Wavelet like basis on manifolds and their applications to harmonic analysis

流形上的类小波基础及其在调和分析中的应用

• 批准号: 14540154
• 负责人: ARAI Hitoshi
• 金额: $ 2.62万
• 依托单位: The University of Tokyo
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2002
• 资助国家: 日本
• 起止时间: 2002 至 2003
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-14540154/
• 关键词: wavelet; multiwavelet; biorthogonal wavelet; BIBO stability; Visual information; visual illusion; Multivariable z transform; system; 視覚情報処理; コイフレット; 多重解像度解析

Arai studied analysis by wavelets of mechanism of appearance of visual illusions. It is widely believed that visual illusions will offer a key to understand how our visual system carries out visual information processing. From this reason, over the past 100 years, many studies of visual illusion have been made. However as for several illusions, their mechanisms are not yet well understood. In this research program Arai studied visual illusions by using both maximal overlap multiresolution analysis with respect to a biorthogonal wavelet and new nonlinear processing. Further, Arai has constructed a computational system modeled after the function of the striate cortex in human's brain. By using this system I did several computer simulations which indicate how our visual system produces visual illusions. by these simulations Arai has been able to explain by mathematical language the mechanism of several visual illusions. Furthermore Arai obtained some theorems related BIBO stability of multidimensional digital systems. Professor Yamada studied seismic waves by wavelet method, Prof. Okada studied nonlinear PDE from view point of numerical analysis via wavelet, and Prof. Nakamura obtained some theorems related to discrete Schrodinger operators.

Arai 研究了视觉错觉出现机制的小波分析。人们普遍认为，视错觉将为理解我们的视觉系统如何进行视觉信息处理提供一把钥匙。出于这个原因，在过去的一百年里，人们对视错觉进行了许多研究。然而，对于几种错觉，其机制尚不清楚。在这个研究项目中，Arai 通过使用双正交小波的最大重叠多分辨率分析和新的非线性处理来研究视错觉。此外，荒井还构建了一个模仿人类大脑纹状皮层功能的计算系统。通过使用这个系统，我做了几次计算机模拟，表明我们的视觉系统如何产生视觉错觉。通过这些模拟，荒井已经能够用数学语言解释几种视觉错觉的机制。此外，Arai还得到了多维数字系统BIBO稳定性的一些定理。山田教授利用小波方法研究了地震波，冈田教授利用小波从数值分析的角度研究了非线性偏微分方程，中村教授获得了一些与离散薛定谔算子相关的定理。

项目成果

S.Maeda, M.Okada: "Hamilton formulation of energy conservative variational equations by wavelet expansions"J.Functional Analysis. 2004(to appear).

S.Maeda，M.Okada：“通过小波展开的能量守恒变分方程的汉密尔顿公式”J.Functional Analysis。

T.Ueno: "Quasi-norms for a double sequence"Interdisciplinary. 8. 157-166 (2002)

T.Ueno：“双序列的准范数”跨学科。

新井 仁之: "フーリエ解析学"朝倉書店(予定)(印刷中). 300

新井仁：《傅立叶分析》朝仓书店（暂定）（印刷中）。

T.Ueno, M.Okada: "A wavelet collocation method for evolution equations with energy conservation properties"Bull.Si.Math.. 127. 180-205 (2003)

T.Ueno, M.Okada：“具有能量守恒性质的演化方程的小波配置方法”Bull.Si.Math.. 127. 180-205 (2003)

新井 仁之: "フーリエ解析学"朝倉書店. 277 (2003)

新井仁：《傅立叶分析》朝仓书店 277 (2003)。