Smooth tight frame wavelets and image microanalyis in the Fourier domain

General results on microlocal analysis and tight frames in R-2 are summarized. To perform microlocal analysis of tempered distributions, orthogonal multiwavelets, whose Fourier transforms consist of characteristic functions of squares or sectors of annuli, are constructed in the Fourier domain and are shown to satisfy a multiresolution analysis with several choices of scaling functions. To have good localization in both the x and Fourier domains, redundant smooth tight wavelet frames, with frame bounds equal to one, called Parseval wavelet frames, are obtained in the Fourier domain by properly tapering the above characteristic functions. These nonorthogonal frame wavelets can be generated by two-scale equations from a multiresolution analysis. A natural formulation of the problem is by means of pseudodifferential operators. Singularities, which are added to smooth images, can be localized in position and direction by means of the frame coefficients of the filtered images computed in the Fourier domain. Using Plancherel's theorem, the frame expansion of the filtered images is obtained in the x domain. Subtracting this expansion from the scarred images restores the original images. (C) 2003 Elsevier Science Ltd. All rights reserved.