On the Bijectivity of Thin-Plate Splines

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.