Injectivity conditions of 2D and 3D uniform cubic B-spline functions

Abstract

Uniform cubic B-spline functions have been used for mapping functions in various areas such as image warping and morphing, 3D deformation, and Volume morphing. The injectivity tone-to-one property) of a mapping function is crucial to obtaining desirable results in these areas. This paper considers the injectivity conditions of 2D and 3D uniform cubic B-spline functions. We propose a geometric interpretation of the injectivity of a uniform cubic B-spline function, with which 2D and 3D cases can be handled in a similar way. Based on our geometric interpretation, we present sufficient conditions for injectivity which are represented in terms of control point displacements. These sufficient conditions can be easily rested and will be useful in guaranteeing the injectivity of mapping functions in application areas. (C) 2000 Academic Press.