A Fast Algorithm to Map Functions Forward

• Authors: Wayne Lawton
• Venue: Multidimensional Systems and Signal Processing
• Year: 1997

Mapping functions forwardis 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-oneonto F(D). Corresponding to a function J :F(D) \rightarrow R, define the function I = J \circF. The forward mapping problem consists of estimatingvalues of J on L_{2} \cap F(D), fromthe values of I and F on L_{1}\cap D. Forward mapping is difficult, because it involvesapproximation 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 easierbecause it involves approximation from regular data (values ofJ on L_{2} \cap D). We develop a fastalgorithm that approximates J by an orthonormalexpansion, using scaling functions related to Daubechies waveletbases. Two techniques for approximating the expansion coefficientsare described and numerical results for a one dimensional problemare used to illustrate the second technique. In contrast to conventionalscattered data interpolation algorithms, the complexity of ouralgorithm is linear in the number of samples.