Surface Quasi-Conformal Mapping by Solving Beltrami Equations

A fast algorithm to solve the Beltrami equation with applications to quasiconformal mappings

MESH GENERATION BY CONFORMAL AND QUASICONFORMAL MAPPINGS

Historically the origin of quasiconformal mappings is connected with developments of the methods of complex functions. Since the power of this concept was first realized, quasiconformal mappings have engaged the attention of many prominent mathematicians and the theory has been greatly expanded to higher dimensions. The fundamental principle of this theory is to interpolate between diffeomorphisms and homeomorphisms [G4]. One difference between conformal and quasiconformal mappings is that the latter need not be differentiable in the usual sense. However, by a theorem due to A. Mori, F. W. Gehring and J. Viiisiilii, every quasiconformal mapping f : n --t Rn is differentiable almost everywhere and its Jacobian determinant .J(x, f) is locally integrable. Moreover, quasiconformality of f can be expressed by the differential inequality

Conformal and their natural generalization to quasi-conformal mappings of surfaces, have been extensively and successfully employed in various tasks of Computer Graphics and Imaging. Due to the intrinsic differences between surfaces and objects from higher dimensions, many important results on surfaces can not be directly generalized to produce desirable mapping of volumes. Moreover, in dimension higher than 2, there are no conformal maps apart from Möbius transformations. Therefore, most of the real-world applications generate only quasi-conformal transformations, which produce some conformal distortion. Hence it is tempting and natural to measure the quality of volume deformation by the amount of a conformal distortion it produces. In this paper we examine theoretical properties of quasi-conformal mappings in 3D. We apply those conclusion to process discrete volumetric data in the terms of conformality. We present numerical methods to measure "the degree of conformality" of a transformation between a given pair of domains, represented by volumetric meshes.

By using methods of integral equations, we investigate problems of conformal and quasiconformal mappings of close domains.

In shape analysis, finding an optimal 1-1 correspondence between surfaces within a large class of admissible bijective mappings is of great importance. Such process is called surface registration. The difficulty lies in the fact that the space of all surface diffeomorphisms is a complicated functional space, making exhaustive search for the best mapping challenging. To tackle this problem, we propose a simple representation of bijective surface maps using Beltrami coefficients (BCs), which are complex-valued functions defined on surfaces with supreme norm less than 1. Fixing any 3 points on a pair of surfaces, there is a 1-1 correspondence between the set of surface diffeomorphisms between them and the set of BCs. Hence, every bijective surface map can be represented by a unique BC. Conversely, given a BC, we can reconstruct the unique surface map associated to it using the Beltrami Holomorphic flow (BHF) method. Using BCs to represent surface maps is advantageous because it is a much simpler functional space, which captures many essential features of a surface map. By adjusting BCs, we equivalently adjust surface diffeomorphisms to obtain the optimal map with desired properties. More specifically, BHF gives us the variation of the associated map under the variation of BC. Using this, a variational problem over the space of surface diffeomorphisms can be easily reformulated into a variational problem over the space of BCs. This makes the minimization procedure much easier. More importantly, the diffeomorphic property is always preserved. We test our method on synthetic examples and real medical applications. Experimental results demonstrate the effectiveness of our proposed algorithm for surface registration.

Topology-preserving image registration with novel multi-dimensional Beltrami regularization

The application of transformation optics to the development of intriguing electromagnetic devices can produce weakly anisotropic or isotropic media with the assistance of quasi-conformal and/or conformal mapping, as opposed to the strongly anisotropic media produced by general mappings; however, it is typically limited to two-dimensional applications. By addressing the conformal mapping between two manifolds embedded in three-dimensional space, we demonstrate that electromagnetic surface waves can be controlled without introducing singularity and anisotropy into the device parameters. Using fruitful surface conformal parameterization methods, a near-perfect conformal mapping between smooth manifolds with arbitrary boundaries can be obtained. Illustrations of concealing and illusions, including surface Luneburg and Eaton lenses and black holes for surface waves, are provided. Our work brings the manipulation of surface waves at microwave and optical wavelengths one step closer.

Quasiconformal maps in the plane are orientation preserving homeomorphisms that satisfy certain distortion inequalities; infinitesimally, they map circles to ellipses of bounded eccentricity. Such maps have many useful geometric distortion properties, and yield a flexible and powerful generalization of conformal mappings. In this work, we study the singularities of these maps, in particular the sizes of the sets where a quasiconformal map can exhibit given stretching and rotation behavior. We improve results by Astala-Iwaniec-Prause-Saksman and Hitruhin to give examples of stretching and rotation sets with non-sigma-finite measure at the appropriate Hausdorff dimension. We also improve this to give examples with positive Riesz capacity at the critical homogeneity, as well as positivity for a broad class of gauged Hausdorff measures at that dimension.

For a given \({\epsilon > 0}\) , we show that there exist two finite index subgroups of \({PSL_2(\mathbb{Z})}\) which are \(({1+\epsilon})\) -quasisymmetrically conjugated and the conjugation homeomorphism is not conformal. This implies that for any \({\epsilon > 0}\) there are two finite regular covers of the Modular once punctured torus T 0 (or just the Modular torus) and a \({(1+\epsilon)}\) -quasiconformal map between them that is not homotopic to a conformal map. As an application of the above results, we show that the orbit of the basepoint in the Teichmüller space T(S p) of the punctured solenoid S p under the action of the corresponding Modular group (which is the mapping class group of S p [6], [7]) has the closure in T(S p) strictly larger than the orbit and that the closure is necessarily uncountable.

Some extremal problems for conformal and quasiconformal mappings

The approximation of three-dimensional, nearly conformal quasiconformal mappings by smooth quasiconformal mappings

Declicated to our teacher Pro-/essor

We prove various inequalities measuring how far from an isometry a local map from a manifold of high curvature to a manifold of low curvature must be. We consider the cases of volume-preserving, conformal and quasi-conformal maps. The proofs relate to a conjectural isoperimetric inequality for manifolds whose curvature is bounded above, and to a higher-dimensional generalization of the Schwarz-Ahlfors lemma.

Landmark constrained genus-one surface Teichmüller map applied to surface registration in medical imaging.