摘要
本論文的研究目標是以影像切割方法之Distance Regularized Level Set Evolution為基礎，從此方法去改良後並可以增強影像切割的應用能力，除了一般真實影像切割之外，也可針對特定較難切割的影像作處理。

論文中提到Distance Regularized Level Set Evolution中演化的過程來自於energy function，以能量最小化做為Level set equation核心定義的式子，其中energy function含內能與外能，我們提出利用改良外能中的edge indicator可以使用在特定較難切割的影像。雖然改良後的外能可以提升影像切割的能力，但因此對於受雜訊影響的物件邊緣部分更為敏感，所以我們也同時提出了配套措施，使用雙邊濾波器平滑雜訊並且保留邊緣、使用canny edge偵測弱邊界的資訊，並在處理影像前改變外能中的edge indicator，如此希望能讓DRLSE發揮最佳的影像切割能力。

我們將研究結果以圖的方式來呈現以比較改良前後差異性，其中包含原本DRLSE處理過後的結果圖，以及用改良edge indicator DRLSE處理過後的結果圖，並且也同時示範了利用canny edge改善此方法之弱邊界的處理結果，最後我們將影像切割結果圖與真正結果圖(ground truth)比對並計算出其錯誤指標，包含ME (misclassification error)、RFAE (relative foreground area error)、NU (region non-uniformity)、MHD (modified Hausdorff distances)、EMM (edge mismatch)和特別針對輪廓準確度的錯誤指標mean error等，也提出該如何使用這些錯誤測量指標。研究最後討論出DRLSE確實可以透過改良edge indicator，處理一些複數個物件互相靠很近或是物件中含有缺口邊緣等類似例子的影像，進而讓DRLSE的影像切割應用能力提升。


關鍵字：影像切割、影像分割、水平集演化、弱邊界、外能、雙邊濾波器、錯誤率

Abstract

In this thesis, the ability of segmentation of Distance Regularized Level Set Evolution (DRLSE) is investigated and improved by changing the definition of edge indicator and foreseeing the weak edge information. The Distance Regularized Level Set Evolution is based on the gradient flow that minimizes an energy functional with a distance regularization term and an external energy term that drives the motion of the zero level set (contour) toward desired locations. The DRLSE has good efficiency and accuracy in locating contours of objects for conventional images by maintaining the regularity of level set function with the distance regularization term. The stable condition is well satisfied by this term. But for some images with multiple objects separated by short distances or having deep jagged edges, the DRLSE may cause the zero level set driven to contours far away from the desired locations. Moreover, the proximity or deep jagged edges of objects affect external energy term such that re-tuning of the parameter is needed to be able to give additional external force to drive the motion of the contour, which then might result in errors in the evolution and even destruction of level set function. We found out that such situations could be remedied by adding more directions (45 and 135 degrees) for computing edge indicator to the DRLSE whose original edge indicator has only two directions (0 and 90 degrees). The change in the definition of edge indicator can not only keep the stable level set function but also drive the motion of the contour more correctly. Like DRLSE, our solution is sensitive to noise corrupting the desired object boundary, therefore we apply some relevant adjustments including bilateral filtering to smooth noise out while preserving edge and foreknowing the weak edge information by canny edge detector.

Several experiments have been conducted to demonstrate and compare the segmentation capability of the original DRLSE and the proposed improved edge indicator DRLSE. In addition to subjective evaluation, some objective measure of errors (MOEs) such as ME (misclassification error)、RFAE (relative foreground area error)、NU (region non-uniformity)、MHD (modified Hausdorff distances)、EMM (edge mismatch) and mean error (contour model especially) are adopted to evaluate the efficacy of the proposed approach. From the experimental results, the proposed approach did provide an easy but efficient improvement to the original DRLSE.



Keywords: image segmentation, level set, weak edge, external energy, edge indicator, bilateral filter, canny edge detector, measure of error

目錄
摘要 i
Abstract ii
誌謝 iv
目錄 v
圖目錄 vi
表目錄 vii
第一章 簡介 1
1.1研究動機 1
1.2文獻探討 1
1.2.1 Snake 1
1.2.2 Level set 3
1.3論文架構 5
第二章 理論與方法 6
2.1傳統Level set 6
2.1.1 level set evolution equation 6
2.1.2 reinitialization 8
2.2 Level Set Evolution Without Re-initialization 10
2.2.1 energy function 10
2.2.2 Gâteaux derivative & formulation 11
2.3 Distance Regularized Level Set Evolution (DRLSE) 17
2.4 改良方法 20
2.4.1 DRLSE的極限 20
2.4.2 方法介紹 20
2.4.3 輔助工具之雙邊濾波器 22
第三章 實驗結果 24
3.1 錯誤測量MOE (Measure of Errors) 24
3.1.1 參數定義 24
3.1.2 使用分類 27
3.2 選取ground truth 28
3.3 結果 29
3.3.1 結果圖與一般錯誤率表 29
3.3.2 Mean Error比較 40
第四章 結論 44
4.1結論 44
4.2未來方向 44
參考文獻 45

[1] J. Canny, "A Computational Approach to Edge Detection," Pattern Analysis and Machine Intelligence, Vols. PAMI-8, no. 6, pp. 679-698, 11 1986.
[2] R. Adams and L. Bischof, "Seeded region growing," Pattern Analysis and Machine Intelligence, vol. 16, no. 6, pp. 641-647, 6 1994.
[3] M. Kass, A. Witkin and D. Terzopoulos, "Snake: active contour models," International journal of computer vision, vol. 1, no. 4, pp. 321-331, 1 1988.
[4] S. Osher and J. A. Sethian, "Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations," Journal of computational physics, vol. 79, no. 1, pp. 12-49, 11 1988.
[5] C. Xu, A. Yezzi, Jr and J. L. Prince, "On the relationship between parametric and geometric active contours," Signals, Systems and Computers, vol. 1, pp. 483-489, 11 2000.
[6] C. Xu and J. L. Prince, "Snakes, shapes, and gradient vector flow," Image Processing, vol. 7, no. 3, pp. 359-369, 3 1998.
[7] S. C. Zhu and A. Yuille, "Region competition: unifying snakes, region growing, and Bayes/MDL for multiband image segmentation," Pattern Analysis and Machine Intelligence, vol. 18, no. 9, pp. 884-900, 9 1996.
[8] L. D. Cohen, "On active contour models and balloons," CVGIP: Image Understanding, vol. 53, no. 2, pp. 211-218, 3 1991.
[9] L. D. Cohen and I. Cohen, "Finite-element methods for active contour models and balloons for 2-D and 3-D images," Pattern Analysis and Machine Intelligence, vol. 15, no. 11, pp. 1131-1147, 11 1993.
[10] C. Xu and J. L. Prince, "Gradient vector flow: A new external force for snakes," Computer Vision and Pattern Recognition, pp. 66-71, 1 1997.
[11] R. Malladi, J. A. Sethian and B. C. Vemuri, "Shape modeling with front propagation: A level set approach," Pattern Analysis and Machine Intelligence, vol. 17, no. 2, pp. 158-175, 2 1995.
[12] T. F. Chan and L. A. Vese, "Active contours without edges," Image Processing, vol. 2, pp. 266-277, 2 2001.
[13] V. Caselles, R. Kimmel and G. Sapiro, "Geodesic active contours," International journal of computer vision, vol. 22, no. 1, pp. 61-79, 2 1997.
[14] S. Kichenassamy, A. Kumar, P. Olver and A. Tannenbaum, "Gradient flows and geometric active contour models," Computer Vision, pp. 810-815, 6 1995.
[15] R. Kimmel, A. Amir and A. M. Bruckstein, "Finding shortest paths on surfaces using level sets propagation," Pattern Analysis and Machine Intelligence, vol. 17, no. 6, pp. 635-640, 6 1995.
[16] C. Samson, L. Blanc-Feraud, G. Aubert and J. Zerubia, "A variational model for image classification and restoration," Pattern Analysis and Machine Intelligence, vol. 22, no. 5, pp. 460-472, 5 2000.
[17] N. Paragios and R. Deriche, "Geodesic active contours and level sets for the detection and tracking of moving objects," Pattern Analysis and Machine Intelligence, vol. 22, no. 3, pp. 266-280, 3 2000.
[18] C. Li, C. Y. Kao, J. C. Gore and Z. Ding, "Minimization of Region-Scalable Fitting Energy for Image Segmentation," Image Processing, vol. 17, no. 10, pp. 1940-1949, 10 2008.
[19] V. Caselles, F. Catté, T. Coll and F. Dibos, "A geometric model for active contours in image processing," Numerische Mathematik, vol. 66, no. 1, pp. 1-31, 1993.
[20] R. Goldenberg, R. Kimmel, E. Rivlin and M. Rudzsky, "Fast Geodesic Active Contours," Image Processing, vol. 10, no. 10, pp. 1467-1475, 2001.
[21] S. Osher and R. Fedkiw, Level set methods and dynamic implicit surfaces, Springer, 2002.
[22] J. A. Sethian, Level Set Methods and Fast Marching Methods: evolving interfaces in computational geometry, fluid mechanics, computer vision, and materials science, Cambridge Univ. Press, 1999.
[23] C. Li, C. Xu and M. D. Fox, "Level set evolution without re-initialization: a new variational formulation," Computer Vision and Pattern Recognition, vol. 1, pp. 430-436, 6 2005.
[24] C. Li, C. Xu, C. Gui and M. D. Fox, "Distance Regularized Level Set Evolution and Its Application to Image Segmentation," Image Processing, vol. 19, no. 12, pp. 3234-3254, 12 2010.
[25] M. Sussman, E. Fatemi, P. Smereka and S. Osher, "An improved level set method for incompressible two-phase flows," Computers & Fluids, vol. 27, no. 5-6, pp. 663-680, 1998.
[26] H. K. Zhao, T. Chan, B. Merriman and S. Osher, "A variational level set approach to multiphase motion," Journal of computational physics, pp. 179-195, 2 1996.
[27] L. C. Evans, Partial Differential Equations, Providence, RI : American Mathematical Society, 2010.
[28] M. Giaquinta and S. Hildebrandt, Calculus of Variations I, Springer, 1996.
[29] G. Gilboa, N. Sochen and Y. Y. Zeevi, "Forward-and-backward diffusion processes for adaptive image enhancement and denoising," Image Processing, vol. 11, no. 7, pp. 689-703, 7 2002.
[30] C. Tomasi and R. Manduchi, "Bilateral filtering for gray and color images," Computer Vision, pp. 839-846, 1 1998.
[31] C. C. Liu and P. C. Chung, "objects extraction algorithm of color image using adaptive forecasting filters created automatically," International Joirnal of Innovative Computing, Information and Control, vol. 7, no. 10, pp. 5771-5757, 10 2010.
[32] M. Sezgin, "Survey over image thresholding techniques and quantitative performance evaluation," Journal of Electronic imaging, vol. 13, no. 1, pp. 146-168, 6 2004.
[33] C. Li, R. Huang, Z. Ding, J. C. Gatenby, D. N. Metaxas and J. C. Gore, "A Level Set Method for Image Segmentation in the Presence of Intensity Inhomogeneities With Application to MRI," Image Processing, vol. 20, no. 7, pp. 2007-2016, 7 2011.
[34] M. Weber, A. Blake and R. Cipolla, "Sparse Finite Elements for Geodesic Contours with Level-Sets," Computer Vision - ECCV 2004, vol. 3022, pp. 391-404, 2004.
[35] J. Gomes and O. Faugeras, "Reconciling Distance Functions and Level Sets," Journal of Visual Communication and Image Representation, vol. 11, no. 2, pp. 209-223, 6 2000.