Multi-wavelet frames and applications to harmonic analysis

本研究项目的主题是(多)小波框架和调和分析。首席研究员Arai研究了小波帧及其在视觉科学中的应用。特别地，Arai构造了适合于研究视觉科学的新的小波帧(与S.Arai)。这些小波框架将为构建人类视觉系统的计算模型提供基础。此外，Arai还研究了图像处理的实际应用。此外，他还构建了颜色知觉的计算模型，并给出了与颜色视觉相关的错觉的计算机模拟。Arai还研究了视觉错觉的出现机制。特别是，他提出了一种新的视觉离散小波分析的一般方案。人们普遍认为，视觉错觉将为我们提供一把钥匙，以便理解我们的视觉系统如何执行视觉信息处理。出于这个原因，在过去的100年里，许多关于视错觉的研究都是…做了更多的EN。然而，对于几种错觉，其机制尚不清楚。在本研究项目中，ARAI利用小波框架得到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模仿人脑中纹状皮质的功能构建了滤波器组，并使用这些系统进行了几次计算机模拟，显示了我们的视觉系统如何产生视觉错觉。通过他的研究，可以用数学统一的方法解释几种视觉错觉的机制。如果一幅图像被输入到Arai的系统中，如果它输出的是一幅具有幻觉的图像，那么我们可以得出结论，这是由V1中的处理引起的。坎津研究了系统正交函数的调和分析。例如，他证明了Cesaro算子在LP和HP上的有界性。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。较少

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