Computing quasi-conformal maps in 3D with applications to geometric modeling and imaging

Abstract

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 Mobius 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.