Wavelet like basis on manifolds and their applications to harmonic analysis

新井研究了用小波分析产生错觉的机制。人们普遍认为，视错觉将提供一个关键，了解我们的视觉系统如何进行视觉信息处理。基于这个原因，在过去的100年里，人们对视错觉进行了许多研究。然而，对于一些错觉，其机制尚未得到很好的理解。在这项研究计划中，新井研究了视觉错觉，使用最大重叠多分辨率分析相对于一个双正交小波和新的非线性处理。此外，新井还构建了一个模拟人脑中纹状体皮层功能的计算系统。通过使用这个系统，我做了几个计算机模拟，表明我们的视觉系统如何产生视觉错觉。通过这些模拟，新井已经能够用数学语言解释几种视觉错觉的机制。此外，Arai还得到了有关多维数字系统BIBO稳定性的一些定理。山田教授用小波方法研究地震波，冈田教授用小波从数值分析的观点研究非线性偏微分方程，中村教授得到了与离散薛定谔算子有关的一些定理。

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.