# American Institute of Mathematical Sciences

February  2014, 8(1): 293-320. doi: 10.3934/ipi.2014.8.293

## A local information based variational model for selective image segmentation

 1 School of Mathematical Sciences, Dalian University of Technology, Dalian, Liaoning, 116024, China 2 Centre for Mathematical Imaging Techniques and Department of Mathematical Sciences, University of Liverpool, Liverpool L69 7ZL, United Kingdom 3 School of Mathematical Science, Dalian University of Technology, Dalian, Liaoning 116024 4 Radiology Department, Royal Liverpool University Hospitals, Prescot Street, Liverpool L7 8XP, United Kingdom

Received  July 2011 Revised  November 2012 Published  March 2014

Many effective models are available for segmentation of an image to extract all homogenous objects within it. For applications where segmentation of a single object identifiable by geometric constraints within an image is desired, much less work has been done for this purpose. This paper presents an improved selective segmentation model, without `balloon' force, combining geometrical constraints and local image intensity information around zero level set, aiming to overcome the weakness of getting spurious solutions by Badshah and Chen's model [8]. A key step in our new strategy is an adaptive local band selection algorithm. Numerical experiments show that the new model appears to be able to detect an object possessing highly complex and nonconvex features, and to produce desirable results in terms of segmentation quality and robustness.
Citation: Jianping Zhang, Ke Chen, Bo Yu, Derek A. Gould. A local information based variational model for selective image segmentation. Inverse Problems & Imaging, 2014, 8 (1) : 293-320. doi: 10.3934/ipi.2014.8.293
##### References:
 [1] D. Adalsteinsson and J. A. Sethian, A fast level set method for propagating interfaces, J. Comput. Phys., 118 (1995), 269-277. doi: 10.1006/jcph.1995.1098.  Google Scholar [2] D. Adalsteinsson and J. A. Sethian, A level set approach to a unified model for etching, deposition, and lithography. II. Three-dimensional simulations, J. Comput. Phys., 122 (1995), 348-366. doi: 10.1006/jcph.1995.1221.  Google Scholar [3] L. Ambrosio and V. Tortorelli, Approximation of functionals depending on jumps by elliptic functionals via $\Gamma$-convergence, Commu. Pure and Applied Math., 43 (1990), 999-1036. doi: 10.1002/cpa.3160430805.  Google Scholar [4] A. Araujo, S. Barbeiro and P. Serranho, Stability of Finite Difference Schemes for Complex Diffusion Processes, Pre-print, Departamento de Matematica da Universidade de Coimbra, DMUC report 10-23, 2010. doi: 10.1137/110825789.  Google Scholar [5] G. Aubert and P. Kornprobst, Mathematical Problems in Image Processing, Springer, New York, 2002.  Google Scholar [6] N. Badshah and K. Chen, Multigrid method for the Chan-Vese model in variational segmentation, Communications in Computational Physics, 4 (2008), 294-316.  Google Scholar [7] N. Badshah and K. Chen, On two multigrid algorithms for modeling variation multiphase image segmentation, IEEE Trans. Image Processing, 18 (2009), 1097-1106. doi: 10.1109/TIP.2009.2014260.  Google Scholar [8] N. Badshah and K. Chen, Image selective segmentation under geometrical constraints using an active contour approach, Commun. Comput. Phys., 7 (2010), 759-778. doi: 10.4208/cicp.2009.09.026.  Google Scholar [9] X. Bresson, S. Esedoglu, P. Vandergheynst, J. Thiran and S. Osher, Fast global minimization of the active contour/snake models, J. Math. Imaging and Vision, 28 (2007), 151-167. doi: 10.1007/s10851-007-0002-0.  Google Scholar [10] E. S. Brown, T. F. Chan and X. Bresson, A convex approach for multi-phase piecewise constant Mumford-Shah image segmentation, Int. J. Computer Vision, 98 (2012), 103-121. doi: 10.1007/s11263-011-0499-y.  Google Scholar [11] E. S. Brown, T. F. Chan and X. Bresson, A convex relaxation method for a class of vector-valued minimization problems with applications to Mumford-Shah segmentation, UCLA CAM report 10-43, 2010. Google Scholar [12] M. Burger, G. Gilboa, S. Osher and J. Xu, Nonlinear inverse scale space methods, Commun. Math. Sci., 4 (2006), 179-212. doi: 10.4310/CMS.2006.v4.n1.a7.  Google Scholar [13] J. F. Canny, Finding Edges and Lines in Images, Technical Report AITR-720, Massachusetts Institute of Technology, Artificial Intelligence Laboratory, 1983. Google Scholar [14] V. Caselles, R. Kimmel and G. Sapiro, Geodesic active contours, Int. J. Computer Vision, 22 (1997), 61-79. doi: 10.1023/A:1007979827043.  Google Scholar [15] T. F. Chan, S. Esedoglu and M. Nikolova, Algorithms for finding global minimizers of image segmentation and denoising models, SIAM J. Applied Mathematics, 66 (2006), 1632-1648. doi: 10.1137/040615286.  Google Scholar [16] T. F. Chan, B. Y. Sandberg and L. A. Vese, Active contours without edges for vector-valued images, J. Visual Commun. Image Representation, 11 (2000), 130-141. doi: 10.1006/jvci.1999.0442.  Google Scholar [17] T. F. Chan and L. A. Vese, An efficient variational multiphase motion for the Mumford-Shah segmentation model, Proc. Asilomar Conf. Signals, Systems, Computers, 1 (2000), 490-494. doi: 10.1109/ACSSC.2000.911004.  Google Scholar [18] T. F. Chan and L. Vese, Active coutours without edges, IEEE Trans. Image Processing, 10 (2001), 266-277. doi: 10.1109/83.902291.  Google Scholar [19] T. F. Chan and J. H. Shen, Image Processing and Analysis: Variational, PDE, Wavelet and Stochastic Methods, SIAM, Philadelphia, 2005. doi: 10.1137/1.9780898717877.  Google Scholar [20] G. Gilboa, N. Sochen and Y. Zeeni, Image enhancement and denoising by complex diffusion processes, IEEE Trans Pattern Anal. Mach. Intell., 26 (2004), 1020-1036. doi: 10.1109/TPAMI.2004.47.  Google Scholar [21] T. Goldstein, X. Bresson and S. Osher, Geometric applications of the split Bregman method: Segmentation and surface reconstruction, J. Sci. Computing, 45 (2010), 272-293. doi: 10.1007/s10915-009-9331-z.  Google Scholar [22] C. Gout, C. Le Guyader and L. A. Vese, Segmentation under geometrical consitions with geodesic active contour and interpolation using level set methods, Numerical Algorithms, 39 (2005), 155-173. doi: 10.1007/s11075-004-3627-8.  Google Scholar [23] C. Le Guyader, N. Forcadel and C. Gout, Image segmentation using a generalized fast marching method, Numerical Algorithms, 48 (2008), 189-212. doi: 10.1007/s11075-008-9183-x.  Google Scholar [24] M. Jeon, M. Alexander, W. Pedrycz and N. Pizzi, Unsupervised hierarchical image segmentation with level set and additive operator splitting, Pattern Recogn. Lett., 26 (2005), 1461-1469. doi: 10.1016/j.patrec.2004.11.023.  Google Scholar [25] M. Kass, A. Witkin and D. Terzopoulos, Snake: Active contour models, Int. J. Computer Vision, 1 (1988), 321-331. doi: 10.1007/BF00133570.  Google Scholar [26] S. Lankton and A. Tannenbaum, Localizing region-based active contours, IEEE Trans. Image Processing, 17 (2008), 2029-2039. doi: 10.1109/TIP.2008.2004611.  Google Scholar [27] C. Li, C. Kao, J. Gore and Z. Ding, Implicit active contours driven by local binary fitting energy, Proceedings of IEEE Conference on Computer Vision and Pattern Recognition (CVPR) (Washington, DC, USA), IEEE Computer Society, (2007), 1-7. doi: 10.1109/CVPR.2007.383014.  Google Scholar [28] F. Li, M. K. Ng and C. Li, Variational fuzzy Mumford-Shah model for image segmentation, SIAM J. Appl. Math., 70 (2010), 2750-2770. doi: 10.1137/090753887.  Google Scholar [29] J. Lie, M. Lysaker and X.-C. Tai, A binary level set model and some applications to Mumford-Shah image segmentation, IEEE Trans. Image Processing, 15 (2006), 1171-1181. doi: 10.1109/TIP.2005.863956.  Google Scholar [30] R. Malladi, J. A. Sethian and B. C. Vemuri, Shape modeling with front propagation: A level set approach, IEEE Trans. Pattern Anal. Mach. Intell., 17 (1995), 158-175. doi: 10.1109/34.368173.  Google Scholar [31] A. Marquina and S. Osher, Explicit algorithms for a new time dependent model based on level set motion for nonlinear deblurring and noise removal, SIAM J. Sci. Computing, 22 (2000), 387-405. doi: 10.1137/S1064827599351751.  Google Scholar [32] H. Mewada and S. Patnaik, Variable kernel based Chan-Vese model for image segmentation, Annual IEEE India Conference (INDICON), (2009), 1-4. doi: 10.1109/INDCON.2009.5409429.  Google Scholar [33] J. Mille, Narrow band region-based active contours and surfaces for 2D and 3D segmentation, Computer Vision and Image Understanding, 113 (2009), 946-965. doi: 10.1016/j.cviu.2009.05.002.  Google Scholar [34] D. Mumford and J. Shah, Optimal approximation by piecewise smooth functions and associated variational problem, Commun. Pure Appl. Math., 42 (1989), 577-685. doi: 10.1002/cpa.3160420503.  Google Scholar [35] S. Osher and R. Fedkiw, Level Set Methods and Dynamic Implicit Surfaces, Springer Verlag, 2005.  Google Scholar [36] S. Osher and 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.  Google Scholar [37] D. P. Peng, B. Merriman, S. Osher, H. K. Zhao and M. Kang, A PDE-Based fast local level set method, J. Comput. Phys., 155 (1999), 410-438. doi: 10.1006/jcph.1999.6345.  Google Scholar [38] J. M. S. Prewitt, Object enhancement and extraction, in Picture Processing and Psychopictorics, (eds. B. S. Lipkin and A. Rosenfeld), New York: Academic, (1970), 75-149. Google Scholar [39] J. A. Sethian, Fast marching methods, SIAM Review, 41 (1999), 199-235. doi: 10.1137/S0036144598347059.  Google Scholar [40] J. H. Shen, $\Gamma$-Convergence approximation to piecewise constant Mumford-Shah segmentation, Advanced Concepts for Intelligent Vision Systems, 3708 (2005), 499-506. doi: 10.1007/11558484_63.  Google Scholar [41] I. Sobel, An isotropic $3\times3$ image gradient operator, Machine Vision for Three-Dimention Scenes, (ed. H. Freeman), (1990), 376-379. Google Scholar [42] M. Sussman, P. Smereka and S. Osher, A level set approach for computing solutions to incompressible two-phase flow, J. Comput. Phys., 114 (1994), 146-159. doi: 10.1006/jcph.1994.1155.  Google Scholar [43] X. C. Tai, O. Christiansen, P. Lin and I. Skjaelaaen, Image segmentation using some piecewise constant level set methods with MBO type of projection, Int. J. Computer Vision, 73 (2007), 61-76. doi: 10.1007/s11263-006-9140-x.  Google Scholar [44] L. A. Vese and T. F. Chan, A multiphase level set framework for image segmentation using the Mumford and Shah model, Int. J. Computer Vision, 50 (2002), 271-293. Google Scholar [45] H. K. Zhao, T. F. Chan, B. Merriman and S. Osher, A variational level set approach to multiphase motion, J. Comput. Phys., 127 (1996), 179-195. doi: 10.1006/jcph.1996.0167.  Google Scholar

show all references

##### References:
 [1] D. Adalsteinsson and J. A. Sethian, A fast level set method for propagating interfaces, J. Comput. Phys., 118 (1995), 269-277. doi: 10.1006/jcph.1995.1098.  Google Scholar [2] D. Adalsteinsson and J. A. Sethian, A level set approach to a unified model for etching, deposition, and lithography. II. Three-dimensional simulations, J. Comput. Phys., 122 (1995), 348-366. doi: 10.1006/jcph.1995.1221.  Google Scholar [3] L. Ambrosio and V. Tortorelli, Approximation of functionals depending on jumps by elliptic functionals via $\Gamma$-convergence, Commu. Pure and Applied Math., 43 (1990), 999-1036. doi: 10.1002/cpa.3160430805.  Google Scholar [4] A. Araujo, S. Barbeiro and P. Serranho, Stability of Finite Difference Schemes for Complex Diffusion Processes, Pre-print, Departamento de Matematica da Universidade de Coimbra, DMUC report 10-23, 2010. doi: 10.1137/110825789.  Google Scholar [5] G. Aubert and P. Kornprobst, Mathematical Problems in Image Processing, Springer, New York, 2002.  Google Scholar [6] N. Badshah and K. Chen, Multigrid method for the Chan-Vese model in variational segmentation, Communications in Computational Physics, 4 (2008), 294-316.  Google Scholar [7] N. Badshah and K. Chen, On two multigrid algorithms for modeling variation multiphase image segmentation, IEEE Trans. Image Processing, 18 (2009), 1097-1106. doi: 10.1109/TIP.2009.2014260.  Google Scholar [8] N. Badshah and K. Chen, Image selective segmentation under geometrical constraints using an active contour approach, Commun. Comput. Phys., 7 (2010), 759-778. doi: 10.4208/cicp.2009.09.026.  Google Scholar [9] X. Bresson, S. Esedoglu, P. Vandergheynst, J. Thiran and S. Osher, Fast global minimization of the active contour/snake models, J. Math. Imaging and Vision, 28 (2007), 151-167. doi: 10.1007/s10851-007-0002-0.  Google Scholar [10] E. S. Brown, T. F. Chan and X. Bresson, A convex approach for multi-phase piecewise constant Mumford-Shah image segmentation, Int. J. Computer Vision, 98 (2012), 103-121. doi: 10.1007/s11263-011-0499-y.  Google Scholar [11] E. S. Brown, T. F. Chan and X. Bresson, A convex relaxation method for a class of vector-valued minimization problems with applications to Mumford-Shah segmentation, UCLA CAM report 10-43, 2010. Google Scholar [12] M. Burger, G. Gilboa, S. Osher and J. Xu, Nonlinear inverse scale space methods, Commun. Math. Sci., 4 (2006), 179-212. doi: 10.4310/CMS.2006.v4.n1.a7.  Google Scholar [13] J. F. Canny, Finding Edges and Lines in Images, Technical Report AITR-720, Massachusetts Institute of Technology, Artificial Intelligence Laboratory, 1983. Google Scholar [14] V. Caselles, R. Kimmel and G. Sapiro, Geodesic active contours, Int. J. Computer Vision, 22 (1997), 61-79. doi: 10.1023/A:1007979827043.  Google Scholar [15] T. F. Chan, S. Esedoglu and M. Nikolova, Algorithms for finding global minimizers of image segmentation and denoising models, SIAM J. Applied Mathematics, 66 (2006), 1632-1648. doi: 10.1137/040615286.  Google Scholar [16] T. F. Chan, B. Y. Sandberg and L. A. Vese, Active contours without edges for vector-valued images, J. Visual Commun. Image Representation, 11 (2000), 130-141. doi: 10.1006/jvci.1999.0442.  Google Scholar [17] T. F. Chan and L. A. Vese, An efficient variational multiphase motion for the Mumford-Shah segmentation model, Proc. Asilomar Conf. Signals, Systems, Computers, 1 (2000), 490-494. doi: 10.1109/ACSSC.2000.911004.  Google Scholar [18] T. F. Chan and L. Vese, Active coutours without edges, IEEE Trans. Image Processing, 10 (2001), 266-277. doi: 10.1109/83.902291.  Google Scholar [19] T. F. Chan and J. H. Shen, Image Processing and Analysis: Variational, PDE, Wavelet and Stochastic Methods, SIAM, Philadelphia, 2005. doi: 10.1137/1.9780898717877.  Google Scholar [20] G. Gilboa, N. Sochen and Y. Zeeni, Image enhancement and denoising by complex diffusion processes, IEEE Trans Pattern Anal. Mach. Intell., 26 (2004), 1020-1036. doi: 10.1109/TPAMI.2004.47.  Google Scholar [21] T. Goldstein, X. Bresson and S. Osher, Geometric applications of the split Bregman method: Segmentation and surface reconstruction, J. Sci. Computing, 45 (2010), 272-293. doi: 10.1007/s10915-009-9331-z.  Google Scholar [22] C. Gout, C. Le Guyader and L. A. Vese, Segmentation under geometrical consitions with geodesic active contour and interpolation using level set methods, Numerical Algorithms, 39 (2005), 155-173. doi: 10.1007/s11075-004-3627-8.  Google Scholar [23] C. Le Guyader, N. Forcadel and C. Gout, Image segmentation using a generalized fast marching method, Numerical Algorithms, 48 (2008), 189-212. doi: 10.1007/s11075-008-9183-x.  Google Scholar [24] M. Jeon, M. Alexander, W. Pedrycz and N. Pizzi, Unsupervised hierarchical image segmentation with level set and additive operator splitting, Pattern Recogn. Lett., 26 (2005), 1461-1469. doi: 10.1016/j.patrec.2004.11.023.  Google Scholar [25] M. Kass, A. Witkin and D. Terzopoulos, Snake: Active contour models, Int. J. Computer Vision, 1 (1988), 321-331. doi: 10.1007/BF00133570.  Google Scholar [26] S. Lankton and A. Tannenbaum, Localizing region-based active contours, IEEE Trans. Image Processing, 17 (2008), 2029-2039. doi: 10.1109/TIP.2008.2004611.  Google Scholar [27] C. Li, C. Kao, J. Gore and Z. Ding, Implicit active contours driven by local binary fitting energy, Proceedings of IEEE Conference on Computer Vision and Pattern Recognition (CVPR) (Washington, DC, USA), IEEE Computer Society, (2007), 1-7. doi: 10.1109/CVPR.2007.383014.  Google Scholar [28] F. Li, M. K. Ng and C. Li, Variational fuzzy Mumford-Shah model for image segmentation, SIAM J. Appl. Math., 70 (2010), 2750-2770. doi: 10.1137/090753887.  Google Scholar [29] J. Lie, M. Lysaker and X.-C. Tai, A binary level set model and some applications to Mumford-Shah image segmentation, IEEE Trans. Image Processing, 15 (2006), 1171-1181. doi: 10.1109/TIP.2005.863956.  Google Scholar [30] R. Malladi, J. A. Sethian and B. C. Vemuri, Shape modeling with front propagation: A level set approach, IEEE Trans. Pattern Anal. Mach. Intell., 17 (1995), 158-175. doi: 10.1109/34.368173.  Google Scholar [31] A. Marquina and S. Osher, Explicit algorithms for a new time dependent model based on level set motion for nonlinear deblurring and noise removal, SIAM J. Sci. Computing, 22 (2000), 387-405. doi: 10.1137/S1064827599351751.  Google Scholar [32] H. Mewada and S. Patnaik, Variable kernel based Chan-Vese model for image segmentation, Annual IEEE India Conference (INDICON), (2009), 1-4. doi: 10.1109/INDCON.2009.5409429.  Google Scholar [33] J. Mille, Narrow band region-based active contours and surfaces for 2D and 3D segmentation, Computer Vision and Image Understanding, 113 (2009), 946-965. doi: 10.1016/j.cviu.2009.05.002.  Google Scholar [34] D. Mumford and J. Shah, Optimal approximation by piecewise smooth functions and associated variational problem, Commun. Pure Appl. Math., 42 (1989), 577-685. doi: 10.1002/cpa.3160420503.  Google Scholar [35] S. Osher and R. Fedkiw, Level Set Methods and Dynamic Implicit Surfaces, Springer Verlag, 2005.  Google Scholar [36] S. Osher and 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.  Google Scholar [37] D. P. Peng, B. Merriman, S. Osher, H. K. Zhao and M. Kang, A PDE-Based fast local level set method, J. Comput. Phys., 155 (1999), 410-438. doi: 10.1006/jcph.1999.6345.  Google Scholar [38] J. M. S. Prewitt, Object enhancement and extraction, in Picture Processing and Psychopictorics, (eds. B. S. Lipkin and A. Rosenfeld), New York: Academic, (1970), 75-149. Google Scholar [39] J. A. Sethian, Fast marching methods, SIAM Review, 41 (1999), 199-235. doi: 10.1137/S0036144598347059.  Google Scholar [40] J. H. Shen, $\Gamma$-Convergence approximation to piecewise constant Mumford-Shah segmentation, Advanced Concepts for Intelligent Vision Systems, 3708 (2005), 499-506. doi: 10.1007/11558484_63.  Google Scholar [41] I. Sobel, An isotropic $3\times3$ image gradient operator, Machine Vision for Three-Dimention Scenes, (ed. H. Freeman), (1990), 376-379. Google Scholar [42] M. Sussman, P. Smereka and S. Osher, A level set approach for computing solutions to incompressible two-phase flow, J. Comput. Phys., 114 (1994), 146-159. doi: 10.1006/jcph.1994.1155.  Google Scholar [43] X. C. Tai, O. Christiansen, P. Lin and I. Skjaelaaen, Image segmentation using some piecewise constant level set methods with MBO type of projection, Int. J. Computer Vision, 73 (2007), 61-76. doi: 10.1007/s11263-006-9140-x.  Google Scholar [44] L. A. Vese and T. F. Chan, A multiphase level set framework for image segmentation using the Mumford and Shah model, Int. J. Computer Vision, 50 (2002), 271-293. Google Scholar [45] H. K. Zhao, T. F. Chan, B. Merriman and S. Osher, A variational level set approach to multiphase motion, J. Comput. Phys., 127 (1996), 179-195. doi: 10.1006/jcph.1996.0167.  Google Scholar
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