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標題: 使用裁縫有限點法來離散影像處理的偏微分模式
An Implementation of Tailored Point Method in Discretization of Partial Differential Model for Image Process
作者: 施因澤
關鍵字: 數學類
摘要: 我們提出一個兩年的研究計畫,主要是研究裁縫有限點法(TFP)藉由一些偏微分方程(PDE) 模式對影像做去雜訊,邊緣偵測與壓縮。 尤其如同有限元素法,裁縫有限點法是適合解多層次PDE模式數值解,因此也相當適合作為影像壓縮。 多層次網格的產生,是由圖像的梯度強度決定。 我們也將比較這種與其他有限單元法解處理偏微分方程的方法。裁縫有限點法是一種新的離散化橢圓型偏微分方程方法是由北京清華大學韓厚德、黃忠儀與南卡羅來納大學R. Bruce Kellogg 於2008 年提出。 我們也與Kellogg 教授和國立中央大學楊肅煜教授分別對TFP 的對流擴散問題有充分多的研究心得,而且最近將結果已發表3 篇論文於頂尖 SCI 期刊。 目前應用TFP 於影像處理上仍有相當大的研究空間。 現在我們考慮TFP 在偏微分方程中的應用於影像處理。在這裡 PDE 將被用來當為模型的是Perona-Malik 的擴散型方程式,擴散對流方程式,Euler-Lagrange 方程式。我們將分為兩階段來作研究。在第一年的研究部分,我們研究了一個格點生成邏輯。 基於我們以前的作品將多層次的網格粗化過程中的梯度準則找到一些合理的指數,以確保每個網格的圖像強度品質。此特殊多層次結構的網格,可用於減少數據存儲。我們研究TFP 對這些離散偏微分方程模型,並比較它與穩定化的有限元素法。因此,通過偏微分方程模型,將去除雜訊和偵測影像邊緣也很好。在本計畫的第二年部分,我們將藉由泰勒展分析和泛函分析的一些技巧對TFP 對偏微分方程模型做錯誤與穩定性分析。許多圖像(包括視頻流)將被檢測,以顯示我們方法的優越性。
Here we propose a two-year research project for considering a tailored finite point method(TFP) for using in image denoising, edge detection and data compression through somepartial differential models. We will study the implementation of TFP in multi-level grid that isdetermined by the gradient of image intensity. We also will compare this technology withother PDE methods.The TFP is a new discretization method for solving an elliptic type PDE proposed by Hung,Huang and Kellogg in 2008. The most advantage for implement TFP in solving PDE sincethere is less limitation on the grid structure and less computation cost in comparing with finiteelement method. We also have adequately studied TFP for the convection-diffusion problemwith Prof. R. Bruce Kellogg of University of South Carolina and Prof. Suh-Yuh Yang ofNational Central University, and the results have published on 3 top SCI papers recently. Nowwe consider the application of TFP in PDE based image process. The PDE models will beconsidered here are the Perona-Malik's type nonlinear diffusion equation, convectiondiffusion equation and Euler-Lagrange type equation.In the first year, we study on some grid point generation logic based on our previous works.The multi-level grid is on some coarsening process to have some reasonable solution gradientcriteria on each mesh to ensure the quality of image intensity, and the special structure of gridcan be used for reducing the data storage. We study the discretization of TFP on these PDEmodels and compare it with stabilized finite element methods. Thus through PDE models, theTFP will de-noise and resolve image edge well too. On the second year project, we do theerror analysis by the Taylor expansion and some functional analysis skill for the TFP on thetime dependent PDE models. Many experimental images included video streams will beexamined to show the robust of our method.
其他識別: NSC100-2115-M005-003
Appears in Collections:應用數學系所



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