Space-Frequency Balance of Biorthogonal Wavelets

This paper shows how to design good biorthogonal FIR filters for wavelet image compression by balancing the space and frequency dispersions of analysis and synthesis lowpass filters. A quality metric is proposed which can be computed directly from the filter coefficients. By optimizing over the space of FIR filter coefficients, a filter bank can be found which minimizes the metric in about 60 seconds on a high performance workstation. The metric contains three parameters which weight the space and frequency dispersions of the low pass analysis and synthesis filters. A series of biorthogonal, symmetric wavelet filters of length 10 was found, each optimized for different weightings. Each of these filter banks was then evaluated by compressing and decompressing five test images at three compression ratios. Selecting each optimum provides fifteen sets of parameters corresponding to filter banks which maximize the PSNR in each case. The average of these parameters was used to define a 'mean' filter bank, which was then evaluated on the test images. Individual images can produce substantially different weightings of the time dispersion at the optimum, but the PSNR of the mean filter is normally close to the optimum. The mean filter also compares favourably with a maximum regularity biorthogonal filter of the same length. The theory of continuous and discrete wavelet transforms [1, 2] has inspired much basic and applied research in signal and image processing, as well as revitalizing the study of sub-band filtering [3, 4, 5]. The Discrete Wavelet Transform (DWT) is obtained by repeated filtering and sub-sampling into two bands with low-and high-pass Finite Impulse Response (FIR) filters called the analysis filters. The inverse process makes use of the synthesis FIR filters, and gives perfect reconstruction if the wavelet is biorthogonal. This is easily shown to be the case [4] if the lowpass analysis filter coefficients {c 0 ,…, c L−1 } and synthesis filter coefficients {u 0 ,…, u L−1 } satisfy Σ n c n u n + 2j = δ j, 0. Wavelets exist only if zero order regularity conditions are satisfied, Σ n c n = √  2, Σ n (−1) n u n = 0, and Σ n (−1) n c n = 0. The lower limit on the time-frequency resolution that can be obtained with wavelet transforms is given theoretically by the Heisenberg uncertainty relation [6] which, applied to the signal f(t) , is expressed by …