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Analysis of Fracture Propagation in the problem of image segmentation

Research Project

Project/Area Number 17540203
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Global analysis
Research InstitutionNihon University

Principal Investigator

YAMAURA Yoshihiko  Nihon University, 文理学部, 教授 (90255597)

Project Period (FY) 2005 – 2008
Project Status Completed (Fiscal Year 2008)
Budget Amount *help
¥1,940,000 (Direct Cost: ¥1,700,000、Indirect Cost: ¥240,000)
Fiscal Year 2008: ¥520,000 (Direct Cost: ¥400,000、Indirect Cost: ¥120,000)
Fiscal Year 2007: ¥520,000 (Direct Cost: ¥400,000、Indirect Cost: ¥120,000)
Fiscal Year 2006: ¥400,000 (Direct Cost: ¥400,000)
Fiscal Year 2005: ¥500,000 (Direct Cost: ¥500,000)
Keywords特異汎関数 / 変分収束 / Gradient Flow / Gradient Flows / Gamma-convergence / Minimizing Movement / Minimizing movement method / Gamma convergence / Alt-Caffarelli functional / Gradient flow / Minimizing Movement method / Berstein Method
Research Abstract

特性関数項を含むエネルギー汎関数の定常、時間発展のそれぞれの問題について、特異項を滑らかな関数によって近似することによって考えられる変分収束近似理論を用いていくつかの性質を導きました。特性関数項には滑らかさおよび凸性がないため、直接既存の変分法的手法が使えません。そこで本研究のメインの手法として考えたのが、変文収束近似法の適用です。この手法は、特性関数項を滑らかな関数によって近似し、いったん第1変分(Euler-Lagrange方程式)での評価を導出し、それに基づいて目的のエネルギー汎関数の最小化関数の正則性を調べようという方法です。この方法はしかしながら近似指数を上げるにつれ、特異性が増すため、近似したエネルギー汎関数に対する評価は近似指数に関して「一様」でなければなりません。そのため、いくつかの工夫が必要になってきます。主要結果として、自由境界付近での近似最小化関数のグラフの非退化性を証明しました。また特性関数項の近似と空間変数の離散化との併用によりエネルギー勾配流の構成を1次元の場合に行いました。さらに時間変数の離散化との併用により一様連続性をもつMinimizing movementの構成に成功しました。

Report

(5 results)
  • 2008 Annual Research Report   Final Research Report ( PDF )
  • 2007 Annual Research Report
  • 2006 Annual Research Report
  • 2005 Annual Research Report
  • Research Products

    (13 results)

All 2009 2008 2007 2006

All Journal Article (6 results) (of which Peer Reviewed: 6 results) Presentation (7 results)

  • [Journal Article] A convergence of the approximated free boundary of regularized functional via Gamma-convergence2008

    • Author(s)
      山浦義彦
    • Journal Title

      Journal of Computational and Applied Mathematics 218/2

      Pages: 473-481

    • Related Report
      2008 Final Research Report
    • Peer Reviewed
  • [Journal Article] A convergence of the approximated free boundary of regularized functionals via Gamma-convergence2008

    • Author(s)
      Yoshihiko Yamaura
    • Journal Title

      Journal of Computational and Applied Mathematics, 未定

      Pages: 9-9

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] A Gradient flow of the one-dimensional Alt-Caffarelli functional2007

    • Author(s)
      山浦義彦
    • Journal Title

      Mediterranean Journal of Mathematics Vol.4

      Pages: 451-482

    • Related Report
      2008 Final Research Report
    • Peer Reviewed
  • [Journal Article] A variational approximation inheriting a non-degeneracy property2007

    • Author(s)
      山浦義彦
    • Journal Title

      International Journal of Pure and Applied Mathematics Vol 40、No.4

      Pages: 473-518

    • Related Report
      2008 Final Research Report
    • Peer Reviewed
  • [Journal Article] A gradient flow of the one-dimensional Alt-Caffarelli functional2007

    • Author(s)
      Yoshihiko Yamaura
    • Journal Title

      Mediterranean Journal of Mathematics 4

      Pages: 451-482

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] A variational approximation inheriting a non-degeneracy property2007

    • Author(s)
      Yoshihiko Yamaura
    • Journal Title

      International Journal of Pure and Applied Mathematics 4

      Pages: 473-518

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Presentation] Variational approximation and the construction of gradient flows2009

    • Author(s)
      山浦義彦
    • Organizer
      研究集会「確率論とPDE」於広島大学
    • Year and Date
      2009-01-20
    • Related Report
      2008 Final Research Report
  • [Presentation] Variational approximation and the construction of gradient flows2009

    • Author(s)
      Yoshihiko Yamaura
    • Organizer
      研究集会「確率論と PDE」
    • Place of Presentation
      広島大学
    • Year and Date
      2009-01-20
    • Related Report
      2008 Annual Research Report
  • [Presentation] 変分収束近似によるgradient flowの構成について2008

    • Author(s)
      山浦義彦
    • Organizer
      研究集会「曲線と曲面の非線型解析」於埼玉大学
    • Year and Date
      2008-12-17
    • Related Report
      2008 Final Research Report
  • [Presentation] 変分収束近似による gradi ent flowの構成について2008

    • Author(s)
      Yoshihiko Yamaura
    • Organizer
      研究集会「曲線と曲面の非線型解析」
    • Place of Presentation
      埼玉大学
    • Year and Date
      2008-12-17
    • Related Report
      2008 Annual Research Report
  • [Presentation] Variational approximation and a construction of gradient flows2008

    • Author(s)
      山浦義彦
    • Organizer
      偏微分方程式研究集会
    • Place of Presentation
      於熊本大学
    • Year and Date
      2008-10-07
    • Related Report
      2008 Final Research Report
  • [Presentation] Variational approximation and a construction of gradient flows2008

    • Author(s)
      Yoshihiko Yamaura
    • Organizer
      偏微分方程式研究集会
    • Place of Presentation
      熊本大学
    • Year and Date
      2008-10-07
    • Related Report
      2008 Annual Research Report
  • [Presentation] A convergence of the approximated free boundary of regularised functional via Gamma-convergence2006

    • Author(s)
      山浦義彦
    • Organizer
      Twelfth International Congress on Computational and applied Mathematics
    • Place of Presentation
      at Katholieke Universiteit Leuven、 in Belgium
    • Year and Date
      2006-07-12
    • Related Report
      2008 Final Research Report

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Published: 2005-04-01   Modified: 2016-04-21  

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