2006 Fiscal Year Final Research Report Summary
Multi-wavelet frames and applications to harmonic analysis
| Project/Area Number |
16340035
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Basic analysis
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| Research Institution | The University of Tokyo |
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Principal Investigator |
ARAI Hitoshi The University of Tokyo, Graduate School of Mathematical Science, Professor, 大学院数理科学研究科, 教授 (10175953)
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| Co-Investigator(Kenkyū-buntansha) |
NAKAMURA Shu The University of Tokyo, Graduate School of Mathematical Science, Professor, 大学院数理科学研究科, 教授 (50183520)
YOSHIDA Nakahiro The University of Tokyo, Graduate School of Mathematical Science, Professor, 大学院数理科学研究科, 教授 (90210707)
KANJIN Yuichi Kanazawa University, Graduate School of Natural Science & Technology, Professor, 大学院自然科学研究科, 教授 (50091674)
TACHIZAWA Kazuya Hokkaido University, Faculty of Science, Associate Professor, 大学院理学研究科, 助教授 (80227090)
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| Project Period (FY) |
2004 – 2006
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| Keywords | wavelet / wavelet frame / harmonic analysis / illusion / visual cortex / color vision / perception / systems orthogonal functions |
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| Research Abstract |
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
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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
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