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Geometric Approach to Estimation of Volumetric Distortions

The problem of measuring geometrical distortions is not trivial for volumetric domains. There exist intrinsic 
 
restrictions and constrains on higher dimensional mappings. Moreover, according to Liouville theorem, most 
 
existing techniques for 2D data can not be directly applied to volumetric objects. In this work we approximate 
 
continuous deformations by piecewise affine functions defined on tetrahedral meshes. Our aim is to study 
 
a few types of geometrical distortions that can be expressed as functions of singular values of a Jacobian. 
 
We employ the proposed methods of estimating conformal and isometric distortions to analyze volumetric 
 
data. In particular, we examine parametrization of tetrahedral models to a ball. Distortions produced by the 
 
resulting spatial mappings depict intrinsic structure of domains, and therefore can be employed in such tasks 
 
as detection of abnormalities and comparison (i.e. similarity assessment) between 3D objects. This geometric 
 
approach and results are highly relevant to various applications in Computer Vision, Computer Graphics, 3D 
 
Printing and Medical Imaging.