| As wavelets are well localized in the time-frequency space and provide good multiresolution analyses of signals, the wavelet transform has many applications in signal processing: Wavelet-based filter banks have been applied to signals and images for data compression, noise reduction, and edge or transient detection, etc. So far, most applications are based on scalar wavelets. The most important orthogonal scalar wavelets with compact support are constructed by Ingrid Daubechies. However, the orthogonal wavelets with compact support are not symmetric. Scalar biorthogonal wavelets can be made symmetric, but still have some limitations: that is, symmetry, short support, high approximation order, and high regularity cannot be combined altogether. Recently, multiwavelet systems have been introduced, where multiple scaling functions (a scaling function vector) and multiple wavelets (a wavelet vector) are involved. A multiwavelet system can combine symmetry and shorter supports with a high approximation order and high regularity, which was not possible with a scalar wavelet system. One of the most important orthogonal multiwavelet systems was developed by Geronimo, Hardin, and Massopust, and more recently biorthogonal multiwavelet systems have been introduced by others.; In this research, we present an application of multiwavelet, analysis to image data, where filter coefficients form matrices. The biorthogonality condition and perfect reconstruction condition of multiwavelet filters are studied. Multiwavelet decomposition and reconstruction algorithms for images are derived, taking into consi6ration the data structure of inputs to the multiwavelet system. As a multiwavelet filter bank has multiple channels of inputs, we investigate the data initialization problem by considering prefilters and postfilters, that may give more efficient representations of the decomposed data. The interpolation postfilter and prefilter are formulated, which are capable to provide a better approximate image at each coarser resolution level. A design process is given to obtain both filters having compact supports, if exist. Image compression performances of some multiwavelet systems are studied in comparison to those of single wavelet systems. |
