRODUCTIONTexture mapping is an effective means in image processing and graphics to achieve improved visual realism. Existing research has largely concentrated on the production of planar parameterization [1]- [7] in order to map a 3D mesh to the planar domain. Most recent works are concerned with texture distortion reduction when mapping a planar image (texture) to a curved surface. Although this is an important issue, in practice the animator is also challenged with other problems in texture mapping. One of them is to register a texture map accurately with the features of a 3D model. For example, if one is to texture map a human face, in addition to reducing texture distortion, one has to ensure the important feature points and lines on the 3D model match those on the texture plane during the mapping process, such as the eyes, nose, eyebrows and lips. In another word, one needs to accurately register the 3D features with their 2D counterparts.With the current production practice, this registration operation is almost completely manual. Once a texture map is generated by animation software, the animator has to painstakingly tweak the unwrapped mesh on the texture plane to align the key feature points on the texture image with the 3D features. He/she has to manually move many vertices around each feature in order to avoid texture distortion being concentrated and visible. This is a timeconsuming task. Manuscript received September 18, 2012; revised December 7, 2012. The authors are with the National Center for Computer Animation, Bournemouth University, Poole, UK. (email:cnyuhc@yahoo.com, xyang@bournemouth.ac.uk, jzhang@bournemouth.ac.uk).Attempts have been made to formulate it as a constrained optimization problem [3]- [5], [7] where the important features in the texture images are to be located correctly on the 3D surface. Despite varying degree of success, a key issue yet to be solved is there is currently no robust solution to controlling the spread of the mesh points such th...

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Section: B Bar-net Based Initial Parameterizationmentioning

confidence: 99%

Constrained Texture Mapping and Foldover-Free Condition

Yu¹,

Yang²,

Zhang³

2013

IJCTE

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“…Yoshizawa et al [132] and Zayer et al [134] also start with a parameterization that minimizes angular distortion. They first use linear, fixedboundary methods to compute one.…”

Section: Trade-off Between Metricsmentioning

confidence: 99%

“…They then compute the stretch of the parameterization; scale the weights used in the linear method by a mitigating factor and solve the linear system again. To scale the weights, Yoshizawa et al [132] used the stretch metric of Sander et al [107], while Zayer et al [134] use a simpler construction based on inverse edge lengths.…”

Section: Trade-off Between Metricsmentioning

confidence: 99%

Mesh Parameterization Methods and Their Applications

Sheffer

Praun

Rose

2006

FNT in Computer Graphics and Vision

369

118

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We present a survey of recent methods for creating piecewise linear mappings between triangulations in 3D and simpler domains such as planar regions, simplicial complexes, and spheres. We also discuss emerging tools such as global parameterization, inter-surface mapping, and parameterization with constraints. We start by describing the wide range of applications where parameterization tools have been used in recent years. We then briefly review the pertinent mathematical background and terminology, before proceeding to survey the existing parameterization techniques. Our survey summarizes the main ideas of each technique and discusses its main properties, comparing it to other methods available. Thus it aims to provide guidance to researchers and developers when assessing the suitability of different methods for various applications. This survey focuses on the practical aspects of the methods available, such as time complexity and robustness and shows multiple examples of parameterizations generated using different methods, allowing the reader to visually evaluate and compare the results.

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“…Variations of harmonic energies were also optimized using discrete Laplace-Beltrami operators in [6], [7], [8], [9], [10], and [11]. More general energy forms can be found in [12], [13], [14], [15], [16], and [17]. Most linear methods apply a convex Dirichlet-type boundary.…”

Section: Introductionmentioning

confidence: 99%

Optimal Surface Parameterization Using Inverse Curvature Map

Yang

Kim

Luo

et al. 2008

IEEE Trans. Visual. Comput. Graphics

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Abstract-Mesh parameterization is a fundamental technique in computer graphics. The major goals during mesh parameterization are to minimize both the angle distortion and the area distortion. Angle distortion can be eliminated by the use of conformal mapping, in principle. Our paper focuses on solving the problem of finding the best discrete conformal mapping that also minimizes area distortion. First, we deduce an exact analytical differential formula to represent area distortion by curvature change in the discrete conformal mapping, giving a dynamic Poisson equation. On a mesh, the vertex curvature is related to edge lengths by the curvature map. Our result shows the map is invertible, i.e., the edge lengths can be computed from the curvature (by integration). Furthermore, we give the explicit Jacobi matrix of the inverse curvature map. Second, we formulate the task of computing conformal parameterizations with least area distortions as a constrained nonlinear optimization problem in curvature space. We deduce explicit conditions for the optima. Third, we give an energy form to measure the area distortions, and show that it has a unique global minimum. We use this to design an efficient algorithm, called free boundary curvature diffusion, which is guaranteed to converge to the global minimum; it has a natural physical interpretation. This result proves the common belief that optimal parameterization with least area distortion has a unique solution and can be achieved by free boundary conformal mapping. Major theoretical results and practical algorithms are presented for optimal parameterization based on the inverse curvature map. Comparisons are conducted with existing methods and using different energies. Novel parameterization applications are also introduced. The theoretical framework of the inverse curvature map can be applied to further study discrete conformal mappings.

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A fast and simple stretch-minimizing mesh parameterization

Abstract: We

Cited by 78 publications

References 20 publications

Constrained Texture Mapping and Foldover-Free Condition

Constrained Texture Mapping and Foldover-Free Condition

Mesh Parameterization Methods and Their Applications

Optimal Surface Parameterization Using Inverse Curvature Map

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