Compression of spectrum with wavelet transform

The advances in microelectronics have greatly enhanced mass storage capacity and processing speed. Archives of information of full spectra rather than only those of absorption peaks become more feasible. However, the demands of huge storage capacity are still somewhat prohibitive for high-resolution spectra. Even if this problem can be resolved, the computer processing speed and the bandwidth of the telephone line or network is still 

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]  

As stated in the previous section, most workers confine their wavelet functions in the Daubechies wavelet series only. For example, we have adopted the Daubechies wavelet function to denoise spectral data from a UV-VIS spectrophotometer [43]. In order to make use of the other available wavelet functions for chemical data analysis, Lu and Mo [44] suggested employing spline wavelets in their work for denoising UV-VIS spectra. The spline wavelet is another commonly used wavelet function in chemical studies. This function has been applied successfully in processing electrochemical signals [9,10] which will be discussed in detail in another chapter of this book. The mth order basis spline (B-spline) wavelet, Nm. is defined as follows [44]  

The symbol denotes a convolution operation between Nm-i and Nm- The k-th term of is given by k 

In spline wavelet computation, two parameters, namely the order of B-spline, m, and truncation frequency, L, which represents the cut-off (or truncation) frequency value between the true signal and noise, need to be optimized. In Lu and Mo s study [44], they found that the best result for denoising UV-VIS spectra with high noise level was obtained with m = 3 and L = 4. Zhao and Wang [45] proposed a technique called the wavelet transform K-factor three-wavelength method to determine simultaneously the concentrations of vanadium, molybdenum and titanium with UV-VIS spectroscopy. In their study, WT was adopted to denoise the spectra acquired. The concentrations of individual ions were determined from the UV-VIS spectra at three selected wavelengths. 

---