MRI图像分割中核聚类算法的研究
摘要
医学图像在临床医学上应用越来越广泛,使得图像分割这一医学图像处理和分析中的基本问题也越来越关键。医学图像分割是指将医学图像分割为一系列相互不交叠的区域,这些区域具有相似的特征,例如灰度、色彩、纹理、局部统计特征或频谱等。分割后的图像可以被广泛应用到多种场合,如组织容积的定量分析、计算机辅助诊断、计算机引导手术、病变组织的定位、局部容积效应的矫正等等。
由于解剖结构复杂多样以及成像设备、成像技术的制约,医学图像不可避免地受到各种退化因素的影响,包括噪声、局部容积效应和有偏场效应等。目前,尽管医学图像分割算法种类繁多,但是仍然没有一种完美的自动分割算法,各种算法都只能针对某些特定的条件或场合取得较佳的分割效果。
本文对二维MRI(Magnetic Resonance Imaging)图像分割中核聚类算法的应用进行了一定研究,对传统FCM(Fuzzy C-means Clustering)算法和KFCM-II(Kernel-based FCM II)算法应用于MRI图像分割加入了灰度有偏场的纠正,进行实验获得实验结果并进行性能分析。实验结果表明,KFCM-II算法对低退化条件的MRI图像的分割任务,结果并不能比FCM算法占优;对于高退化条件的分割任务,则应采用结合灰度有偏场的纠正的算法;若追求算法对不同退化条件下分割的稳定性与准确度的综合平衡,则应该选择结合灰度纠正的KFCM-II算法。
文章首先介绍了医学图像分割的相关背景、MRI成像机理和分割目标,以及分割结果的评估方法。接着阐述了模糊理论与模糊聚类的相关内容,详细介绍了经典的FCM算法,分析了它的性能及缺点。然后引入核方法思想,阐明核方法的优势,重点阐述KFCM-I和KFCM-II算法。继而提出结合灰度有偏场的纠正以改善分割算法性能,介绍了KFCM-III算法原理以及与之相关的聚类典型集和数据分类等概念。最后使用VC++调用Matlab计算引擎对分割算法进行了软件设计并实验。
The application of medical images in clinical medicine is becoming more and more widespread, that makes the image segmentation which is the fundamental problem of medical image processing and analysis in the increasingly critical. Medical image segmentation refers to the process of partitioning observed image data to a serial of non-overlapping regions, which have similar features, such as grayscale, color, texture, local statistical features and spectrum, etc. Segmented images can be widely applied in various applications, such as tissue volume quantitative analysis, computer-aided diagnosis, computer guided surgery, lesion tissue location and partial volume effect correction, etc.
Due to the complexity and varieties of anatomy structures as well as the imperfections of imaging scanner and imaging techniques, obtained medical images will inevitably be affected by lots of corruption factors including additive noises, partial volume effect and intensity bias field. There is still not a perfect auto segmentation algorithm so far, despite the fact that there exist extensive medical image segmenting methods. Various algorithms are only for certain specific conditions or situations to obtain better segmentation.
This dissertation has studied the application of kernel based clustering algorithms for segmenting two-dimensional MRI (Magnetic Resonance Imaging) medical images, adopted correction method for intensity bias field of MRI data to the traditional FCM (Fuzzy C-means Clustering) algorithm and KFCM-II (Kernel-based FCM II) algorithm, and carried out experiments to obtain segmentation results for performance analysis. The experiment results show that, KFCM-II algorithm is not better than FCM algorithm when applied to segment MRI images with low level degraded conditions; As to segmentation tasks with high level degraded conditions, algorithms with correction of intensity bias field should be adopted; If overall balance of accuracy and stability for the segmentation results under different level degrade conditions is the goal, KFCM-II algorithm with correction of intensity bias field should be chosen.
The dissertation first introduces the background of medical image segmentation, MRI imaging mechanism, the segmentation target, and the assessment rules for segmentation results. Then, expounds the theory of fuzzy set and fuzzy clustering, goes into details for the classical FCM algorithm, with a analysis for its performance and shortcomings. The thought of kernel methods is set forth with its advantages clarified, focusing on the KFCM-I and KFCM-II algorithm. And then adopting the correction method of intensity bias field to improve the segmentation algorithm performance is proposed. The KFCM-III and its related concept of typical clustering dataset and data classification are introduced. Finally, using the programming skill of VC + + calling Matlab calculation engine to implement the algorithms and carried out segmentation experiments.
由于解剖结构复杂多样以及成像设备、成像技术的制约,医学图像不可避免地受到各种退化因素的影响,包括噪声、局部容积效应和有偏场效应等。目前,尽管医学图像分割算法种类繁多,但是仍然没有一种完美的自动分割算法,各种算法都只能针对某些特定的条件或场合取得较佳的分割效果。
本文对二维MRI(Magnetic Resonance Imaging)图像分割中核聚类算法的应用进行了一定研究,对传统FCM(Fuzzy C-means Clustering)算法和KFCM-II(Kernel-based FCM II)算法应用于MRI图像分割加入了灰度有偏场的纠正,进行实验获得实验结果并进行性能分析。实验结果表明,KFCM-II算法对低退化条件的MRI图像的分割任务,结果并不能比FCM算法占优;对于高退化条件的分割任务,则应采用结合灰度有偏场的纠正的算法;若追求算法对不同退化条件下分割的稳定性与准确度的综合平衡,则应该选择结合灰度纠正的KFCM-II算法。
文章首先介绍了医学图像分割的相关背景、MRI成像机理和分割目标,以及分割结果的评估方法。接着阐述了模糊理论与模糊聚类的相关内容,详细介绍了经典的FCM算法,分析了它的性能及缺点。然后引入核方法思想,阐明核方法的优势,重点阐述KFCM-I和KFCM-II算法。继而提出结合灰度有偏场的纠正以改善分割算法性能,介绍了KFCM-III算法原理以及与之相关的聚类典型集和数据分类等概念。最后使用VC++调用Matlab计算引擎对分割算法进行了软件设计并实验。
The application of medical images in clinical medicine is becoming more and more widespread, that makes the image segmentation which is the fundamental problem of medical image processing and analysis in the increasingly critical. Medical image segmentation refers to the process of partitioning observed image data to a serial of non-overlapping regions, which have similar features, such as grayscale, color, texture, local statistical features and spectrum, etc. Segmented images can be widely applied in various applications, such as tissue volume quantitative analysis, computer-aided diagnosis, computer guided surgery, lesion tissue location and partial volume effect correction, etc.
Due to the complexity and varieties of anatomy structures as well as the imperfections of imaging scanner and imaging techniques, obtained medical images will inevitably be affected by lots of corruption factors including additive noises, partial volume effect and intensity bias field. There is still not a perfect auto segmentation algorithm so far, despite the fact that there exist extensive medical image segmenting methods. Various algorithms are only for certain specific conditions or situations to obtain better segmentation.
This dissertation has studied the application of kernel based clustering algorithms for segmenting two-dimensional MRI (Magnetic Resonance Imaging) medical images, adopted correction method for intensity bias field of MRI data to the traditional FCM (Fuzzy C-means Clustering) algorithm and KFCM-II (Kernel-based FCM II) algorithm, and carried out experiments to obtain segmentation results for performance analysis. The experiment results show that, KFCM-II algorithm is not better than FCM algorithm when applied to segment MRI images with low level degraded conditions; As to segmentation tasks with high level degraded conditions, algorithms with correction of intensity bias field should be adopted; If overall balance of accuracy and stability for the segmentation results under different level degrade conditions is the goal, KFCM-II algorithm with correction of intensity bias field should be chosen.
The dissertation first introduces the background of medical image segmentation, MRI imaging mechanism, the segmentation target, and the assessment rules for segmentation results. Then, expounds the theory of fuzzy set and fuzzy clustering, goes into details for the classical FCM algorithm, with a analysis for its performance and shortcomings. The thought of kernel methods is set forth with its advantages clarified, focusing on the KFCM-I and KFCM-II algorithm. And then adopting the correction method of intensity bias field to improve the segmentation algorithm performance is proposed. The KFCM-III and its related concept of typical clustering dataset and data classification are introduced. Finally, using the programming skill of VC + + calling Matlab calculation engine to implement the algorithms and carried out segmentation experiments.
引文
[1] Zhuge Y., Udupa J. K., Saha P. K. Vectorial scale-based fuzzy-connected image segmentation[J]. Computer Vision and Image Understanding, 2006(6) 101(3): 177-193
[2]杨润玲,高新波.基于加权模糊C均值聚类的快速图像自动分割算法[J].中国图象图形学报, 2007, 12(12): 2105-2112
[3] Liu Y., Chen K, Liao X. F., et al. A genetic clustering method for intrusion detection[J]. Pattern Recognition, 2004 : 37(5): 927-942
[4] Xu X., Jager J., Kriegel H. P. A fast parallel clustering algorithm for large spatial databases[J]. Data Mining and Knowledge Discovery, 1999, 3(3): 263-290
[5] Xu. C. Y., Prince J. L. Snakes, shapes and gradient vector flow[J]. IEEE Transactions on Image Processing, 1998, 7(3): 359- 369
[6]田捷,韩博闻,王岩等.模糊C均值聚类法在医学图像分析中的应用[J].软件学报, 2001, 12(11): 1623-1629.
[7] Cheng H. D., Chen J.R., Li J. G. Threshold Selection Based on Fuzzy C-Partition Entropy Approach [J]. Pattern Recognition, 1998, 31(7): 857- 870.
[8] Cocosco C. A., Kollokian V., Kwan R. K. S., et al. BrainWeb: Online interface to a 3-D MRI simulated brain database [J]. NeuroImage, 1997, 5(4): 425
[9] Collins D. L., Zijdenbos A. P., Kollokian V., et al. Design and construction of a realistic digital brain phantom [J]. IEEE Transactions on Med Imaging, 1998, 17(3): 463-468
[10]章毓晋.图像分析[M].第二版.北京:清华大学出版社, 2005: 177-188
[11] Jain A. K., Dubes R. C., Algorithms for Clustering Data[M], NJ:Prentice Hall, 1988.
[12] Jain A. K., Murty M. N., Flynn P. J., Data Clustering: a Review[J]. ACM Computing Surveys, 1999, 31(3): 264-323
[13] Han J., Kambr M., Data Ming:Concepts and Techniques[M], Academic Press, 2000
[14] Coleman G. B., Andrews H. C., Image Segmentation by Clustering[J]. Proc.IEEE, 1979, 5(67): 773-785
[15]廖亮.基于核聚类分析的MRI图像分割算法与灰度有偏场估计的研究[D].广州:华南理工大学, 2008
[16] Pal.S.K., A note on quantitative measure of image enhancement through fuzziness[J]. IEEE Trans onPattern Analysis and Machine Intelligence,1982,4(2): 204-208
[17] Pal S. K, King R. A., Image enhancement using smoothing with fuzzy sets[J]. IEEE Trans on System, Man and Cybernetics, 1981, 11(7): 494-501
[18] Clark M C, Hall L O, Goldgof D B, et al.Automatic Tumor Segmentation Using knowledge-Based Technique[J]. IEEE Trans. Medical Imaging,1998,17(1): 187-191.
[19] Bellman R., Kalaba R., Zadeh L. A. Abstraction and Pattern Classification [J]. Journal of Mathematical Analysis and Applications, 1966, 13: 1-7
[20] Liao Liang, Lin Tusheng, Zhang Weidong. MR brain image segmentation based on kernelized fuzzy clustering using fuzzy Gibbs random field model [C] IEEE/ICME International Conference on Complex Medical Engineering, 2007: 524-530
[21] R.Duda and P.Hart, Pattern classification and scene analysis[M].New York:Wiley,1973
[22] J.C.Bezdek, Fuzzy mathematics in pattern classification[D], NY: Cornell Univ., 1973
[23] J.C.Bezdek, Pattern recognition with fuzzy objective function algorithms[M]. NewYork:Plenum,1981
[24] Y.S.Cheung, K.P.Chan. Modified fuzzy ISODATA for the classification of handwritten Chinese characters[A]. Proc.1986, Int.Conf. Chinese Comput[C].Singapore, 1986: 361-364
[25] N.R.Pal and J.C.Bezdek. On cluster validity for the fuzzy c-means model[j]. IEEETrans.Fuzzy Systems,1995,3(3): 370-379
[26] Chen, S. C., Zhang, D. Q. Robust image segmentation using FCM with spatial constraints based on new kernel-induced distance measure [J]. IEEE Transactions on Systems, Man and Cybernetics. 2004, 34(4): 1907-1916
[27] Liao L., Lin T. S., Li B. MRI brain image segmentation and bias field correction based on fast spatially constrained kernel clustering approach [J]. Patter Recognition Letters, 2008, 29(10) : 1580-1588
[28] Liao L., Lin T. S. A fast spatial constrained fuzzy kernel clustering algorithm for MRI brain image segmentation [A]. Proceedings of the International Conference on Wavelet and Pattern Recognition (ICWAPR2007) [C], New York, IEEE press, 2007: 82-87
[29] Liew A.W. C., Yan H., An adaptive spatial fuzzy clustering algorithm for 3-D MR image segmentation [J]. IEEE Transactions on Medical Imaging. 2003, 22(9): 1063-1075
[30] Chiang J. H., Hao P. Y. A new kernel-based fuzzy clustering approach: support vector clustering with cell growing [J]. IEEE Trans on fuzzy system, 2003, 11(4) : 518-527
[31] Pham D. L., Prince J. L. Adaptive fuzzy segmentation of magnetic resonance images [J].IEEE Trans Med Imag, 1999, 18(9): 737–752
[32] Krishnapuram R., Keller J. A possibilistic approach to clustering [J]. IEEE Transactions on Fuzzy System, 1993, 1(2) : 85-110
[33]张莉,周伟达,焦李成.尺度核函数支撑矢量机[J].电子学报, 2002, 30(4):527-529.
[34] Muller K. R., Mika S., Ratsch G. An introduction to kernel-based learning algorithms [J]. IEEE Trans Neural Networks, 2001, 12(2) : 181–202
[35] Sch?lkopf B., Smola A. J., Muller K. R. Nonlinear component analysis as a kernel eigenvalue problem [J]. Neural Computation, 1998, 10(5):1299-1319
[36] Cristianini N., Taylor J. S. An introduction to Support Vector Machines and other kernel-based learning methods [M]. Cambridge: Cambridge University Press, 2000: 27-50
[37] Roth V., Steinhage V. Nonlinear discriminant analysis using kernel functions [J].Advances in Neural Information Processing Systems, 2000, 12: 568-574
[38] Baudat A. F. Generalized discriminant analysis using a kernel approach [J]. Neural Computation, 2000, 12(10): 2385-2404
[39] Sch?lkopf B., Mika S., Burges C. J. C., et al. Input space versus feature space in kernel-based methods [J]. IEEE Trans on Neural Networks, 1999, 10(5): 1000-1017
[40]张莉.支撑矢量机与核方法研究[D].西安:西安电子科技大学,2002年
[41]周伟达.核机器学习方法研究[D].西安:西安电子科技大学, 2003年
[42] Xu R., Wunsch D. Survey of clustering Algorithms [J]. IEEE Trans on Neural networks,2005, 16(3): 645-678
[43] Zhang D. Q., Chen S. C., A comment on "Alternative c-means clustering algorithms[J].Pattern Recognition, 2004, 37(2):173-174
[44]张道强.基于核的联想记忆及聚类算法的研究与应用[D].南京:南京航空航天大学, 2004
[45] Zhang D. Q., Chen S. C. A novel kernelized fuzzy C-means algorithm with application in medical image segmentation [J]. Artificial Intelligence in Medicine, 2004, 32(1):37-50
[46] Baudat A. F. Generalized discriminant analysis using a kernel approach [J]. Neural Computation, 2000, 12(10): 2385-2404
[47] Chapelle O., Vapnik V. N., Bousquet O., et al. Choosing multiple parameters for support vector machines [J]. Machine Learning, 2002, 46(1): 131-159
[48] Zhang D. Q., Chen S. C., Zhou Z. H. Learning the kernel parameters in kernel minimum distance classifier [J]. Pattern Recognition, 2006, 39(1):133-135
[49] Sijbers J., Dekker A. J. D., Scheunders P., et al. Maximum-likelihood estimation of Rician distribution parameters [J], IEEE Trans Med Imag., 1998, 17(3): 357–361
[50] Guillemaud R., Brady M. Estimating the bias field of MR images [J]. IEEE trans on medical imaging, 1997, 16(3): 238-251
[51] Vovk U., Pernu? F., Likar B. A review of methods for correction of intensity inhomogeneity in MRI [J]. IEEE Trans Med Imaging, 2007, 26(3): 405–421
[52] Ahmed M. N., Yamany S. M., Mohamed N, et al. A modified fuzzy c-means algorithm for bias field estimation and segmentation of MRI data [J]. IEEE Transactions on Medical Imaging. 2002, 21(3) 193-199
[2]杨润玲,高新波.基于加权模糊C均值聚类的快速图像自动分割算法[J].中国图象图形学报, 2007, 12(12): 2105-2112
[3] Liu Y., Chen K, Liao X. F., et al. A genetic clustering method for intrusion detection[J]. Pattern Recognition, 2004 : 37(5): 927-942
[4] Xu X., Jager J., Kriegel H. P. A fast parallel clustering algorithm for large spatial databases[J]. Data Mining and Knowledge Discovery, 1999, 3(3): 263-290
[5] Xu. C. Y., Prince J. L. Snakes, shapes and gradient vector flow[J]. IEEE Transactions on Image Processing, 1998, 7(3): 359- 369
[6]田捷,韩博闻,王岩等.模糊C均值聚类法在医学图像分析中的应用[J].软件学报, 2001, 12(11): 1623-1629.
[7] Cheng H. D., Chen J.R., Li J. G. Threshold Selection Based on Fuzzy C-Partition Entropy Approach [J]. Pattern Recognition, 1998, 31(7): 857- 870.
[8] Cocosco C. A., Kollokian V., Kwan R. K. S., et al. BrainWeb: Online interface to a 3-D MRI simulated brain database [J]. NeuroImage, 1997, 5(4): 425
[9] Collins D. L., Zijdenbos A. P., Kollokian V., et al. Design and construction of a realistic digital brain phantom [J]. IEEE Transactions on Med Imaging, 1998, 17(3): 463-468
[10]章毓晋.图像分析[M].第二版.北京:清华大学出版社, 2005: 177-188
[11] Jain A. K., Dubes R. C., Algorithms for Clustering Data[M], NJ:Prentice Hall, 1988.
[12] Jain A. K., Murty M. N., Flynn P. J., Data Clustering: a Review[J]. ACM Computing Surveys, 1999, 31(3): 264-323
[13] Han J., Kambr M., Data Ming:Concepts and Techniques[M], Academic Press, 2000
[14] Coleman G. B., Andrews H. C., Image Segmentation by Clustering[J]. Proc.IEEE, 1979, 5(67): 773-785
[15]廖亮.基于核聚类分析的MRI图像分割算法与灰度有偏场估计的研究[D].广州:华南理工大学, 2008
[16] Pal.S.K., A note on quantitative measure of image enhancement through fuzziness[J]. IEEE Trans onPattern Analysis and Machine Intelligence,1982,4(2): 204-208
[17] Pal S. K, King R. A., Image enhancement using smoothing with fuzzy sets[J]. IEEE Trans on System, Man and Cybernetics, 1981, 11(7): 494-501
[18] Clark M C, Hall L O, Goldgof D B, et al.Automatic Tumor Segmentation Using knowledge-Based Technique[J]. IEEE Trans. Medical Imaging,1998,17(1): 187-191.
[19] Bellman R., Kalaba R., Zadeh L. A. Abstraction and Pattern Classification [J]. Journal of Mathematical Analysis and Applications, 1966, 13: 1-7
[20] Liao Liang, Lin Tusheng, Zhang Weidong. MR brain image segmentation based on kernelized fuzzy clustering using fuzzy Gibbs random field model [C] IEEE/ICME International Conference on Complex Medical Engineering, 2007: 524-530
[21] R.Duda and P.Hart, Pattern classification and scene analysis[M].New York:Wiley,1973
[22] J.C.Bezdek, Fuzzy mathematics in pattern classification[D], NY: Cornell Univ., 1973
[23] J.C.Bezdek, Pattern recognition with fuzzy objective function algorithms[M]. NewYork:Plenum,1981
[24] Y.S.Cheung, K.P.Chan. Modified fuzzy ISODATA for the classification of handwritten Chinese characters[A]. Proc.1986, Int.Conf. Chinese Comput[C].Singapore, 1986: 361-364
[25] N.R.Pal and J.C.Bezdek. On cluster validity for the fuzzy c-means model[j]. IEEETrans.Fuzzy Systems,1995,3(3): 370-379
[26] Chen, S. C., Zhang, D. Q. Robust image segmentation using FCM with spatial constraints based on new kernel-induced distance measure [J]. IEEE Transactions on Systems, Man and Cybernetics. 2004, 34(4): 1907-1916
[27] Liao L., Lin T. S., Li B. MRI brain image segmentation and bias field correction based on fast spatially constrained kernel clustering approach [J]. Patter Recognition Letters, 2008, 29(10) : 1580-1588
[28] Liao L., Lin T. S. A fast spatial constrained fuzzy kernel clustering algorithm for MRI brain image segmentation [A]. Proceedings of the International Conference on Wavelet and Pattern Recognition (ICWAPR2007) [C], New York, IEEE press, 2007: 82-87
[29] Liew A.W. C., Yan H., An adaptive spatial fuzzy clustering algorithm for 3-D MR image segmentation [J]. IEEE Transactions on Medical Imaging. 2003, 22(9): 1063-1075
[30] Chiang J. H., Hao P. Y. A new kernel-based fuzzy clustering approach: support vector clustering with cell growing [J]. IEEE Trans on fuzzy system, 2003, 11(4) : 518-527
[31] Pham D. L., Prince J. L. Adaptive fuzzy segmentation of magnetic resonance images [J].IEEE Trans Med Imag, 1999, 18(9): 737–752
[32] Krishnapuram R., Keller J. A possibilistic approach to clustering [J]. IEEE Transactions on Fuzzy System, 1993, 1(2) : 85-110
[33]张莉,周伟达,焦李成.尺度核函数支撑矢量机[J].电子学报, 2002, 30(4):527-529.
[34] Muller K. R., Mika S., Ratsch G. An introduction to kernel-based learning algorithms [J]. IEEE Trans Neural Networks, 2001, 12(2) : 181–202
[35] Sch?lkopf B., Smola A. J., Muller K. R. Nonlinear component analysis as a kernel eigenvalue problem [J]. Neural Computation, 1998, 10(5):1299-1319
[36] Cristianini N., Taylor J. S. An introduction to Support Vector Machines and other kernel-based learning methods [M]. Cambridge: Cambridge University Press, 2000: 27-50
[37] Roth V., Steinhage V. Nonlinear discriminant analysis using kernel functions [J].Advances in Neural Information Processing Systems, 2000, 12: 568-574
[38] Baudat A. F. Generalized discriminant analysis using a kernel approach [J]. Neural Computation, 2000, 12(10): 2385-2404
[39] Sch?lkopf B., Mika S., Burges C. J. C., et al. Input space versus feature space in kernel-based methods [J]. IEEE Trans on Neural Networks, 1999, 10(5): 1000-1017
[40]张莉.支撑矢量机与核方法研究[D].西安:西安电子科技大学,2002年
[41]周伟达.核机器学习方法研究[D].西安:西安电子科技大学, 2003年
[42] Xu R., Wunsch D. Survey of clustering Algorithms [J]. IEEE Trans on Neural networks,2005, 16(3): 645-678
[43] Zhang D. Q., Chen S. C., A comment on "Alternative c-means clustering algorithms[J].Pattern Recognition, 2004, 37(2):173-174
[44]张道强.基于核的联想记忆及聚类算法的研究与应用[D].南京:南京航空航天大学, 2004
[45] Zhang D. Q., Chen S. C. A novel kernelized fuzzy C-means algorithm with application in medical image segmentation [J]. Artificial Intelligence in Medicine, 2004, 32(1):37-50
[46] Baudat A. F. Generalized discriminant analysis using a kernel approach [J]. Neural Computation, 2000, 12(10): 2385-2404
[47] Chapelle O., Vapnik V. N., Bousquet O., et al. Choosing multiple parameters for support vector machines [J]. Machine Learning, 2002, 46(1): 131-159
[48] Zhang D. Q., Chen S. C., Zhou Z. H. Learning the kernel parameters in kernel minimum distance classifier [J]. Pattern Recognition, 2006, 39(1):133-135
[49] Sijbers J., Dekker A. J. D., Scheunders P., et al. Maximum-likelihood estimation of Rician distribution parameters [J], IEEE Trans Med Imag., 1998, 17(3): 357–361
[50] Guillemaud R., Brady M. Estimating the bias field of MR images [J]. IEEE trans on medical imaging, 1997, 16(3): 238-251
[51] Vovk U., Pernu? F., Likar B. A review of methods for correction of intensity inhomogeneity in MRI [J]. IEEE Trans Med Imaging, 2007, 26(3): 405–421
[52] Ahmed M. N., Yamany S. M., Mohamed N, et al. A modified fuzzy c-means algorithm for bias field estimation and segmentation of MRI data [J]. IEEE Transactions on Medical Imaging. 2002, 21(3) 193-199