Biorthogonal wavelets

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. 

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] ... 

Fault detection on textiles by adapted biorthogonal wavelets... 

ABSTRACT In the discrete wavelet transform approach, a choice of wavelet has a direct impact on the decomposed image, which indicates that the selection of the wavelet is closely related to the detection performance. Since a choice of standard wavelets, e.g. Daubechies wavelets, Coiflets, biorthogonal wavelets etc., may not guarantee efficient discrimination of fabric defects, some researchers suggested methods based on a construction of wavelets adapted to the detection or classification of these defects. We propose a novel method to design adaptive wavelet filters. These filters are constructed to minimize /2-norm of the undecimated discrete wavelet transform of the defect free textile with the aim to enhance the wavelet response in the defect region. Examples show efficiency in the fault detection. 

The discrete biorthogonal wavelet transform is also computationally very efficient, requiring only 0( ) operations, where n is the number of processed data. The most used filters are certainly the 9/7 filters which are by default in the JPEG 2000 norm. 

Other part of the discrete biorthogonal wavelet transform are accomplished in a similar way. 

Sometimes real, symmetric wavelets or wavelets with more vanishing moments and at the same time with smaller support than corresponding orthogonal wavelets or simply more regular", in closed form defined wavelets are required. One way to obtain them is to construct two sets of biorthogonal wavelets r/r and its dual x). One of these two biorthogonal wavelets is consequently used to decompose the signal and the second one to reconstruct it. The numbers M,N of... 

In this paper we use biorthogonal wavelets on the interval constructed by (Cema Finek 2008 Cerna Finek 2009) which outperforms similar construction in the sense of better conditioning of base functions as well as in better conditioning of wavelet transform. The condition number seems to be nearly optimal most especially in the case of cubic spline wavelets. From the viewpoint of numerical stabihty, ideal wavelet bases are orthogonal wavelet bases. However, they are... 

In Fig. 1 is displayed the measured signal and its discrete wavelet transform with the biorthogonal wavelet of order 3 with 5 vanishing moments (CDF35) designed by (Cohen et al. 1992). It seems that oscillations and noise are quite well removed. However as ean be seen on Fig. 2, the disadvantage of this... 

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