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Recent results indicate this nondecimated DWT, together with thresholding, may be the best denoising strategy [link] , [link] , [link] , [link] , [link] , [link] , [link] , [link] . The nondecimated DWT is shift invariant, is less affected by noise,quantization, and error, and has order $Nlog\left(N\right)$ storage and arithmetic complexity. It combines with thresholding to give denoising andcompression superior to the classical Donoho method for many examples. Further discussion of use of the RDWT can be found in Section: Nonlinear Filtering or Denoising with the DWT .
In the case of the redundant discrete wavelet transform just described, an overcomplete expansion system was constructed in such a way as to be atight frame. This allowed a single linear shift-invariant system to describe a very wide set of signals, however, the description was adaptedto the characteristics of the signal. Recent research has been quite successful in constructing expansion systems adaptively soas to give high sparsity and superresolution but at a cost of added computation and being nonlinear. This section will look at some of therecent results in this area [link] , [link] , [link] , [link] .
While use of an adaptive paradigm results in a shift-invariant orthogonal transform, it is nonlinear. It has the property of $DWT\left\{a\phantom{\rule{0.166667em}{0ex}}f\right(x\left)\right\}\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}a\phantom{\rule{0.166667em}{0ex}}DWT\left\{f\right(x\left)\right\}$ , but it does not satisfy superposition, i.e. $DWT\left\{\phantom{\rule{0.166667em}{0ex}}f\right(x)+g(x\left)\right\}\phantom{\rule{0.166667em}{0ex}}\ne \phantom{\rule{0.166667em}{0ex}}DWT\left\{f\right(x\left)\right\}+DWT\left\{g\right(x\left)\right\}$ . That can sometimes be a problem.
Since these finite dimensional overcomplete systems are a frame, a subset of the expansion vectors can be chosen to be a basis while keepingmost of the desirable properties of the frame. This is described well by Chen and Donoho in [link] , [link] . Several of these methods are outlined as follows:
a = pinv(X)*y
. This gives a frame solution, but it is usually not
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