Improved signal representations using rational dilation factors
İlker Bayram · SPIE Newsroom · 2008
Wavelet transforms (WTs) are used in numerous signaland image-processing applications including compression, denoising, deblurring, sharpening, dequantization, interpolation, demosaicking, and waveform classification. WTs are based on signal analysis at several levels of resolution. In most applications, particularly those requiring that the transform is invertible, the resolution is doubled from one level to the next: the dilation factor is two and the WT is ‘dyadic.’ We have recently developed1, 2 a new approach based on rational dilation factors of between one and two, where the resolution increases more gradually from one level to the next. Orthonormal WTs using rational schemes have been explored and developed by a number of groups. However, several issues complicate their effective use compared with their dyadic counterparts. First, the underlying theory3–6 is significantly more complicated. Secondly, the design of good digital filters7, 8 with which to implement a discrete WT is significantly more difficult. In particular, Daubechies’ celebrated construction of short filters with vanishing-moment properties9 cannot be extended to the rational case. In fact, only a few finite-length filters with more than one vanishing moment have been constructed for the latter, which must be done by exact arithmetic computation over several days using Grobner-type methods.1 Moreover, the frequency selectivity and differentiability of the resulting analysis functions are quite poor. Thirdly, the rational WTs associated with using these filters are outperformed by the orthonormal dyadic WTs. However, for overcomplete rational WTs or ‘frames’ the situation is quite different. Frames are transforms that expand an N-point signal to L transform coefficients, with L > N. Frames have become a well-recognized tool10 in signal processing for enhancing the performance of many transform-domain algorithms, such as wavelet-based denoising. Short filters with Figure 1. Analysis functions for the first few levels of (a) a dyadic orthonormal wavelet transform (WT) and (b) a rational wavelet frame (with a dilation factor of 4/3), where the duration and frequency change more gradually from one resolution level to the next.