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Provably good planar mappings

Published: 27 July 2014 Publication History

Abstract

The problem of planar mapping and deformation is central in computer graphics. This paper presents a framework for adapting general, smooth, function bases for building provably good planar mappings. The term "good" in this context means the map has no fold-overs (injective), is smooth, and has low isometric or conformal distortion.
Existing methods that use mesh-based schemes are able to achieve injectivity and/or control distortion, but fail to create smooth mappings, unless they use a prohibitively large number of elements, which slows them down. Meshless methods are usually smooth by construction, yet they are not able to avoid fold-overs and/or control distortion.
Our approach constrains the linear deformation spaces induced by popular smooth basis functions, such as B-Splines, Gaussian and Thin-Plate Splines, at a set of collocation points, using specially tailored convex constraints that prevent fold-overs and high distortion at these points. Our analysis then provides the required density of collocation points and/or constraint type, which guarantees that the map is injective and meets the distortion constraints over the entire domain of interest.
We demonstrate that our method is interactive at reasonably complicated settings and compares favorably to other state-of-the-art mesh and meshless planar deformation methods.

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References

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cover image ACM Transactions on Graphics
ACM Transactions on Graphics  Volume 33, Issue 4
July 2014
1366 pages
ISSN:0730-0301
EISSN:1557-7368
DOI:10.1145/2601097
Issue’s Table of Contents
Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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Publication History

Published: 27 July 2014
Published in TOG Volume 33, Issue 4

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Author Tags

  1. bijective mappings
  2. bounded isometric distortion
  3. conformal distortion
  4. meshless deformation

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