## Semi-Huber potential function for image segmentation |

Optics Express, Vol. 20, Issue 6, pp. 6542-6554 (2012)

http://dx.doi.org/10.1364/OE.20.006542

Acrobat PDF (1034 KB)

### 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.

© 2012 OSA

## 1. Introduction

*H*that, together with an additive noise term

*n*, it operates on an input image

*x*and produces a degraded image

*y*: Given

*y*, some previous knowledge about the degradation function and some knowledge about the additive noise term, the aim is to obtain an estimation

*x̂*of the original image

*x*for a good segmentation of the regions or objects into it. The more we know about

*H*and

*n*, the closer

*x̂*will be to

*x*[1]. The simplest case is when the degradation function

*H*is modeled as a linear function, but it could be non linear in many cases and then the problem becomes more complex. In this work it will be assumed that

*H*is the identity operator and we consider only degradation due to Gaussian noise.

*discontinuity*and

*similarity*[2

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.

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]

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]

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]

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]

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

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]

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

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]

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]

*x*need to be introduced in the estimation process [16

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

*x*given the observations

*y*[22

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]

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]

*semi Huber*MRF as a proposal of a new algorithm for image segmentation. The main advantage of this model lies in the fact that the hyperparameters to be tuned for an adequate result are less than those needed by other models that were taken as a reference to verify the results of the proposed one. Section 2 provides an overview about the Bayesian approach. Section 3 includes a theoretical basis of Markov random fields and describes the MAP estimators used in this work as a reference. A description of the proposed model and its corresponding MAP estimator is provided in section 4. Some results and comments for image segmentation experiments are presented in section 5. Finally, in section 6 some conclusions are given.

## 2. The Bayesian approach

*inverse problems*. A common approach to solve this kind of problems is the Bayesian modeling. A Bayesian model is a statistical description of an estimation problem that consists of three components. The first component, the

*prior model p*(

*x*) that is a probabilistic description of the real world or its properties, that we are trying to estimate, before collecting data. The second component, the

*sensor model p*(

*y|x*), is a description of the behavior of noise or stochastic characteristics that relate the original state

*x*to the sampled input image or sensor values

*y*. These two components can be combined to obtain the third component, the

*posterior model p*(

*x|y*), which is a probabilistic description of the current estimation of the original scene

*x*, given the observed data

*y*. The model is obtained using the Bayes rule: where

*p*(

*y*) is the density function of

*y*and is constant if the observed image is provided [21].

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

*maximum a posteriori*(MAP) estimate, that is, the value of

*x*that maximizes the conditional probability

*p*(

*x|y*). It is one of the more efficient and most used estimators [4

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]

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]

**6**(2), 251–267 (1997). [CrossRef] [PubMed]

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

*g*(

*x*) is a MRF function that models prior information of the phenomena to be estimated as a probability distribution, 𝕏 is the set of pixels capable to maximize

*p*(

*x|y*) and

*p*(

*y|x*) is the likelihood function from

*y*given

*x*[24].

## 3. Markov random fields and MAP estimators

*clique c*is defined as a subset of sites in 𝕊 that consists of a single site

*c*= {

*i*}, a pair of neighboring sites

*c*= {

*i,i*′}, a triple of neighboring sites

*c*= {

*i,i*′,

*i*″}, and so on. All posible cliques for the second order neighborhood system are displayed in Fig. 2 [19, 25]. The Hammersley-Clifford theorem establishes the equivalence between Markov random fields and Gibbs random fields [12

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]

*partition function*and in practice is a normalization constant value.

*T*is the

*temperature*parameter, that controls the sharpness of the distribution [12

**6**, 721–741 (1984). [CrossRef]

*U*(

*x*) is the

*energy function*such that which is determined as a sum of

*clique potentials V*(

_{c}*x*) over all posible cliques in the neighborhood [8

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]

*ρ*(

*λ*(

*x*

_{i}*– x*)) which act on pairs of sites, where

_{j}*λ*is a constant that scales the difference between pixel values.

### 3.1. Generalized Gaussian MRF (GGMRF)

*B*is a symmetric positive definite matrix, named the precision matrix,

*λ*is a constant and

*x*is the transpose of

^{t}*x*. To make this to correspond to a Gibbs distribution with neighborhood system

*∂s*, it is imposed the constraint that

*B*= 0 when

_{sr}*s*is not in

*∂r*and

*s*≠

*r*. This distribution may then be rewritten, to form the log likelihood, as where

*a*= ∑

_{s}_{r∈S}

*B*and

_{sr}*b*= −

_{sr}*B*.

_{sr}*p*, where 1 ≤

*p*≤ 2 and

*λ*is a parameter inversely proportional to the scale of

*x*[5

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]

*a*≥ 0 and

_{s}*b*> 0,

_{sr}*s*is the site of interest,

*r*corresponds to the local neighbors and c is a constant term. In practice it is recommended to take

*a*= 0 for Gaussian noise assumption, thus the unicity of the MAP estimator can be assured, resulting in Here,

_{s}*b*is a constant that depends on the distance between pixels

_{sr}*s*and

*r*. The selected value for power

*p*is determinant, since it constrains the convergence speed of the local or global estimator and the quality of the estimated image [24].

### 3.2. Welsh’s potential function

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

*μ*is the granularity control parameter,

*φ*

_{1}(

*x*) =

*e*

^{2}with

*e*= (

*x*−

_{s}*x*), and

_{r}*k*is a positive scale parameter for edge preservation.

### 3.3. Tukey’s potential function

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

*k*is also a scale parameter and

*μ*provides the granularity control too.

### 3.4. MAP estimators

**2**(3), 296–310 (1993). [CrossRef] [PubMed]

_{s∈𝕊}|

*y*

_{s}*− x*|

_{s}*stands for the term log*

^{q}*p*(

*y|x*), and the term

*σ*∑

^{q}λ^{p}_{{s,r}∈}

*b*−

_{sr}|x_{s}*x*corresponds to the term log

_{r}|^{p}*g*(

*x*) of Eq. (3). This applies for the other models. The minimization problem can be solved from a global or local point of view considering various methods [16

**21**, 345–357 (2003). [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]

*x*. In this work the Levenberg-Marquardt algorithm was used for local minimization, because all the parameters included into the potential functions were chosen heuristically or according to values proposed in references [24]. Thus, local estimation is implemented with the expression where the subset

_{s}*∂s*stands for the sites in the neighborhood. Estimator performance depends on the chosen values for parameters

*p*and

*q*. For example if

*p*=

*q*= 2, we have the Gaussian condition for the potential function and the obtained estimator is similar to the least-square one since the likelihood function is quadratic. Moreover, when

*p*=

*q*= 1, the criterion is absolute and the estimator converges to the median one; nevertheless, this criterion is not differentiable at zero and it causes instability in the minimization process [24]. The form of the first term in Eq. (12) depends on the type of noise regarded. For all experiments made in this work we assumed that noise has a Gaussian distribution with mean value

*μ*and variance

_{n}*q*was set at 2.

**21**, 345–357 (2003). [CrossRef]

**21**, 345–357 (2003). [CrossRef]

## 4. Semi-Huber proposal

*g*(

*x*) in Eq. (3), we introduce the Huber-like norm or semi-Huber potential function, which has been used in one dimensional robust estimation problems [27

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]

*s*is the site of interest,

*r*corresponds to the local neighbors, c is a constant term and Here Δ

_{0}> 0 is a constant value and

*φ*

_{1}(

*x*) =

*e*

^{2}with

*e*= (

*x*−

_{s}*x*).

_{r}_{0}= 1. Near zero the function is quadratic and for values beyond ±1, the function is almost linear. This linear region of the function allows sharp edges, while convexity makes MAP estimate efficient to compute.

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]

## 5. Experiments and results

*X*

_{0}, to start the search of the solution. It was observed that the final result depended on the choice of this value, which adds one additional hyperparameter to the segmentation process.

*n*∼ (0,

_{0}in this case, keeping

*λ*= 1 (two if one takes into account the initial value in the optimization stage). Therefore, the number of times that it was necessary to run the segmentation process was significantly lower than with other models.

## 6. Conclusion

## Acknowledgments

## References and links

1. | R. C. Gonzalez, R. E. Woods, and S. L. Eddins, |

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. |

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. |

5. | C. Bouman and K. Sauer, “A generalized Gaussian image model for edge-preserving MAP estimation,” IEEE Trans. Image Process. |

6. | K. Held, E. R. Kops, B. J. Krause, W. M. Wells III, R. Kikinis, and H. W. Müller-Gärtner, “Markov random field segmentation of brain MR images,” IEEE Trans. Med. Imaging |

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 |

9. | S. Krishnamachari and R. Chellappa, “Multiresolution Gauss-Markov random field models for texture segmentation,” IEEE Trans. Image Process. |

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. |

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. |

12. | S. Geman and C. Geman, “Stochastic relaxation, Gibbs distribution, and the Bayesian restoration of images,” IEEE Trans. Pattern Anal. Mach. Intell. |

13. | J. E. Besag, “On the statistical analysis of dirty pictures,” J. Roy. Stat. Soc. B |

14. | S. Z. Li, “MAP image restoration and segmentation by constrained optimization,” IEEE Trans. Image Process. |

15. | R. Pan and S. J. Reeves, “Efficient Huber-Markov edge-preserving image restoration,” IEEE Trans. Image Process. |

16. | M. Rivera and J. L. Marroquin, “Efficent half-quadratic regularization with granularity control,” Image Vision Comput. |

17. | M. Rivera, O. Ocegueda, and J. L. Marroquin, “Entropy-controlled quadratic Markov measure field models for efficient image segmentation,” IEEE Trans. Image Process. |

18. | M. Mignotte, “A segmentation-based regularization term for image deconvolution,” IEEE Trans. Image Process. |

19. | H. Deng and D. A. Clausi, “Unsupervised image segmentation using a simple MRF model with a new implementation scheme,” Pattern Recogn. |

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. |

22. | J. Marroquin, S. Mitter, and t. Poggio, “Probabilistic solution of ill-posed problems in computational vision,” J. Amer. Statist. Assoc. |

23. | R. Szeliski, “Bayesian modeling of uncertainty in low-level vision,” Int. J. Comput. Vision |

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, |

26. | J. E. Besag, “Spatial interaction and the statistical analysis of lattice systems,” J. Roy. Stat. Soc. B |

27. | J. I. de la Rosa and G. Fleury, “Bootstrap methods for a measurement estimation problem,” IEEE Trans. Instrum. Meas. |

28. | M. Nikolova and R. Chan, “The equivalence of half-quadratic minimization and the gradient linearization iteration,” IEEE Trans. Image Process. |

29. | T. F. Chan, S. Esedoglu, and M. Nikolova, “Algorithms for finding global minimizers of image segmentation and denoising models,” SIAM J. Appl. Math. |

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). |

**OCIS Codes**

(100.0100) Image processing : Image processing

(100.2000) Image processing : Digital image processing

**ToC Category:**

Image Processing

**History**

Original Manuscript: September 23, 2011

Revised Manuscript: December 24, 2011

Manuscript Accepted: February 27, 2012

Published: March 6, 2012

**Citation**

Osvaldo Gutiérrez, Ismael de la Rosa, Jesús Villa, Efrén González, and Nivia Escalante, "Semi-Huber potential function for image segmentation," Opt. Express **20**, 6542-6554 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-6-6542

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### References

- R. C. Gonzalez, R. E. Woods, and S. L. Eddins, Digital Image Processing Using MATLAB, (Prentice Hall, 2004).
- 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]
- 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.
- K. Sauer and C. Bouman, “Bayesian estimation of transmission tomograms using segmentation based optimization,” IEEE Trans. Nucl. Sci.39(4), 1144–1152 (1992). [CrossRef]
- 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. 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. Imaging16(6), 878–886 (1997). [CrossRef]
- 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).
- 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. Imaging20(1), 45–57 (2001). [CrossRef] [PubMed]
- 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]
- 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]
- 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]
- 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]
- J. E. Besag, “On the statistical analysis of dirty pictures,” J. Roy. Stat. Soc. B48, 259–302 (1986).
- S. Z. Li, “MAP image restoration and segmentation by constrained optimization,” IEEE Trans. Image Process.7(12), 1730–1735 (1998). [CrossRef]
- R. Pan and S. J. Reeves, “Efficient Huber-Markov edge-preserving image restoration,” IEEE Trans. Image Process.15(12), 3728–3735 (2006). [CrossRef] [PubMed]
- M. Rivera and J. L. Marroquin, “Efficent half-quadratic regularization with granularity control,” Image Vision Comput.21, 345–357 (2003). [CrossRef]
- 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. Mignotte, “A segmentation-based regularization term for image deconvolution,” IEEE Trans. Image Process.15(7), 1973–1984 (2006). [CrossRef] [PubMed]
- 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).
- 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).
- 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).
- 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]
- R. Szeliski, “Bayesian modeling of uncertainty in low-level vision,” Int. J. Comput. Vision5(3), 271–301 (1990). [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).
- S. Z. Li, Markov Random Field Modeling in Image Analysis (Springer-Verlag, 2009).
- J. E. Besag, “Spatial interaction and the statistical analysis of lattice systems,” J. Roy. Stat. Soc. B36, 192–236 (1974).
- 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]
- 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).

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