| General results on Fourier Transforms, tight frame wavelets and pseudodifferential operators are presented to provide a theoretical framework for the applications. Known and new tight frame wavelets that are characteristic and tapered characteristic functions in Fourier Space are constructed in Cartesian and polar Fourier Space with frame bound 1. These wavelets are used to localize singularities in images.; A review of the use of the diffusion equation in de-noising images is presented. A new method, which applies a multi-directional diffusion in Fourier Space is given. Properties of this new product filter method are described and the product filter's de-noising ability is evaluated. The product filter algorithm is also compared to other techniques, including MATLAB's built-in filters and two recent wavelet techniques.; Finally, a summary of the results of a study of the de-noising abilities of third-order partial differential equations is presented. |
