Multi-wavelet frames and applications to harmonic analysis

本课题的研究主题是多小波帧和谐波分析。首席研究员Arai研究了小波帧及其在视觉科学中的应用。特别是，Arai构建了新的小波帧，适合于研究视觉科学（与S. Arai）。这些小波帧将为构建人类视觉系统的计算模型提供基础。此外，新井还研究了图像处理的实际应用。此外，他还构建了色彩感知的计算模型，并对与色觉相关的错觉进行了计算机模拟。新井还研究了视觉错觉出现的机制。特别地，他提出了一种新的视觉离散小波分析的通用方案。人们普遍认为，视错觉将为我们理解视觉系统如何进行视觉信息处理提供一把钥匙。基于这个原因，在过去的100年里，人们对视觉错觉进行了许多研究。然而，对于一些错觉，它们的机制还没有被很好地理解。在这个研究项目中，Arai利用小波帧获得了早期视觉的非线性模型，并研究了某种错觉出现的数学机制。更具体地说，Arai根据人类大脑中纹状皮层的功能构建了过滤器库，并利用这些系统进行了几次计算机模拟，以表明我们的视觉系统是如何产生视觉错觉的。通过他的研究，几种视错觉的机制可以用数学统一的方式来解释。如果一个图像被输入到Arai的系统，如果它输出一个带有错觉的图像，那么我们可以得出结论，这是由V1处理引起的。Kanjin研究了系统正交函数的谐波分析。例如，他证明了Lp和Hp上的Cesaro算子的有界性。Nakamura研究了解析波前集的局部平滑效应。吉田研究了随机微分方程的抽样问题。Tachizawa从小波理论的角度研究了一些加权函数空间。少

Main themes of this research project are (multi-) wavelet frames and harmonic analysis. The head investigator, Arai, studied wavelet frames and applications to vision science. In particular Arai constructed new wavelet frames which are appropriate to studying vision science (with S. Arai). These wavelet frames will give basis of constructing computational models of human's visual system. In addition, Arai studied also practical applications to image processing. Furthermore he has constructed a computational model of color perception, and gave computer simulation of illusions related color vision. Arai has studied also the mechanism of appearance of visual illusions. In particular, he proposed a new general scheme for discrete wavelet analysis of vision. It is widely believed that visual illusions will provide us a key in order 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 be … More en made. However as for several illusions, their mechanisms are not yet well understood. In this research program Arai obtained nonlinear models of the early vision by using wavelet frames, and investigated mathematical mechanism of appearance of a certain illusion. More specifically, Arai constructed filter banks modeled after the function of the striate cortex in human's brain, and using these systems Arai made several computer simulations which indicate how our visual system produces visual illusions. By his research the mechanisms of several visual illusions can be explained by a mathematical unified way. If an image is inputted to Arai's system and if it outputs an image with illusion, then we can conclude that it is (they are) caused by processing in V1. Kanjin studied harmonic analysis of systems orthogonal functions. For example he proved boundedness of Cesaro operators on Lp and Hp. Nakamura studied local smoothing effect of analytic wave front sets. Yoshida studied sampling problem for stochastic differential equations. Tachizawa studied some weighted function spaces from viewpoint of wavelet theory. Less