快速且精確的影像分割一直都是電腦視覺領域中很大的挑戰，原因在於自然和醫學影像普遍受到雜訊以及像素強度不均的干擾，為了克服這些因素，模糊分群法被大量應用在影像分割中，尤其在加入核心概念後，能夠有效地處理影像上的雜訊及異值點，並透過水平集函數的實作，使影像上的曲線能夠收斂到物體的輪廓，但是水平集函數的計算複雜度較高，也需要透過再初始化或能量限制項維持函數的平滑程度及穩定性。
本研究中提出演算法架構是將格子波爾茲曼方法應用在解開核心模糊分群的水平集函數中，格子波爾茲曼方法透過每個像素獨立運算，獲得水平集函數迭代式的解答，並同時維持水平集函數的平滑性質，這也代表著能夠高度地被圖形處理器平行化，因此這個演算法能提供快速、穩定並且不易受到初始輪廓影像的準確結果，我們在實驗中，藉由合成和自然影像的分割結果，證實了演算法的穩定以及準確度，並透過圖形處理器的執行時間，展現演算法的效率。

Fast and accurate image segmentation is still a challenging task in computer vision because real-world images are often distorted by noise and intensity inhomogeneity. In order to overcome these problems, fuzzy clustering is extensively applied to image segmentation because of the strong ability to reject local minimum. It also incorporates with kernel metrics to enhance robustness against noise and outliers and construct a nonlinear energy function based on a variational level set framework. However, level set implementation often costs a lot of CPU times, and needs “re-initialization”or regularizing terms to keep level set function smooth and stable.
Our research provides the algorithm which solve level set equation of kernel fuzzy active contour model by using Lattice Boltzmann Method solver. Lattice Boltzmann Method can recover the level set PDE by computing pixel-by-pixel independently and maintain the stability and smoothness of level set function in the meanwhile. Therefore, this algorithm is fast, stable and independent to the position of the initial curve. Experiments on synthetic and real-world images demonstrate the stability and performance of the proposed method, and also show the efficiency of algorithm by using graphics processing unit.

1 前言 p.6
2 分群法 (Clustering) p.9
2.1 K-平均分群法 (K-means clustering) p.9
2.2 模糊C-平均分群法 (Fuzzy C-means clustering, FCM) p.10
2.3 調整後的模糊C-平均分群法 (Modified Fuzzy C-means clustering) p.11
3 動態輪廓模型 (Active Contour Model) p.16
3.1 邊界型 (Edge-based) p.16
3.2 區域型 (Region-based) p.17
3.2.1 Chan-Vese模型 p.17
3.2.2 調整後的Chan-Vese模型 p.19
3.3 水平集方法 (Level Set Method) p.23
4 格子波爾茲曼方法 (Lattice Boltzmann Method, LBM) p.28
4.1 圖形處理器 (Graphic Processing Unit, GPU)平行化 p.28
4.2 用格子波爾茲曼方法解偏微分方程 (Lattice Boltzmann Method for PDE Solver) p.29
5 核心模糊能量結合格子波爾茲曼方法 (Kernel Fuzzy Energy Lattice Boltzmann Method, KFELBM) p.33
5.1 構築模型(Model Construction) p.33
5.2 最小化能量 (Energy Minimization) p.36
5.3 選擇參數 (Parameter Selection) p.38
6 實驗結果與討論 p.41
7 結論與未來工作 p.55

[1] J. MacQueen et al., “Some methods for classification and analysis of multivariate observations,” in Proceedings of the fifth Berkeley symposium on mathematical statistics and probability, vol. 1, pp. 281–297, Oakland, CA, USA., 1967.

[2] J. C. Bezdek, L. O. Hall, and L. P. Clarke, “Review of mr image segmentation techniques using pattern recognition,” Medical Physics, vol. 20, no. 4, pp. 1033–1048, 1993.

[3] J. C. Bezdek, Pattern recognition with fuzzy objective function algorithms. Springer Science & Business Media, 2013.

[4] S. Chen and D. Zhang, “Robust image segmentation using fcm with spatial constraints based on new kernel-induced distance measure,” IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics), vol. 34, pp. 1907–1916, Aug 2004.

[5] M. Kass, A. Witkin, and D. Terzopoulos, “Snakes: Active contour models,” International Journal of Computer Vision, vol. 1, no. 4, pp. 321–331, 1988.

[6] T. F. Chan and L. A. Vese, “Active contours without edges,” IEEE Transactions on Image Processing, vol. 10, pp. 266–277, Feb 2001.

[7] 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, 1988.57

[8] L. Wang and C. Pan, “Robust level set image segmentation via a local correntropy-based k-means clustering,” Pattern Recognition, vol. 47, no. 5, pp. 1917 – 1925, 2014.

[9] S. Krinidis and V. Chatzis, “Fuzzy energy-based active contours,” IEEE Transactions on Image Processing, vol. 18, pp. 2747–2755, Dec 2009.

[10] Y. Wu, W. Ma, M. Gong, H. Li, and L. Jiao, “Novel fuzzy active contour model with kernel metric for image segmentation,” Applied Soft Computing, vol. 34, pp. 301 – 311, 2015.

[11] C. Li, C. Xu, C. Gui, and M. D. Fox, “Level set evolution without re-initialization: a new variational formulation,” in 2005 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR’05), vol. 1, pp. 430–436 vol. 1, June 2005.

[12] C. Li, C. Xu, C. Gui, and M. D. Fox, “Distance regularized level set evolution and its application to image segmentation,” IEEE Transactions on Image Processing, vol. 19, pp. 3243–3254, Dec 2010.

[13] S. Balla-Arab￿, X. Gao, and B. Wang, “A fast and robust level set method for image segmentation using fuzzy clustering and lattice boltzmann method,” IEEE Transactions on Cybernetics, vol. 43, pp. 910–920, June 2013.

[14] Y. Zhao, “Lattice boltzmann based pde solver on the gpu,” The Visual Computer, vol. 24, no. 5, pp. 323–333, 2008.

[15] D. MacKay, “An example inference task: clustering,” Information theory, inference and learning algorithms, vol. 20, pp. 284–292, 2003.

[16] L. Zadeh, “Fuzzy sets,” Information and Control, vol. 8, no. 3, pp. 338 – 353, 1965.

[17] E. H. Ruspini, “A new approach to clustering,” Information and Control, vol. 15, no. 1, pp. 22 – 32, 1969.

[18] J. C. Bezdek, “A convergence theorem for the fuzzy isodata clustering algorithms,” IEEE Transactions on Pattern Analysis and Machine Intel ligence, vol. PAMI-2, pp. 1–8, Jan 1980.

[19] J. C. Dunn, “A fuzzy relative of the isodata process and its use in detecting compact well-separated clusters,” Journal of Cybernetics, vol. 3, no. 3, pp. 32–57, 1973.

[20] W. M. Wells, W. E. L. Grimson, R. Kikinis, and F. A. Jolesz, “Adaptive segmentation of mri data,” IEEE Transactions on Medical Imaging, vol. 15, pp. 429–442, Aug 1996.

[21] D. L. Pham and J. L. Prince, “Adaptive fuzzy segmentation of magnetic resonance images,” IEEE Transactions on Medical Imaging, vol. 18, pp. 737–752, Sept 1999.

[22] W. Chen and M. L. Giger, “A fuzzy c-means (fcm) based algorithm for intensity inhomogeneity correction and segmentation of mr images,” in 2004 2nd IEEE International Symposium on Biomedical Imaging: Nano to Macro (IEEE Cat No. 04EX821), pp. 1307–1310 Vol. 2, April 2004.

[23] D. L. Pham, “Spatial models for fuzzy clustering,” Computer Vision and Image Understanding, vol. 84, no. 2, pp. 285 – 297, 2001.

[24] A. W. C. Liew, S. H. Leung, and W. H. Lau, “Segmentation of color lip images by spatial fuzzy clustering,” IEEE Transactions on Fuzzy Systems, vol. 11, pp. 542–549, Aug 2003.

[25] Y. A. Tolias and S. M. Panas, “Image segmentation by a fuzzy clustering algorithm using adaptive spatially constrained membership functions,” IEEE Transactions on Systems, Man, and Cybernetics - Part A: Systems and Humans, vol. 28, pp. 359–369, May 1998.

[26] K.-S. Chuang, H.-L. Tzeng, S. Chen, J. Wu, and T.-J. Chen, “Fuzzy c-means clustering with spatial information for image segmentation,” Computerized Medical Imaging and Graphics, vol. 30, no. 1, pp. 9 – 15, 2006.

[27] M. N. Ahmed, S. M. Yamany, N. Mohamed, A. A. Farag, and T. Moriarty, “A modified fuzzy c-means algorithm for bias field estimation and segmentation of mri data,” IEEE Transactions on Medical Imaging, vol. 21, pp. 193–199, March 2002.

[28] J. Łęski, “An ε-insensitive approach to fuzzy clustering,” Int. J. Appl. Math. Comput. Sci, vol. 11, no. 4, pp. 993–1007, 2001.

[29] R. J. Hathaway, J. C. Bezdek, and Y. Hu, “Generalized fuzzy c-means clustering strategies using lp norm distances,” IEEE Transactions on Fuzzy Systems, vol. 8, pp. 576–582, Oct 2000.

[30] P. V. Gehler and B. Schölkopf, “An introduction to kernel learning algorithms,” Kernel methods for remote sensing data analysis, pp. 25–45, 2009.

[31] M. Gong, Y. Liang, J. Shi, W. Ma, and J. Ma, “Fuzzy c-means clustering with local information and kernel metric for image segmentation,” IEEE Transactions on Image Processing, vol. 22, pp. 573–584, Feb 2013.

[32] V. Caselles, R. Kimmel, and G. Sapiro, “Geodesic active contours,” International Journal of Computer Vision, vol. 22, no. 1, pp. 61–79, 1997.

[33] K. Zhang, L. Zhang, H. Song, and W. Zhou, “Active contours with selective local or global segmentation: A new formulation and level set method,” Image and Vision Computing, vol. 28, no. 4, pp. 668 – 676, 2010.

[34] D. Mumford and J. Shah, “Optimal approximations by piecewise smooth functions and associated variational problems,” Communications on Pure and Applied Mathematics, vol. 42, no. 5, pp. 577–685, 1989.

[35] C. Li, C. Y. Kao, J. C. Gore, and Z. Ding, “Minimization of region-scalable fitting energy for image segmentation,” IEEE Transactions on Image Processing, vol. 17, pp. 1940–1949, Oct 2008.

[36] X.-F. Wang, H. Min, L. Zou, and Y.-G. Zhang, “A novel level set method for image segmentation by incorporating local statistical analysis and global similarity measurement,” Pattern Recognition, vol. 48, no. 1, pp. 189 – 204,2015.

[37] 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,” IEEE Transactions on Image Processing,vol. 20, pp. 2007–2016, July 2011

[38] C. Li, J. C. Gore, and C. Davatzikos, “Multiplicative intrinsic component (mico) for mri bias field estimation and tissue segmentation,” Magnetic Resonance Imaging, vol. 32, no. 7, pp. 913 – 923, 2014.

[39] C. Feng, D. Zhao, and M. Huang, “Image segmentation and bias correction using local inhomogeneous intensity clustering (linc): A region-based level set method,” Neurocomputing, vol. 219, pp. 107 – 129, 2017.

[40] W. Liu, P. P. Pokharel, and J. C. Principe, “Correntropy: Properties and applications in non-gaussian signal processing,” IEEE Transactions on Signal Processing, vol. 55, pp. 5286–5298, Nov 2007.

[41] S. Osher and R. P. Fedkiw, “Level set methods: An overview and some recent results,” Journal of Computational Physics, vol. 169, no. 2, pp. 463 – 502,2001.

[42] M. Hajiaghayi, E. M. Groves, H. Jafarkhani, and A. Kheradvar, “A 3-d active contour method for automated segmentation of the left ventricle from magnetic resonance images,” IEEE Transactions on Biomedical Engineering, vol. 64, pp. 134–144, Jan 2017.

[43] S. Osher and R. Fedkiw, Level set methods and dynamic implicit surfaces, vol. 153. Springer Science & Business Media, 2006.

[44] S. Succi, The lattice Boltzmann equation: for fluid dynamics and beyond. Oxford university press, 2001.

[45] P. L. Bhatnagar, E. P. Gross, and M. Krook, “A model for collision processes in gases. i. small amplitude processes in charged and neutral one-component systems,” Phys. Rev., vol. 94, pp. 511–525, May 1954.

[46] J. M. Buick and C. A. Greated, “Gravity in a lattice boltzmann model,” Phys.Rev. E, vol. 61, pp. 5307–5320, May 2000.

[47] Y. Chen, Z. Yan, and Y. Chu, “Cellular automata based level set method for image segmentation,” in 2007 IEEE/ICME International Conference on Complex Medical Engineering, pp. 171–174, May 2007.

[48] X. Sun, Z. Wang, and G. Chen, “Parallel active contour with lattice boltzmann scheme on modern gpu,” in 2012 19th IEEE International Conference on Image Processing, pp. 1709–1712, Sept 2012.

[49] S. Balla-Arabe, B. Wang, and X. Gao, “Level set region based image segmentation using lattice boltzmann method,” in 2011 Seventh International Conference on Computational Intel ligence and Security, pp. 1159–1163, Dec2011.

[50] A. Hagan and Y. Zhao, Paral lel 3D Image Segmentation of Large Data Sets on a GPU Cluster, pp. 960–969. Berlin, Heidelberg: Springer Berlin Heidelberg,2009.

[51] B. Song and T. Chan, “A fast algorithm for level set based optimization,” UCLA Cam Report, vol. 2, no. 68, 2002.

[52] V.-T. Pham, T.-T. Tran, K.-K. Shyu, C. Lin, P.-C. Wang, and M.-T. Lo,

“Shape collaborative representation with fuzzy energy based active contour model,” Engineering Applications of Artificial Intel ligence, vol. 56, pp. 60 – 74, 2016.

[53] J. Rosen, “The gradient projection method for nonlinear programming. partii. nonlinear constraints,” Journal of the Society for Industrial and Applied Mathematics, vol. 9, no. 4, pp. 514–532, 1961.

[54] P. Getreuer, “Chan-Vese Segmentation,” Image Processing On Line, vol. 2, pp. 214–224, 2012.

[55] D. Martin, C. Fowlkes, D. Tal, and J. Malik, “A database of human segmented natural images and its application to evaluating segmentation algorithms and measuring ecological statistics,” in Proc. 8th Int’l Conf. Computer Vision, vol. 2, pp. 416–423, July 2001.