Microlocal filtering with multiwavelet frames

证明了能在Hilbert空间L^2(R^n)(1&lt；p&lt；∞)及其他相关空间中对函数进行微局部分解的标准多小波基在L^p(R^n)(1&lt；p&lt；Lt)及其他相关空间中是“逐级”无条件基.作为证明的一部分，给出了L^p(R^n)型采样定理无条件收敛的初等证明。最后，给出了一些分布作为全纯函数边值之和的表示的应用，在富里叶域中构造了其傅里叶变换由平方或环面扇形的特征函数组成的正交多小波，并证明了它满足具有多种尺度函数选择的多分辨分析。得到了冗余的光滑紧小波框架，这些非正交框架小波可由多分辨分析的双尺度方程产生。奇异点可以在位置和方向上定位，并可以从伤痕图像中恢复原始图像。

Our orthonormal multiwavelet bases, which can decompose functions in the Hilbert space L^2(R^n) microlocally, are shown to be a "stepwise" unconditional basis in L^p(R^n) (1<p<∞) and other related spaces. As part of the proof, an elementary proof of the L^p(R^n) version of the sampling theorem with unconditional convergence is given. Finally, an application is given to the expression of some distributions as sums of boundary values of holomorphic functions.Orthogonal multiwavelets, whose Fourier transforms consist of characteristic functions of squares or sectors of annuli, are constructed in the Fourier domain and are shown to satisfy a multiresolution analysis with several choices of scaling functions. Redundant smooth tight wavelet frames are obtained and these nonorthogonal frame wavelets can be generated by two-scale equations from, a multiresolution analysis. Singularities can be localized in position and direction and the original images can be restored from the scarred images.