A Fast Algorithm to Solve the Beltrami Equation with Applications to Quasiconformal Mappings

Conformal maps are widely used in geometry processing applications. They are smooth, preserve angles, and are locally injective by construction. However, conformal maps do not allow for boundary positions to be prescribed. A natural extension to the space of conformal maps is the richer space of quasiconformal maps of bounded conformal distortion. Extremal quasiconformal maps, that is, maps minimizing the maximal conformal distortion, have a number of appealing properties making them a suitable candidate for geometry processing tasks. Similarly to conformal maps, they are guaranteed to be locally bijective; unlike conformal maps however, extremal quasiconformal maps have sufficient flexibility to allow for solution of boundary value problems. Moreover, in practically relevant cases, these solutions are guaranteed to exist, are unique and have an explicit characterization. We present an algorithm for computing piecewise linear approximations of extremal quasiconformal maps for genus-zero surfaces with boundaries, based on Teichmüller's characterization of the dilatation of extremal maps using holomorphic quadratic differentials. We demonstrate that the algorithm closely approximates the maps when an explicit solution is available and exhibits good convergence properties for a variety of boundary conditions.

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“…in [Dar93]. [ZLYG09] proposes a method for computing quasiconformal maps from Beltrami coefficients on general multiply connected domains.…”

Section: Related Workmentioning

confidence: 99%

Computing Extremal Quasiconformal Maps

Weber

Myles

Zorin

2012

Computer Graphics Forum

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“…The algorithm in [9] for computingP is defined in terms of a circular N × M grid and its computational complexity is O(N M log 2 M ). In order to use this algorithm it is necessary to find an approximation for h z , given the values of h on a circular grid.…”

Section: Case Z ∈ C \ B(0 R)mentioning

confidence: 99%

On Numerical Algorithms for the Solution of a Beltrami Equation

Gaidashev¹,

Khmelev²

2008

SIAM J. Numer. Anal.

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Abstract. The paper concerns numerical algorithms for solving the Beltrami equation fz(z) = µ(z)fz (z) for a compactly supported µ.First, we study an efficient algorithm that has been proposed in the literature, and present its rigorous justification. We then propose a different scheme for solving the Beltrami equation which has a comparable speed and accuracy, but has the virtue of a greater simplicity of implementation.

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“…A fast algorithm using FFT for solving the Beltrami equation has been proposed by Daripa et al [3], [4], [5]. Based on Daripa's algorithm, a slightly modified version was given in [6].…”

Section: Algorithm Of Constructing the Quasiconformal Mapmentioning

confidence: 99%

“…In Section 4, we will describe a numerical scheme to solve the Beltrami equation. This algorithm is a combination of the fast algorithm developed by Daripa et al [3], [4], [5] (also see [6]) and the NUFFT (nonuniform fast Fourier transform) [7]. The rest of the paper is organized as follows.…”

Section: Introductionmentioning

confidence: 99%