0.7 Generalizations of the basic multiresolution wavelet system  (Page 19/28)

A method to create a linear, shift-invariant DWT is to construct a frame from the orthogonal DWT supplemented by shifted orthogonal DWTs using theideas from the previous section. If you do this, the result is a frame and, because of the redundancy, is called the redundant DWT or RDWT.

The typical wavelet based signal processing framework consists of the following three simple steps,1) wavelet transform; 2) point-by-point processing of the wavelet coefficients (e.g. thresholding for denoising,quantization for compression); 3) inverse wavelet transform. The diagram of the framework is shown in [link] . As mentioned before, the wavelet transform is not translation-invariant,so if we shift the signal, perform the above processing, and shift the output back, then the results are different for different shifts.Since the frame vectors of the RDWT consist of the shifted orthogonal DWT basis, if we replace the forward/inverse wavelet transform

in the above framework by the forward/inverseRDWT, then the result of the scheme in [link] is the same as the average of all the processing results using DWTs withdifferent shifts of the input data. This is one of the main reasons that RDWT-based signal processing tends to be more robust.

Still another view of this new transform can be had by looking at the Mallat-derived filter bank described in  Chapter: The Scaling Function and Scaling Coefficients, Wavelet and Wavelet Coefficients and Chapter: Filter Banks and Transmultiplexers . The DWT filter banks illustrated in  Figure: Two-Stage Two-Band Analysis Tree and Figure: Two-Band Synthesis Bank can be modified by removing the decimators between each stage to give the coefficients of the tight frame expansion(the RDWT) of the signal. We call this structure the undecimated filterbank. Notice that, without the decimation, the number of terms in the DWTis larger than $N$ . However, since these are the expansion coefficients in our new overcomplete frame, that is consistent. Also, notice that thisidea can be applied to M-band wavelets and wavelet packets in the same way.

These RDWTs are not precisely a tight frame because each scale has a different redundancy. However, except for this factor, the RDWT andundecimated filter have the same characteristics of a tight frame and, they support a form of Parseval's theorem or energy partitioning.

If we use this modified tight frame as a dictionary to choose a particular subset of expansion vectors as a new frame or basis, we can tailor thesystem to the signal or signal class. This is discussed in the next section on adaptive systems.

This idea of RDWT was suggested by Mallat [link] , Beylkin [link] , Shensa [link] , Dutilleux [link] , Nason [link] , Guo [link] , [link] , Coifman, and others. This redundancy comes at a price of the new RDWT having $O(Nlog(N\left)\right)$ arithmetic complexity rather than $O\left(N\right)$ . Liang and Parks [link] , [link] , Bao and Erdol [link] , [link] , Marco and Weiss [link] , [link] , [link] , Daubechies [link] , and others [link] have used some form of averaging or “best basis" transform to obtain shift invariance.