Retinal vessel segmentation using a finite element based binary level set method

Previous Article
Compressive optical deflectometric tomography: A constrained total-variation minimization approach
IPI Home
This Issue
Next Article
A Rellich type theorem for discrete Schrödinger operators

May 2014, 8(2): 459-473. doi: 10.3934/ipi.2014.8.459

Retinal vessel segmentation using a finite element based binary level set method

Zhenlin Guo ^1, , Ping Lin ^2, , Guangrong Ji ^3, and Yangfan Wang ^4,

1.	Department of Mathematics and Mechanics, University of Science and Technology Beijing (USTB), Beijing, 100083, China
2.	Division of Mathematics, University of Dundee, Dundee, DD1 4HN, United Kingdom
3.	College of Information Science and Engineering, Ocean University of China, Qingdao, 266071, China
4.	College of Marine Life Science, Ocean University of China, Qingdao, 266071, China

Received July 2012 Revised October 2013 Published May 2014

Abstract
Related Papers

In this paper we combine a few techniques to label blood vessels in the matched filter (MF) response image by using a finite element based binary level set method. An operator-splitting method is applied to numerically solve the Euler-Lagrange equation from minimizing an energy functional. Unlike the traditional MF methods, where a threshold is difficult to be selected, our method can automatically get more precise blood vessel segmentation using an enhanced edge information. In order to demonstrate the good performance, we compare our method with a few other methods when they are applied to a publicly available standard database of coloured images (with manual segmentations available too).

Keywords: operator-splitting., Retinal vessel segmentation, binary level set method.

Mathematics Subject Classification: Primary: 65M60, 65J22; Secondary: 94A08, 53C3.

Citation: Zhenlin Guo, Ping Lin, Guangrong Ji, Yangfan Wang. Retinal vessel segmentation using a finite element based binary level set method. Inverse Problems & Imaging, 2014, 8 (2) : 459-473. doi: 10.3934/ipi.2014.8.459

References:

[1]	X. Cai, R. Chan, S. Morigi and F. Sgallari, Framelet-based algorithm for segmentation of tubular structures,, Lecture Notes in Computer Science, 6667* (2012), 411. doi: 10.1007/978-3-642-24785-9_35. Google Scholar*
[2]	A. Can, H. Shen, J. N. Turner, H. L. Tanenbaum and B. Roysam, Rapid automated tracing and feature extraction from retinal fundus images using direct exploratory algorithms,, IEEE Trans. Inform. Technol. Biomed., 3 (1999), 125. Google Scholar
[3]	V. Caselles, F. Catte, T. Coll and F. Dibos, A geometric model for active contours in image processing,, Numer. Math., 66 (1993), 1. doi: 10.1007/BF01385685. Google Scholar
[4]	T. Chan and L. Vese, Active contours without edges,, IEEE Image Proc., 10 (2001), 266. doi: 10.1109/83.902291. Google Scholar
[5]	S. Chaudhuri, S. Chatterjee, N. Katz, M. Nelson and M. Goldbaum, Detection of blood vessel in retinal images using two-dimensional matched filter,, IEEE Trans. Med. Imag., 8 (1989), 263. doi: 10.1109/42.34715. Google Scholar
[6]	N. Cheung, K. Donaghue, G. Liew, L. Rogers, J. Wang, S. Lim, A. Jenkins, W. Hsu, L. Lee and T. Wong, Quantitative Assessment of Early Diabetic Retinopathy Using Fractal Analysis,, Diabetes Care, 32 (2009), 106. Google Scholar
[7]	J. Chen and A. Amini, Quantifying 3D vascular structures in MRA images using hybrid PDE and geometric deformable models,, IEEE Trans. Med. Imag., 10 (2004), 1251. Google Scholar
[8]	O. Chutatape, L. Zheng and S. Krishman, Retinal blood vessel detection and tracking by matched Gaussian and Kalman filters,, in Proc. IEEE Int. Conf. Eng. Biol. Soc., 6 (1998), 3144. doi: 10.1109/IEMBS.1998.746160. Google Scholar
[9]	B. Dong, A. Chien and Z. Shen, Frame based segmentation for medical images,, Commun.Math. Sci., 9 (2011), 551. doi: 10.4310/CMS.2011.v9.n2.a10. Google Scholar
[10]	A. F. Frangi, W. J. Niessen, R. M. Hoogeveen, T. van Walsum and M. A. Viergever, Model-based quantitation of 3-D magnetic resonance angiographic images,, IEEE Trans. Med. Imag. , 18 (1999), 946. doi: 10.1109/42.811279. Google Scholar
[11]	L. Gang, O. Chutatape and S. M. Krishnan, Detection and measurement of retinal vessels in fundus images using amplitude modified second-order Gaussian filter,, IEEE. Trans. Biomed. Eng., 49 (2002), 168. Google Scholar
[12]	R. Glowinski, P. Lin and X. Pan, An operator-splitting method for a liquid crystal model,, Comp Phys. Comm., 152 (2003), 242. doi: 10.1016/S0010-4655(02)00823-8. Google Scholar
[13]	R. Glowinski, P. Lin and X. Pan, A three-stage operator-splitting/finite element method for the numerical simulation of liquid crystal flow,, Int. J. Numer. Anal. Mod., 6 (2009), 440. Google Scholar
[14]	R. Glowinski and P. Tallec, Augmented Lagrangian and Operator-Splitting Methods in Nonlinear Mechanics,, SIAM, (1989). doi: 10.1137/1.9781611970838. Google Scholar
[15]	A. Hoover, V. Kouznetsova and M. Goldbaum, Locating blood vessels in retinal images by piecewise threshold probing of a matched filter response,, IEEE Trans. Med. Imag., 19 (2000), 203. Google Scholar
[16]	J. Hua, P. Lin, C. Liu and Q. Wang, Energy law preserving C0 finite element schemes for phase field models in two-phase flow computations,, J. Comput. Phys., 230 (2011), 7115. doi: 10.1016/j.jcp.2011.05.013. Google Scholar
[17]	X. Jiang and D. Mojon, Adaptive local thresholding by verification based multithreshold probing with application to vessel detection in retinal images,, IEEE Trans. Pattern Anal. Mach. Intell., 25 (2003), 131. Google Scholar
[18]	M. Kass, A. Witkin and D. Terzopoulos, Snakes: Active contour models,, Int. J. Comput. Vis., 1 (1987), 321. doi: 10.1007/BF00133570. Google Scholar
[19]	C. Kirbas and F. K. H. Quek, A review of vessel extraction techniques and algorithms,, ACM Comput. Surv., 36 (2004), 81. doi: 10.1145/1031120.1031121. Google Scholar
[20]	J. Lie, M. Lysaker and X. Tai, A variant of the levelset method and applications to image segmentation,, UCLA CAM 03- 50, (2003). Google Scholar
[21]	J. Lie, M. Lysaker and X. Tai, A binary level set metod and some application to image processing,, UCLA CAM 04-31, (2004), 04. Google Scholar
[22]	J. Lie, M. Lysaker and X. Tai, Piecewise constant level set methods and image segmentation. In Scale Space and PDE Methods in Computer Vision,, Lectures notes in Computer Sciences, 3459 (2005), 573. Google Scholar
[23]	P. Lin and C. Liu, Simulation of singularity dynamics in liquid crystal flows: a C0 finite element approach,, J. Comp. Phys., 215 (2006), 348. doi: 10.1016/j.jcp.2005.10.027. Google Scholar
[24]	C. Liu and J. Shen, A phase field model for the mixture of two incompressible fluids and its approximation by a Fourier-Spectral method,, Phys. D, 179 (2003), 211. doi: 10.1016/S0167-2789(03)00030-7. Google Scholar
[25]	L. M. Lorigo, O. Faugeras, W. E. L. Grimson, R. Keriven, R. Kikins, A. Nabavi and C.-F. Westin, CURVES: Curve evolution for vessel segmentation,, Med. Image. Anal., 5 (2001), 195. doi: 10.1016/S1361-8415(01)00040-8. Google Scholar
[26]	C. Lupascu, D. Tegolo and E. Trucco, FABC: Retinal vessel segmentation using adaBoost,, IEEE Trans. Inf. Technol. Biomed., 14 (2010), 1267. doi: 10.1109/TITB.2010.2052282. Google Scholar
[27]	T. McInerney and D. Terzopoulos, T-snakes: Topology adaptive snakes,, Med. Imag. Anal. , 4 (2000), 73. doi: 10.1016/S1361-8415(00)00008-6. Google Scholar
[28]	T. McInerney and D. Terzopoulos, Deformable models in medical image analysis: A survey,, Med. Image Anal., 1 (1996), 91. doi: 10.1016/S1361-8415(96)80007-7. Google Scholar
[29]	A. M. Mendonca and A. Campilho, Segmentation of Retinal Blood Vessels by Combining the Detection of Centerlines and Morphological Reconstruction,, IEEE Trans. Med. Imag., 25 (2006), 1200. doi: 10.1109/TMI.2006.879955. Google Scholar
[30]	C. E. Metz, Basic principles of ROC analysis,, Seminars Nucl. Med., 8 (1978), 283. doi: 10.1016/S0001-2998(78)80014-2. Google Scholar
[31]	D. Mumford and J. Shah, Optimal approximation by piecewise smooth functions and associated variational problems,, Commun. Pure Appl. Math., 42 (1989), 577. doi: 10.1002/cpa.3160420503. Google Scholar
[32]	M. Niemeijer, J. Staal, B. Ginneken, M. Long and M. D. Abramoff, Comparative study of retinal vessel segmentation methods on a new publicly available database,, Proc. SPIE Med. Imag., 5370* (2004), 648. doi: 10.1117/12.535349. Google Scholar*
[33]	S. Osher and J. A. Sethian, Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi Formulation,, J. Comput. Phys., 79 (1988), 12. doi: 10.1016/0021-9991(88)90002-2. Google Scholar
[34]	Q. Sheng, Recent trends in splitting, adaptive and hybrid numerical methods for differential equations,, Neural, 16 (2008), 283. Google Scholar
[35]	C. Sinthanayothin, J. F. Boyce, T. H. Williamson, H. L. Cook, E. Mensah, S. Lal and D. Usher, Automated detection of diabetic retinopathy on digital fundus images,, Diabetic Med., 19 (2002), 105. doi: 10.1046/j.1464-5491.2002.00613.x. Google Scholar
[36]	J. Soares, J. Leandro, J. Cesar, H. Jelinek and M. Cree, Retinal vessel segmentation using the 2-d gabor wavelet and supervised classification,, IEEE Trans. Med. Imag., 25 (2006), 1214. doi: 10.1109/TMI.2006.879967. Google Scholar
[37]	J. Staal, M. Abramoff, M. Viergever and B. Ginneken, Ridge based vessel segmentation in color images of the retina,, IEEE Trans. Med. Imag., 23 (2004), 501. doi: 10.1109/TMI.2004.825627. Google Scholar
[38]	K. Sum and P. Cheung, Vessel extraction under non-uniform illumination: A level set approach,, IEEE Trans. Biomed. Eng., 55 (2008), 358. doi: 10.1109/TBME.2007.896587. Google Scholar
[39]	X. Tai, O. Christiansen, P. Lin and I. Skjaelaaen, A remark on the MBO scheme and some piecewise constant level set methods,, Int. J. Comput. Vis., 73 (2007), 61. Google Scholar
[40]	T. Walter and J. C. Klein, Segmentation of color fundus images of the human retina: Detection of the optic disc and the vascular tree using morphological techniques,, in Medical Data Analysis, 2199 (2001), 282. doi: 10.1007/3-540-45497-7_43. Google Scholar
[41]	L. Wang, A. Bhalerao and R. Wilson, Analysis of retinal vasculature using a multiresolution hermite model,, IEEE Trans. Med. Imag., 26 (2007), 137. doi: 10.1109/TMI.2006.889732. Google Scholar
[42]	Y. Wang, G. Ji, P. Lin and E. Trucco, Retinal vessel segmentation using matched filter with multiwavelet kernels and multiscale hierachical decomposition,, Pattern Recog., 46 (2013), 2117. Google Scholar
[43]	C. Wu and X. Tai, Augmented Lagrangian method, Dual methods and Split-Bregman Iterations for ROF, vectorial TV and higher order models,, SIAM J. Imag. Sci., 3 (2010), 300. doi: 10.1137/090767558. Google Scholar
[44]	F. Zana and J. C. Klein, Segmentation of vessel-like patterns using mathematical morphology and curvature evaluation,, IEEE Trans. Imag. Proc., 10 (2001), 1010. doi: 10.1109/83.931095. Google Scholar
[45]	D. Zonoobi, A. Kassim and W. Shen, Vasculature segmentation in MRA images using gradient compensated geodesic active contours,, J. Sign. Process. Syst., 54 (2009), 171. doi: 10.1007/s11265-008-0216-4. Google Scholar

show all references

References:

[1]	X. Cai, R. Chan, S. Morigi and F. Sgallari, Framelet-based algorithm for segmentation of tubular structures,, Lecture Notes in Computer Science, 6667* (2012), 411. doi: 10.1007/978-3-642-24785-9_35. Google Scholar*
[2]	A. Can, H. Shen, J. N. Turner, H. L. Tanenbaum and B. Roysam, Rapid automated tracing and feature extraction from retinal fundus images using direct exploratory algorithms,, IEEE Trans. Inform. Technol. Biomed., 3 (1999), 125. Google Scholar
[3]	V. Caselles, F. Catte, T. Coll and F. Dibos, A geometric model for active contours in image processing,, Numer. Math., 66 (1993), 1. doi: 10.1007/BF01385685. Google Scholar
[4]	T. Chan and L. Vese, Active contours without edges,, IEEE Image Proc., 10 (2001), 266. doi: 10.1109/83.902291. Google Scholar
[5]	S. Chaudhuri, S. Chatterjee, N. Katz, M. Nelson and M. Goldbaum, Detection of blood vessel in retinal images using two-dimensional matched filter,, IEEE Trans. Med. Imag., 8 (1989), 263. doi: 10.1109/42.34715. Google Scholar
[6]	N. Cheung, K. Donaghue, G. Liew, L. Rogers, J. Wang, S. Lim, A. Jenkins, W. Hsu, L. Lee and T. Wong, Quantitative Assessment of Early Diabetic Retinopathy Using Fractal Analysis,, Diabetes Care, 32 (2009), 106. Google Scholar
[7]	J. Chen and A. Amini, Quantifying 3D vascular structures in MRA images using hybrid PDE and geometric deformable models,, IEEE Trans. Med. Imag., 10 (2004), 1251. Google Scholar
[8]	O. Chutatape, L. Zheng and S. Krishman, Retinal blood vessel detection and tracking by matched Gaussian and Kalman filters,, in Proc. IEEE Int. Conf. Eng. Biol. Soc., 6 (1998), 3144. doi: 10.1109/IEMBS.1998.746160. Google Scholar
[9]	B. Dong, A. Chien and Z. Shen, Frame based segmentation for medical images,, Commun.Math. Sci., 9 (2011), 551. doi: 10.4310/CMS.2011.v9.n2.a10. Google Scholar
[10]	A. F. Frangi, W. J. Niessen, R. M. Hoogeveen, T. van Walsum and M. A. Viergever, Model-based quantitation of 3-D magnetic resonance angiographic images,, IEEE Trans. Med. Imag. , 18 (1999), 946. doi: 10.1109/42.811279. Google Scholar
[11]	L. Gang, O. Chutatape and S. M. Krishnan, Detection and measurement of retinal vessels in fundus images using amplitude modified second-order Gaussian filter,, IEEE. Trans. Biomed. Eng., 49 (2002), 168. Google Scholar
[12]	R. Glowinski, P. Lin and X. Pan, An operator-splitting method for a liquid crystal model,, Comp Phys. Comm., 152 (2003), 242. doi: 10.1016/S0010-4655(02)00823-8. Google Scholar
[13]	R. Glowinski, P. Lin and X. Pan, A three-stage operator-splitting/finite element method for the numerical simulation of liquid crystal flow,, Int. J. Numer. Anal. Mod., 6 (2009), 440. Google Scholar
[14]	R. Glowinski and P. Tallec, Augmented Lagrangian and Operator-Splitting Methods in Nonlinear Mechanics,, SIAM, (1989). doi: 10.1137/1.9781611970838. Google Scholar
[15]	A. Hoover, V. Kouznetsova and M. Goldbaum, Locating blood vessels in retinal images by piecewise threshold probing of a matched filter response,, IEEE Trans. Med. Imag., 19 (2000), 203. Google Scholar
[16]	J. Hua, P. Lin, C. Liu and Q. Wang, Energy law preserving C0 finite element schemes for phase field models in two-phase flow computations,, J. Comput. Phys., 230 (2011), 7115. doi: 10.1016/j.jcp.2011.05.013. Google Scholar
[17]	X. Jiang and D. Mojon, Adaptive local thresholding by verification based multithreshold probing with application to vessel detection in retinal images,, IEEE Trans. Pattern Anal. Mach. Intell., 25 (2003), 131. Google Scholar
[18]	M. Kass, A. Witkin and D. Terzopoulos, Snakes: Active contour models,, Int. J. Comput. Vis., 1 (1987), 321. doi: 10.1007/BF00133570. Google Scholar
[19]	C. Kirbas and F. K. H. Quek, A review of vessel extraction techniques and algorithms,, ACM Comput. Surv., 36 (2004), 81. doi: 10.1145/1031120.1031121. Google Scholar
[20]	J. Lie, M. Lysaker and X. Tai, A variant of the levelset method and applications to image segmentation,, UCLA CAM 03- 50, (2003). Google Scholar
[21]	J. Lie, M. Lysaker and X. Tai, A binary level set metod and some application to image processing,, UCLA CAM 04-31, (2004), 04. Google Scholar
[22]	J. Lie, M. Lysaker and X. Tai, Piecewise constant level set methods and image segmentation. In Scale Space and PDE Methods in Computer Vision,, Lectures notes in Computer Sciences, 3459 (2005), 573. Google Scholar
[23]	P. Lin and C. Liu, Simulation of singularity dynamics in liquid crystal flows: a C0 finite element approach,, J. Comp. Phys., 215 (2006), 348. doi: 10.1016/j.jcp.2005.10.027. Google Scholar
[24]	C. Liu and J. Shen, A phase field model for the mixture of two incompressible fluids and its approximation by a Fourier-Spectral method,, Phys. D, 179 (2003), 211. doi: 10.1016/S0167-2789(03)00030-7. Google Scholar
[25]	L. M. Lorigo, O. Faugeras, W. E. L. Grimson, R. Keriven, R. Kikins, A. Nabavi and C.-F. Westin, CURVES: Curve evolution for vessel segmentation,, Med. Image. Anal., 5 (2001), 195. doi: 10.1016/S1361-8415(01)00040-8. Google Scholar
[26]	C. Lupascu, D. Tegolo and E. Trucco, FABC: Retinal vessel segmentation using adaBoost,, IEEE Trans. Inf. Technol. Biomed., 14 (2010), 1267. doi: 10.1109/TITB.2010.2052282. Google Scholar
[27]	T. McInerney and D. Terzopoulos, T-snakes: Topology adaptive snakes,, Med. Imag. Anal. , 4 (2000), 73. doi: 10.1016/S1361-8415(00)00008-6. Google Scholar
[28]	T. McInerney and D. Terzopoulos, Deformable models in medical image analysis: A survey,, Med. Image Anal., 1 (1996), 91. doi: 10.1016/S1361-8415(96)80007-7. Google Scholar
[29]	A. M. Mendonca and A. Campilho, Segmentation of Retinal Blood Vessels by Combining the Detection of Centerlines and Morphological Reconstruction,, IEEE Trans. Med. Imag., 25 (2006), 1200. doi: 10.1109/TMI.2006.879955. Google Scholar
[30]	C. E. Metz, Basic principles of ROC analysis,, Seminars Nucl. Med., 8 (1978), 283. doi: 10.1016/S0001-2998(78)80014-2. Google Scholar
[31]	D. Mumford and J. Shah, Optimal approximation by piecewise smooth functions and associated variational problems,, Commun. Pure Appl. Math., 42 (1989), 577. doi: 10.1002/cpa.3160420503. Google Scholar
[32]	M. Niemeijer, J. Staal, B. Ginneken, M. Long and M. D. Abramoff, Comparative study of retinal vessel segmentation methods on a new publicly available database,, Proc. SPIE Med. Imag., 5370* (2004), 648. doi: 10.1117/12.535349. Google Scholar*
[33]	S. Osher and J. A. Sethian, Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi Formulation,, J. Comput. Phys., 79 (1988), 12. doi: 10.1016/0021-9991(88)90002-2. Google Scholar
[34]	Q. Sheng, Recent trends in splitting, adaptive and hybrid numerical methods for differential equations,, Neural, 16 (2008), 283. Google Scholar
[35]	C. Sinthanayothin, J. F. Boyce, T. H. Williamson, H. L. Cook, E. Mensah, S. Lal and D. Usher, Automated detection of diabetic retinopathy on digital fundus images,, Diabetic Med., 19 (2002), 105. doi: 10.1046/j.1464-5491.2002.00613.x. Google Scholar
[36]	J. Soares, J. Leandro, J. Cesar, H. Jelinek and M. Cree, Retinal vessel segmentation using the 2-d gabor wavelet and supervised classification,, IEEE Trans. Med. Imag., 25 (2006), 1214. doi: 10.1109/TMI.2006.879967. Google Scholar
[37]	J. Staal, M. Abramoff, M. Viergever and B. Ginneken, Ridge based vessel segmentation in color images of the retina,, IEEE Trans. Med. Imag., 23 (2004), 501. doi: 10.1109/TMI.2004.825627. Google Scholar
[38]	K. Sum and P. Cheung, Vessel extraction under non-uniform illumination: A level set approach,, IEEE Trans. Biomed. Eng., 55 (2008), 358. doi: 10.1109/TBME.2007.896587. Google Scholar
[39]	X. Tai, O. Christiansen, P. Lin and I. Skjaelaaen, A remark on the MBO scheme and some piecewise constant level set methods,, Int. J. Comput. Vis., 73 (2007), 61. Google Scholar
[40]	T. Walter and J. C. Klein, Segmentation of color fundus images of the human retina: Detection of the optic disc and the vascular tree using morphological techniques,, in Medical Data Analysis, 2199 (2001), 282. doi: 10.1007/3-540-45497-7_43. Google Scholar
[41]	L. Wang, A. Bhalerao and R. Wilson, Analysis of retinal vasculature using a multiresolution hermite model,, IEEE Trans. Med. Imag., 26 (2007), 137. doi: 10.1109/TMI.2006.889732. Google Scholar
[42]	Y. Wang, G. Ji, P. Lin and E. Trucco, Retinal vessel segmentation using matched filter with multiwavelet kernels and multiscale hierachical decomposition,, Pattern Recog., 46 (2013), 2117. Google Scholar
[43]	C. Wu and X. Tai, Augmented Lagrangian method, Dual methods and Split-Bregman Iterations for ROF, vectorial TV and higher order models,, SIAM J. Imag. Sci., 3 (2010), 300. doi: 10.1137/090767558. Google Scholar
[44]	F. Zana and J. C. Klein, Segmentation of vessel-like patterns using mathematical morphology and curvature evaluation,, IEEE Trans. Imag. Proc., 10 (2001), 1010. doi: 10.1109/83.931095. Google Scholar
[45]	D. Zonoobi, A. Kassim and W. Shen, Vasculature segmentation in MRA images using gradient compensated geodesic active contours,, J. Sign. Process. Syst., 54 (2009), 171. doi: 10.1007/s11265-008-0216-4. Google Scholar

[1]	Liejune Shiau, Roland Glowinski. Operator splitting method for friction constrained dynamical systems. Conference Publications, 2005, 2005 (Special) : 806-815. doi: 10.3934/proc.2005.2005.806
[2]	Esther Klann, Ronny Ramlau, Wolfgang Ring. A Mumford-Shah level-set approach for the inversion and segmentation of SPECT/CT data. Inverse Problems & Imaging, 2011, 5 (1) : 137-166. doi: 10.3934/ipi.2011.5.137
[3]	Zhili Ge, Gang Qian, Deren Han. Global convergence of an inexact operator splitting method for monotone variational inequalities. Journal of Industrial & Management Optimization, 2011, 7 (4) : 1013-1026. doi: 10.3934/jimo.2011.7.1013
[4]	Lijian Jiang, Craig C. Douglas. Analysis of an operator splitting method in 4D-Var. Conference Publications, 2009, 2009 (Special) : 394-403. doi: 10.3934/proc.2009.2009.394
[5]	Wangtao Lu, Shingyu Leung, Jianliang Qian. An improved fast local level set method for three-dimensional inverse gravimetry. Inverse Problems & Imaging, 2015, 9 (2) : 479-509. doi: 10.3934/ipi.2015.9.479
[6]	Kangkang Deng, Zheng Peng, Jianli Chen. Sparse probabilistic Boolean network problems: A partial proximal-type operator splitting method. Journal of Industrial & Management Optimization, 2019, 15 (4) : 1881-1896. doi: 10.3934/jimo.2018127
[7]	Yunmei Chen, Xianqi Li, Yuyuan Ouyang, Eduardo Pasiliao. Accelerated bregman operator splitting with backtracking. Inverse Problems & Imaging, 2017, 11 (6) : 1047-1070. doi: 10.3934/ipi.2017048
[8]	Ye Yuan, Yan Ren, Xiaodong Liu, Jing Wang. Approach to image segmentation based on interval neutrosophic set. Numerical Algebra, Control & Optimization, 2019, 0 (0) : 0-0. doi: 10.3934/naco.2019028
[9]	Denis Dmitriev, Jonathan Jedwab. Bounds on the growth rate of the peak sidelobe level of binary sequences. Advances in Mathematics of Communications, 2007, 1 (4) : 461-475. doi: 10.3934/amc.2007.1.461
[10]	Bin Dong, Aichi Chien, Yu Mao, Jian Ye, Fernando Vinuela, Stanley Osher. Level set based brain aneurysm capturing in 3D. Inverse Problems & Imaging, 2010, 4 (2) : 241-255. doi: 10.3934/ipi.2010.4.241
[11]	Manh Hong Duong, Yulong Lu. An operator splitting scheme for the fractional kinetic Fokker-Planck equation. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 5707-5727. doi: 10.3934/dcds.2019250
[12]	Ji-Woong Jang, Young-Sik Kim, Sang-Hyo Kim. New design of quaternary LCZ and ZCZ sequence set from binary LCZ and ZCZ sequence set. Advances in Mathematics of Communications, 2009, 3 (2) : 115-124. doi: 10.3934/amc.2009.3.115
[13]	Xiaohui Liu, Jinhua Wang, Dianhua Wu. Two new classes of binary sequence pairs with three-level cross-correlation. Advances in Mathematics of Communications, 2015, 9 (1) : 117-128. doi: 10.3934/amc.2015.9.117
[14]	Jing Xu, Xue-Cheng Tai, Li-Lian Wang. A two-level domain decomposition method for image restoration. Inverse Problems & Imaging, 2010, 4 (3) : 523-545. doi: 10.3934/ipi.2010.4.523
[15]	Petra Csomós, Hermann Mena. Fourier-splitting method for solving hyperbolic LQR problems. Numerical Algebra, Control & Optimization, 2018, 8 (1) : 17-46. doi: 10.3934/naco.2018002
[16]	Kai Wang, Lingling Xu, Deren Han. A new parallel splitting descent method for structured variational inequalities. Journal of Industrial & Management Optimization, 2014, 10 (2) : 461-476. doi: 10.3934/jimo.2014.10.461
[17]	Xiao Ding, Deren Han. A modification of the forward-backward splitting method for maximal monotone mappings. Numerical Algebra, Control & Optimization, 2013, 3 (2) : 295-307. doi: 10.3934/naco.2013.3.295
[18]	Hao Chen, Kaitai Li, Yuchuan Chu, Zhiqiang Chen, Yiren Yang. A dimension splitting and characteristic projection method for three-dimensional incompressible flow. Discrete & Continuous Dynamical Systems - B, 2019, 24 (1) : 127-147. doi: 10.3934/dcdsb.2018111
[19]	Chaabane Djamal, Pirlot Marc. A method for optimizing over the integer efficient set. Journal of Industrial & Management Optimization, 2010, 6 (4) : 811-823. doi: 10.3934/jimo.2010.6.811
[20]	Henning Struchtrup. Unique moment set from the order of magnitude method. Kinetic & Related Models, 2012, 5 (2) : 417-440. doi: 10.3934/krm.2012.5.417

2018 Impact Factor: 1.469

American Institute of Mathematical Sciences