Theory And Construction Of Matrix-Valued Wavelets

Wavelet has been studied extensively in both theories and applicatons during the last decade. The main advantage of wavelets is their time-frequency localization property. Many signals in areas like music, image, etc, can be efficiently represented by wavelets that are translations and dilations of a single function called mother wavelet with bandpass property. But by the development of study, it is found that there is a limitation for the time-frequency localization of a single mother wavelet, that is, if it is very localized in the time domain then it will not be very localized in the frequency domain. It is also known that an orthogonal wavelet function with compact support and certain regularity can not have any symmetry. In order to overcome these shortcomings of wavelet, multiwavelets which have no less than two functions was proposed, and researchers have constructed orthogonal multiwavelets that is continuous, short support, and of certain symmetry. When several mother wavelets are used in an expansion, better properties, such as energy compaction, can be achieved over single wavelets. But using multiwavelets to process scalar-valued signals is still a challenging problem. Prefilter or balance multiwavelets are requested. So, matrix-valued wavelets or vector-valued multiwavelets to process matrix-valued signals are first discussed by Xia XG. Multiwavelets and matrix-valued wavelets are different in the following sense. Matrix-valued wavelets can be used to decorrelate a vector-valued signal not only in the time domain but also between the components of matrix for a fixed time. The construction of multiwavelets focuses only on the decorrelation of signals in time domain. Another difference is between their discrete implementations. Prefiltering is usually required for discrete multiwavelet transforms but not necessary for discrete matrix-valued transforms. This paper mainly discussed theories and constructions of matrix-valued wavelets. Firstly, we studied M -band orthogonal vector-valued multiwavelets using bi-infinite matrix. Proved the existence of M -band orthogonal vector-valued multiwavelets, give orthogonality in form of symbols, then we discussed the relationship between vector-valued multiwavelets and multiwavelets and the construction of symbols of vector-valued multiwavelets, furthermore, fast vector-valued multiwavelets transform are investigated. Secondly, we discussed biorthogonal matrix-valued wavelets. Sufficient and necessary conditions for the existence of biorthogonal matrix-valued wavelets are proved. An algorithm to construct biorthogonal matrix-valued wavelets with which highpass fiters can be expressed explicity by lowpass filters are proposed, then two examples are showed.