May  2015, 9(2): 551-578. doi: 10.3934/ipi.2015.9.551

A fast edge detection algorithm using binary labels

1. 

Department of Mathematics and Physics, North China Electric Power University, Beijing, 102206

2. 

Institute for Infocomm Research, Agency for Science, Technology and Research (A*STAR), 138632, Singapore

3. 

Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, 637371

4. 

Department of Mathematics, University of Bergen, P. O. Box 7800, N-5020, Bergen, Norway

Received  April 2013 Revised  April 2014 Published  March 2015

Edge detection (for both open and closed edges) from real images is a challenging problem. Developing fast algorithms with good accuracy and stability for noisy images is difficult yet and in demand. In this work, we present a variational model which is related to the well-known Mumford-Shah functional and design fast numerical methods to solve this new model through a binary labeling processing. A pre-smoothing step is implemented for the model, which enhances the accuracy of detection. Ample numerical experiments on grey-scale as well as color images are provided. The efficiency and accuracy of the model and the proposed minimization algorithms are demonstrated through comparing it with some existing methodologies.
Citation: Yuying Shi, Ying Gu, Li-Lian Wang, Xue-Cheng Tai. A fast edge detection algorithm using binary labels. Inverse Problems & Imaging, 2015, 9 (2) : 551-578. doi: 10.3934/ipi.2015.9.551
References:
[1]

L. Alvarez, P. L. Lions and J. M. Morel, Image selective smoothing and edge detection by nonlinear diffusion II,, SIAM J. Numer. Anal., 29 (1992), 845.  doi: 10.1137/0729052.  Google Scholar

[2]

L. Ambrosio and V. M. Tortorelli, Approximation of functions depending on jumps by elliptic functions via $\Gamma$-convergence,, Comm. Pure Appl. Math., 43 (1990), 999.  doi: 10.1002/cpa.3160430805.  Google Scholar

[3]

L. Ambrosio and V. M. Tortorelli, On the Approximation of Functionals Depending on Jumps by Quadratic, Elliptic Functionals,, Boll. Un. Mat. Ital., (1992).   Google Scholar

[4]

N. Badshah and K. Chen, Image selective segmentation under geometrical constraints using an active contour approach,, Commun. Comput. Phys., 7 (2010), 759.  doi: 10.4208/cicp.2009.09.026.  Google Scholar

[5]

S. Basu, D. P. Mukherjee and S. T. Acton, Implicit evolution of open ended curves,, in IEEE International Conference on Image Processing, (2007), 261.  doi: 10.1109/ICIP.2007.4378941.  Google Scholar

[6]

B. Berkels, A. Rätz, M. Rumpf and A. Voigt, Extracting grain boundaries and macroscopic deformations from images on atomic scale,, J. Sci. Comput., 35 (2008), 1.  doi: 10.1007/s10915-007-9157-5.  Google Scholar

[7]

A. Braides, Approximation of Free-Discontinuity Problems,, Springer-Verlag, (1998).   Google Scholar

[8]

L. M. Bregman, The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming,, (Russian) v'Z. Vyv'cisl. Mat. i Mat. Fiz., 7 (1967), 620.   Google Scholar

[9]

X. Bresson and T. F. Chan, Fast dual minimization of the vectorial total variation norm and applications to color image processing,, Inverse Probl. Imag., 2 (2008), 455.  doi: 10.3934/ipi.2008.2.455.  Google Scholar

[10]

A. Brook, R. Kimmel and N. A. Sochen, Variational restoration and edge detection for color images,, J. Math. Imaging Vis., 18 (2003), 247.  doi: 10.1023/A:1022895410391.  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, (2010), 10.   Google Scholar

[12]

J. F. Cai, R. H. Chan and M. Nikolova, Two-phase approach for deblurring images corrupted by impulse plus Gaussian noise,, Inverse Probl. Imag., 2 (2008), 187.  doi: 10.3934/ipi.2008.2.187.  Google Scholar

[13]

J. F. Cai, S. Osher and Z. W. Shen, Split Bregman methods and frame based image restoration,, Multiscale Model. Sim., 8 (2009), 337.  doi: 10.1137/090753504.  Google Scholar

[14]

X. Cai, R. Chan and T. Zeng, A two-stage image segmentation method using a convex variant of the Mumford-Shah model and thresholding,, SIAM J. Imaging Sci., 6 (2013), 368.  doi: 10.1137/120867068.  Google Scholar

[15]

J. Canny, A computational approach to edge detection,, IEEE T. Pattern Anal., PAMI-8 (1986), 679.  doi: 10.1109/TPAMI.1986.4767851.  Google Scholar

[16]

V. Caselles, R. Kimmel and G. Sapiro, Geodesic active contours,, Computer Vision, (1995), 694.  doi: 10.1109/ICCV.1995.466871.  Google Scholar

[17]

F. Catté, P. L. Lions, J. M. Morel and T. Coll, Image selective smoothing and edge detection by nonlinear diffusion,, SIAM J. Numer. Anal., 29 (1992), 182.  doi: 10.1137/0729012.  Google Scholar

[18]

A. Chambolle, Image segmentation by variational methods: Mumford and shah functional and the discrete approximations,, SIAM J. Appl. Math., 55 (1995), 827.  doi: 10.1137/S0036139993257132.  Google Scholar

[19]

A. Chambolle, An algorithm for total variation minimization and applications,, J. Math. Imaging Vis., 20 (2004), 89.  doi: 10.1023/B:JMIV.0000011321.19549.88.  Google Scholar

[20]

R. H. Chan, M. Tao and X. M. Yuan, Constrained total variation deblurring models and fast algorithms based on alternating direction method of multipliers,, SIAM J. Imaging Sci., 6 (2013), 680.  doi: 10.1137/110860185.  Google Scholar

[21]

T. F. Chan, S. Esedoglu and M. Nikolova, Algorithms for finding global minimizers of image segmentation and denoising models,, SIAM J. Appl. Math., 66 (2006), 1632.  doi: 10.1137/040615286.  Google Scholar

[22]

G. Dal Maso, Introduction to $\Gamma$-Convergence,, Birkhäuser, (1993).  doi: 10.1007/978-1-4612-0327-8.  Google Scholar

[23]

R. Deriche, Using Canny's criteria to derive a recursively implemented optimal edge detector,, Int. J. Comput. Vis., 1 (1987), 167.  doi: 10.1007/BF00123164.  Google Scholar

[24]

R. T. Farouki and C. A. Neff, Analytic properties of plane offset curves,, Computer Aided Geometric Design, 7 (1990), 83.  doi: 10.1016/0167-8396(90)90023-K.  Google Scholar

[25]

A. Giacomini, Ambrosio-Tortorelli approximation of quasi-static evolution of brittle fractures,, Calc. Var. Partial Differ. Equ., 22 (2005), 129.  doi: 10.1007/s00526-004-0269-6.  Google Scholar

[26]

T. Goldstein, X. Bresson and S. Osher, Geometric applications of the split Bregman method: Segmentation and surface reconstruction,, J. Sci. Comput., 45 (2010), 272.  doi: 10.1007/s10915-009-9331-z.  Google Scholar

[27]

T. Goldstein and S. Osher, The split Bregman method for L1 regularized problems,, SIAM J. Imaging Sci., 2 (2009), 323.  doi: 10.1137/080725891.  Google Scholar

[28]

W. Guo and M. J. Lai, Box spline wavelet frames for image edge analysis,, SIAM J. Imaging Sci., 6 (2013), 1553.  doi: 10.1137/120881348.  Google Scholar

[29]

Y. Huang, D. Lu and T. Zeng, Two-step approach for the restoration of images corrupted by multiplicative noise,, SIAM J. Sci. Comput., 35 (2013).  doi: 10.1137/120898693.  Google Scholar

[30]

M. Kass, A. P. Witkin and D. Terzopoulos, Snakes: Active contour models,, Int. J. Comput. Vis., 1 (1988), 321.  doi: 10.1007/BF00133570.  Google Scholar

[31]

R. Lai and T. F. Chan, A framework for intrinsic image processing on surfaces,, Comput. Vis. Image Underst., 115 (2011), 1647.  doi: 10.1016/j.cviu.2011.05.011.  Google Scholar

[32]

S. Lee, H. Lee, P. Abbeel and A. Y. Ng, Efficient $l_1$ regularized logistic regression,, in Proceedings of the National Conference on Artificial Intelligence, (1999).   Google Scholar

[33]

S. Leung and H. Zhao, A grid based particle method for evolution of open curves and surfaces,, J. Comput. Phys., 228 (2009), 7706.  doi: 10.1016/j.jcp.2009.07.017.  Google Scholar

[34]

B. Llanas and S. Lantarón, Edge detection by adaptive splitting,, J. Sci. Comput., 46 (2011), 485.  doi: 10.1007/s10915-010-9416-8.  Google Scholar

[35]

W. Y. Ma and B. S. Manjunath, Edgeflow: A technique for boundary detection and image segmentation,, IEEE T. Image Process., 9 (2000), 1375.  doi: 10.1109/83.855433.  Google Scholar

[36]

R. March and M. Dozio, A variational method for the recovery of smooth boundaries,, Image Vis. Comput., 15 (1997), 705.  doi: 10.1016/S0262-8856(97)00002-4.  Google Scholar

[37]

E. Meinhardt, E. Zacur, A. F. Frangi and V. Caselles, 3D edge detection by selection of level surface patches,, J. Math. Imaging Vis., 34 (2009), 1.  doi: 10.1007/s10851-008-0118-x.  Google Scholar

[38]

C. A. Micchelli, L. X. Shen and Y. S. Xu, Proximity algorithms for image models: Denoising,, Inverse Probl. Imag., 27 (2011).  doi: 10.1088/0266-5611/27/4/045009.  Google Scholar

[39]

C. A. Micchelli, L. X. Shen, Y. S. Xu and X. Y. Zeng, Proximity algorithms for the L1/TV image denoising model,, Adv. Comput. Math., 38 (2013), 401.  doi: 10.1007/s10444-011-9243-y.  Google Scholar

[40]

R. Mohieddine and L. A. Vese, An open level set framework for image segmentation and restoration using the Mumford and Shah model,, in Proc. SPIE 7873, (7873).  doi: 10.1117/12.872457.  Google Scholar

[41]

J. J. Moreau, Fonctions convexes duales et points proximaux dans un espace hilbertien,, C.R. Acad. Sci. Paris Sér. A Math., 255 (1962), 2897.   Google Scholar

[42]

J. J. Moreau, Proximité et dualité dans un espace hilbertien,, Bull. Soc. Math. France, 93 (1965), 273.   Google Scholar

[43]

D. Mumford and J. Shah, Optimal approximations by piecewise smooth functions and associated variational problems,, Comm. Pure Appl. Math, 42 (1989), 577.  doi: 10.1002/cpa.3160420503.  Google Scholar

[44]

M. Nikolova and M. Ng, Fast image reconstruction algorithms combining half-quadratic regularization and preconditioning,, in Proceedings of 2001 International Conference on Image Processing, (2001), 277.  doi: 10.1109/ICIP.2001.959007.  Google Scholar

[45]

M. Nikolova and M. Ng, Analysis of half-quadratic minimization methods for signal and image recovery,, SIAM J. Sci. Comput., 27 (2005), 937.  doi: 10.1137/030600862.  Google Scholar

[46]

S. Osher, M. Burger, D. Goldfarb, J. Xu and W. Yin, An iterative regularization method for total variation-based image restoration,, Multiscale Modeling $&$ Simulation, 4 (2005), 460.  doi: 10.1137/040605412.  Google Scholar

[47]

S. Osher and J. A. Sethian, Fronts propagating with curvature dependent speed: Algorithms based on Hamilton-Jacobi formulations,, J. Comput. Phys., 79 (1988), 12.  doi: 10.1016/0021-9991(88)90002-2.  Google Scholar

[48]

P. Perona and J. Malik, Scale-space and edge-detection using anisotropic diffusion,, IEEE T. Pattern Anal., 12 (1990), 629.  doi: 10.1109/34.56205.  Google Scholar

[49]

T. Pock, D. Cremers, H. Bischof and A. Chambolle, An algorithm for minimizing the Mumford-Shah functional,, in 12th International Conference on Computer Vision, (2009), 1133.  doi: 10.1109/ICCV.2009.5459348.  Google Scholar

[50]

R. Ramlau and W. Ring, A Mumford-Shah level-set approach for the inversion and segmentation of X-ray tomography data,, J. Comput. Phys., 221 (2007), 539.  doi: 10.1016/j.jcp.2006.06.041.  Google Scholar

[51]

L. I. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms,, Physica D: Nonlinear Phenomena, 60 (1992), 259.  doi: 10.1016/0167-2789(92)90242-F.  Google Scholar

[52]

G. Sapiro and D. L. Ringach, Anisotropic diffusion of multivalued images with applications to color filtering,, IEEE T. Image Process., 5 (1996), 1582.  doi: 10.1109/83.541429.  Google Scholar

[53]

H. Schaeffer and L. Vese, Active contours with free endpoints,, J. Math. Imaging Vis., 49 (2014), 20.  doi: 10.1007/s10851-013-0437-4.  Google Scholar

[54]

S. Setzer, Split Bregman algorithm, Douglas-Rachford splitting and frame shrinkage,, in Scale Space and Variational Methods in Computer Vision, (5567), 464.  doi: 10.1007/978-3-642-02256-2_39.  Google Scholar

[55]

J. Shah, A common framework for curve evolution, segmentation and anisotropic diffusion,, in IEEE Conference on Computer Vision and Pattern Recognition, (1996), 136.  doi: 10.1109/CVPR.1996.517065.  Google Scholar

[56]

J. Shen, A stochastic-variational model for soft Mumford-Shah segmentation,, Int. J. Biome., 2006 (2006), 2.  doi: 10.1155/IJBI/2006/92329.  Google Scholar

[57]

Y. Y. Shi, L. L. Wang and X. C. Tai, Geometry of total variation regularized $L^p$-model,, J. Comput. Appl. Math., 236 (2012), 2223.  doi: 10.1016/j.cam.2011.09.043.  Google Scholar

[58]

P. Smereka, Spiral crystal growth,, Physica D: Nonlinear Phenomena, 138 (2000), 282.  doi: 10.1016/S0167-2789(99)00216-X.  Google Scholar

[59]

S. M. Smith, Edge Thinning Used in the SUSAN Edge Detector,, Technical Report, (1995).   Google Scholar

[60]

Y. Sun, P. Wu, G. W. Wei and G. Wang, Evolution-operator-based single-step method for image processing,, Int. J. Biomed. Imaging, 2006 (2006), 1.  doi: 10.1155/IJBI/2006/83847.  Google Scholar

[61]

Y. Suzuki, T. Takayama, I. N. Motoike and T. Asai, A reaction-diffusion model performing stripe-and spot-image restoration and its LSI implementation,, Electronics and Communications in Japan (Part III: Fundamental Electronic Science), 90 (2007), 20.  doi: 10.1002/ecjc.20243.  Google Scholar

[62]

X. C. Tai and C. Wu, Augmented Lagrangian method, dual methods and split Bregman iteration for ROF model,, in Scale Space and Variational Methods in Computer Vision, (5567), 502.  doi: 10.1007/978-3-642-02256-2_42.  Google Scholar

[63]

B. Tang, G. Sapiro and V. Caselles, Color image enhancement via chromaticity diffusion,, IEEE T. Image Process., 10 (2001), 701.  doi: 10.1109/83.918563.  Google Scholar

[64]

W. Tao, F. Chang, L. Liu, H. Jin and T. Wang, Interactively multiphase image segmentation based on variational formulation and graph cuts,, Pattern Recogn., 43 (2010), 3208.  doi: 10.1016/j.patcog.2010.04.014.  Google Scholar

[65]

Z. Tari, J. Shah and H. Pien, Extraction of shape skeletons from grayscale images,, Computer Vis. Image Und., 66 (1997), 133.  doi: 10.1006/cviu.1997.0612.  Google Scholar

[66]

V. Toponogov, Differential Geometry of Curves and Surfaces: A Concise Guide,, Birkhäuser, (2006).   Google Scholar

[67]

M. Upmanyu, R. W. Smith and D. J. Srolovitz, Atomistic simulation of curvature driven grain boundary migration,, Interface Sci., 6 (1998), 41.   Google Scholar

[68]

L. L. Wang, Y. Y. Shi and X. C. Tai, Robust edge detection using Mumford-Shah model and binary level set method,, in Scale Space and Variational Methods in Computer Vision, (6667), 291.  doi: 10.1007/978-3-642-24785-9_25.  Google Scholar

[69]

Y. L. Wang, J. F. Yang, W. T. Yin and Y. Zhang, A new alternating minimization algorithm for total variation image reconstruction,, SIAM J. Imaging Sci., 1 (2008), 248.  doi: 10.1137/080724265.  Google Scholar

[70]

G. W. Wei and Y. Q. Jia, Synchronization-based image edge detection,, EPL (Europhysics Letters), 59 (2002), 814.  doi: 10.1209/epl/i2002-00115-8.  Google Scholar

[71]

C. Wu and X. C. Tai, Augmented Lagrangian method, dual methods, and split Bregman iteration for ROF, vectorial TV, and high order models,, SIAM J. Imaging Sci., 3 (2010), 300.  doi: 10.1137/090767558.  Google Scholar

show all references

References:
[1]

L. Alvarez, P. L. Lions and J. M. Morel, Image selective smoothing and edge detection by nonlinear diffusion II,, SIAM J. Numer. Anal., 29 (1992), 845.  doi: 10.1137/0729052.  Google Scholar

[2]

L. Ambrosio and V. M. Tortorelli, Approximation of functions depending on jumps by elliptic functions via $\Gamma$-convergence,, Comm. Pure Appl. Math., 43 (1990), 999.  doi: 10.1002/cpa.3160430805.  Google Scholar

[3]

L. Ambrosio and V. M. Tortorelli, On the Approximation of Functionals Depending on Jumps by Quadratic, Elliptic Functionals,, Boll. Un. Mat. Ital., (1992).   Google Scholar

[4]

N. Badshah and K. Chen, Image selective segmentation under geometrical constraints using an active contour approach,, Commun. Comput. Phys., 7 (2010), 759.  doi: 10.4208/cicp.2009.09.026.  Google Scholar

[5]

S. Basu, D. P. Mukherjee and S. T. Acton, Implicit evolution of open ended curves,, in IEEE International Conference on Image Processing, (2007), 261.  doi: 10.1109/ICIP.2007.4378941.  Google Scholar

[6]

B. Berkels, A. Rätz, M. Rumpf and A. Voigt, Extracting grain boundaries and macroscopic deformations from images on atomic scale,, J. Sci. Comput., 35 (2008), 1.  doi: 10.1007/s10915-007-9157-5.  Google Scholar

[7]

A. Braides, Approximation of Free-Discontinuity Problems,, Springer-Verlag, (1998).   Google Scholar

[8]

L. M. Bregman, The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming,, (Russian) v'Z. Vyv'cisl. Mat. i Mat. Fiz., 7 (1967), 620.   Google Scholar

[9]

X. Bresson and T. F. Chan, Fast dual minimization of the vectorial total variation norm and applications to color image processing,, Inverse Probl. Imag., 2 (2008), 455.  doi: 10.3934/ipi.2008.2.455.  Google Scholar

[10]

A. Brook, R. Kimmel and N. A. Sochen, Variational restoration and edge detection for color images,, J. Math. Imaging Vis., 18 (2003), 247.  doi: 10.1023/A:1022895410391.  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, (2010), 10.   Google Scholar

[12]

J. F. Cai, R. H. Chan and M. Nikolova, Two-phase approach for deblurring images corrupted by impulse plus Gaussian noise,, Inverse Probl. Imag., 2 (2008), 187.  doi: 10.3934/ipi.2008.2.187.  Google Scholar

[13]

J. F. Cai, S. Osher and Z. W. Shen, Split Bregman methods and frame based image restoration,, Multiscale Model. Sim., 8 (2009), 337.  doi: 10.1137/090753504.  Google Scholar

[14]

X. Cai, R. Chan and T. Zeng, A two-stage image segmentation method using a convex variant of the Mumford-Shah model and thresholding,, SIAM J. Imaging Sci., 6 (2013), 368.  doi: 10.1137/120867068.  Google Scholar

[15]

J. Canny, A computational approach to edge detection,, IEEE T. Pattern Anal., PAMI-8 (1986), 679.  doi: 10.1109/TPAMI.1986.4767851.  Google Scholar

[16]

V. Caselles, R. Kimmel and G. Sapiro, Geodesic active contours,, Computer Vision, (1995), 694.  doi: 10.1109/ICCV.1995.466871.  Google Scholar

[17]

F. Catté, P. L. Lions, J. M. Morel and T. Coll, Image selective smoothing and edge detection by nonlinear diffusion,, SIAM J. Numer. Anal., 29 (1992), 182.  doi: 10.1137/0729012.  Google Scholar

[18]

A. Chambolle, Image segmentation by variational methods: Mumford and shah functional and the discrete approximations,, SIAM J. Appl. Math., 55 (1995), 827.  doi: 10.1137/S0036139993257132.  Google Scholar

[19]

A. Chambolle, An algorithm for total variation minimization and applications,, J. Math. Imaging Vis., 20 (2004), 89.  doi: 10.1023/B:JMIV.0000011321.19549.88.  Google Scholar

[20]

R. H. Chan, M. Tao and X. M. Yuan, Constrained total variation deblurring models and fast algorithms based on alternating direction method of multipliers,, SIAM J. Imaging Sci., 6 (2013), 680.  doi: 10.1137/110860185.  Google Scholar

[21]

T. F. Chan, S. Esedoglu and M. Nikolova, Algorithms for finding global minimizers of image segmentation and denoising models,, SIAM J. Appl. Math., 66 (2006), 1632.  doi: 10.1137/040615286.  Google Scholar

[22]

G. Dal Maso, Introduction to $\Gamma$-Convergence,, Birkhäuser, (1993).  doi: 10.1007/978-1-4612-0327-8.  Google Scholar

[23]

R. Deriche, Using Canny's criteria to derive a recursively implemented optimal edge detector,, Int. J. Comput. Vis., 1 (1987), 167.  doi: 10.1007/BF00123164.  Google Scholar

[24]

R. T. Farouki and C. A. Neff, Analytic properties of plane offset curves,, Computer Aided Geometric Design, 7 (1990), 83.  doi: 10.1016/0167-8396(90)90023-K.  Google Scholar

[25]

A. Giacomini, Ambrosio-Tortorelli approximation of quasi-static evolution of brittle fractures,, Calc. Var. Partial Differ. Equ., 22 (2005), 129.  doi: 10.1007/s00526-004-0269-6.  Google Scholar

[26]

T. Goldstein, X. Bresson and S. Osher, Geometric applications of the split Bregman method: Segmentation and surface reconstruction,, J. Sci. Comput., 45 (2010), 272.  doi: 10.1007/s10915-009-9331-z.  Google Scholar

[27]

T. Goldstein and S. Osher, The split Bregman method for L1 regularized problems,, SIAM J. Imaging Sci., 2 (2009), 323.  doi: 10.1137/080725891.  Google Scholar

[28]

W. Guo and M. J. Lai, Box spline wavelet frames for image edge analysis,, SIAM J. Imaging Sci., 6 (2013), 1553.  doi: 10.1137/120881348.  Google Scholar

[29]

Y. Huang, D. Lu and T. Zeng, Two-step approach for the restoration of images corrupted by multiplicative noise,, SIAM J. Sci. Comput., 35 (2013).  doi: 10.1137/120898693.  Google Scholar

[30]

M. Kass, A. P. Witkin and D. Terzopoulos, Snakes: Active contour models,, Int. J. Comput. Vis., 1 (1988), 321.  doi: 10.1007/BF00133570.  Google Scholar

[31]

R. Lai and T. F. Chan, A framework for intrinsic image processing on surfaces,, Comput. Vis. Image Underst., 115 (2011), 1647.  doi: 10.1016/j.cviu.2011.05.011.  Google Scholar

[32]

S. Lee, H. Lee, P. Abbeel and A. Y. Ng, Efficient $l_1$ regularized logistic regression,, in Proceedings of the National Conference on Artificial Intelligence, (1999).   Google Scholar

[33]

S. Leung and H. Zhao, A grid based particle method for evolution of open curves and surfaces,, J. Comput. Phys., 228 (2009), 7706.  doi: 10.1016/j.jcp.2009.07.017.  Google Scholar

[34]

B. Llanas and S. Lantarón, Edge detection by adaptive splitting,, J. Sci. Comput., 46 (2011), 485.  doi: 10.1007/s10915-010-9416-8.  Google Scholar

[35]

W. Y. Ma and B. S. Manjunath, Edgeflow: A technique for boundary detection and image segmentation,, IEEE T. Image Process., 9 (2000), 1375.  doi: 10.1109/83.855433.  Google Scholar

[36]

R. March and M. Dozio, A variational method for the recovery of smooth boundaries,, Image Vis. Comput., 15 (1997), 705.  doi: 10.1016/S0262-8856(97)00002-4.  Google Scholar

[37]

E. Meinhardt, E. Zacur, A. F. Frangi and V. Caselles, 3D edge detection by selection of level surface patches,, J. Math. Imaging Vis., 34 (2009), 1.  doi: 10.1007/s10851-008-0118-x.  Google Scholar

[38]

C. A. Micchelli, L. X. Shen and Y. S. Xu, Proximity algorithms for image models: Denoising,, Inverse Probl. Imag., 27 (2011).  doi: 10.1088/0266-5611/27/4/045009.  Google Scholar

[39]

C. A. Micchelli, L. X. Shen, Y. S. Xu and X. Y. Zeng, Proximity algorithms for the L1/TV image denoising model,, Adv. Comput. Math., 38 (2013), 401.  doi: 10.1007/s10444-011-9243-y.  Google Scholar

[40]

R. Mohieddine and L. A. Vese, An open level set framework for image segmentation and restoration using the Mumford and Shah model,, in Proc. SPIE 7873, (7873).  doi: 10.1117/12.872457.  Google Scholar

[41]

J. J. Moreau, Fonctions convexes duales et points proximaux dans un espace hilbertien,, C.R. Acad. Sci. Paris Sér. A Math., 255 (1962), 2897.   Google Scholar

[42]

J. J. Moreau, Proximité et dualité dans un espace hilbertien,, Bull. Soc. Math. France, 93 (1965), 273.   Google Scholar

[43]

D. Mumford and J. Shah, Optimal approximations by piecewise smooth functions and associated variational problems,, Comm. Pure Appl. Math, 42 (1989), 577.  doi: 10.1002/cpa.3160420503.  Google Scholar

[44]

M. Nikolova and M. Ng, Fast image reconstruction algorithms combining half-quadratic regularization and preconditioning,, in Proceedings of 2001 International Conference on Image Processing, (2001), 277.  doi: 10.1109/ICIP.2001.959007.  Google Scholar

[45]

M. Nikolova and M. Ng, Analysis of half-quadratic minimization methods for signal and image recovery,, SIAM J. Sci. Comput., 27 (2005), 937.  doi: 10.1137/030600862.  Google Scholar

[46]

S. Osher, M. Burger, D. Goldfarb, J. Xu and W. Yin, An iterative regularization method for total variation-based image restoration,, Multiscale Modeling $&$ Simulation, 4 (2005), 460.  doi: 10.1137/040605412.  Google Scholar

[47]

S. Osher and J. A. Sethian, Fronts propagating with curvature dependent speed: Algorithms based on Hamilton-Jacobi formulations,, J. Comput. Phys., 79 (1988), 12.  doi: 10.1016/0021-9991(88)90002-2.  Google Scholar

[48]

P. Perona and J. Malik, Scale-space and edge-detection using anisotropic diffusion,, IEEE T. Pattern Anal., 12 (1990), 629.  doi: 10.1109/34.56205.  Google Scholar

[49]

T. Pock, D. Cremers, H. Bischof and A. Chambolle, An algorithm for minimizing the Mumford-Shah functional,, in 12th International Conference on Computer Vision, (2009), 1133.  doi: 10.1109/ICCV.2009.5459348.  Google Scholar

[50]

R. Ramlau and W. Ring, A Mumford-Shah level-set approach for the inversion and segmentation of X-ray tomography data,, J. Comput. Phys., 221 (2007), 539.  doi: 10.1016/j.jcp.2006.06.041.  Google Scholar

[51]

L. I. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms,, Physica D: Nonlinear Phenomena, 60 (1992), 259.  doi: 10.1016/0167-2789(92)90242-F.  Google Scholar

[52]

G. Sapiro and D. L. Ringach, Anisotropic diffusion of multivalued images with applications to color filtering,, IEEE T. Image Process., 5 (1996), 1582.  doi: 10.1109/83.541429.  Google Scholar

[53]

H. Schaeffer and L. Vese, Active contours with free endpoints,, J. Math. Imaging Vis., 49 (2014), 20.  doi: 10.1007/s10851-013-0437-4.  Google Scholar

[54]

S. Setzer, Split Bregman algorithm, Douglas-Rachford splitting and frame shrinkage,, in Scale Space and Variational Methods in Computer Vision, (5567), 464.  doi: 10.1007/978-3-642-02256-2_39.  Google Scholar

[55]

J. Shah, A common framework for curve evolution, segmentation and anisotropic diffusion,, in IEEE Conference on Computer Vision and Pattern Recognition, (1996), 136.  doi: 10.1109/CVPR.1996.517065.  Google Scholar

[56]

J. Shen, A stochastic-variational model for soft Mumford-Shah segmentation,, Int. J. Biome., 2006 (2006), 2.  doi: 10.1155/IJBI/2006/92329.  Google Scholar

[57]

Y. Y. Shi, L. L. Wang and X. C. Tai, Geometry of total variation regularized $L^p$-model,, J. Comput. Appl. Math., 236 (2012), 2223.  doi: 10.1016/j.cam.2011.09.043.  Google Scholar

[58]

P. Smereka, Spiral crystal growth,, Physica D: Nonlinear Phenomena, 138 (2000), 282.  doi: 10.1016/S0167-2789(99)00216-X.  Google Scholar

[59]

S. M. Smith, Edge Thinning Used in the SUSAN Edge Detector,, Technical Report, (1995).   Google Scholar

[60]

Y. Sun, P. Wu, G. W. Wei and G. Wang, Evolution-operator-based single-step method for image processing,, Int. J. Biomed. Imaging, 2006 (2006), 1.  doi: 10.1155/IJBI/2006/83847.  Google Scholar

[61]

Y. Suzuki, T. Takayama, I. N. Motoike and T. Asai, A reaction-diffusion model performing stripe-and spot-image restoration and its LSI implementation,, Electronics and Communications in Japan (Part III: Fundamental Electronic Science), 90 (2007), 20.  doi: 10.1002/ecjc.20243.  Google Scholar

[62]

X. C. Tai and C. Wu, Augmented Lagrangian method, dual methods and split Bregman iteration for ROF model,, in Scale Space and Variational Methods in Computer Vision, (5567), 502.  doi: 10.1007/978-3-642-02256-2_42.  Google Scholar

[63]

B. Tang, G. Sapiro and V. Caselles, Color image enhancement via chromaticity diffusion,, IEEE T. Image Process., 10 (2001), 701.  doi: 10.1109/83.918563.  Google Scholar

[64]

W. Tao, F. Chang, L. Liu, H. Jin and T. Wang, Interactively multiphase image segmentation based on variational formulation and graph cuts,, Pattern Recogn., 43 (2010), 3208.  doi: 10.1016/j.patcog.2010.04.014.  Google Scholar

[65]

Z. Tari, J. Shah and H. Pien, Extraction of shape skeletons from grayscale images,, Computer Vis. Image Und., 66 (1997), 133.  doi: 10.1006/cviu.1997.0612.  Google Scholar

[66]

V. Toponogov, Differential Geometry of Curves and Surfaces: A Concise Guide,, Birkhäuser, (2006).   Google Scholar

[67]

M. Upmanyu, R. W. Smith and D. J. Srolovitz, Atomistic simulation of curvature driven grain boundary migration,, Interface Sci., 6 (1998), 41.   Google Scholar

[68]

L. L. Wang, Y. Y. Shi and X. C. Tai, Robust edge detection using Mumford-Shah model and binary level set method,, in Scale Space and Variational Methods in Computer Vision, (6667), 291.  doi: 10.1007/978-3-642-24785-9_25.  Google Scholar

[69]

Y. L. Wang, J. F. Yang, W. T. Yin and Y. Zhang, A new alternating minimization algorithm for total variation image reconstruction,, SIAM J. Imaging Sci., 1 (2008), 248.  doi: 10.1137/080724265.  Google Scholar

[70]

G. W. Wei and Y. Q. Jia, Synchronization-based image edge detection,, EPL (Europhysics Letters), 59 (2002), 814.  doi: 10.1209/epl/i2002-00115-8.  Google Scholar

[71]

C. Wu and X. C. Tai, Augmented Lagrangian method, dual methods, and split Bregman iteration for ROF, vectorial TV, and high order models,, SIAM J. Imaging Sci., 3 (2010), 300.  doi: 10.1137/090767558.  Google Scholar

[1]

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

[2]

Li Shen, Eric Todd Quinto, Shiqiang Wang, Ming Jiang. Simultaneous reconstruction and segmentation with the Mumford-Shah functional for electron tomography. Inverse Problems & Imaging, 2018, 12 (6) : 1343-1364. doi: 10.3934/ipi.2018056

[3]

Koichi Osaki, Hirotoshi Satoh, Shigetoshi Yazaki. Towards modelling spiral motion of open plane curves. Discrete & Continuous Dynamical Systems - S, 2015, 8 (5) : 1009-1022. doi: 10.3934/dcdss.2015.8.1009

[4]

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

[5]

Matteo Novaga, Enrico Valdinoci. Closed curves of prescribed curvature and a pinning effect. Networks & Heterogeneous Media, 2011, 6 (1) : 77-88. doi: 10.3934/nhm.2011.6.77

[6]

Minoru Murai, Waichiro Matsumoto, Shoji Yotsutani. Representation formula for the plane closed elastic curves. Conference Publications, 2013, 2013 (special) : 565-585. doi: 10.3934/proc.2013.2013.565

[7]

David L. Finn. Convexity of level curves for solutions to semilinear elliptic equations. Communications on Pure & Applied Analysis, 2008, 7 (6) : 1335-1343. doi: 10.3934/cpaa.2008.7.1335

[8]

Antonin Chambolle, Francesco Doveri. Minimizing movements of the Mumford and Shah energy. Discrete & Continuous Dynamical Systems - A, 1997, 3 (2) : 153-174. doi: 10.3934/dcds.1997.3.153

[9]

Feng Luo. On non-separating simple closed curves in a compact surface. Electronic Research Announcements, 1995, 1: 18-25.

[10]

Miroslav KolÁŘ, Michal BeneŠ, Daniel ŠevČoviČ. Area preserving geodesic curvature driven flow of closed curves on a surface. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3671-3689. doi: 10.3934/dcdsb.2017148

[11]

Peter Birkner, Nicolas Thériault. Efficient halving for genus 3 curves over binary fields. Advances in Mathematics of Communications, 2010, 4 (1) : 23-47. doi: 10.3934/amc.2010.4.23

[12]

Monika Muszkieta. A variational approach to edge detection. Inverse Problems & Imaging, 2016, 10 (2) : 499-517. doi: 10.3934/ipi.2016009

[13]

Giovanna Citti, Maria Manfredini, Alessandro Sarti. Finite difference approximation of the Mumford and Shah functional in a contact manifold of the Heisenberg space. Communications on Pure & Applied Analysis, 2010, 9 (4) : 905-927. doi: 10.3934/cpaa.2010.9.905

[14]

Audric Drogoul, Gilles Aubert. The topological gradient method for semi-linear problems and application to edge detection and noise removal. Inverse Problems & Imaging, 2016, 10 (1) : 51-86. doi: 10.3934/ipi.2016.10.51

[15]

Liming Zhang, Tao Qian, Qingye Zeng. Edge detection by using rotational wavelets. Communications on Pure & Applied Analysis, 2007, 6 (3) : 899-915. doi: 10.3934/cpaa.2007.6.899

[16]

Anton Stolbunov. Constructing public-key cryptographic schemes based on class group action on a set of isogenous elliptic curves. Advances in Mathematics of Communications, 2010, 4 (2) : 215-235. doi: 10.3934/amc.2010.4.215

[17]

Gian-Italo Bischi, Laura Gardini, Fabio Tramontana. Bifurcation curves in discontinuous maps. Discrete & Continuous Dynamical Systems - B, 2010, 13 (2) : 249-267. doi: 10.3934/dcdsb.2010.13.249

[18]

Stefano Marò. Relativistic pendulum and invariant curves. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 1139-1162. doi: 10.3934/dcds.2015.35.1139

[19]

Carlos Munuera, Alonso Sepúlveda, Fernando Torres. Castle curves and codes. Advances in Mathematics of Communications, 2009, 3 (4) : 399-408. doi: 10.3934/amc.2009.3.399

[20]

Vladimir Georgiev, Eugene Stepanov. Metric cycles, curves and solenoids. Discrete & Continuous Dynamical Systems - A, 2014, 34 (4) : 1443-1463. doi: 10.3934/dcds.2014.34.1443

2018 Impact Factor: 1.469

Metrics

  • PDF downloads (12)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]