Minimum-energy Wavelet Frames

The theory of frames comes from signal processing firstly. It was introduced for Hilbertspace by R.Duffin and A.Schaffer to deal with problems in nonharmonic Fourier series.After I.Daubechies defined affine frames (wavelet frames) by combining the theory of con-tinuous wavelets transforms and frames, people start to research the theory and applicationof frames. Wavelets frames not only have time-frequency localization and shift-invarianceproperty, but also can make up for wavelets’ shortcomings by its redundancy which makesit is more numerically stable to reconstruction the signal using the frames. Now, waveletsframes has been widely applied in many kinds of theoretical research and application do-mains, including signal analysis, image processing, numerical calculation, Banach spacetheory, and Besove space theory. In2000, C.K.Chui and W.He proposed the concept ofminimum-energy wavelet frame, then the further studies are carried out by many other re-searchers. The work for the minimum-energy wavelet frames of this paper are as follows:(1) Based on the thought of minimum-energy wavelet frames in L2(R), we define theconcept of minimum-energy interval wavelet frames with arbitrary integer dilation factor awhich is greater than and equal to2, and present its necessary and sufficient conditions. Thealgorithm of minimum-energy wavelet frames on the interval with arbitrary integer support γis designed, and the structure of matrices in the algorithm is presented constructively. Finally,the decomposition and reconstruction formulas of minimum-energy interval wavelet framesare deduced, and several numerical examples are given.(2) Combining the goodness of multi-wavelet and the significant properties of minimum-energy frames, the theory system of minimum-energy multi-wavelet frames and minimum-energy interval multi-wavelet frames were established in the paper. Their equivalent char-acterizations have been presented. We design the algorithm for minimum-energy (inter-val) multi-wavelet frames. And necessary condition and a series of sufficient condition forminimum-energy multi-wavelet frames are get. Finally, the decomposition and reconstruc-tion formulas of minimum-energy (interval) multi-wavelet frames are deduced, and severalnumerical examples are given.(3) The thought of minimum-energy frames been brought in the research of high di-mensional signal. We researched the construction of minimum-energy multivariate waveletframes with arbitrary dilation matrix systematically. The equivalent characterization andsufficient condition, necessary condition for that have been presented. We also give somenumerical examples in the finally.