There is a decision about which wavelet is best for each application and each input image/signal, since the type of wavelet chosen affects the performance of the algorithm. In the past, researchers have chosen the wavelet shape based on (a) ease of use, (b) input signal properties, (c) a 'library' search of possible shapes, and/or (d) their own experience and intuition. We demonstrate a technique that determines the best wavelet for each image from within the class of all orthogonal wavelets (tight frames) with a fixed number of coefficients. In our technique, we compress the input with a particular wavelet, calculate the PSNR, then adapt or adjust the wavelet coefficients dynamically to achieve the best PSNR. This 'feedback-based' approach is based on traditional adaptive filtering algorithms. The problem of building an adaptable or feedback-based wavelet filter was simplified when Lai and Roach developed an explicit parameterization of the wavelet scaling functions of short support (more specifically, a parameterization of all tight frames). The representation has one parameter for length-4 wavelets, two free parameters for length-6 wavelets, and multiple parameters for longer wavelets. As the parameter(s) are perturbed, the scaling function’s shape is also perturbed. However, it changes in such a way that the wavelet constraints are still fulfilled.
We have applied the feedback-based approach using the parameterized wavelets in an image compression scheme. For short wavelet filters (length-4 up to length-10), we have confirmed that there is a wide range of performance as the wavelet shape is varied and the feedback procedure indeed converges towards optimal orthogonal wavelet filters for a given support size and a chosen image.
In all current Fourier transform processing systems, which we call conventional Fourier transform (CFT) processors, no matter what kind of filter is used, its filter function can be expressed as a diagonal matrix, if in the view of digital image processing. We have presented a generalized Fourier transform (GFT) processor by extending the diagonal filter matrix into a nondiagonal matrix. It includes CFT as a special case, and still retains the space/time- invariance property. In this paper, we present a method based on genetic algorithms for finding an optimal filter of GFT processor. The behavior of the optimal filter in GFT processor and its advantages over that in the CFT processor are illustrated by the satisfied test results. An optimal generalized Teoplitz matrix for the GFT processor filter based on the figure of merit--the Manhatten error norm is also proposed.
Fourier processing method is very popular in signal and image analysis. In this paper, a generalized Fourier transform (GFT) processor is introduced. It is generalized in that it contains the conventional Fourier transform (CFT) processor as a very limited special case by extending the diagonal filter matrix in CFT to a nondiagonal one. Thus, obviously, the GFT processor should be more powerful than the CFT processor. A computer demonstration shows this.