连续切波变换的高维奇异性分析

在高维数据处理过程中,确定高维平方可积函数的奇异性有着重要的意义,它可作为模式识别、数据挖掘、频谱分析、大型机械故障诊断、航空航天、遥感与控制以及三维图像处理的基础.本文首先给出高维平方可积函数的连续切波变换重构公式;其次研究几种特殊函数的切波系数的衰减性质;最后运用重构公式中的切波系数刻画平方可积函数的奇异支撑集.本文的结果推广了Kutyniok和Dahlke等人给出的一些已知结果.

High-dimensional singularity analysis using the continuous shearlet transform

In the processing process of high-dimensional data, it is of great significance to identify the singularity of high-dimensional square integrable functions, which can be used as foundation of pattern recognition, data mining, spectrum analysis, fault diagnosis of large machinery, aerospace, remote sensing and control as well as the 3D image processing. Firstly, in the paper, the reconstruction formula of high-dimensional square integrable functions is given by using the continuous shearlet transform; secondly, the decay property of shearlet coefficients of several special functions is studied; finally, the singular support of square integrable functions is characterized by using the shearlet coefficients of reconstruction formula. Our results improve some known ones given by Kutyniok and Dahlke, et al.