Authors:
Alexander Naitsat
1
;
Emil Saucan
2
and
Yehoshua Y. Zeevi
1
Affiliations:
1
Technion, Israel
;
2
Max Planck Institute for Mathematics in the Sciences, Germany
Keyword(s):
Conformal Geometry, Isometric Distortion, Volume Parameterization, Volumetric Meshes, Geometric Computing, Computer Graphics.
Related
Ontology
Subjects/Areas/Topics:
Computer Vision, Visualization and Computer Graphics
;
Fundamental Methods and Algorithms
;
Geometric Computing
;
Geometry and Modeling
;
Modeling and Algorithms
Abstract:
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 h
ighly relevant to various applications in Computer Vision, Computer Graphics, 3D
Printing and Medical Imaging.
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