Optimal ℒ1 approximation of the Gaussian kernel with application to scale-space construction

Scale-space construction based on Gaussian filtering requires convolving signals with a large bank of Gaussian filters with different widths. We propose an efficient way for this purpose by L₁ optimal approximation of the Gaussian kernel in terms of linear combinations of a small number of basis functions. Exploring total positivity of the Gaussian kernel, the method has the following properties: 1) the optimal basis functions are still Gaussian and can be obtained analytically; 2) scale-spaces for a continuum of scales can be computed easily; 3) a significant reduction in computation and storage costs is possible. Moreover, this work sheds light on some issues related to use of Gaussian models for multiscale image processing