Compression of individual signals

Signals such as, for instance IR spectra, have in a wavelet domain sparse representation. It means that in wavelet domain there are many wavelet 

The main idea of MDL approach can be summarized as follows the MDL cost function represents two conflicting requirements, we would like to compress signal to the highest possible degree, but simultaneously we would like to have as small reconstruction error as possible. These requirements are represented by two terms the first term describes the reconstruction error depending on the number of retained coefficients, and the second one is a penalty function, increasing with the number of the retained coefficients  

The basic assumption of this approach, discussed in detail by Saito [13], is that for real signals, the MDL cost function reaches a minimum, which indicates the optimal number of retained wavelet coefficients (Fig. 1). 

The optimal number of retained wavelet coefficients, and the reconstruction error strongly depend on the applied filter. It means that the MDL cost function can be used for the filter optimization. Filter for which the MDL achieves minimal value or the minimal number of retained wavelet coefficients can be considered the optimal one for data compression. 

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