Signal analysis using a multiresolution form of the singular value decomposition

This paper proposes a multiresolution form of the singular value decomposition (SVD) and shows how it may be used for signal analysis and approximation. It is well-known that the SVD has optimal decorrelation and subrank approximation properties. The multiresolution form of SVD proposed here retains those properties, and moreover, has linear computational complexity. By using the multiresolution SVD, the following important characteristics of a signal may be measured, at each of several levels of resolution: isotropy, sphericity of principal components, self-similarity under scaling, and resolution of mean-squared error into meaningful components. Theoretical calculations are provided for simple statistical models to show what might be expected. Results are provided with real images to show the usefulness of the SVD decomposition.