On the bijectivity of thin-plate splines

Research output: Chapter in Book/Report/Conference proceedingBook chapter


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.


Research areas and keywords

Subject classification (UKÄ) – MANDATORY

  • Mathematics
Original languageEnglish
Title of host publicationAnalysis for Science, Engineering and Beyond, The Tribute Workshop in Honour of Gunnar Sparr held in Lund, May 8-9, 2008
EditorsKarl Åström, Lars-Erik Persson, Sergei Silvestrov
ISBN (Print)978-3-642-20236-0, 978-3-642-20235-3 (print)
StatePublished - 2012
Publication categoryResearch

Publication series