A Fast Algorithm to Map Functions Forward

Abstract

Mapping functions forward is required in image warping and other signal processing applications. The problem is described as follows: specify an integer d \geq 1, a compact domain D \subset R^{d}, lattices L_{1}, L_{2} \subset R^{d}, and a deformation function F : D \rightarrow R^{d} that is continuously differentiable and maps D one-to-one onto F(D). Corresponding to a function J : F(D) \rightarrow R, define the function I = J \circ F. The forward mapping problem consists of estimating values of J on L_{2} \cap F(D), from the values of I and F on L_{1} \cap D. Forward mapping is difficult, because it involves approximation from scattered data (values of I\circ F^{-1} on the set F(L_{1} \cap D)), whereas backward mapping (computing I from J) is much easier because it involves approximation from regular data (values ofJ on L_{2} \cap D). We develop a fast algorithm that approximates J by an orthonormal expansion, using scaling functions related to Daubechies wavelet bases. Two techniques for approximating the expansion coefficients are described and numerical results for a one dimensional problem are used to illustrate the second technique. In contrast to conventional scattered data interpolation algorithms, the complexity of our algorithm is linear in the number of samples.