Computing Extremal Quasiconformal Maps

Abstract

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.

A Splitting Method for Diffeomorphism Optimization Problem Using Beltrami Coefficients

“…Note that when p = 2, the minimizer of the least square problem (12) is called the leastsquare quasi-conformal map (LSQC) associated to ν, which has also been studied in [35,42]. In this paper, we will apply the method in [38] to cope with a subproblem in the alternating minimization algorithm for solving the DOP.…”

“…Various efficient algorithms have been introduced to compute the quasiconformal map [23,24,[36][37][38]41]. In this paper, we approximate the quasi-conformal map f with a Beltrami coefficient μ by directy solving the Beltrami's equation (35) as in [38]. The details of the computation of the quasi-conformal map will be explained in Sect.…”

“…We can then look for a quasi-conformal map f with BC μ. It is equivalent to looking for f satisfying: (35) subject to the boundary constraints that f | ∂ S 1 = ∂ S 2 . In [1], the quasi-conformal map is obtained using the integration formula (10) or (11).…”

Integer-grid maps for reliable quad meshing

“…Obviously, if the relaxed minimizer of Eq positions several singular vertices close to the same integer lattice point these singularities are likely to snap onto each other and thus induce a degenerate map. While the first issue can be satisfactorily handled by the stiffening approach of [Bommes et al 2009] or recent parametrization approaches that bound the quasi-conformal distortion [Lipman 2012;Weber et al 2012], the second integer related one cannot. Moreover it seems to be very difficult if not impossible to design a greedy snapping strategy which guarantees a valid map while respecting the linear interdependency caused by the transition functions.…”

“…Therefore all of the above methods do not provide the required reliability or cannot be applied to our setting. In the area of quasiconformal maps, however, two recent approaches [Weber et al 2012;Lipman 2012] are lifting the variational setting to a new level. While the nonlinear approach of [Weber et al 2012] usually succeeds but cannot guarantee to find a bijective mapping, the powerful concept of "Bounded Distortion Mapping Spaces" [Lipman 2012] is closely related to one aspect of our setting.…”