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A Rellich type theorem for discrete Schrödinger operators
Retinal vessel segmentation using a finite element based binary level set method
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 |
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. |
[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. |
[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. |
[4] |
T. Chan and L. Vese, Active contours without edges,, IEEE Image Proc., 10 (2001), 266.
doi: 10.1109/83.902291. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[14] |
R. Glowinski and P. Tallec, Augmented Lagrangian and Operator-Splitting Methods in Nonlinear Mechanics,, SIAM, (1989).
doi: 10.1137/1.9781611970838. |
[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. |
[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. |
[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. |
[18] |
M. Kass, A. Witkin and D. Terzopoulos, Snakes: Active contour models,, Int. J. Comput. Vis., 1 (1987), 321.
doi: 10.1007/BF00133570. |
[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. |
[20] |
J. Lie, M. Lysaker and X. Tai, A variant of the levelset method and applications to image segmentation,, UCLA CAM 03- 50, (2003). |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[30] |
C. E. Metz, Basic principles of ROC analysis,, Seminars Nucl. Med., 8 (1978), 283.
doi: 10.1016/S0001-2998(78)80014-2. |
[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. |
[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. |
[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. |
[34] |
Q. Sheng, Recent trends in splitting, adaptive and hybrid numerical methods for differential equations,, Neural, 16 (2008), 283.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
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. |
[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. |
[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. |
[4] |
T. Chan and L. Vese, Active contours without edges,, IEEE Image Proc., 10 (2001), 266.
doi: 10.1109/83.902291. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[14] |
R. Glowinski and P. Tallec, Augmented Lagrangian and Operator-Splitting Methods in Nonlinear Mechanics,, SIAM, (1989).
doi: 10.1137/1.9781611970838. |
[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. |
[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. |
[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. |
[18] |
M. Kass, A. Witkin and D. Terzopoulos, Snakes: Active contour models,, Int. J. Comput. Vis., 1 (1987), 321.
doi: 10.1007/BF00133570. |
[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. |
[20] |
J. Lie, M. Lysaker and X. Tai, A variant of the levelset method and applications to image segmentation,, UCLA CAM 03- 50, (2003). |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[30] |
C. E. Metz, Basic principles of ROC analysis,, Seminars Nucl. Med., 8 (1978), 283.
doi: 10.1016/S0001-2998(78)80014-2. |
[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. |
[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. |
[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. |
[34] |
Q. Sheng, Recent trends in splitting, adaptive and hybrid numerical methods for differential equations,, Neural, 16 (2008), 283.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
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