Biorthogonality property

The biorthogonality property (12.4) then allows one to easily evaluate scalar products of each excess intensity RK) with basis axis vectors R ) or R ). The scalar product of extensive Xj) with (12.76) gives... 

Chau and his co-workers have proposed some wavelet-based methods to compress UV-VIS spectra [24,37]. In their work, a UV-VIS spectrum was processed with the Daubechies wavelet function, Djfi. Then, all the Cj elements and selected Dj coefficients at different) resolution levels were stored as the compressed spectral data. A hard-thresholding method was adopted for the selection of coefficients from Dj. A compression ratio up to 83% was achieved. As mentioned in the previous section, the choice of mother wavelets is vast in WT, so one can select the best wavelet function for different applications. Flowever, most workers restrict their choices to the orthogonal wavelet bases such as Daubechies wavelet. Chau et al. chose the biorthogonal wavelet for UV VIS spectral data compression in another study [37]. Unlike the orthogonal case, which needs only one mother wavelet (p(t), the biorthogonal one requires two mother wavelets. (p(t) and (p(t), which satisfy the following biorthogonal property [38] ... 

Stanton JF, Bartlett RJ (1993) The equation of motion coupled-cluster method - a systematic biorthogonal approach to molecular-excitation energies, transition-probabilities, and excited-state properties. J Chem Phys 98 7029... 

Coupled-Cluster Method. A Systematic Biorthogonal Approach to Molecular Excitation Energies, Transition Probabilities, and Excited State Properties. 

Much of the literature on wavelets tends to be biased towards discussions on orthogonal wavelets because they are convenient and simple to implement. However, we feel that is necessary to make the reader aware that wavelets need not be orthogonal and that wavelets with other properties can be quite useful too. In this section we discuss biorthogonal wavelets as one alternative to orthogonal wavelets. We direct the reader to [1,7,10,14,15] for more information on other kinds of wavelets. 

Briefly, when using orthogonal compactly supported wavelets it is not straightforward to obtain a wavelet which has symmetrical properties [7,12] and allows for an exact reconstruction. That is of course with the exception of the trivial Haar wavelet. Biorthogonal wavelets relax the assumptions of orthogonality, and allow for a perfect reconstruction with symmetrical wavelets. 

There is a whole family of different wavelet methods available depending on the signal properties and the type of information that is to be extracted. However, this chapter will only focus on the fast wavelet transform (FWT) which is based on Mallat s algorithm [39.40]. It should be mentioned that the described methods to achieve parsimonious models are not dependent on one particular type of wavelet transform. Other types of wavelet transforms can be used. FWT is not always optimal for all types of problems and other techniques such as wavelet packets [41], continuous transforms [42,43] and biorthogonal transforms [37] should be considered. Some of the properties of the FWT that makes it an attractive transform are ... 

Stanton, J. E, Bartlett, R. J. (1993). The Equation of Motion Coupled-Cluster Method. A Systematic Biorthogonal Approach to Molecular Excitation Enei es, Transition Probabilities, and Excited State Properties,/. Chem. Phys., 98, 7029-7039. 

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