On the bijectivity of thin-plate splines

Eriksson, Anders P; Åström, Karl (2012). On the bijectivity of thin-plate splines In . Åström, Karl; Persson, Lars-Erik; Silvestrov, Sergei (Eds.). Analysis for Science, Engineering and Beyond, The Tribute Workshop in Honour of Gunnar Sparr held in Lund, May 8-9, 2008, 6,, 93 - 141: Springer
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Book Chapter | Published | English
Eriksson, Anders P ; Åström, Karl
Åström, Karl ; Persson, Lars-Erik ; Silvestrov, Sergei
Mathematics (Faculty of Engineering)
Centre for Mathematical Sciences
Mathematical Imaging Group
Research Group:
Mathematical Imaging Group
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.
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