Abstract

In this work, a novel model of Markov Random Field (MRF) is introduced. Such a model is based on a proposed Semi-Huber potential function and it is applied successfully to image segmentation in presence of noise. The main difference with respect to other half-quadratic models that have been taken as a reference is, that the number of parameters to be tuned in the proposed model is smaller and simpler. The idea is then, to choose adequate parameter values heuristically for a good segmentation of the image. In that sense, some experimental results show that the proposed model allows an easier parameter adjustment with reasonable computation times.

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  1. R. C. Gonzalez, R. E. Woods, and S. L. Eddins, Digital Image Processing Using MATLAB, (Prentice Hall, 2004).
  2. X. Cufí, X. Muñoz, J. Freixenet, and J. Martí, “A review on image segmentation thechniques integrating region and boundary information,” Adv. Imag. Elect. Phys. 120, 1–39 (Elsevier, 2003).
    [CrossRef]
  3. M. M. Fernández, “Contribuciones al análisis automático y semiautomático de ecografía fetal tridimensional mediante campos aleatorios de Markov y contornos activos. Ayudas al diagnóstico precoz de malformaciones,” PhD Thesis, Escuela Técnica Superior de Ingenieros de Telecomunicación, Universidad de Valladolid, November2001.
  4. K. Sauer and C. Bouman, “Bayesian estimation of transmission tomograms using segmentation based optimization,” IEEE Trans. Nucl. Sci. 39(4), 1144–1152 (1992).
    [CrossRef]
  5. C. Bouman and K. Sauer, “A generalized Gaussian image model for edge-preserving MAP estimation,” IEEE Trans. Image Process. 2(3), 296–310 (1993).
    [CrossRef] [PubMed]
  6. K. Held, E. R. Kops, B. J. Krause, W. M. Wells, R. Kikinis, and H. W. Müller-Gärtner, “Markov random field segmentation of brain MR images,” IEEE Trans. Med. Imaging 16(6), 878–886 (1997).
    [CrossRef]
  7. L. Cordero-Grande, P. Casaseca-de-la-Higuera, M. Martín-Fernández, and C. Alberola-López, “Endocardium and epicardium contour modeling based on Markov random fields and active contours,” in Proc. of IEEE EMBS Annu. Int. Conf., 928–931 (2006).
  8. Y. Zhang, M. Brady, and S. Smith, “Segmentation of brain MR images through a hidden Markov random field model and the expectation-maximization algorithm,” IEEE Trans. Med. Imaging 20(1), 45–57 (2001).
    [CrossRef] [PubMed]
  9. S. Krishnamachari and R. Chellappa, “Multiresolution Gauss-Markov random field models for texture segmentation,” IEEE Trans. Image Process. 6(2), 251–267 (1997).
    [CrossRef] [PubMed]
  10. D. A. Clausi and B. Yue, “Comparing cooccurrence probabilities and Markov random fields for texture analysis of SAR sea ice imagery,” IEEE Trans. Geosci. Remote Sens. 42(1), 215–228 (2004).
    [CrossRef]
  11. Y. Li and P. Gong, “An efficient texture image segmentation algorithm based on the GMRF model for classification of remotely sensed imagery,” Int. J. Remote Sens. 26(22), 5149–5159 (2005).
    [CrossRef]
  12. S. Geman and C. Geman, “Stochastic relaxation, Gibbs distribution, and the Bayesian restoration of images,” IEEE Trans. Pattern Anal. Mach. Intell. 6, 721–741 (1984).
    [CrossRef]
  13. J. E. Besag, “On the statistical analysis of dirty pictures,” J. Roy. Stat. Soc. B 48, 259–302 (1986).
  14. S. Z. Li, “MAP image restoration and segmentation by constrained optimization,” IEEE Trans. Image Process. 7(12), 1730–1735 (1998).
    [CrossRef]
  15. R. Pan and S. J. Reeves, “Efficient Huber-Markov edge-preserving image restoration,” IEEE Trans. Image Process. 15(12), 3728–3735 (2006).
    [CrossRef] [PubMed]
  16. M. Rivera and J. L. Marroquin, “Efficent half-quadratic regularization with granularity control,” Image Vision Comput. 21, 345–357 (2003).
    [CrossRef]
  17. M. Rivera, O. Ocegueda, and J. L. Marroquin, “Entropy-controlled quadratic Markov measure field models for efficient image segmentation,” IEEE Trans. Image Process. 16(12), 3047–3057 (2007).
    [CrossRef] [PubMed]
  18. M. Mignotte, “A segmentation-based regularization term for image deconvolution,” IEEE Trans. Image Process. 15(7), 1973–1984 (2006).
    [CrossRef] [PubMed]
  19. H. Deng and D. A. Clausi, “Unsupervised image segmentation using a simple MRF model with a new implementation scheme,” Pattern Recogn. 37, 2323–2335 (2004).
  20. O. Lankoande, M. M. Hayat, and B. Santhanam, “Segmentation of SAR images based on Markov random field model,” in Proc. of IEEE Int. Conf. on Systems, Man, and Cybernetics, 2956–2961 (2005).
  21. X. Lei, Y. Li, N. Zhao, and Y. Zhang, “Fast segmentation approach for SAR image based on simple Markov random field,” J. Syst. Eng. Electron. 21(1), 31–36 (2010).
  22. J. Marroquin, S. Mitter, and t. Poggio, “Probabilistic solution of ill-posed problems in computational vision,” J. Amer. Statist. Assoc. 82(397), 76–89 (1987).
    [CrossRef]
  23. R. Szeliski, “Bayesian modeling of uncertainty in low-level vision,” Int. J. Comput. Vision 5(3), 271–301 (1990).
    [CrossRef]
  24. J. I. de la Rosa, J. J. Villa, and Ma. A. Araiza, “Markovian random fields and comparison between different convex criteria optimization in image restoration,” in Proc. XVII Int. Conf. on Electronics, Communications and Computers, 9 (CONIELECOMP, 2007).
  25. S. Z. Li, Markov Random Field Modeling in Image Analysis (Springer-Verlag, 2009).
  26. J. E. Besag, “Spatial interaction and the statistical analysis of lattice systems,” J. Roy. Stat. Soc. B 36, 192–236 (1974).
  27. J. I. de la Rosa and G. Fleury, “Bootstrap methods for a measurement estimation problem,” IEEE Trans. Instrum. Meas. 55(3), 820–827 (2006).
    [CrossRef]
  28. M. Nikolova and R. Chan, “The equivalence of half-quadratic minimization and the gradient linearization iteration,” IEEE Trans. Image Process. 16(6), 1623–1627 (2007).
    [CrossRef] [PubMed]
  29. T. F. Chan, S. Esedoglu, and M. Nikolova, “Algorithms for finding global minimizers of image segmentation and denoising models,” SIAM J. Appl. Math. 66(5), 1632–1648 (2006).
    [CrossRef]
  30. M. Nikolova, “Functionals for signal and image reconstruction: properties of their minimizers and applications,” Research report to obtain the Habilitation à diriger des recherches, Centre de Mathématiques et de Leurs Applications (CMLA), Ecole Normale Supérieure de Cachan (2006).

2010 (1)

X. Lei, Y. Li, N. Zhao, and Y. Zhang, “Fast segmentation approach for SAR image based on simple Markov random field,” J. Syst. Eng. Electron. 21(1), 31–36 (2010).

2007 (2)

M. Nikolova and R. Chan, “The equivalence of half-quadratic minimization and the gradient linearization iteration,” IEEE Trans. Image Process. 16(6), 1623–1627 (2007).
[CrossRef] [PubMed]

M. Rivera, O. Ocegueda, and J. L. Marroquin, “Entropy-controlled quadratic Markov measure field models for efficient image segmentation,” IEEE Trans. Image Process. 16(12), 3047–3057 (2007).
[CrossRef] [PubMed]

2006 (5)

M. Mignotte, “A segmentation-based regularization term for image deconvolution,” IEEE Trans. Image Process. 15(7), 1973–1984 (2006).
[CrossRef] [PubMed]

R. Pan and S. J. Reeves, “Efficient Huber-Markov edge-preserving image restoration,” IEEE Trans. Image Process. 15(12), 3728–3735 (2006).
[CrossRef] [PubMed]

L. Cordero-Grande, P. Casaseca-de-la-Higuera, M. Martín-Fernández, and C. Alberola-López, “Endocardium and epicardium contour modeling based on Markov random fields and active contours,” in Proc. of IEEE EMBS Annu. Int. Conf., 928–931 (2006).

T. F. Chan, S. Esedoglu, and M. Nikolova, “Algorithms for finding global minimizers of image segmentation and denoising models,” SIAM J. Appl. Math. 66(5), 1632–1648 (2006).
[CrossRef]

J. I. de la Rosa and G. Fleury, “Bootstrap methods for a measurement estimation problem,” IEEE Trans. Instrum. Meas. 55(3), 820–827 (2006).
[CrossRef]

2005 (2)

Y. Li and P. Gong, “An efficient texture image segmentation algorithm based on the GMRF model for classification of remotely sensed imagery,” Int. J. Remote Sens. 26(22), 5149–5159 (2005).
[CrossRef]

O. Lankoande, M. M. Hayat, and B. Santhanam, “Segmentation of SAR images based on Markov random field model,” in Proc. of IEEE Int. Conf. on Systems, Man, and Cybernetics, 2956–2961 (2005).

2004 (2)

H. Deng and D. A. Clausi, “Unsupervised image segmentation using a simple MRF model with a new implementation scheme,” Pattern Recogn. 37, 2323–2335 (2004).

D. A. Clausi and B. Yue, “Comparing cooccurrence probabilities and Markov random fields for texture analysis of SAR sea ice imagery,” IEEE Trans. Geosci. Remote Sens. 42(1), 215–228 (2004).
[CrossRef]

2003 (2)

M. Rivera and J. L. Marroquin, “Efficent half-quadratic regularization with granularity control,” Image Vision Comput. 21, 345–357 (2003).
[CrossRef]

X. Cufí, X. Muñoz, J. Freixenet, and J. Martí, “A review on image segmentation thechniques integrating region and boundary information,” Adv. Imag. Elect. Phys. 120, 1–39 (Elsevier, 2003).
[CrossRef]

2001 (1)

Y. Zhang, M. Brady, and S. Smith, “Segmentation of brain MR images through a hidden Markov random field model and the expectation-maximization algorithm,” IEEE Trans. Med. Imaging 20(1), 45–57 (2001).
[CrossRef] [PubMed]

1998 (1)

S. Z. Li, “MAP image restoration and segmentation by constrained optimization,” IEEE Trans. Image Process. 7(12), 1730–1735 (1998).
[CrossRef]

1997 (2)

S. Krishnamachari and R. Chellappa, “Multiresolution Gauss-Markov random field models for texture segmentation,” IEEE Trans. Image Process. 6(2), 251–267 (1997).
[CrossRef] [PubMed]

K. Held, E. R. Kops, B. J. Krause, W. M. Wells, R. Kikinis, and H. W. Müller-Gärtner, “Markov random field segmentation of brain MR images,” IEEE Trans. Med. Imaging 16(6), 878–886 (1997).
[CrossRef]

1993 (1)

C. Bouman and K. Sauer, “A generalized Gaussian image model for edge-preserving MAP estimation,” IEEE Trans. Image Process. 2(3), 296–310 (1993).
[CrossRef] [PubMed]

1992 (1)

K. Sauer and C. Bouman, “Bayesian estimation of transmission tomograms using segmentation based optimization,” IEEE Trans. Nucl. Sci. 39(4), 1144–1152 (1992).
[CrossRef]

1990 (1)

R. Szeliski, “Bayesian modeling of uncertainty in low-level vision,” Int. J. Comput. Vision 5(3), 271–301 (1990).
[CrossRef]

1987 (1)

J. Marroquin, S. Mitter, and t. Poggio, “Probabilistic solution of ill-posed problems in computational vision,” J. Amer. Statist. Assoc. 82(397), 76–89 (1987).
[CrossRef]

1986 (1)

J. E. Besag, “On the statistical analysis of dirty pictures,” J. Roy. Stat. Soc. B 48, 259–302 (1986).

1984 (1)

S. Geman and C. Geman, “Stochastic relaxation, Gibbs distribution, and the Bayesian restoration of images,” IEEE Trans. Pattern Anal. Mach. Intell. 6, 721–741 (1984).
[CrossRef]

1974 (1)

J. E. Besag, “Spatial interaction and the statistical analysis of lattice systems,” J. Roy. Stat. Soc. B 36, 192–236 (1974).

Alberola-López, C.

L. Cordero-Grande, P. Casaseca-de-la-Higuera, M. Martín-Fernández, and C. Alberola-López, “Endocardium and epicardium contour modeling based on Markov random fields and active contours,” in Proc. of IEEE EMBS Annu. Int. Conf., 928–931 (2006).

Araiza, Ma. A.

J. I. de la Rosa, J. J. Villa, and Ma. A. Araiza, “Markovian random fields and comparison between different convex criteria optimization in image restoration,” in Proc. XVII Int. Conf. on Electronics, Communications and Computers, 9 (CONIELECOMP, 2007).

Besag, J. E.

J. E. Besag, “On the statistical analysis of dirty pictures,” J. Roy. Stat. Soc. B 48, 259–302 (1986).

J. E. Besag, “Spatial interaction and the statistical analysis of lattice systems,” J. Roy. Stat. Soc. B 36, 192–236 (1974).

Bouman, C.

C. Bouman and K. Sauer, “A generalized Gaussian image model for edge-preserving MAP estimation,” IEEE Trans. Image Process. 2(3), 296–310 (1993).
[CrossRef] [PubMed]

K. Sauer and C. Bouman, “Bayesian estimation of transmission tomograms using segmentation based optimization,” IEEE Trans. Nucl. Sci. 39(4), 1144–1152 (1992).
[CrossRef]

Brady, M.

Y. Zhang, M. Brady, and S. Smith, “Segmentation of brain MR images through a hidden Markov random field model and the expectation-maximization algorithm,” IEEE Trans. Med. Imaging 20(1), 45–57 (2001).
[CrossRef] [PubMed]

Casaseca-de-la-Higuera, P.

L. Cordero-Grande, P. Casaseca-de-la-Higuera, M. Martín-Fernández, and C. Alberola-López, “Endocardium and epicardium contour modeling based on Markov random fields and active contours,” in Proc. of IEEE EMBS Annu. Int. Conf., 928–931 (2006).

Chan, R.

M. Nikolova and R. Chan, “The equivalence of half-quadratic minimization and the gradient linearization iteration,” IEEE Trans. Image Process. 16(6), 1623–1627 (2007).
[CrossRef] [PubMed]

Chan, T. F.

T. F. Chan, S. Esedoglu, and M. Nikolova, “Algorithms for finding global minimizers of image segmentation and denoising models,” SIAM J. Appl. Math. 66(5), 1632–1648 (2006).
[CrossRef]

Chellappa, R.

S. Krishnamachari and R. Chellappa, “Multiresolution Gauss-Markov random field models for texture segmentation,” IEEE Trans. Image Process. 6(2), 251–267 (1997).
[CrossRef] [PubMed]

Clausi, D. A.

D. A. Clausi and B. Yue, “Comparing cooccurrence probabilities and Markov random fields for texture analysis of SAR sea ice imagery,” IEEE Trans. Geosci. Remote Sens. 42(1), 215–228 (2004).
[CrossRef]

H. Deng and D. A. Clausi, “Unsupervised image segmentation using a simple MRF model with a new implementation scheme,” Pattern Recogn. 37, 2323–2335 (2004).

Cordero-Grande, L.

L. Cordero-Grande, P. Casaseca-de-la-Higuera, M. Martín-Fernández, and C. Alberola-López, “Endocardium and epicardium contour modeling based on Markov random fields and active contours,” in Proc. of IEEE EMBS Annu. Int. Conf., 928–931 (2006).

Cufí, X.

X. Cufí, X. Muñoz, J. Freixenet, and J. Martí, “A review on image segmentation thechniques integrating region and boundary information,” Adv. Imag. Elect. Phys. 120, 1–39 (Elsevier, 2003).
[CrossRef]

de la Rosa, J. I.

J. I. de la Rosa and G. Fleury, “Bootstrap methods for a measurement estimation problem,” IEEE Trans. Instrum. Meas. 55(3), 820–827 (2006).
[CrossRef]

J. I. de la Rosa, J. J. Villa, and Ma. A. Araiza, “Markovian random fields and comparison between different convex criteria optimization in image restoration,” in Proc. XVII Int. Conf. on Electronics, Communications and Computers, 9 (CONIELECOMP, 2007).

Deng, H.

H. Deng and D. A. Clausi, “Unsupervised image segmentation using a simple MRF model with a new implementation scheme,” Pattern Recogn. 37, 2323–2335 (2004).

Eddins, S. L.

R. C. Gonzalez, R. E. Woods, and S. L. Eddins, Digital Image Processing Using MATLAB, (Prentice Hall, 2004).

Esedoglu, S.

T. F. Chan, S. Esedoglu, and M. Nikolova, “Algorithms for finding global minimizers of image segmentation and denoising models,” SIAM J. Appl. Math. 66(5), 1632–1648 (2006).
[CrossRef]

Fernández, M. M.

M. M. Fernández, “Contribuciones al análisis automático y semiautomático de ecografía fetal tridimensional mediante campos aleatorios de Markov y contornos activos. Ayudas al diagnóstico precoz de malformaciones,” PhD Thesis, Escuela Técnica Superior de Ingenieros de Telecomunicación, Universidad de Valladolid, November2001.

Fleury, G.

J. I. de la Rosa and G. Fleury, “Bootstrap methods for a measurement estimation problem,” IEEE Trans. Instrum. Meas. 55(3), 820–827 (2006).
[CrossRef]

Freixenet, J.

X. Cufí, X. Muñoz, J. Freixenet, and J. Martí, “A review on image segmentation thechniques integrating region and boundary information,” Adv. Imag. Elect. Phys. 120, 1–39 (Elsevier, 2003).
[CrossRef]

Geman, C.

S. Geman and C. Geman, “Stochastic relaxation, Gibbs distribution, and the Bayesian restoration of images,” IEEE Trans. Pattern Anal. Mach. Intell. 6, 721–741 (1984).
[CrossRef]

Geman, S.

S. Geman and C. Geman, “Stochastic relaxation, Gibbs distribution, and the Bayesian restoration of images,” IEEE Trans. Pattern Anal. Mach. Intell. 6, 721–741 (1984).
[CrossRef]

Gong, P.

Y. Li and P. Gong, “An efficient texture image segmentation algorithm based on the GMRF model for classification of remotely sensed imagery,” Int. J. Remote Sens. 26(22), 5149–5159 (2005).
[CrossRef]

Gonzalez, R. C.

R. C. Gonzalez, R. E. Woods, and S. L. Eddins, Digital Image Processing Using MATLAB, (Prentice Hall, 2004).

Hayat, M. M.

O. Lankoande, M. M. Hayat, and B. Santhanam, “Segmentation of SAR images based on Markov random field model,” in Proc. of IEEE Int. Conf. on Systems, Man, and Cybernetics, 2956–2961 (2005).

Held, K.

K. Held, E. R. Kops, B. J. Krause, W. M. Wells, R. Kikinis, and H. W. Müller-Gärtner, “Markov random field segmentation of brain MR images,” IEEE Trans. Med. Imaging 16(6), 878–886 (1997).
[CrossRef]

Kikinis, R.

K. Held, E. R. Kops, B. J. Krause, W. M. Wells, R. Kikinis, and H. W. Müller-Gärtner, “Markov random field segmentation of brain MR images,” IEEE Trans. Med. Imaging 16(6), 878–886 (1997).
[CrossRef]

Kops, E. R.

K. Held, E. R. Kops, B. J. Krause, W. M. Wells, R. Kikinis, and H. W. Müller-Gärtner, “Markov random field segmentation of brain MR images,” IEEE Trans. Med. Imaging 16(6), 878–886 (1997).
[CrossRef]

Krause, B. J.

K. Held, E. R. Kops, B. J. Krause, W. M. Wells, R. Kikinis, and H. W. Müller-Gärtner, “Markov random field segmentation of brain MR images,” IEEE Trans. Med. Imaging 16(6), 878–886 (1997).
[CrossRef]

Krishnamachari, S.

S. Krishnamachari and R. Chellappa, “Multiresolution Gauss-Markov random field models for texture segmentation,” IEEE Trans. Image Process. 6(2), 251–267 (1997).
[CrossRef] [PubMed]

Lankoande, O.

O. Lankoande, M. M. Hayat, and B. Santhanam, “Segmentation of SAR images based on Markov random field model,” in Proc. of IEEE Int. Conf. on Systems, Man, and Cybernetics, 2956–2961 (2005).

Lei, X.

X. Lei, Y. Li, N. Zhao, and Y. Zhang, “Fast segmentation approach for SAR image based on simple Markov random field,” J. Syst. Eng. Electron. 21(1), 31–36 (2010).

Li, S. Z.

S. Z. Li, “MAP image restoration and segmentation by constrained optimization,” IEEE Trans. Image Process. 7(12), 1730–1735 (1998).
[CrossRef]

S. Z. Li, Markov Random Field Modeling in Image Analysis (Springer-Verlag, 2009).

Li, Y.

X. Lei, Y. Li, N. Zhao, and Y. Zhang, “Fast segmentation approach for SAR image based on simple Markov random field,” J. Syst. Eng. Electron. 21(1), 31–36 (2010).

Y. Li and P. Gong, “An efficient texture image segmentation algorithm based on the GMRF model for classification of remotely sensed imagery,” Int. J. Remote Sens. 26(22), 5149–5159 (2005).
[CrossRef]

Marroquin, J.

J. Marroquin, S. Mitter, and t. Poggio, “Probabilistic solution of ill-posed problems in computational vision,” J. Amer. Statist. Assoc. 82(397), 76–89 (1987).
[CrossRef]

Marroquin, J. L.

M. Rivera, O. Ocegueda, and J. L. Marroquin, “Entropy-controlled quadratic Markov measure field models for efficient image segmentation,” IEEE Trans. Image Process. 16(12), 3047–3057 (2007).
[CrossRef] [PubMed]

M. Rivera and J. L. Marroquin, “Efficent half-quadratic regularization with granularity control,” Image Vision Comput. 21, 345–357 (2003).
[CrossRef]

Martí, J.

X. Cufí, X. Muñoz, J. Freixenet, and J. Martí, “A review on image segmentation thechniques integrating region and boundary information,” Adv. Imag. Elect. Phys. 120, 1–39 (Elsevier, 2003).
[CrossRef]

Martín-Fernández, M.

L. Cordero-Grande, P. Casaseca-de-la-Higuera, M. Martín-Fernández, and C. Alberola-López, “Endocardium and epicardium contour modeling based on Markov random fields and active contours,” in Proc. of IEEE EMBS Annu. Int. Conf., 928–931 (2006).

Mignotte, M.

M. Mignotte, “A segmentation-based regularization term for image deconvolution,” IEEE Trans. Image Process. 15(7), 1973–1984 (2006).
[CrossRef] [PubMed]

Mitter, S.

J. Marroquin, S. Mitter, and t. Poggio, “Probabilistic solution of ill-posed problems in computational vision,” J. Amer. Statist. Assoc. 82(397), 76–89 (1987).
[CrossRef]

Müller-Gärtner, H. W.

K. Held, E. R. Kops, B. J. Krause, W. M. Wells, R. Kikinis, and H. W. Müller-Gärtner, “Markov random field segmentation of brain MR images,” IEEE Trans. Med. Imaging 16(6), 878–886 (1997).
[CrossRef]

Muñoz, X.

X. Cufí, X. Muñoz, J. Freixenet, and J. Martí, “A review on image segmentation thechniques integrating region and boundary information,” Adv. Imag. Elect. Phys. 120, 1–39 (Elsevier, 2003).
[CrossRef]

Nikolova, M.

M. Nikolova and R. Chan, “The equivalence of half-quadratic minimization and the gradient linearization iteration,” IEEE Trans. Image Process. 16(6), 1623–1627 (2007).
[CrossRef] [PubMed]

T. F. Chan, S. Esedoglu, and M. Nikolova, “Algorithms for finding global minimizers of image segmentation and denoising models,” SIAM J. Appl. Math. 66(5), 1632–1648 (2006).
[CrossRef]

M. Nikolova, “Functionals for signal and image reconstruction: properties of their minimizers and applications,” Research report to obtain the Habilitation à diriger des recherches, Centre de Mathématiques et de Leurs Applications (CMLA), Ecole Normale Supérieure de Cachan (2006).

Ocegueda, O.

M. Rivera, O. Ocegueda, and J. L. Marroquin, “Entropy-controlled quadratic Markov measure field models for efficient image segmentation,” IEEE Trans. Image Process. 16(12), 3047–3057 (2007).
[CrossRef] [PubMed]

Pan, R.

R. Pan and S. J. Reeves, “Efficient Huber-Markov edge-preserving image restoration,” IEEE Trans. Image Process. 15(12), 3728–3735 (2006).
[CrossRef] [PubMed]

Poggio, t.

J. Marroquin, S. Mitter, and t. Poggio, “Probabilistic solution of ill-posed problems in computational vision,” J. Amer. Statist. Assoc. 82(397), 76–89 (1987).
[CrossRef]

Reeves, S. J.

R. Pan and S. J. Reeves, “Efficient Huber-Markov edge-preserving image restoration,” IEEE Trans. Image Process. 15(12), 3728–3735 (2006).
[CrossRef] [PubMed]

Rivera, M.

M. Rivera, O. Ocegueda, and J. L. Marroquin, “Entropy-controlled quadratic Markov measure field models for efficient image segmentation,” IEEE Trans. Image Process. 16(12), 3047–3057 (2007).
[CrossRef] [PubMed]

M. Rivera and J. L. Marroquin, “Efficent half-quadratic regularization with granularity control,” Image Vision Comput. 21, 345–357 (2003).
[CrossRef]

Santhanam, B.

O. Lankoande, M. M. Hayat, and B. Santhanam, “Segmentation of SAR images based on Markov random field model,” in Proc. of IEEE Int. Conf. on Systems, Man, and Cybernetics, 2956–2961 (2005).

Sauer, K.

C. Bouman and K. Sauer, “A generalized Gaussian image model for edge-preserving MAP estimation,” IEEE Trans. Image Process. 2(3), 296–310 (1993).
[CrossRef] [PubMed]

K. Sauer and C. Bouman, “Bayesian estimation of transmission tomograms using segmentation based optimization,” IEEE Trans. Nucl. Sci. 39(4), 1144–1152 (1992).
[CrossRef]

Smith, S.

Y. Zhang, M. Brady, and S. Smith, “Segmentation of brain MR images through a hidden Markov random field model and the expectation-maximization algorithm,” IEEE Trans. Med. Imaging 20(1), 45–57 (2001).
[CrossRef] [PubMed]

Szeliski, R.

R. Szeliski, “Bayesian modeling of uncertainty in low-level vision,” Int. J. Comput. Vision 5(3), 271–301 (1990).
[CrossRef]

Villa, J. J.

J. I. de la Rosa, J. J. Villa, and Ma. A. Araiza, “Markovian random fields and comparison between different convex criteria optimization in image restoration,” in Proc. XVII Int. Conf. on Electronics, Communications and Computers, 9 (CONIELECOMP, 2007).

Wells, W. M.

K. Held, E. R. Kops, B. J. Krause, W. M. Wells, R. Kikinis, and H. W. Müller-Gärtner, “Markov random field segmentation of brain MR images,” IEEE Trans. Med. Imaging 16(6), 878–886 (1997).
[CrossRef]

Woods, R. E.

R. C. Gonzalez, R. E. Woods, and S. L. Eddins, Digital Image Processing Using MATLAB, (Prentice Hall, 2004).

Yue, B.

D. A. Clausi and B. Yue, “Comparing cooccurrence probabilities and Markov random fields for texture analysis of SAR sea ice imagery,” IEEE Trans. Geosci. Remote Sens. 42(1), 215–228 (2004).
[CrossRef]

Zhang, Y.

X. Lei, Y. Li, N. Zhao, and Y. Zhang, “Fast segmentation approach for SAR image based on simple Markov random field,” J. Syst. Eng. Electron. 21(1), 31–36 (2010).

Y. Zhang, M. Brady, and S. Smith, “Segmentation of brain MR images through a hidden Markov random field model and the expectation-maximization algorithm,” IEEE Trans. Med. Imaging 20(1), 45–57 (2001).
[CrossRef] [PubMed]

Zhao, N.

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Figures (8)

Fig. 1
Fig. 1

Neighborhood sets for a single site of a lattice.

Fig. 2
Fig. 2

Cliques for the second order neighborhood system.

Fig. 3
Fig. 3

Semi-Huber cost function.

Fig. 4
Fig. 4

Top row: original brain image, brain image corrupted by Gaussian noise and the segmentation result using the semi-Huber MRF. Bottom row: segmentation result using the GGMRF, segmentation result using the Welsh’s MRF and segmentation result using the Tukey’s MRF.

Fig. 5
Fig. 5

Top row: original dike image, dike image corrupted by Gaussian noise and the segmentation result using the semi-Huber MRF. Bottom row: segmentation result using the GGMRF, segmentation result using the Welsh’s MRF and segmentation result using the Tukey’s MRF.

Fig. 6
Fig. 6

Top row: original satellite image of Villahermosa, Tabasco, image corrupted by Gaussian noise and the segmentation result using the semi-Huber MRF. Bottom row: segmentation result using the GGMRF, segmentation result using the Welsh’s MRF and segmentation result using the Tukey’s MRF.

Fig. 7
Fig. 7

Synthetic image, original and degraded by noise.

Fig. 8
Fig. 8

Segmentation results of the chessboard synthetic image corresponding to each model.

Tables (5)

Tables Icon

Table 1 Computation times taken by each model of MRF for segmentation of brain image.

Tables Icon

Table 2 List of parameter values for segmentation results in Fig. 4.

Tables Icon

Table 3 Computation times taken by each model of MRF for segmentation of dike image.

Tables Icon

Table 4 List of parameter values for segmentation results in Fig. 5.

Tables Icon

Table 5 Numerical results of the error measures for the segmentation of the synthetic image with each model.

Equations (25)

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y = H x + n .
p ( x | y ) = p ( y | x ) p ( x ) p ( y ) ,
x ^ MAP = arg max x 𝕏 { p ( x | y ) } = arg max x 𝕏 { log p ( y | x ) + log g ( x ) } ,
= { i | i 𝕊 } ,
g ( x ) = 1 Z exp ( 1 T U ( x ) ) ,
Z = x 𝕏 exp ( 1 T U ( x ) ) ,
U ( x ) = c 𝔺 V c ( x ) ,
g ( x ) = λ 2 ( 2 π ) N / 2 | B | 1 / 2 exp ( λ 2 x t B x ) ,
log g ( x ) = λ 2 ( s 𝕊 a s x s 2 + { s , r } 𝔺 b s r | x s x r | 2 ) ,
log g ( x ) = λ p ( s 𝕊 a s x s p + { s , r } 𝔺 b s r | x s x r | p ) + c ,
log g ( x ) = λ p ( { s , r } 𝔺 b s r | x s x r | p ) + c .
log g ( x ) = λ ( μ { s , r } 𝔺 b s r φ 1 ( x ) + ( 1 μ ) { s , r } 𝔺 b s r ρ 2 ( x ) ) + c ,
ρ 2 ( x ) = 1 1 2 k exp ( k φ 1 ( x ) ) ,
log g ( x ) = λ ( μ { s , r } 𝔺 b s r φ 1 ( x ) + ( 1 μ ) { s , r } 𝔺 b s r ρ 3 ( x ) ) + c ,
ρ 3 ( x ) = { 1 ( 1 ( 2 e / k ) 2 ) 3 , for | e / k | < 1 / 2 , 1 , in other case ,
x ^ MAP g g = arg min x 𝕏 { s 𝕊 | y s x s | q + σ q λ p { s , r } 𝔺 b s r | x s x r | p } ,
x ^ s g g = arg min x 𝕏 { | y s x s | q + σ q λ p r s b s r | x s x r | p } ,
x ^ MAP wel = arg min x 𝕏 { s 𝕊 | y s x s | 2 + λ ( μ { s , r } 𝔺 b s r φ 1 ( x ) + ( 1 μ ) { s , r } 𝔺 b s r ρ 2 ( x ) ) } .
x ^ s wel = arg min x 𝕏 { | y s x s | 2 + λ ( μ r s b s r φ 1 ( x ) + ( 1 μ ) r s b s r ρ 2 ( x ) ) } .
x ^ MAP tuk = arg min x 𝕏 { s 𝕊 | y s x s | 2 + λ ( μ { s , r } 𝔺 b s r φ 1 ( x ) + ( 1 μ ) { s , r } 𝔺 b s r ρ 3 ( x ) ) } ,
x ^ s tuk = arg min x 𝕏 { | y s x s | 2 + λ ( μ r s b s r φ 1 ( x ) + ( 1 μ ) r s b s r ρ 3 ( x ) ) } .
log g ( x ) = λ ( { s , r } 𝔺 b s r ρ 1 ( x ) ) + c ,
ρ 1 ( x ) = Δ 0 2 2 ( 1 + 4 φ 1 ( x ) Δ 0 2 1 ) .
x ^ MAP qh = arg min x 𝕏 { s 𝕊 | y s x s | 2 + λ { s , r } 𝔺 b s r ρ 1 ( x ) } .
x ^ s qh = arg min x 𝕏 { | y s x s | 2 + λ r s b s r ρ 1 ( x ) } ,

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