טכניון מכון טכנולוגי לישראל
הטכניון מכון טכנולוגי לישראל - בית הספר ללימודי מוסמכים  
M.Sc Thesis
M.Sc StudentNaitsat Alexander
SubjectQuasi-Conformal Mappings for Volumetric Deformations in
Geometric Modeling
DepartmentDepartment of Applied Mathematics
Supervisors Professor Emeritus Yehoshua Zeevi
Dr. Emil Saucan


Abstract

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Conformal and their natural generalization to quasi-conformal mappings of surfaces have been extensively and successfully employed in various computer graphics and imaging tasks.

In view of the successful results in 2D, it is tempting and natural to attempt to extend this approach to 3 and higher dimensions. However, because of the intrinsic differences between surfaces and higher dimensional objects, some important results regarding surfaces cannot be extended to volumetric domains. Most significantly, there exist no conformal volumetric maps other than Möbius transformations. Therefore, it is natural to consider instead quasi-conformal mappings.

Unfortunately, such mappings between a given pair of domains are generally difficult to compute, and resulting functions often are “far from conformality.”

We propose first to highlight and illustrate these theoretical drawbacks of quasi-conformal mappings in space. Then, we apply the conclusions to process discrete volumetric data in the terms of conformality.

We introduce methods for assessing the extent of the local and global volumetric deformation by means of the amount of conformal distortion produced. To this end, we first illustrate various 3D quasi-conformal deformations between a given pair of domains, represented by volumetric meshes. We highlight theoretical issues associated with spatial quasi-conformal mappings and the relation that exists between the geometry of the domain and conformal distortion.

Although sometimes stated explicitly, it is frequently overlooked that existing volume parameterization methods produce only quasi-conformal maps, which may be “far from conformality.”

Therefore, we present basic parameterization techniques for volumes and measure “the degree of conformality” of the deformations that are produced by these parameterizations.