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On the bijectivity of thin-plate splines

Eriksson, Anders P LU and Åström, Karl LU (2012) In Analysis for Science, Engineering and Beyond, The Tribute Workshop in Honour of Gunnar Sparr held in Lund, May 8-9, 2008 6. p.93-141
Abstract
The thin-plate spline (TPS) has been widely used in a number of areas

such as image warping, shape analysis and scattered data interpolation. Introduced

by Bookstein (IEEE Trans. Pattern Anal. Mach. Intell. 11(6):567–585 1989), it is a

natural interpolating function in two dimensions, parameterized by a finite number

of landmarks. However, even though the thin-plate spline has a very intuitive

interpretation as well as an elegant mathematical formulation, it has no inherent

restriction to prevent folding, i.e. a non-bijective interpolating function. In this

chapter we discuss some of the properties of the set of parameterizations that form

bijective thin-plate splines,... (More)
The thin-plate spline (TPS) has been widely used in a number of areas

such as image warping, shape analysis and scattered data interpolation. Introduced

by Bookstein (IEEE Trans. Pattern Anal. Mach. Intell. 11(6):567–585 1989), it is a

natural interpolating function in two dimensions, parameterized by a finite number

of landmarks. However, even though the thin-plate spline has a very intuitive

interpretation as well as an elegant mathematical formulation, it has no inherent

restriction to prevent folding, i.e. a non-bijective interpolating function. In this

chapter we discuss some of the properties of the set of parameterizations that form

bijective thin-plate splines, such as convexity and boundness. Methods for finding

sufficient as well as necessary conditions for bijectivity are also presented. The

methods are used in two settings (a) to register two images using thin-plate spline

deformations, while ensuring bijectivity and (b) group-wise registration of a set of

images, while enforcing bijectivity constraints. (Less)
Please use this url to cite or link to this publication:
author
organization
publishing date
type
Chapter in Book/Report/Conference proceeding
publication status
published
subject
in
Analysis for Science, Engineering and Beyond, The Tribute Workshop in Honour of Gunnar Sparr held in Lund, May 8-9, 2008
editor
Åström, Karl; Persson, Lars-Erik and Silvestrov, Sergei
volume
6
pages
93 - 141
publisher
Springer
external identifiers
  • Scopus:84893550201
ISBN
978-3-642-20236-0
978-3-642-20235-3 (print)
DOI
10.1007/978-3-642-20236-0_5
language
English
LU publication?
yes
id
a57fecda-f6bb-4a56-9281-091d52684c49 (old id 3437196)
date added to LUP
2013-09-06 17:30:29
date last changed
2016-11-18 10:05:51
@misc{a57fecda-f6bb-4a56-9281-091d52684c49,
  abstract     = {The thin-plate spline (TPS) has been widely used in a number of areas<br/><br>
such as image warping, shape analysis and scattered data interpolation. Introduced<br/><br>
by Bookstein (IEEE Trans. Pattern Anal. Mach. Intell. 11(6):567–585 1989), it is a<br/><br>
natural interpolating function in two dimensions, parameterized by a finite number<br/><br>
of landmarks. However, even though the thin-plate spline has a very intuitive<br/><br>
interpretation as well as an elegant mathematical formulation, it has no inherent<br/><br>
restriction to prevent folding, i.e. a non-bijective interpolating function. In this<br/><br>
chapter we discuss some of the properties of the set of parameterizations that form<br/><br>
bijective thin-plate splines, such as convexity and boundness. Methods for finding<br/><br>
sufficient as well as necessary conditions for bijectivity are also presented. The<br/><br>
methods are used in two settings (a) to register two images using thin-plate spline<br/><br>
deformations, while ensuring bijectivity and (b) group-wise registration of a set of<br/><br>
images, while enforcing bijectivity constraints.},
  author       = {Eriksson, Anders P and Åström, Karl},
  editor       = {Åström, Karl and Persson, Lars-Erik and Silvestrov, Sergei},
  isbn         = {978-3-642-20236-0},
  language     = {eng},
  pages        = {93--141},
  publisher    = {ARRAY(0xb865218)},
  series       = {Analysis for Science, Engineering and Beyond, The Tribute Workshop in Honour of Gunnar Sparr held in Lund, May 8-9, 2008},
  title        = {On the bijectivity of thin-plate splines},
  url          = {http://dx.doi.org/10.1007/978-3-642-20236-0_5},
  volume       = {6},
  year         = {2012},
}