0.10 Wavelet-based signal processing and applications  (Page 9/13)

Mallat proposes a scheme for computing an approximation of the continuous wavelet transform [link] that turns out to be equivalent to the method described above. This has been realized and proved byShensa [link] . Moreover, Shensa shows that Mallat's algorithm exhibits the same structure as the so-called algorithm à trous.Interestingly, Mallat's intention in [link] was not in particular to overcome the shift variance of the DWT but to get an approximation ofthe continuous wavelet transform.

In the following, we shall refer to the algorithm for computing the SIDWT as the Beylkin algorithm However, it should be noted that Mallat published his algorithm earlier. since this is the one we have implemented. Alternative algorithms for computing a shift-invariantwavelet transform [link] are based on the scheme presented in [link] . They explicitly or implicitly try to find an optimal, signal-dependent shift of the input signal. Thus, the transform becomesshift-invariant and orthogonal but signal dependent and, therefore, nonlinear. We mention that the generalization of the Beylkin algorithm tothe multidimensional case, to an $M$ -band multiresolution analysis, and to wavelet packets is straightforward.

Combining the shensa-beylkin-mallat-à trous algorithms and wavelet denoising

It was Coifman who suggested that the application of Donoho's method to several shifts of the observation combined with averagingyields a considerable improvement. A similar remark can be found in [link] , p. 53. This statement first lead us to the following algorithm: 1) apply Donoho's method not onlyto “some” but to all circular shifts of the input signal 2) average the adjusted output signals. As has beenshown in the previous section, the computation of all possible shifts can be effectively done using Beylkin's algorithm. Thus,instead of using the algorithm just described, one simply applies thresholding to the SIDWT of the observation and computes theinverse transform.

Before going into details, we want to briefly discuss the differences between using the traditional orthogonal and the shift-invariant wavelettransform. Obviously, by using more than $N$ wavelet coefficients, we introduce redundancy. Several authors stated that redundant wavelettransforms, or frames, add to the numerical robustness [link] in case of adding white noise in the transform domain; e.g., by quantization. Thisis, however, different from the scenario we are interested in, since 1) we have correlated noise due to the redundancy, and 2) we try to remove noisein the transform domain rather than considering the effect of adding some noise [link] , [link] .

Performance analysis

The analysis of the ideal risk for the SIDWT is similar to that by Guo [link] . Define the sets $A$ and $B$ according to

and an ideal diagonal projection estimator, or oracle,

The pointwise estimation error is then

In the following, a vector or matrix indexed by $A$ (or $B$ ) indicates that only those rows are kept that have indices out of $A$ (or $B$ ). All others are set to zero. With these definitions and [link] , the ideal risk for the SIDWT can be derived