標題: Applications of Partial Diffusion Equations for Image Processing偏微方程式於影像處理的應用 作者: 施因澤 關鍵字: 數學類應用研究 摘要: 在這個計劃裡我們將採用偏微分方程式來處理影像雜訊的去除、邊界偵測以及資料壓縮。首先我們研究一個二階偏微分方程式— 對流擴散方程式。將之與目前已被廣汎使用於去除高斯白雜訊的非等向性擴散方程式作一比較。我們將研究它如何有效率地同時地去除高斯雜訊與鹽和胡椒雜訊。我與Elman 所建議的修正流線擴散有限單元法將用來離散方程式而且能防止粗格子化時，影像與不連續邊界所造成內部層(數值誤差)。透過多重網格法，它能有效率地減少計算成本以及減少所需儲存空間。我們將對誤差沽算及其收斂情形作些數學理論分析。最後藉由數值實驗來證明對流擴散模式能有效率地去除高斯雜訊與鹽和胡椒混合雜訊且有相當好地保持邊界。另外我們將研究將四階偏微分方程式以去除雜訊且邊界偵測時以避免塊狀效應當得到良好品質影像。You & Maveh 藉由最小化一個垂直於影像亮度的拉普拉斯函數的決對值而推導出一個四階(雙和諧)偏微分方程式模式，他們利用差分法展開此方程式模式以成功地處理影像。我們將使用高階有限單元法與多層網網格法來解此四階偏微分方程式，最後藉由數值實驗來證明此模式也能有效率地去除高斯雜訊與鹽和胡椒混合雜訊且有相當好地保持邊界。Here we propose some partial differential equations for processing the image noise removaledge preservation and compression. First of all, we focus on a 2nd order PDE model -convection-diffusion equation. We will compare it with a popular nonlinear anisotropicdiffusion model which has been widely implemented in image denoising for Gaussian whitenoise. We study our convection-diffusion model to see how it removes noisy effectively,especially for the mixture of salt-and-pepper noise and Gaussian noise. Modified streamlinediffusion finite element method [Shih & Elman] will be implemented for the discretization ofthe convection-diffusion model to prevent from internal layers around the discontinuitieswhile using the coarsening algorithm for image compression. The computation cost throughmulti-level coarsening processing can be reduced dramatically. We will examine somenumerical experiments to show the efficiency of our convection-diffusion filtering model forremoving the mixture of Gaussian and salt-and-pepper noises, efficiently and reliably inpreserving edges quite satisfactorily.In additional, we will study 4-th order partial differential equation to avoid blocky effectswhile achieving good quality in both noise removal and edge preservation. [You & Maveh]proposed a 4th order PDE model that seeks to minimize a functional proportional to theabsolute value of Laplacian of image intensity and it has been successively implemented byusing a finite difference method. We will use a bicubic Hermite finite element method todiscretize this model and using multi-level grid concept to save the storage space as well asspeeding up the computation time while using Krylov subspace method for the linear systemsolver. URI: http://hdl.handle.net/11455/49753 其他識別: NSC97-2115-M005-002 文章連結: http://grbsearch.stpi.narl.org.tw/GRB/result.jsp?id=1663489&plan_no=NSC97-2115-M005-002&plan_year=97&projkey=PA9709-0290&target=plan&highStr=*&check=0&pnchDesc=%E5%81%8F%E5%BE%AE%E6%96%B9%E7%A8%8B%E5%BC%8F%E6%96%BC%E5%BD%B1%E5%83%8F%E8%99%95%E7%90%86%E7%9A%84%E6%87%89%E7%94%A8 顯示於類別： 應用數學系所

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