Chapter 12 - Spatial Interpolants for Warping

Abstract

The basic idea of the approach to warping that is presented in this chapter is to deform the whole space in which the object to be warped is embedded, so that the object is warped at the same time. In the physical world, for example, the object might be an image drawn on an elastic skin. The elastic skin may be deformed, thus causing a deformation of the image. The deformation of the elastic skin may be specified by moving some of its points into a new position. Then the skin interpolates the new position of any other point from these displacements. The scattered data methods discussed in the chapter include the distance-weighted methods, methods based on radial basis functions, simplex-based methods, and natural neighbor interpolation. Distance-weighted methods for interpolation use a weighted average of the data values at the data points with weights depending on the distance of the observed point from the given data points. Another common interpolation method is to determine a triangulation of the data points and to perform linear or cubic interpolation within each of the triangles. The advantage of simplex-based methods is their strong locality, which gives a good computational performance. Like triangle-based interpolation, natural neighbor interpolation uses a local coordinate system. However, instead of barycentric coordinates, natural neighbor coordinates are used.