Summary: 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 \(F:D\to R^d\) that is continuously differentiable and maps \(D\) one-to-one onto \(F(D)\). Corresponding to a function \(J:F(D)\to R\), 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 of \(J\) 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.