# American Institute of Mathematical Sciences

August  2011, 5(3): 645-657. doi: 10.3934/ipi.2011.5.645

## A fast algorithm for global minimization of maximum likelihood based on ultrasound image segmentation

 1 Department of Applied Mathematics, Nanjing University of Science and Technology, Nanjing, Jiangsu 210094, China, China 2 Department of Mathematics, University of Florida, Gainesville, FL 32611

Received  November 2010 Revised  April 2011 Published  August 2011

This paper presents a novel variational model for ultrasound image segmentation that uses a maximum likelihood estimator based on Fisher-Tippett distribution of the intensities of ultrasound images. A convex relaxation method is applied to get a convex model of the subproblem with fixed distribution parameters. The relaxed subproblem, which is convex, can be fast solved by using a primal-dual hybrid gradient algorithm. The experimental results on simulated and real ultrasound images indicate the effectiveness of the method presented.
Citation: Jie Huang, Xiaoping Yang, Yunmei Chen. A fast algorithm for global minimization of maximum likelihood based on ultrasound image segmentation. Inverse Problems and Imaging, 2011, 5 (3) : 645-657. doi: 10.3934/ipi.2011.5.645
##### References:
 [1] V. Caselles, R. Kimmel and G. Sapiro, On geodesic active contours, Int. J. Comput. Vis., 22 (1997), 61-79. doi: 10.1023/A:1007979827043. [2] D. Mumford and J. Shah, Optimal approximation by piecewise smooth functions and associated variational problems, Commun. Pure Appl. Math, 42 (1989), 577-685. doi: 10.1002/cpa.3160420503. [3] S. C. Zhu and A. Yuille, Region competition: Unifying snakes, region growing, and Bayes/MDL for multiband image segmentation, IEEE PAMI, 18 (1996), 884-900. doi: 10.1109/34.537343. [4] N. Paragios and R. Deriche, Geodesic active regions and level set methods for supervised texture segmentation, Int. J. Computer Vision, 46 (2002), 223-247. doi: 10.1023/A:1014080923068. [5] T. F. Chan and L. A. Vese, Active contoure without edges, IEEE Trans. Image Processing, 10 (2001), 266-277. doi: 10.1109/83.902291. [6] I. B. Ayed, C. Vazquez, A. Mitiche and Z. Belhadj, SAR image segmentation with active contours and level sets, Proceedings of IEEE Intl. Conf. Image Process (ICIP), 4 (2004), 2717-2720. [7] Zhong Tao and H. D. Tagare, Evaluation of four probability distribution models for speckle in clinical cardic ultrasound images, IEEE Trans. Medical Imaging, 25 (2006), 1483-1491. doi: 10.1109/TMI.2006.881376. [8] A. Sarti, C. Corsi and E. Mazzini, Maximum likelihood segmentation of ultrasound images with Rayleigh distribution, IEEE Trans. Ultrasonics Ferroelectrics and Frequency Control, 52 (2005), 947-960. doi: 10.1109/TUFFC.2005.1504017. [9] J. M. Thijssen, Ultrasonic speckle formation, analysis and processing applied to tissue characterization, Pattern Recognition Letters, 24 (2003), 659-675. doi: 10.1016/S0167-8655(02)00173-3. [10] S. Osher, J. Sethian, Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations, J. Comput. Phys., 79 (1988), 12-49. doi: 10.1016/0021-9991(88)90002-2. [11] T. F. Chan, S. Esedoḡlu and M. Nikolova, Algorithms for finding global minimizers of image segmentation and denoising models, SIAM J. Appl. Math, 66 (2006), 1632-1648. doi: 10.1137/040615286. [12] X. Bresson, S. Esedoḡlu, P. Vandergheynst, J.-P. Thiran and S. Osher, Fast global minimization of the active contour/snake model, J. Math Imaging Vis., 28 (2007), 151-167. doi: 10.1007/s10851-007-0002-0. [13] T. Goldstein, X. Bresson and S. Osher, Geometric application of the split Bregman method: Segmentation and surface reconstruction, Journal of Scientific Computing, 45 (2010), 272-293. [14] J. Yuan, E. Bae and Xue-Cheng Tai, A study on continuous max-flow and min-cut approaches, 2010 IEEE Conference on Computer Vision and Pattern Recognition, June 2010, 2217-2224. [15] E. S. Brown, T. F. Chan and X. Bresson, "Globally Convex Chan-Vese Image Segmentation," CAM Report, 10-44, UCLA, 2010. [16] E. Bae, J. Yuan and Xue-Cheng Tai, Global minimization for continuous multiphase partitioning problems using a dual approach, Int. J. Comput. Vis., 92 (2011), 112-129. doi: 10.1007/s11263-010-0406-y. [17] Mingqiang Zhu and T. F. Chan, "An Efficient Primal-Dual Hybrid Gradient Algorithm for TV Image Restoration," CAM Report, 8-34, UCLA, 2008. [18] C. B. Burckhardt, Speckle in ultrasound B-mode scans, IEEE Transaction on Sonics and Ultrasonics, 25 (1978), 1-6. [19] R. F. Wanger, S. W. Smith and J. M. Sandrik, Statistics of speckle in ultrasound B-scans, IEEE Transaction on Sonics and Ultrasonics, 30 (1983), 156-163. doi: 10.1109/T-SU.1983.31404. [20] V. Dutt and J. Greenleaf, Statistics of the log-compression envelope, Journal of Acoustical Society of America, 99 (1996), 3817-3825. [21] J. M. Thijssen, B. J. Oosterveld and R. F. Wanger, Gray level transforms and lesion detectabivity in echographic images, Utrason. Imag, 10 (1988), 171-195. [22] E. Esser, Xiaoqun Zhang and T. F. Chan, "A General Framework for a Class of First Order Primal-Dual Algorithms for TV Minimization," CAM Report, 9-67, UCLA, 2009. [23] Zhang Xu, A unified primal-dual algorithm based on l1 and Bregman iteration, private communication, April 2009. [24] J. A. Jensen, Field: A program for simulating ultrasound systems, Biological Engineering and Computing, 34 (1996), 351-353. [25] J. A. Jensen and N. B. Svendsen, Calculation of pressure fields from arbitrarily shaped, apodized, and excited ultrasound transducers, IEEE Trans. Ultrason., Ferroelec., Freq. Contr., 39 (1992), 262-267.

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##### References:
 [1] V. Caselles, R. Kimmel and G. Sapiro, On geodesic active contours, Int. J. Comput. Vis., 22 (1997), 61-79. doi: 10.1023/A:1007979827043. [2] D. Mumford and J. Shah, Optimal approximation by piecewise smooth functions and associated variational problems, Commun. Pure Appl. Math, 42 (1989), 577-685. doi: 10.1002/cpa.3160420503. [3] S. C. Zhu and A. Yuille, Region competition: Unifying snakes, region growing, and Bayes/MDL for multiband image segmentation, IEEE PAMI, 18 (1996), 884-900. doi: 10.1109/34.537343. [4] N. Paragios and R. Deriche, Geodesic active regions and level set methods for supervised texture segmentation, Int. J. Computer Vision, 46 (2002), 223-247. doi: 10.1023/A:1014080923068. [5] T. F. Chan and L. A. Vese, Active contoure without edges, IEEE Trans. Image Processing, 10 (2001), 266-277. doi: 10.1109/83.902291. [6] I. B. Ayed, C. Vazquez, A. Mitiche and Z. Belhadj, SAR image segmentation with active contours and level sets, Proceedings of IEEE Intl. Conf. Image Process (ICIP), 4 (2004), 2717-2720. [7] Zhong Tao and H. D. Tagare, Evaluation of four probability distribution models for speckle in clinical cardic ultrasound images, IEEE Trans. Medical Imaging, 25 (2006), 1483-1491. doi: 10.1109/TMI.2006.881376. [8] A. Sarti, C. Corsi and E. Mazzini, Maximum likelihood segmentation of ultrasound images with Rayleigh distribution, IEEE Trans. Ultrasonics Ferroelectrics and Frequency Control, 52 (2005), 947-960. doi: 10.1109/TUFFC.2005.1504017. [9] J. M. Thijssen, Ultrasonic speckle formation, analysis and processing applied to tissue characterization, Pattern Recognition Letters, 24 (2003), 659-675. doi: 10.1016/S0167-8655(02)00173-3. [10] S. Osher, J. Sethian, Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations, J. Comput. Phys., 79 (1988), 12-49. doi: 10.1016/0021-9991(88)90002-2. [11] T. F. Chan, S. Esedoḡlu and M. Nikolova, Algorithms for finding global minimizers of image segmentation and denoising models, SIAM J. Appl. Math, 66 (2006), 1632-1648. doi: 10.1137/040615286. [12] X. Bresson, S. Esedoḡlu, P. Vandergheynst, J.-P. Thiran and S. Osher, Fast global minimization of the active contour/snake model, J. Math Imaging Vis., 28 (2007), 151-167. doi: 10.1007/s10851-007-0002-0. [13] T. Goldstein, X. Bresson and S. Osher, Geometric application of the split Bregman method: Segmentation and surface reconstruction, Journal of Scientific Computing, 45 (2010), 272-293. [14] J. Yuan, E. Bae and Xue-Cheng Tai, A study on continuous max-flow and min-cut approaches, 2010 IEEE Conference on Computer Vision and Pattern Recognition, June 2010, 2217-2224. [15] E. S. Brown, T. F. Chan and X. Bresson, "Globally Convex Chan-Vese Image Segmentation," CAM Report, 10-44, UCLA, 2010. [16] E. Bae, J. Yuan and Xue-Cheng Tai, Global minimization for continuous multiphase partitioning problems using a dual approach, Int. J. Comput. Vis., 92 (2011), 112-129. doi: 10.1007/s11263-010-0406-y. [17] Mingqiang Zhu and T. F. Chan, "An Efficient Primal-Dual Hybrid Gradient Algorithm for TV Image Restoration," CAM Report, 8-34, UCLA, 2008. [18] C. B. Burckhardt, Speckle in ultrasound B-mode scans, IEEE Transaction on Sonics and Ultrasonics, 25 (1978), 1-6. [19] R. F. Wanger, S. W. Smith and J. M. Sandrik, Statistics of speckle in ultrasound B-scans, IEEE Transaction on Sonics and Ultrasonics, 30 (1983), 156-163. doi: 10.1109/T-SU.1983.31404. [20] V. Dutt and J. Greenleaf, Statistics of the log-compression envelope, Journal of Acoustical Society of America, 99 (1996), 3817-3825. [21] J. M. Thijssen, B. J. Oosterveld and R. F. Wanger, Gray level transforms and lesion detectabivity in echographic images, Utrason. Imag, 10 (1988), 171-195. [22] E. Esser, Xiaoqun Zhang and T. F. Chan, "A General Framework for a Class of First Order Primal-Dual Algorithms for TV Minimization," CAM Report, 9-67, UCLA, 2009. [23] Zhang Xu, A unified primal-dual algorithm based on l1 and Bregman iteration, private communication, April 2009. [24] J. A. Jensen, Field: A program for simulating ultrasound systems, Biological Engineering and Computing, 34 (1996), 351-353. [25] J. A. Jensen and N. B. Svendsen, Calculation of pressure fields from arbitrarily shaped, apodized, and excited ultrasound transducers, IEEE Trans. Ultrason., Ferroelec., Freq. Contr., 39 (1992), 262-267.
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