Microlocal filtering with multiwavelets

R^n中的超函数直观地被认为是定义在C^n中的无穷小楔形中的全纯函数的边值和。微局部分析将超函数的解析性方向对应于其傅立叶变换的指数衰减方向。正交多小波是正交单小波的推广，它通过多个尺度函数产生多分辨率分析。滤波是一种数值方法，在数字图像处理中起着核心作用。提出了一种适用于微局部滤波的多小波系统。本文的主要结果如下：从函数或信号的多小波展开中得到其微局部含量的粗略估计，提出了一种多小波微局部滤波的快速算法，并给出了数字图像处理中的几个数值例子。

Hyperfunctions in R^n are intuitively considered as sums of boundary values of holomorphic functions defined in infinitesimal wedges in C^n. Microlocal analysis corresponds the direction of analyticity of hyperfunctions to the direction of exponential decay of their Fourier transforms. Orthonormal multiwavelets, which are a generalization of orthonormal single wavelets, generate a multiresolution analysis by means of several scaling functions. Filtering is one of numerical methods and play central roles in digital image processing. We propose a multiwavelet system adapted to microlocal filtering. The main results are the following.A rough estimate of the microlocal content of functions or signals is obtained from their multiwavelet expansions.A fast algorithm for multiwavelet microlocal filtering is presented.Several numerical examples in digital image processing are considered.