2021 Fiscal Year Final Research Report
New Development of Geometric and Microlocal Analysis
Project/Area Number |
19K03569
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Research Category |
Grant-in-Aid for Scientific Research (C)
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Allocation Type | Multi-year Fund |
Section | 一般 |
Review Section |
Basic Section 12020:Mathematical analysis-related
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Research Institution | University of the Ryukyus |
Principal Investigator |
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Project Period (FY) |
2019-04-01 – 2022-03-31
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Keywords | フーリエ積分作用素 / 超局所解析 / ラドン変換 / X線変換 / 波面集合 / 医用画像 |
Outline of Final Research Achievements |
I studied microlocal analysis of integral geometry, and geometric tomography. Firstly I studied two-dimensional hyperbolic and paraboloc Radon transforms arising in seismology, and obtained the refinement of the inversion formula and the generalization for higher dimensions. These results already published in two papers. Secondly I introduced a Fourier integral operator with a complex phase function on the space of hyperplanes. This might be used for the microlocalization of the images of the Radon transform corresponding the measurements of CT scanners. Thirdly I studied the microlocal analysis of the d-plane transform on the Euclidean space. More precisely I described the canonical relation of the d-plane transform as a Fourier integral operator using my original coordinatization of the affine Grassmannians. Moreover, I studied the metal streaking artifacts using the canonical relation, and proved that for any two convex metal regions, the artifact is a conomal distribution.
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Free Research Field |
解析学
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Academic Significance and Societal Importance of the Research Achievements |
d平面変換の超局所解析の結果は従来の2次元に限られていたCTスキャナーに関連する超局所解析を現代の3次元CTスキャナーに関連する超局所解析の基盤を築くものであり、今後のさらなる様々な応用が期待される。一方、一般のアフィン・グラスマン多様体ではなく平面のなす空間に限定ではあるが、CTスキャナーで言うところの観測データに相当する超関数の超局所化の方法を構成したことになるが、超局所解析と深層学習を利用するCT関連の最先端の応用数学では観測データの超局所解析に踏み込めておらず、本研究はその意味で応用される可能性がある。
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