The geometry and the analytic properties of isotropic multiresolution analysis

In this paper we investigate Isotropic Multiresolution Analysis (IMRA), isotropic refinable functions, and wavelets. The main results are the characterization of IMRAs in terms of the Lax–Wiener Theorem, and the characterization of isotropic refinable functions in terms of the support of their Fourier transform. As an immediate consequence of these results, there are no compactly supported (in the space domain) isotropic refinable functions in many dimensions. Next we study the approximation properties of IMRAs. Finally, we discuss the application of IMRA wavelets to 2D and 3D-texture segmentation in natural and biomedical images.