R. Szeliski, "Bayesian modeling of uncertainty in low-level vision," Int. J. Comput. Vis., vol. 5, pp. 271--301, Dec. 1990.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....the surface must be known a priori. This approach is in general not realistic for an autonomous robot that cannot afford to perform large motions without attending to the scene and thus running the risk of a critical collision. The second approach to depth estimation is the iconic depth estimator [30, 46, 60, 70] in which all pixels contribute a depth estimate. This approach is more suitable for a navigating robot as it lends itself to small motions between viewpoints. Thus, the image and depth correspondence problems are locally constrained and facilitated, and a dense disparity field is obtained. The ....

....A novel approach to comparing these confidence values in the context of data accumulation will be introduced. A hybrid optical flow estimator will be constructed from the different flow estimation algorithms considered. The second part of the thesis draws heavily from previous work by Szeliski [60] and Matthies et al. 46] who discuss a Bayesian formulation for weighted accumulation of information using a maximal estimation framework [48] Szeliski has shown that Bayesian modeling can be used for low level vision systems. As such, measurements can successfully be represented as a mean and ....

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Szeliski, R., Bayesian Modeling of Uncertainty in Low-Level Vision, International Journal of Computer Vision, 5:3, pp. 271-301, 1990.

....that uses completed computations, before moving on to partial results. The estimate x c (of the state of the world x) results from using visual cue c. Probabilistic approaches to estimation use models for noise introduced in the sensing process to describe the uncertainty in the estimate x c [26]. Maximum a posteriori estimation yields the state that maximizes the posterior p(x y c )given the noisy source of information y c : x c = argmax p(x y c ) 2) We can think of y c as resulting from a series of n computational steps (iterations of an optimization algorithm) Y c ....

Szeliski, R.: Bayesian modeling of uncertainty in low-level vision. IJCV 5 (1990) 271--302

....ODDR E MURI funded through ARO Grant DAAD19 00 1 0466. E. B. Sudderth was partially funded by an NDSEG fellowship. I. INTRODUCTION Gaussian processes play an important role in a wide range of practical, large scale statistical estimation problems. For example, in such fields as computer vision [3, 4] and oceanography [5] Gaussian priors are commonly used to model the statistical dependencies among hundreds of thousands of random variables. Since direct linear algebraic methods are intractable for very large problems, families of structured statistical models, and associated classes of ....

R. Szeliski. Bayesian modeling of uncertainty in low--level vision. International Journal of Computer Vision, 5(3):271--301, 1990.

....well structured environments consisting of man made objects, but lack representative power in complex domains. Pixel based models represent depth at each pixel in the image. Statistical formulations of pixelbased depth estimation, using random field models of the depth map, have been presented in [14, 19, 26]. Pixelbased depth models promise more generality than feature based models; however, much remains to be done on both mathematical and system aspects of this approach. The central system issue is how to find stereo correspondences efficiently and reliably. There are two types of approach: ....

....in the context of multi sensor systems and incremental construction of 3D scene descriptions. The formulation used here, which implicitly models the depth estimate at each pixel as statistically independent from other pixels, is a special case of more general random field models of the depth map [15, 26] and joint Bayesian formulations of the stereo matching problem [15, 16] Whereas previous Bayesian approaches to stereo have obtained the prior density from heuristic, surface smoothness considerations [15] the bootstrap operation has the conceptual and practica advantage that the prior density ....

R. Szeliski. Bayesian Modeling of Uncertainty in Low Level Vision. PhD thesis, Carnegie Mellon University, August 1988.

....constraint to produce more accurate flow fields. Most previous attempts at detecting discontinuities from motion have focused on an analysis of this flow field using region growing (Potter 1980) or edge detection techniques (Thompson, Mutch, Berzius 1982) A variation on this approach (Mutch Thompson 1988) computes accretion deletion regions using cor relation techniques. Another approach, which we will also exploit, uses information about the distribution of flow vectors in a neighborhood about a point to decide if a discontinuity is present (Spoerri Ullman 1987) These previous approaches ....

....of the constraints, which is admittedly somewhat simple. We are currently working on a Bayesian interpretation for our constraints. A conditional probability for a discontinuity can be obtained from each constraint and these can then be combined. The Bayesian model of the uncertainty developed in (Szeliski 1988) for flowfield computation provides hope that such a rigorous treatment is possible. Figure 9: Experiments. a) Threshold of potential field using Cs, CS,A and CF, C, b) initial snake positions. c) final positions. There is also work to be done extending the constraints themselves; in ....

R. S. Szeliski. Bayesian Modeling of Uncertainty in Low-Level Vision. PhD thesis, Carnegie Mellon University, 1988.

....seem to be a theoretical justification for using GCV for arbitrary data, although it has some nice asymptotic properties. A different approach, which is closer to ours, is that of Bayesian model selection which, to the best of our knowledge, was first suggested in the pioneering work of Szeliski [24]. There, the following question is posed: given the data D, what is the most probable value of J More recent work in this direction was done by MacKay [18] This article suggests a different approach, namely, computing the probability distribution by directly integrating over the ....

.... 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 ....

R. Szeliski. Bayesian Modeling of Uncertainty in Low-Level Vision. Kluwer, 1989.

.... Minimizing (2) with the constraint (3) or variations of this problem, where, for example, the hard constraint (3) is relaxed and replaced by a quadratic penalty on the di#erence between the two sides of this equation, is a classic variational problem [152] Alternatively, as discussed in [312, 223, 111] (see also Section 6.2.1) optimization problems such as this can also be interpreted as estimation problems with fractal priors. Computing the optimal estimates for such problems involves solving partial di#erential equations [152] a computationally intensive, but not overwhelming task in ....

....interpretation of the second term requires some care , but, if we take the perspective that all we are attempting to do is to capture the intent of this regularization penalty, we can use a simple observation to replace it with a very simple MR quadtree model. In particular, as pointed out in [312, 223], the second term in (52) can be thought of as a (Gaussian) fractal prior. For example, in 1 D each of the functions in Figure 16 yields identical values for the 1 D version of the second term in (52) and thus are equally likely under this implied prior. Alternatively, as argued in [223, 111] ....

R. Szeliski. Bayesian Modeling of Uncertainty in Low-level Vision. Kluwer, Norwell MA, 1989.

....material, albedo or motion. For such an estimation task, the representation of the quantities to be estimated can be critical. Here we propose and describe a new scene representation with appealing qualities for estimation. Typically, these scene properties might be represented as a bitmap (eg [14]) or as a series expansion in a basis set of surface deformations (eg [10] To represent accurately the details of realworld shapes or textures requires either full resolution images or very high order series expansions. Having to infer high dimensional quantities makes the estimation ....

....deformations (eg [10] To represent accurately the details of realworld shapes or textures requires either full resolution images or very high order series expansions. Having to infer high dimensional quantities makes the estimation intrinsically difficult [3] Strong priors are often needed [14], leading to arbitrary choices that can give unrealistic shape reconstructions. The approach we propose is to let the image itself bear as much of the representational burden as possible. We assume that the image is always available and we describe the underlying scene in reference to the image. ....

R. Szeliski. Bayesian modeling of uncertainty in low-level vision. International Journal of Computer Vision, 5(3):271301, 1990.

....beliefs about shape and image disparities. Examples include the use of SSD based likelihoods [Matthies, 1992] window based correlation [Kanade and Okutomi, 1994] ordering constraints on correspondence [Cox et al. 1996; Ishikawa and Geiger, 1998] shape priors [Belhumeur, 1996] as well as MRF [Szeliski, 1990; Geiger and Girosi, 1991] and parametric models [Kanatani and Ohta, 1999; Torr et al. 2001] But these approaches do not estimate the probability that a region is empty or occupied. This is because image consistency is not equivalent to occupancy, nor is it the opposite of emptiness (Figure 1a) ....

R. Szeliski, \Bayesian modeling of uncertainty in low-level vision," Int. J. Computer Vision, 5(3):271-301, 1990.

....and analyzes experimental modes is available from whitechapel.media.mit.edu in the file u ftp misc facerecognition.tar. Z. 3. 4 The connection The connection between physical and experimental modes comes from the close relationship between me chanical and probabilistic prior models [9][10]. The mechanical viewpoint is the one we have used above, modeling the elastic field by a stiffness matrix K and then minimizing the deformation energy which is a function both of the displacements and of K (Equation 6) As mentioned, given K we can analytically compute the modes of variation, ....

R. Szeliski. Bayesian Modeling of Uncertainty in Low-level Vision. Kluwer Academic Publishers, 1989.

....include photometric effects, occlusions, sensor and discretization noise. The problem with occlusion exists commonly in all three matching methods and will be discussed later. The last cause (mainly noise) can be partially overcome by statistical optimal estimations, for example, MAP methods [8] [24] [4] which are well suited to solve in general ill posed problems in low level vision. Photometric effects can partly be eliminated by some adequate techniques, such as using a spatial coherent multiplier in the matching process [10] Thus the intensity based methods are also efficient and useful ....

R.Szeliski, Bayesian modeling of uncertainty in low-level vision, Int. Journal of Computer Vision, Vol.5, 1990 pp. 271-301

....techniques do not compute optical flow as a necessary intermediate step, nor establish correspondence between points in two images. But the results are still very sensitive to noise and unreliable, because only the information about two image frames is used. There are many works [MSK89] [Sze89], SW90] Hl91a] which use the Kalman filters to improve the motion and depth estimation from image sequences. Because their stochastic dynamic models rely on the full optical flow, all of them need to first extract the full optical flow by the correlation methods, which is computationally ....

.... optical flow: x u k Gamma1 (x) 9) with u k Gamma1 (x) A(x) k Gamma1 (x) u Thus we predict: k Gamma1 (x) d = f k Gamma1 (x) d and P k Gamma1 (x) Q k Gamma1 (x) 11) where f k Gamma1 (x) 1 xW y Gamma yW x Gamma T z : 12) As in [Sze89] suggested, we simply assume that the error covariance of the estimate uniformly increases by a small factor ffl instead of assuming that the uncertainty exists in the transit process due to additive Gaussian noise and other multiplicative errors: k Gamma1 (x) 1 ffl) 13) 4 ....

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R.Szeliski, Bayesian modeling of uncertainty in low-level vision, Kluwer Academic, Boston, 1989

.... surface reconstruction [8,53,98,17,99,24,43] edge detection [103,131,44] texture analysis [23,30,40,34,35] optical ow [64,63,78,118,60] shape from X [9,68] active contours [74,3,123] deformable templates [97,95,70] data fusion [26] visual integration, and perceptual organization [2,125]. The use of MRFs in high level, such as for object matching and recognition, has also emerged in recent years [100,29,51,4,42,76,28,87,90,91] MRF theory tells us how to model the a priori probabilityofcontextual dependent patterns, such as textures and object features. A particular MRF model ....

....from these sources of knowledge is the Bayes labeling. The maximum a pos31 Fig. 5. Twochoices of cost functions. terior (MAP) solution, as a special case in the Bayes framework, is sought in many image analysis algorithms. The MAP MRF framework is advocated by Geman and Geman (1984) and others [48,35,47,38,15,125,45]. Since the paper of [46] numerous problems have been formulated in this framework. This section reviews related concepts and derives involved probabilistic distributions and energies in MAP MRF labeling. For more detailed materials on Bayes theory, the reader is referred to books like [129] ....

R. Szeliski. Bayesian modeling of uncertainty in low-level vision. Kluwer, 1989.

....sites are processed sequentially, and for each site the label which gives the largest increase of the energy function is chosen. This algorithm is very sensitive to the initial labeling. An alternative is to use discrete relaxation labeling methods; this has been done by many authors, including [13,36,39]. In relaxation labeling, combinatorial optimization is converted into real optimization with linear constraints. Then some form of gradient descent which gives the solution satisfying the constraints is used. 1.6 Contributions and prior publications The main contribution of this thesis lies in ....

R.S. Szeliski. Bayesian modeling of uncertainty in low-level vision. International Journal of Computer Vision, 5(3):271-302, 1990.

....d, which is related to f by means of the likelihood function P (djf ) The most popular way to estimate an MRF is maximum a posteriori (MAP) estimation. The MAP MRF framework was popularized in vision by Geman and Geman [20] and has been studied by many since (see for example Besag [5] Szeliski [38]) MAP estimation consists of maximizing the posterior probability p(f jd) From the point of view of Bayes estimation, the MAP estimate minimizes the risk under the zero one cost function. Using Bayes rule, the posterior probability can be written as p(f jd) p(djf)p(f) p(d) 19 Thus the ....

R. Szeliski. Bayesian modeling of uncertainty in low-level vision. Kluwer Academic Publishers, 1989. 90

....may be reasonable to approximate # w Pr( f D,w)Pr(w D) dw by approximating the integrand with a rectangular function around w max . However, the distribution Pr(w D) can be complicated and A Full Bayesian Approach to Curve and Surface Reconstruction 29 this approximation will then fail [24, 33]; see also an example of such a data set and the corresponding probability distribution it induces on the weights, in this paper (Figs. 12, 13) In this paper, it will be shown how to find the function f maximizing # w Pr( f D,w)Pr(w D) dw.Two other problems which are addressed are computing ....

....a few methods for choosing the smoothing parameter are analyzed. Bayesian model selection is another approach for choosing an optimal smoothing parameter. To the best of our knowledge, it was first suggested to apply Bayesian model selection to regularization in the pioneering work of Szeliski [33]. There, the following question is posed: given the data D, what is the most probable value of the smoothing parameter # More recent work in this direction was done by MacKay in [24] and [23] which contains an extensive study on approximations to the ideal Bayesian approach, which, as the ....

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R. Szeliski, Bayesian Modeling of Uncertainty in Low-Level Vision, Kluwer, 1989.

....Variational techniques use the Euler equations, which are guaranteed to hold at a local minimum. 4 To apply these algorithms to actual imagery, of course, requires discretization. Another alternative is to use discrete relaxation labeling methods; this has been done by many authors, including [12, 36, 41]. In relaxation labeling, combinatorial optimization is converted into continuous optimization with linear constraints. Then some form of gradient descent which gives the solution satisfying the constraints is used. Relaxation labeling techniques are actually more general than energy minimization ....

R.S. Szeliski. Bayesian modeling of uncertainty in low-level vision. International Journal of Computer Vision, 5(3):271-302, December 1990.

.... data collection (optimal experiments [26] active data selection [53] autonomous exploration [86] ffl For the purposes of model selection (Bayesian Ockam Razor ) to score alternative models according to both their goodness of fit to the data and their relative complexity [31] 51] 73] [77]. 2.3.3. Model Matching. The next logical step in MBOR is matching the models recovered from the data with those stored in a previously built database. Matching means establishing, ideally, a one to one correspondence between the models recovered from the data and nominal models built on ....

R. Szeliski. Bayesian Modeling of Uncertainty in Low--Level Vision. Kluwer Academic Publishers, Boston, MA, USA., 1989.

....of interpolating noisy data. The Bayesian framework I will describe for these tasks is due to Gull and Skilling [5, 6, 8, 17, 18] who have used Bayesian methods to achieve the state of the art in image reconstruction. The same approach to regularisation has also been developed in part by Szeliski [22]. Bayesian model comparison is also discussed by Bretthorst [2] who has used Bayesian methods to push back the limits of NMR signal detection. As the quantities of data collected throughout science and engineering continue to increase, and the computational power and techniques available to ....

R. Szeliski (1989). Bayesian modeling of uncertainty in low level vision, Kluwer.

.... at the conditional mean of the Gaussian, E[f jg] What s more, E[f jg] is the Bayes least squares estimate of f , according to the distribution induced by the modeling equations (6) Thus, one can view the problem of minimizing (4) from the perspective of optimization or of statistical estimation [9, 10, 16]. The main advantage of the Bayesian interpretation is that it casts the problem into a probabilistic framework in which it is natural to examine the accuracy of the resulting estimates. This is especially relevant in scientific applications such as remote sensing, in which one may be, for ....

....of the image, and the second is that the associated estimation algorithm passes information everywhere along the tree so as to produce globally optimal estimates of the lifted fields. A. Multiscale Models for Segmentation Consider first, the model used for the estimation of s. As discussed in [9, 10, 12, 13, 16], the smoothness penalty associated with the gradient, used for example in (2) and (3) corresponds to a fractal penalty in that it is roughly equivalent to a 1=f like prior spectrum for the random field being modeled. This type of spectrum has a natural scaling law; namely, the variances of ....

R. Szeliski, Bayesian modeling of uncertainty in low-level vision, Kluwer Academic Publishers, Boston, 1989.

....If four line segments meet at a common point, and two are parallel, then those two segments are presumed to result from a surface bend. 5.1. 2 Vision Modules It is common to hypothesize modules specific to particular visual tasks, the outputs of which are integrated at higher processing levels [8, 10, 28, 72, 90, 105]. Such integration is needed to analyze images such as Fig. 1 1. Both Adelson [1] and Knill and Kersten [56, 55] have described the importance of contours and other contextual information on the perception of lightness and transparency. They demonstrate this with illusions which simultaneous ....

R. Szeliski. Bayesian Modeling of Uncertainty in Low-level Vision. Kluwer Academic Publishers, Boston, 1989.

.... well suited to modeling data sampled from a Markov Random Field (MRF) with Gaussian noise added [8] The same principles hold true for regularization [3, 24] where the energies of a physical model can be related directly with measurement and prior probabilities used in Bayesian estimation [23]. Assume that the distribution on shape parameters for a particular shape category can be modeled as a multidimensional, unimodal Gaussian distribution. The distribution can be characterized by the mean and covariance . The likelihood of a pattern is given by: ....

R. Szeliski. Bayesian Modeling of Uncertainty in Low-Level Vision. Kluwer Academic, 1989.

.... well suited to modeling data sampled from a Markov Random Field (MRF) with Gaussian noise added [11] The same principles hold true for regularization [6, 33] where the energies of a physical model can be related directly with measurement and prior probabilities used in Bayesian estimation [30]. Rather than modeling the system as an elastic material, we can instead assume nothing about it, collect data samples of the displacements of each node, and then perform a principal components analysis [9, 20, 21] The principal directions are defined in terms of the eigenvectors and eigenvalues ....

R. Szeliski. Bayesian Modeling of Uncertainty in Low-Level Vision. Kluwer, 1989.

....we show that for certain vision problems, once can de ne local updates that are guaranteed to (1) use only as many iterations as are needed for the data to reach all nodes and (2) avoid local minima. This is done by viewing the cost functional in equation 1 as the log of a posterior probability [21] and then using the machinery of Bayesian Belief Propagation [13] to derive the local updates. The organization of this paper is as follows. Section 1 illustrates the distinction between traditional relaxation and Bayesian belief propagation (BBP) on the simple example of 1D interpolation. ....

R. Szeliski. Bayesian modeling of uncertainty in low-level vision. Kluwer, 1989.

....and correlation methods. We analyze both linear and non linear estimation techniques, as well as some robust methods. In some of our analyses we do not make either a Gaussian or an asymptotic assumption. 1.2 Bias There has been previous work on optical flow that analyzes error. Examples are [22, 41, 48, 54]. However, it has not been widely noticed by the computer vision community that optical flow estimates can be biased. It has been pointed out in [37] that optical flow estimated using gradient methods is biased: estimates tend to be underestimates. As we shall show here, even the estimates of the ....

R. Szeliski. Bayesian modeling of uncertainty in low-level vision. International Journal of Computer Vision, 5:271--301, 1990.

....everywhere and may lead to poor results at object boundaries. Energy functions that do not have this problem are called discontinuity preserving and are based on robust # functions [119, 16, 97] Geman and Geman s seminal paper [47] gave a Bayesian interpretation of these kinds of energy functions [110] and proposed a discontinuity preserving energy function based on Markov Random Fields (MRFs) and additional line processes. Black and Rangarajan [16] show how line processes can be often be subsumed by a robust regularization framework. The terms in E can also be made to depend on the ....

R. Szeliski. Bayesian Modeling of Uncertainty in Low-Level Vision. Kluwer Academic Publishers, Boston, MA, 1989.

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R. Szeliski, Bayesian modeling of uncertainty in low-level vision. Ph.D. thesis, Carnegie Mellon University, 1988.

....disparities d(x, y) and ZP is a normalizing factor. The potential function itself is the sum of clique potentials that only involve neighboring sites in the field. The simplest such field is a first order field, where EP (d) # i,j #P (d(x ,y) d(x, y) #P (d(x, y ) d(x, y) 2) see [38, 31] for generalizations to higher order fields) When #(x) is a quadratic, #(x) x 2 , the field is a Gauss MRF, and corresponds in a probabilistic sense to a first order regularized (membrane) surface model [38, 31] When #(x) is a unit impulse, #(x) 1 #(x) it corresponds to a MRF that favors ....

.... EP (d) # i,j #P (d(x ,y) d(x, y) #P (d(x, y ) d(x, y) 2) see [38, 31] for generalizations to higher order fields) When #(x) is a quadratic, #(x) x 2 , the field is a Gauss MRF, and corresponds in a probabilistic sense to a first order regularized (membrane) surface model [38, 31]. When #(x) is a unit impulse, #(x) 1 #(x) it corresponds to a MRF that favors fronto parallel surfaces [13] In between these two extremes are functions derived from robust statistics, which behave much like surface models with discontinuities [5] The second part of a Bayesian model is the ....

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R. Szeliski. Bayesian Modeling of Uncertainty in Low-Level Vision. Kluwer Academic Publishers, Boston, Massachusetts, 1989.

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R. Szeliski, "Bayesian modeling of uncertainty in low-level vision," Int. J. Comput. Vis., vol. 5, pp. 271--301, Dec. 1990.

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R. Szeliski. Bayesian modeling of uncertainty in lowlevel vision. Int. J. Comput. Vis., 5(3):271--301, December 1990.

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R. Szeliski. Bayesian modeling of uncertainty in low-level vision. Int. J. Comput. Vis., 5(3):271-- 301, December 1990.

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R. Szeliski. Bayesian modeling of uncertainty in low-level vision. International Journal of Computer Vision, 5(3):271--301, 1990.

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R. Szeliski. Bayesian modeling of uncertainty in low-level vision. International Journal of Computer Vision, 5(3):271--301, 1990.

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R. Szeliski, Bayesian Modeling of Uncertainty in Low Level Vision. Kluwer, 1989.

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SZELISKI R.: Bayesian Modeling of Uncertainty in Low Level Vision. Kluwer, 1989. 5

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Szeliski, R.: 1989, Bayesian Modeling of Uncertainty in Low-level Vision. Boston: Kluwer Academic Publishers.

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R. Szeliski. Bayesian Modeling of Uncertainty in Low-level Vision. Kluwer Academic Publishers, Boston, 1989.

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R. Szeliski. Bayesian Modeling of Uncertainty in Low-level Vision. Kluwer Academic Publishers, Boston, 1989.

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R. Szeliski. Bayesian Modeling of Uncertainty in Low-Level Vision. Int. J.of Comp. Vision, 5(3):271 -- 301, December 1990. 61

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R. Szeliski. Bayesian modeling of uncertainty in low-level vision. International Journal of Computer Vision, 5(3):271-- 302, December 1990.

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R. Szeliski. Bayesian modeling of uncertainty in low-level vision. International Journal of Computer Vision, 5(3):271-- 302, December 1990.

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R. Szeliski, Bayesian Modeling of Uncertainty in Low-Level Vision, Kluwer, 1989.

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SZELISKI R.: Bayesian Modeling of Uncertainty in Low Level Vision. Kluwer, 1989. 5

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R. Szeliski. Bayesian modeling of uncertainty in low-level vision. Intl. J. Comp. Vis., 5(3):271--301, 1990.

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R. Szeliski, Bayesian Modeling of Uncertainty in Low Level Vision. Kluwer, 1989.

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R. Szeliski. Bayesian modeling of uncertainty in low-level vision. International Journal of Computer Vision, 5(3):271--301, 1990.

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R. Szeliski. Bayesian Modeling of Uncertainty in Low-Level Vision. Int. J.of Comp. Vision, 5(3):271 -- 301, December 1990. 61

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R. Szeliski. Bayesian Modeling of Uncertainty in Low-level Vision. Kluwer Academic Publishers, Boston, 1989.

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R. Szeliski. Bayesian Modeling of Uncertainty in Low-level Vision. Kluwer Academic Publishers, Boston, 1989.

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R. Szeliski, Bayesian Modeling of Uncertainty in Low-Level Vision. Academic, MA, 1989.

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