S. Geman and D. Geman. Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images. IEEE Trans. on Pattern Analysis and Machine Intelligence, 6(6):721--741, 1984.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....enjoyed considerable success in Bayesian image reconstruction[4] and restoration[1] However, MRF approaches are typically limited to modeling very local interactions in images. Several MRF potential functions have been proposed that provide good edge preservation without explicitly modeling edges[5, 6, 7, 8, 9, 10, 11]. In comparison to MRF priors, multiresolution methods can improve reconstruction quality and o#er fast and robust estimation algorithms[12, 13, 14, 15, 16, 17] Multiresolution models better account for long range interactions and can more easily be designed to separately account for edges, ....

D. Geman and G. Reynolds. Constrained restoration and the recovery of discontinuities. IEEE Trans. on Pattern Analysis and Machine Intelligence, 14(3):367--383, March 1992.

.... and image segmentation, relying on succes sive refinement to converge to a stable interpretation of the scene [O, 10] Examples of this approach include Markov Random Field models incorporating line processes to de couple motion estimation across boundaries, and brittle membrane models [5]. These techniques tend to be slow to converge and cumbersome to apply to practical problems. Also, this approach cannot directly help with problems such as transparency. In such situations every point has two motions, and no spatial segmentation can separate them. Another approach avoids the ....

S. Geman and D. Geman. Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images. IEEE Trans. on Pattern Analysis and Machine Intelligence, 6:721-741, November 1984.

....depth estimate and the generic model have been aligned, the boundaries where there are sharp depth discontinuities are identified from the generic model. Each vertex of the triangular mesh representing the model is assigned a binary variable (defined as a line process, following the terminology of [29]) depending upon whether or not it is part of a depth boundary. Within each region inside the boundaries, the trend in the values of the 3D estimate is considered and any appreciable deviations are smoothed using an energy function minimization process. The energy function consists of two terms ....

....sudden changes, and is calculated on the basis of the generic mesh, since it is free from errors. For each of the N vertices, we assign a binary number indicating whether or not it is part of the line process. This concept of the line process is borrowed from the seminal work of Geman and Geman [29] on stochastic relaxation algorithms in image restoration. The optimization function we propose is N N d 2 d 2 i:1 i:1 (1 li) fi fj)2 lds#dg, i:1 jEJfi (14) where li 1 if the ith vertex is part of a line process and is a combining factor which controls the extent of the ....

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S. Geman, D. Geman, Stochastic relaxation, gibbs distributions, and the bayesian restoration of images, IEEE Trans. on Pattern Analysis and Machine Intelligence 6 (6) (1984) 721-741. 30

....due to the fact that there is a strong correlation (or anti correlation) between color bands [Wan95, HK84] 2. 1 Isotropic smoothness Model Previous approaches for the reconstruction of a single band image used the Markov Random Field (MRF) model for introducing the prior probability P (I) [GG84, MMP87]. With this model, natural images are assumed to be isotropically smooth with probability conforming to the Gibbs distribution: e #(I) where Z is a normalization constant and #(I) is a smoothness measure of the image I. Popular smoothness measures are the first order (membrane) and the ....

....textured areas where the smoothness assumption is incorrect. For this end, several alternatives were suggested, all of them model a typical image as a piecewise constant or a piecewise linear function. Anisotropic di#usion [Pm90] robust statistics [MMRK91] and regularization with line processes [GG84], were developed to treat image interpolation and reconstruction with discontinuities. In recent studies [BZ87, BR96, BSmH97] all the above approaches were shown to be theoretically equivalent. This work is similar in spirit to these approaches, namely, interpolation is performed along edges and ....

S. Geman and D. Geman. Stochastic relaxation, gibbs distribution, and the bayesian restoration of images. IEEE Trans. Pattern Analysis and Machine Intelligence, 6:721--741, 1984.

....segmentation, and by Geiger and Yuille [1] to image segmentation. H erault and Horaud s comparison of annealing techniques [4] concludes that on its domain of application mean field annealing is the most efficient annealing technique. It is on the order of 40 times faster than simulated annealing [2]. Since we focus on figure ground segmentation, we take the performance of mean field annealing as the gold standard that we challenge with our approach, both in efficiency as well as in quality of the solution. 3 Figure ground optimization problem In [4] the problem of figure ground segmentation ....

S. Geman and D. Geman. Stochastic relaxation, Gibbs distribution, and Baysian restoration of images. IEEE Trans. on Pattern Analysis and Machine Intelligence, 6:721--741, 1984.

.... max Pt(D f )Pt(f M) 3 Regularization, for instance, can be formalized in this way because i xt Litx:yl 1 Pt(D f) 27r)ey n i=1 (assuming uncorrelated Gaussian noise of constant variance) and the prior distribution is Pt(f) o exp( A [f (v) 2dv) which resembles the Boltzmann distribution [24, 6, 11, 12, 23, 21, 19, 17, 25, 20]. Multiplying, we get that the f chosen from M should maximize exp( M(f) or minimize M(f) This simple analysis shows how regularization is consistent with Bayes rule for choosing the MAP estimate, given A. Now, what if a few models are possible As explained before, the first step is to ....

S. Geman and D.Geman. Stochastic relaxation, gibbs distribution, and the bayesian restorat ion of images. IEEE Trans. on Pattern Analysis and Machine Intelligence, 6:721-741, June 1984.

....R being the conjugate of R, and for i = 1: K we have 1 = 0: 28) We alternate resolutions of (27) and (28) until the solution set (u; Phi 1 ; Phi K ) does not evolve anymore. The resolution of (27) is done thanks to the half quadratic regularization method [6, 11] based on the introduction of an auxiliary variable related to the discontinuity set. System (28) is embeded in a dynamical scheme as for the resolution of (22) with the dynamical system (23) 6 Experimental results We made some experiments on both synthetic and real images. Some of the noisy ....

S. Geman and G. Reynolds. iConstrained restoration and the recovery of discontinuitiesj. IEEE Trans. on Pattern Analysis and Machine Intelligence, 14(3):367383, 1992.

....is clearly defined and can be computed. The control algorithm we use [9, 32] is based on this statement. It treats the search for I as a combinatorial optimization problem (see below) and solves it by means of iterative optimization methods, e.g. simulated annealing [20] stochastic relaxation [13], and genetic algorithms [14] By using iterative methods the any time capability is provided, since after each iteration step a (sub )optimal solution is always available and can be improved if necessary by performing more iterations. Another advantage is that the algorithm allows an easy ....

S. Geman and D. Geman. Stochastic relaxation, gibbs distributions, and the bayesian restoration of images. IEEE Trans. on Pattern Analysis and Machine Intelligence, 6(6):721--741, November 1984.

....converges (in distribution) to the target distribution, i.e. the distribution to be integrated, is generated. Then, the expec tation is calculated through Monte Carlo integration over the obtained samples. As sampling strategy, in our work we used the Gibbs sampling scheme, proposed originally in [13]. This method allows to derive, together with the point estimates, the confidence intervals of the variables; moreover, with respect to standard Kalman filtering, the method allows to estimate the process prior statistics, namely a, a. For a more detailed discussion see [7] and [9] for technical ....

S. Geman and D. Geman, Stochastic relaxation, Gibbs distributions, and the bayesian restoration of images, IEEE Trans. on Pattern Analysis and Machine Intelligence, vol. 6, pp. 721-741, 1984.

.... p X (x) P fx 2 Xg at an optimal level p which is chosen in such a way that E V X has a volume close to the expected volume of X , see [24] This mean also has some good properties but is not suitable for random sets with zero volume (such as point processes) In Bayesian image analysis [11] the computation of a best posterior image is related to the choice of error criterion. It is traditional to calculate the MAP (maximum posterior probability) image, which is the mode of the posterior probability distribution for the true image, and minimises a trivial measure of error. Many ....

S. Geman and D. Geman. Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images. IEEE Trans. Pattern Analysis and Machine Intelligence, 6:721--741, 1984. 18

....to control the impact of the prior on the properties desired for the solution. Because of their ability to model global properties using local constraints, Markov Random Fields (MRFs) are very popular priors. Several optimization algorithm converging either toward a global minimum of the energy [1] or a local one [2] 3] are now well dened. But accurate estimation of the parameters is still an open issue. Indeed, the partition function (normalization constant) leads to intractable computation. Parameter estimation methods are then either devoted to very specic models [4] 5] or based on ....

....on approximations such as Maximum Pseudo Likelihood [6] 7] Unfortunately, these approximations lead to inaccurate estimators for the prior parameters. Markov Chain Monte Carlo algorithms (MCMC) 8] are very popular in image processing to derive optimization methods when using a Markovian prior [1]. In fact, MCMC algorithms can be developped for other purposes such as estimation. The partition function of Gibbs Fields can be estimated using an MCMC procedure. A Maximum Likelihood estimation using an MCMC algorithm is proposed in [9] This method can be applied to a wide range of models such ....

S. Geman, D. Geman. Stochastic relaxation, Gibbs distribution, and the Bayesian restoration of images. IEEE trans. on Pattern Analysis and Machine Intelligence, 6(6):721741, 1984.

....model for generative models. This gure is from S. Roweis and Z. Ghahramani. Figure 7: A Markov Random Field for low level vision. X i is the hidden state of the world at grid position i, Y i is the corresponding observed state. This gure is from [FP00] 7 In low level vision problems (e.g. GG84, FPC00] the X i s are usually hidden, and each X i node has its own private observation node Y i , as in Figure 7. The potential (x i ; y i ) P (y i jx i ) encodes the local likelihood; this is often a conditional Gaussian, where Y i is the image intensity of pixel i, and X i is the ....

S. Geman and D. Geman. Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images. IEEE Trans. on Pattern Analysis and Machine Intelligence, 6(6), 1984.

....areaand feature based matching is obtained, but occlusions are not managed by the algorithm. In [22] an MRF model was designed to take into account occlusions defining a specific dual field, so that they can be estimated in a similar way as in a classic line process similar to that presented in [12]. In our approach, for each pixel, di#erent windows are (ideally) considered to estimate the Sum of Squared Di#erences (SSD) values and related disparities. When using area based matching, the disparity is correct only if the area covered by the matching window has constant depth. The idea is ....

.... as a realization of the discrete stochastic process in which pixel i is associated to a random variable D i (being d i its realization) and where, owing to the Markov property, the conditional probability P # d i d I i # depends only on the value on the neighboring set of i, N i (see [12]) The Hammersley Cli#ord theorem establishes the Markov Gibbs equivalence between MRFs and Gibbs Random Fields [18] so the probability distribution takes the following form: P (d) Z 1 e #U(d) 1) where Z is a normalization factor called partition function, # is a parameter called ....

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S. Geman and D. Geman. Stochastic relaxation, Gibbs distribution, and Bayesian restoration of images. IEEE Trans. on Pattern Analysis and Machine Intelligence, 6(6):721--741, 1984.

....Petri net specification. ffl Pattern recognition applications, depending on whether they focus on one dimensional signals, such as for speech recognition, or multidimensional ones, as in image analysis and understanding, frequently rely on hidden Markov models (HMMs) 30] or Markov random fields [12, 10]. Both classes of models have proved quite successful in their respective application areas. In particular, the best speech recognition systems currently available are based on HMMs. The nonintrusive appliance load monitoring problem described recently in [16] represents another interesting ....

....J of , we must solve a fixed point equation of the form (3.24) which prohibits incremental simulation. 2 As a side remark, note that fixed point equations of the form (3. 24) can be solved iteratively by employing stochastic relaxation methods such as the Metropolis algorithm or the Gibbs sampler [12]. However, such schemes fall outside the scope of the incremental simulation procedures described here. Next, since most compound systems of the form (3.22) usually give rise to execution graphs which contain either undirected branches or cycles, it is of interest to develop ....

S. Geman and D. Geman, Stochastic relaxation, Gibbs distribution, and the Bayesian restoration of images, IEEE Trans. on Pattern Analysis and Machine Intelligence, 6 (1984), pp. 721--741.

.... to a MAP solution with probability p, where p approaches one as the number of iterations approaches infinity [2] Moreover, it can be shown that if the temperature T (t) decreases towards zero slowly Boltzmann machines 5 enough, the convergence is almost certain even with a finite time process [6]. Unfortunately, for a theoretically guaranteed convergence, a computationally infeasible exponential number of iterations is needed. Although in practice good results are sometimes obtained with a relatively small number of iterations, the method can be excruciatingly slow. In the following, we ....

Geman, S. and Geman, D., Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images. IEEE Trans. on Pattern Analysis and Machine Intelligence 6 (1984), 721--741. REFERENCES 12

.... 2 is an overrelaxation parameter that is used to overcorrect the estimate of u (n 1) at stage n 1. The first partial derivative of the robust flow equation (7) is simply: E u s = X s2S [I x (I x u I y v I t ; oe 1 ) X n2G s (u s Gamma un ; oe 2 ) 3 Geman and Reynolds [3] showed that this approach can be generalized to analog line processes that assume continuous nonnegative values. 4 Only the equations for the horizontal component of the flow are show; the treatment of the vertical component is identical. where (x) ae= x. The term T (u s ) is an upper ....

D. Geman and G. Reynolds. Constrained restoration and the recovery of discontinuities. IEEE Trans. on Pattern Analysis and Machine Intelligence, 14(3):376--383, March 1992.

....sites as interacting particles taking discrete states (the continuous part has been studied in Sec. 3) The framework for studying large amounts of interacting particles is statistical physics. This framework has proved to be of great interest in the elds of image analysis and neural networks [4, 5]. We now refer to [4] and introduce the formalism of Markov random elds used in image restoration. Let be the set of possible site con gurations (j j = 5) as described in the rst paragraph and S denotes the nite set of potential sites. S is the set of all con gurations over the whole ....

....particles taking discrete states (the continuous part has been studied in Sec. 3) The framework for studying large amounts of interacting particles is statistical physics. This framework has proved to be of great interest in the elds of image analysis and neural networks [4, 5] We now refer to [4] and introduce the formalism of Markov random elds used in image restoration. Let be the set of possible site con gurations (j j = 5) as described in the rst paragraph and S denotes the nite set of potential sites. S is the set of all con gurations over the whole radio network. Let = ....

S. Geman and D. Geman, Stochastic relaxation, Gibbs distributions, and the bayesian restoration of images, IEEE Trans. on Pattern Analysis and Machine Intelligence, vol. 6, no. 6, pp. 721741, November 1984;

....b a Figure 2: Example of double well potential W . 4 Algorithm Let describe the strategy we have adopted to minimize (2) with respect to f . To overcome the diOEculty related to the nonlinearity of the Euler Lagrange equations associated to (2) we use the half quadratic regularization method [5, 8]. Doing so, if the function veries certain conditions, mainly ( p (t) strictly concave, we get (t) inf b (bt 2 (b) where b is the dual variable associated to f and is a strictly convex decreasing function. The variable b is related to the discontinuities of f such that b 0 on ....

....ffl 0) get a labelling. 5.2 Experimental results All results were obtained on a 166 MHZ computer. We assume that parameters , oe and the number M of classes are known (preliminary estimation or parameters given by an expert) The selected function is the one proposed by Geman and Mc Clure [8]: t) t 2 1 t 2 . More tests have been conducted on synthetic and real data in [18] Synthetic image: this image is of size 256 Theta 256 pixels ( gdr image) and the noisy version is such that the SNR is equal to 10 dB (see Fig. 3) We can notice that for these data the diOEculties ....

D. Geman and G. Reynolds. iConstrained restoration and the recovery of discontinuitiesj. IEEE Trans. on Pattern Analysis and Machine Intelligence, 14(3):367383, 1992.

....of the data, if and only if the PF # in (1.3) is nonsmooth at zero. Such a behavior is local in two di#erent senses: it is independent of the shape of # beyond 0 and it is exhibited by almost any strict local minimizer of E y . Among the most popular nonsmooth at zero PFs, we cite the following [23, 17, 19, 27, 2]: modulus: #(t) t , 1.4) concave: #(t) # t (1 # t ) 1.5) 0 1 : #(t) 1 if t #= 0, #(0) 0. 1.6) In order to simplify the presentation, we suppose # is twice di#erentiable everywhere except at 0. However, our results can be extended to more general objective functions combining ....

....1 norm of the derivatives of the unknown signal. Such regularizations have been observed to produce blocky 2 Recall that ln P (y x) # # Ax y# 2 when n is white Gaussian noise. LOCAL STRONG HOMOGENEITY USING REGULARIZATION 635 estimates [15, 11] The concave PF (1. 5) is shown in [19] to give rise to a step shaped estimate from ramp shaped data, and this PF is called strictly noninterpolating. Our study can be seen as an attempt to understand what regularization using nonsmooth at zero PFs accomplishes on the estimate, in comparison with smoothat zero PFs. It provides some ....

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D. Geman and G. Reynolds, Constrained restoration and recovery of discontinuities, IEEE Trans. Pattern Analysis and Machine Intelligence, PAMI-14 (1992), pp. 367--383.

....class of theories is more conned. Our second aim is to point out that our dynamic scale space paradigm can be applied and supplement mathematical morphology [23, 37, 14, 16, 17] Morphological ltering by means of size density estimators [41, 42] statistical and Bayesian morphological scale spaces [13, 51] and mathematical morphological scale spaces based on watershed methods [48, 29, 6, 7, 25] and based on parabolic dilations [44, 45] are described, substantiated as well as generalised by means of our paradigm. In section 2 an image is physically and mathematically modelled and its formation is ....

....free energy F is still conserved. These dioeerences are related to the invariance conditions imposed on the ltering scheme or better the physical objects considered as we ll demonstrate in section 2.4.2. Our approach distinguishes itself considerably from methods based on simulated annealing [24, 13, 51, 5]. There is no change in the total free energy and the convergence to a global minimum of the free energy is governed by a postulated exchange principle in which the scale variables can be conceived as inverted temperatures. These scales can also be considered as volume measures of the image over ....

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S. Geman and D. Geman. Stochastic relaxation, gibbs distributions, and the bayesian restoration of images. IEEE Trans. Pattern Analysis and Machine Intelligence, 6:721741, 1984.

....a mean to quantify the image formation in terms of curvatures and equivalences. Secondly, these curvatures and equivalences can be subjected to (categorical) AEows on the basis of exchange principles for themselves. The latter principles strongly deviate from statistical annealing principles [6]. Furthermore, our paradigm allows a straightforward generalisation of mathematical morphological paradigms [18] in particular if gauge invariance is required. Keywords: modern geometry, dynamic scale space paradigm 1 Introduction Our aim will be to present a modern geometric and statistical ....

....free energy F is still conserved. These dioeerences are related to the invariance conditions imposed on the ltering scheme or better the physical objects considered as we ll demonstrate in section 2.4.2. Our approach distinguishes itself considerably from methods based on simulated annealing [6]. There is no change in the total free energy and the convergence to a global minimum of the free energy is governed by a postulated exchange principle in which the scale variables can be conceived as inverted temperatures. These scales can also be considered as volume measures of the image over ....

S. Geman and D. Geman. Stochastic relaxation, gibbs distributions, and the bayesian restoration of images. IEEE Trans. Pattern Analysis and Machine Intelligence, 6:721741, 1984.

....Imaging in Radio Astronomy via an Expectation Maximization Algorithm for Structured Covariance Estimation 13 FIGURE 5. Results of 1000 iterations of the EM algorithm using Silverman s roughness penalty with # (NM) 0.002 (left column) and # (NM) 0. 005 (right column) image reconstruction [GR92] which could be tried here. For instance, the use of the square in (1.20) and (1.24) results in reconstructions that, while less noisy, have smoothed edges. Employing powers less than two results in a penalty that tends to smooth continuous regions while better preserving edges. For simple ....

D. Geman and G. Reynolds. Constrained restoration and the recovery of discontinuities. IEEE Trans. on Pattern Analysis and Machine Intelligence, 14(3):367--383, March 1992.

....widersprechende Interpretationen entstehen. Um eine konsistente Interpretation der Daten zu erreichen, m ussen diese Hypothesen in einem globaleren Kontext bewertet werden, wozu jeder Hypothese eine Signifikanz zugewiesen wird. Dies erfolgt mittels eines Markov Random Field (siehe z.B. CJ93, GG84] Jede Gruppierungshypothese bildet einen Knoten eines ungerichteten Graphen, die durch unterst utzende oder konkurrierende Nachbarschaftsbeziehungen verbunden sind. Auch die Kontursegmente der initialen Segmentierung, die Element einer Gruppierungshypothese sind, werden als Knoten in den ....

....Gruppieren f ur die 1D Ebene der Gruppierungshierarchie aus Bild 1 dargestellt. Weitere Einzelheiten sind in [Sch95] zu finden. 3. 1 Nachbarschaftsrelationen Die Nachbarschaftsrelationen im Markov Random Field modellieren in unserer Anwendung keine direkte r aumliche Nachbarschaft (wie etwa in [GG84] sondern die Kompatibilit at von Knoten des Graphen, die als Teilinterpretationen der Bilddaten aufgefat werden k onnen. Daher werden diese Nachbarschaftsbeziehungen in sich gegenseitig unterst utzende und einander widersprechende Gruppierungen unterschieden. Diese Nachbarschaftstypen finden ....

S. Geman und D. Geman. Stochastic Relaxation, Gibbs Distributions, and the Bayesian Restoration of Images. Trans. on Pattern Analysis and Machine Intelligence (PAMI), 6:721--741, 1984.

.... processing, the usual guideline is rst the design of an energy composed by a data driven term and a regularization term, and second the minimization of this energy performed either by deterministic algorithms (Iterated Conditional Modes (ICM) 1] or by stochastic ones (Simulated Annealing (SA) [4]) Since this energy corresponds to the Hamiltonian of some Gibbs eld, both terms are sums of clique potentials. When the state space of the random eld includes a small number of values (binary images, labels for segmentation) the usual regularization terms are Ising potentials, which is already ....

....of the regularization term to the global energy in the case of high ratio of noise results in the loss of small objects and ne structures such as lines. The usual way to overcome the problem of localization of edges is the consideration of an additional line process dened on the dual lattice [4], which involves the manipulation of two interacting models and results in complex parameter estimations. Moreover, this approach does not preserve lines and ne structures. Lines can be recovered using structural models or features dictionnary [5] but then, some paticular knowledge about the scene ....

S. Geman D. Geman. Stochastic relaxation, Gibbs distribution, and the Bayesian restoration of images. IEEE trans. on Pattern Analysis and Machine Intelligence, 6(6):721741, 1984.

....(1) where k is a normalisation constant that is independent of x. In the general case that p(zjx) is multi modal p(xjz) cannot be evaluated simply in closed form: instead iterative sampling techniques can be used. The first use of such an iterative solution was proposed by Geman and Geman [11] for restoration of an image represented by mixed variables, both continuous (pixels) and discrete (the line process ) Sampling methods for recovery of a parametric curve x by sampling [24, 14, 25] have generally used spatial Markov processes as the underlying probabilistic model p(x) The basic ....

Stuart Geman and Donald Geman. Stochastic Relaxation, Gibbs Distributions, and the Bayesian Restoration of Images. IEEE Trans. Pattern Analysis and Machine Intelligence, 6(6):721--741, 1984.

....been characterized analytically, the idea of including corners in completion shapes is not new. For example, the functionals of Kass et al. 8] and Mumford and Shah[11] permit orientation discontinuities accompanied by large (but fixed size) penalties. This follows work by Blake[12] and others[13, 14, 15, 16] on interpolation of smooth surfaces with creases from sparse depth (or brightness) measurements. More recently, Belheumer[17] working with stereo pairs) used a similiar functional for interpolation of disparity along epipolar lines. Belheumer s approach is especially related because he derives ....

Geman, S. and D. Geman, Stochastic Relaxation, Gibbs Distributions, and Bayesian Restoration of Images, IEEE Trans. Pattern Analysis and Machine Intelligence 6, pp. 721-741, 1984.

....to control the impact of the prior on the properties desired for the solution. Because of their ability to model global properties using local constraints, Markov Random Fields (MRFs) are very popular priors. Several optimization algorithms converging either toward a global minimum of the energy [1] or a local one [2] 3] are now well dened. But accurate estimation of the parameters is still an open issue. Indeed, the partition function (normalization constant) leads to intractable computation. Parameter estimation methods are then either devoted to very specic models [4] 5] or based on ....

....required to get homogeneous realizations. Thus, MLEs should improve image segmentation and image restoration algorithms based on Markovian priors. Markov Chain Monte Carlo algorithms (MCMC) 10] are very popular in image processing to derive optimization methods when using a Markovian prior [2] [1]. In fact, MCMC algorithms can be developed for other purposes than Bayesian inference. Indeed, they can be used to derive MLE. The partition function of Gibbs Fields can be estimated using an MCMC procedure. INRIA MCMCML 5 A Maximum Likelihood estimation using an MCMC algorithm is proposed in ....

S. Geman, D. Geman. Stochastic relaxation, Gibbs distribution, and the Bayesian restoration of images. IEEE trans. on Pattern Analysis and Machine Intelligence, 6(6):721741, 1984.

....strength of the soft modeling approach is its versatility. For example, the smoothing prior proposed by Geman and McClure [9] and [10] for tomography reconstruction has been successfully applied to many disparate areas of image restoration including infrared image enhancement [6] image deblurring [7], movie restoration [11] and Hubble telescope image enhancement [16] The primary strength of the hard modeling approach is in the ability to directly estimate certain parameters which might be of scientific importance without actually restoring the image. In the case of soft model image ....

D. Geman and G. Reynolds. Constrained restoration and the recovery of discontinuities. IEEE Trans. Pattern Analysis and Machine Intelligence, 14(3), 1992.

....in formulating the problem are involved and smaller number of sites are needed in the MAP search procedure. As a result, a substantially reduced cost in the MFT based optimization procedure is achieved. Within our formulation, we will argue that the line field (LF) which is first introduced in [10] and employed extensively in the literature ever since, can be discarded in the process of estimating motion vectors. Instead, the discontinuity problem is taken care of by a truncation function. As such, only two MRF s are involved in our MAP search process, namely, the motion vector field and ....

....by most real world images. The MRF lends us a powerful tool in modeling the local property for images. However, for the purpose of image processing, the computation based on the pdf s for each site is prohibitively intensive. Due to the equivalence of the MRF to Gibbs Random Field (GRF) [10, 12], a method to characterize the global feature of a random field from the local features is provided. With this equivalence, the pdf P (f) for a configuration f , which is a joint event, has the following close form expression: P (f) exp[ GammafiU (f ) Z; 6) where U(f) X c2C V c (f) 7) ....

[Article contains additional citation context not shown here]

S. Geman and D. Geman. Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images. IEEE Trans. on Pattern Analysis and Machine Intelligence, 6(6):721--741, 1984.

....deformable contour models as snakes [10] or balloons [4] provide one possible way to perform the edge fragment grouping. 2) Area based methods, such as region growing and splitting merging techniques. 1] 11] 3) Global Optimization methods, such energy functionals [17] Gibbs fields [15] [8] or combinations of them. Local methods using discontinuities to distinguish different regions provide a good starting point for image segmentation. Their major drawback is that they do not use information cues occurring on larger scale, thus ignoring available information of regions. Neglecting ....

S. Geman and D. Geman. Stochastic relaxation, gibbs distributions, and the bayesian restoration of images. IEEE Trans. on Pattern Analysis and Machine Intelligence, 6(6):721--741, Nov. 1984.

....been characterized analytically, the idea of including corners in completion shapes is not new. For example, the functionals of Kass et al. 11] and Mumford and Shah[13] permit orientation discontinuities accompanied by large (but fixed size) penalties. This follows work by Blake[2] and others[6, 7, 12, 15] on interpolation of smooth surfaces with creases from sparse depth (or brightness) measurements. More recently, Belheumer[1] working with stereo pairs) used a similiar functional for interpolation of disparity along epipolar lines. Belheumer s approach is especially related because he derives ....

Geman, S. and D. Geman, Stochastic Relaxation, Gibbs Distributions, and Bayesian Restoration of Images, IEEE Trans. Pattern Analysis and Machine Intelligence 6, pp. 721-741, 1984.

....is estimated using any of the above approaches, it is relatively straightforward to find the flow discontinuities (Thompson, 1985) The second approach is to make discontinuities explicit, thus avoiding the need to deal with mixed distributions. All of these methods follow the basic structure of Geman and Geman (1984) or Blake and Zisserman (1987) The most common approach utilizes a Markov random field (MRF) formulation with explicit line processes (Murray and Buxton, 1987; Gamble and Poggio 1987; Hutchinson et al. 1988; Koch et al. 1989; Konrad and Dubois, 1992; Heitz and Bouthemy, 1993) Interacting ....

S. Geman and D. Geman. Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images. IEEE Trans. on Pattern Analysis and Machine Intelligence, PAMI-6:721--741, November 1984.

.... p X (x) P fx 2 Xg at an optimal level p which is chosen in such a way that E V X has a volume close to the expected volume of X , see [24] This mean also has some good properties but is not suitable for random sets with zero volume (such as point processes) In Bayesian image analysis [11] the computation of a best posterior image is related to the choice of error criterion. It is traditional to calculate the MAP (maximum posterior probability) image, which is the mode of the posterior probability distribution for the true image, and minimises a trivial measure of error. Many ....

S. Geman and D. Geman. Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images. IEEE Trans. Pattern Analysis and Machine Intelligence, 6:721--741, 1984. 18

....takes advantage of the neighborhood system for the formation of local interacting grouping hypotheses. In the following subsection we give a brief introduction to MRFs and the notation we use in the rest of the paper. For a thorough treatment of MRFs, especially applied to computer vision see [11, 23]. In the subsequent subsections we describe our use of MRFs for perceptual grouping. 6.1 Markov Random Fields To represent a given problem with a MRF, we define a set of entities or sites S = fs 1 ; s N g. In image analysis sites typically correspond to pixels or image primitives like ....

S. Geman and D. Geman. Stochastic relaxation, gibbs distributions, and the bayesian restoration of images. Trans. on Pattern Analysis and Machine Intelligence (PAMI), 6:721--741, 1984.

....on two clouds in the lighting space. A second distribution describes the lighting variation as a smooth process, in the sense the lighting at a pixel must be very close to at least one of its neighbors. Formally we represent this local constraints using a Markov random field (MRF) model [6]. We define the clique energy at pixel x due to lighting change as c l (x) min y2N (x) kl(x) l(y)k l (2) with N (x) describing all the four neighbors of pixel x. The probability of observing lighting l given its neighborhood is given by the Gibbs distribution g(ljN ) ke c l (x) T ....

D Geman and Geman S. Stochastic relaxation, gibbs distribution, and the bayesian restoration of images. IEEE Trans. Pattern Analysis and Machine Intelligence, 6:721-- 741, 1984.

....on two clouds in the lighting space. A second distribution describes the lighting variation as a smooth process, in the sense the lighting at a pixel must be very close to at least one of its neighbors. Formally we represent this local constraints using a Markov random field (MRF) model [6]. We define the clique energy at pixel x due to lighting change as c l (x) min y#N (x) #l(x) l(y)# 2 # 2 l (2) with N (x) describing all the four neighbors of pixel x. The probability of observing lighting l given its neighborhood is given by the Gibbs distribution g(l N ) ke c ....

D Geman and Geman S. Stochastic relaxation, gibbs distribution, and the bayesian restoration of images. IEEE Trans. Pattern Analysis and Machine Intelligence, 6:721-- 741, 1984.

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S. Geman and D. Geman. Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images. IEEE Trans. on Pattern Analysis and Machine Intelligence, 6(6):721--741, 1984.

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S. Geman and D. Geman. Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images. IEEE Trans. on Pattern Analysis and Machine Intelligence, 6(6):721--741, 1984.

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S. Geman and D. Geman. Stochastic relaxation, gibbs distribution, and the bayesian retoration of images. IEEE Trans. Pattern Analysis and Machine Intelligence, 6:721--41, Nov. 1984.

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S. Geman and D. Geman. Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images. IEEE Trans. on Pattern Analysis and Machine Intelligence, 6(6), 1984.

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S. Geman and D. Geman. Stochastic relaxation, gibbs distributions, and the bayesian restoration of images. IEEE Trans. on Pattern Analysis and Machine Intelligence, 6(6):721--741, November 1984.

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S. Geman and D. Geman. Stochastic relaxation, gibbs distributions, and the bayesian restoration of images. IEEE Trans. on Pattern Analysis and Machine Intelligence, 6(6):721--741, November 1984.

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Geman S. and Geman, D. (1984), "Stochastic Relaxation, Gibbs Distributions, and the Bayesian Restoration of Images, IEEE Trans. Pattern Analysis and Machine Intelligence, PAMI-6, pp. 721-741.

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S. Geman and D. Geman. Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images. IEEE Trans. on Pattern Analysis and Machine Intelligence, 6(6):721--741, 1984.

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S. Geman and D. Geman. Stochastic relaxation, gibbs distributions, and the bayesian restoration of images. IEEE Trans. on Pattern Analysis and Machine Intelligence, 6:1721--741, 1984.

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S. Geman and D. Geman. Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images. IEEE Trans. on Pattern Analysis and Machine Intelligence, 6(6), 1984.

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S. Geman and D. Geman. Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images. IEEE Trans. on Pattern Analysis and Machine Intelligence, 6(6):721--741, 1984.

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Geman, S. and Geman, D.: #Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images," IEEE Trans. on Pattern Analysis and Machine Intelligence,Vol. PAMI #November 1984# 721-741.

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Geman S. and Geman D., \Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images", IEEE Trans. on Pattern Analysis and Machine Intelligence 6, 721-741, 1984.

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Geman, S., Geman, D., IEEE Trans. on Pattern Analysis and Machine Intelligence, 6(6): 721-741, 1984.

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