Sequential linear interpolation of multidimensional functions

We introduce a new approach that we call sequential linear interpolation (SLI) for approximating multidimensional nonlinear functions. The SLI is a partially separable grid structure that allows us to allocate more grid points to the regions where the function to be interpolated is more nonlinear. This approach reduces the mean squared error (MSE) between the original and approximated function while retaining much of the computational advantage of the conventional uniform grid interpolation. To obtain the optimal grid point placement for the SLI structure, we appeal to an asymptotic analysis similar to the asymptotic vector quantization (VQ) theory. In the asymptotic analysis, we assume that the number of interpolation grid points is large and the function to be interpolated is smooth. Closed form expressions for the MSE of the interpolation are obtained from the asymptotic analysis. These expressions are used to guide us in designing the optimal SLI structure. For cases where the assumptions underlying the asymptotic theory are not satisfied, we develop a postprocessing technique to improve the MSE performance of the SLI structure. The SLI technique is applied to the problem of color printer characterization where a highly nonlinear multidimensional function must be efficiently approximated. Our experimental results show that the appropriately designed SLI structure can greatly improve the MSE performance over the conventional uniform grid.