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.