Microlocal filtering with multiwavelet frames

多小波帧的微局部滤波

• 批准号: 13640171
• 负责人: ASHINO Ryuichi
• 金额: $ 2.18万
• 依托单位: Osaka Kyoiku University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2001
• 资助国家: 日本
• 起止时间: 2001 至 2002
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-13640171/
• 关键词: microlocal analysis; wavelet frame; multiwavelet; filter; time frequency analysis; wavelet analysis; image processing; フレーム; ウェーブレット

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.

我们的标准正交多小波基可以在Hilbert空间L^2（R^n）微局部分解函数，在L^p(R^n) （1<p<∞）和其他相关空间中证明是“逐步”无条件基。作为证明的一部分，给出了具有无条件收敛性的抽样定理的L^p（R^n）版本的初等证明。最后，给出了用全纯函数边值和表示某些分布的一个应用。正交多小波的傅里叶变换是由环空的正方形或扇形的特征函数构成的，在傅里叶域中构造了正交多小波，并证明了它可以满足多种尺度函数选择的多分辨率分析。得到了冗余光滑紧小波帧，这些非正交小波帧可由多分辨率分析的双尺度方程生成。奇异点可以定位在位置和方向上，并可以从疤痕图像中恢复原始图像。

项目成果

R.ASHINO (共著): "Wavelet bases for microlocal filtering and the sampling theorem in L_p(R^n)"Applicable Anal.. 82. 1-24 (2003)

R.ASHINO（合著者）：“微局域滤波的小波基和 L_p(R^n) 中的采样定理”Applicable Anal.. 82. 1-24 (2003)

R.ASHINO (共著): "Microlocal analysis and multiwavelets"Geometry, Analysis and Applications (Varanasi, 2000). 293-302 (2001)

R.ASHINO（合著者）：“微局域分析和多小波”几何、分析和应用（瓦拉纳西，2000 年）。

R. Ashino, S. J. Desjardins, C. Heil, M. Nagase, R. Vaillancourt: "Microlocal analysis, smooth frames and denoising in Fourier space"Asian Information-Science-Life. 1. 153-160 (2002)

R. Ashino、S. J. Desjardins、C. Heil、M. Nagase、R. Vaillancourt：“傅里叶空间中的微局域分析、平滑框架和去噪”亚洲信息科学生活。

R.ASHINO (共著): "Multiwavelets, pseudodifferential operators and microlocal analysis"AMS/IP Stud.Adv.Math.. 25. 9-20 (2002)

R.ASHINO（合著者）：“多小波、伪微分算子和微局域分析”AMS/IP Stud.Adv.Math.. 25. 9-20 (2002)

S. Sakakibara, T. Mandai, R. Ashino: "Wavelets and other orthogonal systems, (translation into Japanese)"Tokyo Denkidai University Press. 308 (2001)

S. Sakakibara、T. Mandai、R. Ashino：“小波和其他正交系统，（译成日文）”东京电机大学出版社。