Efficient computation of constrained parameterizations on parallel platforms

Athanasiadis, Theodoros; Zioupos, Georgios; Fudos, Ioannis

doi:10.1016/j.cag.2013.05.023

Cited by 3 publications

(3 citation statements)

References 35 publications

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“…The advantages of our implementation are described below: Initial parameterization : Our implementation allows an initial random parameterization U 0 for iterating equation (9), providing consistent parameterizations in all our test cases. This is superior to most nonlinear-gradient algorithms which require an initial valid parameterization (estimated by a linear parameterization algorithm) to proceed, such as in Athanasiadis et al (2013), Liu et al (2008) and Smith and Schaefer (2015). Hessian estimator : The LM algorithm proposes to estimate the Hessian matrix as ℋ[ F ] ≈ ∇ F · ∇ F T . This approach leads to a dense Hessian matrix.…”

Section: Methodsmentioning

confidence: 99%

“…However, the ASAP/ARAP method requires a post-processing step owing to triangle flips occurring in the resulting parameterization. The Constrained Parameterization on Parallel Platforms algorithm (Athanasiadis et al , 2013) minimizes an energy function similar to the MIPS energy, using GPU resources to improve the computational efficiency. These nonlinear-gradient methods allow to parameterize surfaces with holes as opposed to previous methods.…”

Section: Literature Reviewmentioning

confidence: 99%

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Weighted area/angle distortion minimization for Mesh Parameterization

Mejía

Acosta

Ruiz-Salguero

2017

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Purpose Mesh Parameterization is central to reverse engineering, tool path planning, etc. This work synthesizes parameterizations with un-constrained borders, overall minimum angle plus area distortion. This study aims to present an assessment of the sensitivity of the minimized distortion with respect to weighed area and angle distortions. Design/methodology/approach A Mesh Parameterization which does not constrain borders is implemented by performing: isometry maps for each triangle to the plane Z = 0; an affine transform within the plane Z = 0 to glue the triangles back together; and a Levenberg–Marquardt minimization algorithm of a nonlinear F penalty function that modifies the parameters of the first two transformations to discourage triangle flips, angle or area distortions. F is a convex weighed combination of area distortion (weight: α with 0 ≤ α ≤ 1) and angle distortion (weight: 1 − α). Findings The present study parameterization algorithm has linear complexity [𝒪(n), n = number of mesh vertices]. The sensitivity analysis permits a fine-tuning of the weight parameter which achieves overall bijective parameterizations in the studied cases. No theoretical guarantee is given in this manuscript for the bijectivity. This algorithm has equal or superior performance compared with the ABF, LSCM and ARAP algorithms for the Ball, Cow and Gargoyle data sets. Additional correct results of this algorithm alone are presented for the Foot, Fandisk and Sliced-Glove data sets. Originality/value The devised free boundary nonlinear Mesh Parameterization method does not require a valid initial parameterization and produces locally bijective parameterizations in all of our tests. A formal sensitivity analysis shows that the resulting parameterization is more stable, i.e. the UV mapping changes very little when the algorithm tries to preserve angles than when it tries to preserve areas. The algorithm presented in this study belongs to the class that parameterizes meshes with holes. This study presents the results of a complexity analysis comparing the present study algorithm with 12 competing ones.

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Section: Methodsmentioning

confidence: 99%

Section: Literature Reviewmentioning

confidence: 99%

Weighted area/angle distortion minimization for Mesh Parameterization

Mejía

Acosta

Ruiz-Salguero

2017

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“…For instance, “MIPS” [HG00] energy is inversely proportional to the area in the parameter‐space element. Other energies, such as “stretch energy” [SSGH01] which is used to measure “texture stretch” and the area‐preserving version of “MIPS energy” presented in [AZF13], naturally possess parameter‐space area term as barrier function and thus have similar effects as well. The methods [YLY*12, SHSF13] restrict the mapping to be the composition of a series of special mappings.…”

Section: Related Workmentioning

confidence: 99%

Remeshing‐assisted Optimization for Locally Injective Mappings

Jin

Huang

Tong

2014

Computer Graphics Forum

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Constructing locally injective mappings for 2D triangular meshes is vital in applications such as deformations. In such a highly constrained optimization, the prescribed tessellation may impose strong restriction on the solution. As a consequence, the feasible region may be too small to contain an ideal solution, which leads to problems of slow convergence, poor solution, or even that no solution can be found. We propose to integrate adaptive remeshing into interior point method to solve this issue. We update the vertex positions via a parameter‐free relaxation enhanced geometry optimization, and then use edge‐flip operations to reduce the residual and keep a reasonable condition number for better convergence. For more robustness, when the iteration of interior point method terminates but leaves the positional constraints unsatisfied, we estimate the edges in the current tessellation that block vertices moving based on the convergence information of the optimization, and then split neighboring edges to break the restriction. The results show that our method has better performance than the solely geometric optimization approaches, especially for extreme deformations.

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