多重散乱情報をもちいる次世代イメージング手法の数理解析

• Project/Area Number: 20H01821
• Research Category: Grant-in-Aid for Scientific Research (B)
• Allocation Type: Single-year Grants
• Section: 一般
• Review Section: Basic Section 12040:Applied mathematics and statistics-related
• Research Institution: Kyoto University
• Principal Investigator: 藤原 宏志 京都大学, 情報学研究科, 准教授 (00362583)
• Co-Investigator(Kenkyū-buntansha): 川越 大輔 京都大学, 情報学研究科, 助教 (30848073) ; 大石 直也 京都大学, 医学研究科, 特定准教授 (40526878) ; 吉川 仁 京都大学, 総合生存学館, 特定教授 (90359836)
• Project Period (FY): 2020-04-01 – 2024-03-31
• Project Status: Completed (Fiscal Year 2023)
• Budget Amount: ¥16,380,000 (Direct Cost: ¥12,600,000、Indirect Cost: ¥3,780,000); Fiscal Year 2023: ¥3,900,000 (Direct Cost: ¥3,000,000、Indirect Cost: ¥900,000); Fiscal Year 2022: ¥3,250,000 (Direct Cost: ¥2,500,000、Indirect Cost: ¥750,000); Fiscal Year 2021: ¥7,020,000 (Direct Cost: ¥5,400,000、Indirect Cost: ¥1,620,000); Fiscal Year 2020: ¥2,210,000 (Direct Cost: ¥1,700,000、Indirect Cost: ¥510,000)
• Keywords: 逆問題 / トモグラフィ / 輻射輸送方程式 / 数値解析 / 数値計算 / 部分観測 / イメージング / エックス線トモグラフィ / 陽電子断層撮影 / 再構成

Outline of Research at the Start

医用イメージングは，画像診断を支える基盤技術である．代表的なものに，X線をもちいた計算機トモグラフィ(断層撮影法)があるが，これはX線の直進性を本質的に利用しており，高効率な手法が開発と高画質化に成功し，実用に寄与している．一方，先進的なイメージング手法では，信号が必ずしも直進せず，生体内で散乱を受けるものが利用されており，画像の劣化や信頼性の低下といった問題点が指摘されている．本研究では，これら散乱信号をもちいるイメージング手法の信頼性と高精度化のために数理科学と数値解析学の視点から研究をおこない，新たなトモグラフィ方法論の確立に取り組む．

Outline of Annual Research Achievements

まず，陽電子断層トモグラフィ (Positron Emission Tomography) の部分観測の数理モデルとして，輻射輸送方程式の境界からの流出境界条件の一部を既知する非斉次項決定逆問題を考察した．前年までに数学解析による定性的な不安定性の解明，数値計算例は発表していたが，不安定性があるにも関わらず数値計算において誤差の増大が深刻でない理由が不明であった．これに対して，数値計算過程で現れるCauchy型の特異核をもつ積分方程式を離散化した連立一次方程式の条件数の評価が高々多項式増大であることを証明し，不安定性の定量的説明を与え，数値計算結果に対して数値解析理論から計算結果の信頼性について説明を与えた．同時に，正則化をおこなった場合は条件数が一様有界になることも説明された．
次に，ひかりトモグラフィの数理モデルのひとつである輻射輸送方程式の減衰係数決定逆問題に対して，分担者が得ていた理論を２次元の場合に数値計算をおこない，減衰係数の定性的性質，特に不連続性が再構成されることを示し，論文として発表した．さらに前半の理論とあわせて，部分観測の場合にも減衰係数が部分的に再構成される数値計算結果も得た．これらの数値計算では観測データは数値実験によるデータを利用しており，前年度までに得ていた解の不連続性を検出し得るスキーム，および本助成で購入した並列計算機を利用した大規模数値実験をおこなったものである．３次元についても同様の数値計算結果を得るに至った．
これらはいずれも，伝播中に多重散乱される信号からの再構成の数学解析・数値解析としての結果であり，次世代イメージングの基礎理論に寄与するものと研究期間を総括した．

Research Progress Status

令和5年度が最終年度であるため、記入しない。

Strategy for Future Research Activity

令和5年度が最終年度であるため、記入しない。

Research Products

• [Int'l Joint Research] University of Central Florida(米国)
• [Int'l Joint Research] Johann Radon Institute(オーストリア)
• [Journal Article] DETERMINATION OF CRACKS USING LASSO FOR QNDE WITH TOPOLOGICAL DERIVERTIVES (2024)
  • Author(s): YOSHIOKA Tomoki、YOSHIKAWA Hitoshi
  • Journal Title: Japanese Journal of JSCE Volume: 80 Issue: 15 Pages: n/a
  • DOI: 10.2208/jscejj.23-15005
  • ISSN: 2436-6021
  • Peer Reviewed / Open Access
• [Journal Article] The Algebraic Range of the Planar X-Ray Transform of Symmetric Tensors and Applications to Noise Reduction (2023)
  • Author(s): Fujiwara Hiroshi、Sadiq Kamran、Tamasan Alexandru
  • Journal Title: Proceedings of Practical Inverse Problems and Their Prospects, Mathematics for Industry Volume: 37 Pages: 47-68
  • DOI: 10.1007/978-981-99-2408-0_4
  • ISBN: 9789819924073, 9789819924080
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Tomography from Scattered Signals Obeying the Stationary Radiative Transport Equation (2023)
  • Author(s): Chen I-Kun、Fujiwara Hiroshi、Kawagoe Daisuke
  • Journal Title: Proceedings of Practical Inverse Problems and Their Prospects, Mathematics for Industry Volume: 37 Pages: 27-46
  • DOI: 10.1007/978-981-99-2408-0_3
  • ISBN: 9789819924073, 9789819924080
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Numerical Reconstruction of Radiative Sources from Partial Boundary Measurements (2023)
  • Author(s): Fujiwara Hiroshi、Sadiq Kamran、Tamasan Alexandru
  • Journal Title: SIAM Journal on Imaging Sciences Volume: 16 Issue: 2 Pages: 948-968
  • DOI: 10.1137/22m1507449
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] FreeFEM での境界要素法の数値計算と PDF 出力機能拡張 (2023)
  • Author(s): 藤原宏志
  • Journal Title: 計算数理工学会レビュー Volume: 2023-2 Pages: 11-35
• [Journal Article] 解の不連続性を利用する3次元散乱信号からのトモグラフィの数値的試み (2023)
  • Author(s): 藤原宏志、川越大輔、大石直也
  • Journal Title: 第28回計算工学講演会講演論文集 Volume: 28
• [Journal Article] トポロジー導関数を用いた3次元スカラー波動散乱場のクラック決定解析 (2023)
  • Author(s): 吉川 仁, 林 正磨
  • Journal Title: 計算数理工学論文集 Volume: 23 Pages: 129-136
  • Peer Reviewed
• [Journal Article] Denoising with Graphics Processing Units and Deep Learning in Non-invasive Medical Imaging (2023)
  • Author(s): Oishi Naoya
  • Journal Title: Proceedings of Practical Inverse Problems and Their Prospects, Mathematics for Industry Volume: 37 Pages: 15-26
  • DOI: 10.1007/978-981-99-2408-0_2
  • ISBN: 9789819924073, 9789819924080
  • Peer Reviewed
• [Journal Article] Correlation between trait emotional intelligence and prefrontal activation during a verbal fluency task: A functional near-infrared spectroscopy study (2023)
  • Author(s): Fukumoto T, Amitani H, Wada M, Oishi N, *Asakawa A
  • Journal Title: Medicine Volume: 102 Issue: 29 Pages: e34418-e34418
  • DOI: 10.1097/md.0000000000034418
  • Peer Reviewed / Open Access
• [Journal Article] Voxel‐based clustered imaging by multiparameter diffusion tensor images for predicting the grade and proliferative activity of meningioma (2023)
  • Author(s): Takahashi Yuki、Oishi Naoya、Yamao Yukihiro、Kunieda Takeharu、Kikuchi Takayuki、Fukuyama Hidenao、Miyamoto Susumu、Arakawa Yoshiki
  • Journal Title: Brain and Behavior Volume: 13 Issue: 10
  • DOI: 10.1002/brb3.3201
  • Peer Reviewed / Open Access
• [Journal Article] Cauchy型積分による部分観測のもとでのX線計算機断層撮影法 (2022)
  • Author(s): 藤原宏志，大石直也，SADIQ Kamran, TAMASAN Alexandru
  • Journal Title: 計算数理工学会レビュー Volume: 2022-2 Pages: 33-48
  • Int'l Joint Research
• [Journal Article] A pictorial review of structural neuroimaging at 7T. (2022)
  • Author(s): *Okada T, Fujimoto K, Fushimi Y, Akasaka T, Thuy DHD, Shima A, Sawamoto N, Oishi N, Zhang Z, Funaki T, Nakamoto Y, Murai T, Miyamoto S, Takahashi R, Isa T.
  • Journal Title: Quantitative Imaging in Medicine and Surgery Volume: 12 Issue: 6 Pages: 3406-3435
  • DOI: 10.21037/qims-21-969
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] Combining multiple indices of diffusion tensor imaging can better differentiate patients with traumatic brain injury from healthy subjects. (2022)
  • Author(s): *Abdelrahman H, Ubukata S, Ueda K, Fujimoto G, Oishi N, Aso T, Murai T.
  • Journal Title: Neuropsychiatr Dis Treat. Volume: 18 Pages: 1801-1814
  • DOI: 10.2147/ndt.s354265
  • Peer Reviewed / Open Access
• [Journal Article] 境界要素法により数値的に求められた散乱音場と頭部伝達関数によるバイノーラル信号の合成 (2022)
  • Author(s): 川井郁哉, 吉川仁
  • Journal Title: 計算数理工学論文集 Volume: 22 Pages: 131-138
  • Peer Reviewed / Open Access
• [Journal Article] レーザ干渉計による計測データを用いた弾性材料の非破壊評価のためのトポロジー導関数を用いた欠陥決定解析 (2022)
  • Author(s): 中園昂志, 吉川仁
  • Journal Title: 計算数理工学論文集 Volume: 22 Pages: 171-178
  • Peer Reviewed / Open Access
• [Journal Article] Partial inversion of the 2D attenuated X-ray transform with data on an arc (2022)
  • Author(s): Fujiwara Hiroshi、Sadiq Kamran、Tamasan Alexandru
  • Journal Title: Inverse Problems & Imaging Volume: 16 Issue: 1 Pages: 215-228
  • DOI: 10.3934/ipi.2021047
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Machine learning trained with quantitative susceptibility mapping to detect mild cognitive impairment in Parkinson's disease (2022)
  • Author(s): Shibata Haruto、Uchida Yuto、Inui Shohei、Kan Hirohito、Sakurai Keita、Oishi Naoya、Ueki Yoshino、Oishi Kenichi、Matsukawa Noriyuki
  • Journal Title: Parkinsonism and Related Disorders Volume: 94 Pages: 104-110
  • DOI: 10.1016/j.parkreldis.2021.12.004
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Web-based Visualization Framework on FreeFEM (2021)
  • Author(s): Lee, Yu-Hsun, 藤原宏志
  • Journal Title: 計算数理工学論文集 Volume: 21 Pages: 57-62
  • Peer Reviewed / Open Access
• [Journal Article] 部分観測による X 線計算機断層撮影法の数値的実現 (2021)
  • Author(s): 藤原宏志，大石直也, SADIQ, Kamran, TAMASAN, Alexandru
  • Journal Title: 計算数理工学論文集 Volume: 21 Pages: 37-43
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] Numerical regularity map for simple one-dimensional fractional differential equations with H\"{o}lder continuous solutions (2021)
  • Author(s): Mana Kato, Hiroshi Fujiwara and Hitoshi Imai
  • Journal Title: Adv. Math. Sci. Appl. Volume: 30 Pages: 499-506
  • Peer Reviewed
• [Journal Article] A two dimensional source reconstruction method in radiative transport using boundary data measured on an arc (2021)
  • Author(s): H.Fujiwara, K.Sadiq, and A.Tamasan
  • Journal Title: Inverse Problems Volume: 37-11 Issue: 11 Pages: 115005-115005
  • DOI: 10.1088/1361-6420/ac2d75
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] Development of three-dimensional integral-type reconstruction formula for magnetic resonance elastography (2021)
  • Author(s): Takeda Tasuku、Fujiwara Hiroshi、Suga Mikio
  • Journal Title: International Journal of Computer Assisted Radiology and Surgery Volume: 16 Issue: 11 Pages: 1947-1956
  • DOI: 10.1007/s11548-021-02517-8
  • Peer Reviewed
• [Journal Article] Multiple-Precision Arithmetic of Biot-Savart Integrals for Reconnections of Vortex Filaments (2021)
  • Author(s): Lee Yu-Hsun、Fujiwara Hiroshi
  • Journal Title: Lecture Notes in Computer Science book series (LNTCS) Volume: 12953 Pages: 191-201
  • DOI: 10.1007/978-3-030-86976-2_13
  • ISBN: 9783030869755, 9783030869762
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Piecewise Constant Upwind Approximations to the Stationary Radiative Transport Equation (2020)
  • Author(s): Hiroshi Fujiwara
  • Journal Title: Mathematics for Industry Volume: 34 Pages: 35-45
  • DOI: 10.1007/978-981-15-6062-0_3
  • ISBN: 9789811560613, 9789811560620
  • Peer Reviewed
• [Journal Article] 境界積分による X 線計算機断層撮影法の正則化 (2020)
  • Author(s): 藤原宏志，大石直也
  • Journal Title: 計算数理工学論文集 Volume: 20 Pages: 75-79
  • Peer Reviewed / Open Access
• [Journal Article] 時間域BIEMを用いたゲームエンジンによる3次元空間を移動する受音点のリアルタイム可聴化 (2020)
  • Author(s): 吉川 仁，鈴木 賢人
  • Journal Title: 計算数理工学論文集 Volume: 20 Pages: 89-93
  • Peer Reviewed / Open Access
• [Journal Article] <p>Spatial Patterns of Amyloid Deposition in Patients with Chronic Focal or Diffuse Traumatic Brain Injury Using ¹⁸F-FPYBF-2 PET</p> (2020)
  • Author(s): Ubukata Shiho、Oishi Naoya、Higashi Tatsuya、Kagawa Shinya、Yamauchi Hiroshi、Okuyama Chio、Watanabe Hiroyuki、Ono Masahiro、Saji Hideo、Aso Toshihiko、Murai Toshiya、Ueda Keita
  • Journal Title: Neuropsychiatric Disease and Treatment Volume: Volume 16 Pages: 2719-2732
  • DOI: 10.2147/ndt.s268504
  • Peer Reviewed / Open Access
• [Journal Article] Intact in vivo visualization of telencephalic microvasculature in medaka using optical coherence tomography (2020)
  • Author(s): Suzuki Takashi、Ueno Tomohiro、Oishi Naoya、Fukuyama Hidenao
  • Journal Title: Scientific Reports Volume: 10 Issue: 1 Pages: 19831-19831
  • DOI: 10.1038/s41598-020-76468-6
  • NAID: 120006898162
  • Peer Reviewed / Open Access
• [Presentation] Numerical challenges in inverse and ill-posed problems (2024)
  • Author(s): Hiroshi Fujiwara
  • Organizer: HKUST and Kyoto University Joint Symposium on Informatics
  • Int'l Joint Research
• [Presentation] Numerical feasibility of optical tomography by the stationary radiative transport equation (2024)
  • Author(s): Hiroshi Fujiwara
  • Organizer: 2024 Japan-Taiwan Joint Workshop on Numerical Analysis and Inverse Problems
  • Int'l Joint Research / Invited
• [Presentation] On numerical instability of a singular integral equation in x-ray computerized tomography with partial measurement (2024)
  • Author(s): Hiroshi Fujiwara, Naoya Oishi, Kamran Sadiq, Alexandru Tamasan
  • Organizer: RIMS workshop on "Some approaches on ill-posed problems -theory and practice-"
  • Int'l Joint Research / Invited
• [Presentation] 微小脳構造を核とした安静時機能結合MRI (2024)
  • Author(s): 大石 直也
  • Organizer: 第26回日本ヒト脳機能マッピング学会
  • Invited
• [Presentation] On numerical analysis of x-ray computerized tomography with partial measurement (2023)
  • Author(s): Hiroshi Fujiwara
  • Organizer: NCTS-Kyoto Mathematics Symposium
  • Int'l Joint Research / Invited
• [Presentation] Numerical reconstruction of sources in transport equations and their applications (2023)
  • Author(s): Hiroshi Fujiwara
  • Organizer: The 5th workshop on recent development of mathematical fluid dynamics and hyperbolic conservation laws
  • Int'l Joint Research
• [Presentation] Numerical challenges to optical tomography by the stationary radiative transport equatio (2023)
  • Author(s): I-Kun Chen, Hiroshi Fujiwara, Daisuke Kawagoe
  • Organizer: ICIAM2023
  • Int'l Joint Research
• [Presentation] Some remarks on BemTool and treatment of PDF files in FreeFEM (2023)
  • Author(s): Hiroshi Fujiwara
  • Organizer: Variational problem in engineering and numerical tools
  • Int'l Joint Research
• [Presentation] 有限要素法による音場解析手法の構築とそ の可聴化 (2023)
  • Author(s): 樫山 和男, 深澤 一志, 宮内 暖季, 吉川 仁
  • Organizer: 土木学会第26回応用力学シンポジウム
• [Presentation] ボイドとクラックが混在する材料に対するトポロジー導関数を用いた非破壊評価 (2023)
  • Author(s): 小金丸一輝, 吉川仁
  • Organizer: 機械学会第36回計算力学講演会
• [Presentation] On a Cauchy-type singular integral equation for x-ray computerized tomography with partial measurement (2023)
  • Author(s): 藤原宏志, Kamran Sadiq, Alexandru Tamasan
  • Organizer: 日本数学会 2023年度年会 応用数学分科会
• [Presentation] ハイパースペクトルイメージングの脳領域への応用 (2023)
  • Author(s): 大石直也
  • Organizer: 第25回日本ヒト脳機能マッピング学会
  • Invited
• [Presentation] A remark on the generalized convexity condition and propagation of boundary-induced discontinuity in stationary radiative transfer (2023)
  • Author(s): 川越大輔
  • Organizer: RIMS 共同研究 (公開型) 逆問題と医用イメージングとその周辺
  • Int'l Joint Research / Invited
• [Presentation] ひかりトモグラフィのための定常輻射輸送方程式の解の不連続性の解析, (2022)
  • Author(s): 川越大輔, 藤原宏志, 陳逸昆
  • Organizer: 2022年度応用数学合同研究集会
• [Presentation] Numerical investigation of the 3D regularized Biot-Savart model towards vortex reconnection (2022)
  • Author(s): Yoshifumi Kimura, Hiroshi Fujiwara, Yu-Hsun Lee
  • Organizer: American Physical Society, 75th Annual Meeting of the Division of Fluid Dynamics
  • Int'l Joint Research
• [Presentation] Cauchy型境界積分によるエックス線断層撮影法の安定化パラメータの選択 (2022)
  • Author(s): 藤原宏志
  • Organizer: 第14回最適化シンポジウム
• [Presentation] Heat convections in the horizontal layer with non-uniform heat supply. Stommel model (2022)
  • Author(s): 西田孝明, 藤原宏志
  • Organizer: 日本数学会 2022年度秋季総合分科会
• [Presentation] FreeFEMにおける2次元数値計算結果のPDFファイル出力の機能拡張 (2022)
  • Author(s): 藤原宏志
  • Organizer: 日本応用数理学科2022年度年会
• [Presentation] 不連続性に基く散乱信号からのトモグラフィの数値的実現 (2022)
  • Author(s): 藤原宏志, 川越大輔、Chen, I-Kun
  • Organizer: 第27回計算工学講演会
• [Presentation] Time Domain Sound Field Analysis Using the Finite Element Method and the Fast Multipole Boundary Element Method (2022)
  • Author(s): Kazushi Fukazawa, Hitoshi Yoshikawa, Kazuo Kashiyama
  • Organizer: WCCM-APCOM YOKOHAMA, APACM, IACM, JSCES, JACM
  • Int'l Joint Research
• [Presentation] 3次元スカラー波動問題におけるトポロジー導関数を用いたクラック決定解析 (2022)
  • Author(s): 林正磨, 吉川仁
  • Organizer: 第14回最適化シンポジウム2022, 機械学会
• [Presentation] 特異積分方程式による部分観測でのＸ線計算機トモグラフィ (2022)
  • Author(s): 藤原宏志, Kamran Sadiq, Alexandru Tamasan
  • Organizer: 日本応用数理学会2022年度年会
• [Presentation] On feasibility of numerical reconstruction of the attenuation coefficient in the stationary radiative transport equation (2022)
  • Author(s): Hiroshi Fujiwara
  • Organizer: Practical inverse problems and their prospects
  • Int'l Joint Research / Invited
• [Presentation] Denoising in non-invasive medical imaging (2022)
  • Author(s): Naoya Oishi
  • Organizer: Practical inverse problems and their prospects
  • Int'l Joint Research / Invited
• [Presentation] Propagation of boundary-induced discontinuity in stationary radiative transfer and its application to the optical tomography (2022)
  • Author(s): Daisuke Kawagoe
  • Organizer: Practical inverse problems and their prospects
  • Int'l Joint Research / Invited
• [Presentation] An Efficient Numerical Scheme for Near-Infrared Light Propagation in Turbid Media (2021)
  • Author(s): Hiroshi Fujiwara
  • Organizer: IUTAM symposium computational methods for large-scale and complex wave problems
  • Int'l Joint Research / Invited
• [Presentation] 輸送方程式のソース項同定逆問題による X 線計算機断層撮影の数値的実現 (2021)
  • Author(s): 藤原宏志
  • Organizer: 第26回計算工学講演会
• [Presentation] 有限要素解析ツール FreeFem++ のファイル出力機能拡張と可視化の可能性 (2020)
  • Author(s): 藤原宏志
  • Organizer: 第25回計算工学講演会
• [Presentation] Theoretical numerical analysis and computation of light propagation in biomedical tissue (2020)
  • Author(s): 藤原宏志
  • Organizer: 第6回理論応用力学シンポジウム
  • Invited
• [Presentation] PDF output extensions of 2D Computations in FreeFEM (2020)
  • Author(s): Hiroshi Fujiwara
  • Organizer: FreeFEM Days 2020
  • Int'l Joint Research
• [Remarks] extend precision floating-point arithmetic library
  • URL: https://www-an.acs.i.kyoto-u.ac.jp/~fujiwara/exflib
• [Remarks] FreeFEM Programs
  • URL: https://www-an.acs.i.kyoto-u.ac.jp/~fujiwara/ff
• [Remarks] Exflib information
  • URL: http://www-an.acs.i.kyoto-u.ac.jp/~fujiwara/exflib/
• [Remarks] FreeFEM Programs
  • URL: http://www-an.acs.i.kyoto-u.ac.jp/~fujiwara/ff/
• [Remarks] Exflib information
  • URL: http://www-an.acs.i.kyoto-u.ac.jp/~fujiwara/exflib
• [Remarks] FreeFEM Programs
  • URL: http://www-an.acs.i.kyoto-u.ac.jp/~fujiwara/ff
• [Remarks] exflib home page
  • URL: http://www-an.acs.i.kyoto-u.ac.jp/~fujiwara/exflib