C. Bouman and K. Sauer, "A generalized Gaussian image model for edge-preserving MAP estimation," IEEE Trans. Image Processing, vol. 2, pp. 296--310, 1993.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....x i or in some cases G i x = x i 1 2x i x i 1 . Typically, for all . r , we have # i = 0 and # i reads # i (z) #(#z#) r , 4) where # : IR IR is an increasing function, often called potential function. Several functions #, among the most popular, are the following [20, 5, 21, 29, 22, 33, 11, 35, 7]: Lorentzian #(t) #t (1 #t Concave #(t) # t (1 # t ) Gaussian #(t) 1 exp ( #t Truncated quadratic #(t) min #t Huber #(t) t # #(# 2 t # ) if #. 5) Objective functions as specified above are based either on PDE s [29, 33, 2, 13, 12, ....

C. Bouman and K. Sauer, A generalized Gaussian image model for edge-preserving map estimation, IEEE Transactions on Image Processing, IP-2 (1993), pp. 296--310. 21

....#(g i x) 3) where g IR, for i = 1, r, yield the di#erences between neighboring samples of x and # : IR is a potential function. We will denote by G the r p matrix whose ith row is g i , for every i = 1, r. Several frequently used , convex potential functions # are [4, 16, 7, 19, 6, 25] , 1 2, 4) # t , 5) #(t) 1 t # log (1 t #) 6) #(t) log(cosh (t #) 7) #t 2 if t # 1 #, t 1 (2#) if 1 #, 8) where # 0 is a parameter. These functions are known to be edge preserving since the minimizer x can involve large di#erences ....

.... t #) 6) #(t) log(cosh (t #) 7) #t 2 if t # 1 #, t 1 (2#) if 1 #, 8) where # 0 is a parameter. These functions are known to be edge preserving since the minimizer x can involve large di#erences g i x at the locations of edges in the original signal or image [7, 19, 6, 11, 25]. The equivalent form (2) now corresponds to Q y (z) i (z y) 9) where b i is the ith row of the matrix B = GA 1 , for every i = 1, r. We suppose that B does not contain zero valued columns. If # is , symmetric and convex as all the functions # in (4) 8) ....

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C. Bouman and K. Sauer, A generalized Gaussian image model for edge-preserving map estimation, IEEE Transactions on Image Processing, IP-2 (1993), pp. 296--310.

....between neighboring samples of x. Typically, for all i . r , we have # i = 0 and # i reads # i (z) #(#z#) #i # 1, r , 3) where # : IR IR is an increasing function, often called potential function. Several functions #, among the most popular, are the following [6, 1, 7, 9, 8, 11, 3, 12, 2]: #(t) 2, Lorentzian #(t) #t (1 #t Concave #(t) # t (1 # t ) Gaussian #(t) 1 exp ( #t Truncated quadratic #(t) min #t Huber #(t) t t # #(# 2 t # ) if #. 4) # CMLA UMR 8536 ENS de Cachan, 61 av. President Wilson, 94235 Cachan ....

C. Bouman and K. Sauer, A generalized Gaussian image model for edge-preserving map estimation, IEEE Transactions on Image Processing, IP-2 (1993), pp. 296--310.

....C2 is needed for gradientbased iterations. C3 ensures convexity of Phi. C5 ensures that the proposed iterations are well defined, and C4 is central to the convergence proofs. A large class of functions satisfy the above conditions. One notable exception is the Generalized Gaussian class [9] (t) jtj for p 2, due to either C2 (for p 1) or C5 (for 1 p 2) Newton type methods seem to be poorly suited to that class, and line search methods appear necessary. Another exception is the entropy functions of the form (t) t log t. Also excluded is (t) jtj for p 2, due to ....

.... Gamma c] i (1 Gamma ff) Bv Gamma c] i ) ff i ( Bu Gamma c] i ) 1 Gamma ff) i ( Bv Gamma c] i ) ff Phi(u) 1 Gamma ff) Phi(v) We consider convexity to be desirable since non convex objective functions lead to estimators that are discontinuous functions of the measurements [9]. Algorithms for the convex case are also of use for problems that are non convex since graduated nonconvexity and deterministic annealing methods are often applied. III. HUBER S ALGORITHM In [4] Huber derived an algorithm for minimizing objective functions of the form (5) in the context of ....

C Bouman and K Sauer. A generalized Gaussian image model for edge-preserving MAP estimation. IEEE Tr. Im. Proc., 2(3):296-- 310, July 1993.

....problem (with #=0) is poorly conditioned or even under determined, so some regularization is required to ensure a stable solution. Gradient based iterative methods generally converge only to local minima for non convex regularizing functions, so we focus here on convex penalty functions [31]. The following general form [32] expresses most of the convex penalty functions that have been proposed for regularization of imaging problems: R(x) # k ( Cx c] k ) 3) where C is a K p matrix and c # ,forsome user defined number K of soft constraints of the form [Cx] k # c k . ....

....Regularization methods are generally designed to ensure such positive definiteness. It suffices to have G WG # DMD where D is diagonal and M is approximately block circulant. K # 2p and K # 4p for first and second order neighborhoods respectively. This assumption precludes the choice [31] k (t) t for p 2 which has unbounded second derivative. For certain ROI quantification tasks the quadratic penalty is useful [30] and even outperforms nonquadratic penalties [34] Since this paper focuses on comparing various preconditioners, for simplicity we ignore any nonnegativity ....

C. Bouman and K. Sauer, "A generalized Gaussian image model for edge-preserving MAP estimation," IEEE Tr. Im. Proc., vol. 2, no. 3, pp. 296--310, July 1993.

....the original object to be restored and of the noise. A simple noise model is often satisfactory, while finding a well adapted a priori model of the object is more complicated. To model homogeneous regions separated by sharp discontinuities, Markov Gibbs random fields (MRFs) have often been chosen [2] because they are based upon local pixel interactions. Moreover, Gibbs potentials used in MRFs can be customized to achieve sharper restoration than with the classical quadratic (i.e. Gaussian) a priori models. But, the MRFs designed to restore sharp discontinuities often yield nonconvex criteria ....

C. Bouman and K. Sauer, "A generalized Gaussian image model for edge-preserving MAP estimation," IEEE Trans. Image Processing, vol. 2, no. 3, pp. 296-310, July 1993.

.... Courant Institute of Mathematical Science, New York University, New York, NY 10012, USA. This work was partly supported by the AFOSR MURI grant F49620 96 1 0028 and the NSF grant IIS 0114391. Centre National d Etudes Spatiales, 18 Av. Edouard Belin, 31055 Toulouse, France. estimators [2, 3]. A classical approach to deconvolution is to write the estimation as a minimization problem that incorporates a fidelity term to the observed data and an a priori measure that regularizes the estimation. This approach can also be casted as a Bayesian estimation which minimizes a posterior ....

C.A. Bouman and K. D. Sauer, "A generalized Gaussian image model for edge-preserving," IEEE Trans. on Image Processing, vol. 2, no. 3, pp. 296--310, July 1993.

....theorc,n ensures convexity of R0 ( I) for any function R0 which is simultaneously convex and increasing on . Theorem 3. 1 [5, 5] Let g be a convex]unction from to JR, and] be a convex ]unction from to f o is convex on L functions pP (1 p 2) commonly used in im age reconstruction [6] and Lz functions like Vf p 2, also used in edge preserving image restoration [7] satisfy the conditions of theorem 3.1 and increase more slowly than the Lz function p. Therefore, R0 is chosen among those functions. We now describe the minimization stage of 7(X) Sacchi et al. 2] have used a ....

....optimization Mgorithm. 3.2. Gibbslan penalization Separable regularizatlon leads to a significant gain of res olution, but it does not favor retrieval of smooth spectral components (see Fig. b) c) To strengthen these fea tures, a classicM technique consists in appending a Gibbsinn term [10, 6] to 7(X) If one processes as in image restoration, one can add to R0 a penalization on the first differences. Since we eventually take interest in restoring the power spectrum rather than the complex spectral amplitudes, we penalize the first differences of the modulus rather than those of the ....

C. Bouman and K. Sauer, "A generalized Gaussian image model for edge-preserving MAP estimation", IEEE Trans. Image Processing, vol. IP-2, no. 3, pp. 296-310, July 1993.

....: R(x) R(#) with # p = and # : 4a) stric6L cric ; 4b) cb)f6LE6 fR di#erentiable (C ) 4c in#nite at in#nity ; i:e: lim R(x) # : 4d) As a c#6LfRG#MEf J isstric6E cric as a sum of cff6L andstric3M cic3 terms. Then, the minimizer x is unique andcTE6 LfRz w.r.t. the data [4]; this guarantees the well posedness ofthe regularized problem [22] Constraints (4b) 4d) make thecfT LMEfR6 of x feasible by manydeterministic descrm method(suc as gradient based methods, IRLS, etc,f The maincnf3333MfRz ofthis paper is to propose aspecL3 cec ofblocRzz6Mzfc#M descoc (BCD) ....

C.A. Bouman, K.D. Sauer, A generalized Gaussian image model for edge-preserving MAP estimation, IEEE Trans. ImageProcf36 2 (3) (July 1993) 296--310.

....the original object to be restored and of the noise. A simple noise model is often satisfactory, while finding a well adapted a priori model of the object is more complicated. To model homogeneous regions separated by sharp discontinuities, Markov Gibbs random fields (MRFs) have often been chosen [2] because they are based upon local pixel interactions. Moreover, Gibbs potentials used in MRFs can be customized to achieve sharper restora tion than with the classical quadratic (i.e. Gaussian) a priori models. But, the MRFs designed to restore sharp discontinuities often yield nonconvex ....

C. Bournart and K. Sauer, "A generalized Gaussian image model for edge-preserving MAP estimation," IEEE 2ans. Image Processing, vol. 2, no. 3, pp. 296-310, July 1993.

....where denotes the set of sites s and H is the set of 367 horizontal neighbour sites. The functions Px and p account respectively for the attenuation homogeneity along the lateral directions and for the domination of steel in the object: they can be chosen among classical regularisation functions [2] [7] The bivariate function pm(u,r) enables to model both the spikiness of the reflectivity and the link between the attenuation and the reflectivity: it is the core of the joint potential definition. Since the main contribution of this paper deals with the linking between the attenuation and ....

C. Bouman, K. Sauer. A generalized gaussian image model for edge-preserving MAP estimation. IEEE Trans. Image Processing, vol.2:296-310, 1993.

....improve the numerical e#ciency of MRF based image restoration methods, Geman et al. 9] 2] proposed simulated annealing techniques based on convex duality principles for maximizing the objective function. However, the computational cost still remains high. For similar purposes, Bouman and Sauer [10] proposed to use special convex potential MRFs that can still provide adequate edge preservation while yielding a convex criterion, thereby simplifying the optimization considerably. Nonetheless, in both cases, the numerical cost may depend directly on the spatial extent of the linear distortion. ....

....can be easily introduced. Besides, local minimization is well suited to MRFs because there is no long range interaction between voxels. The sphere of influence of a given voxel is thus determined by the size of the PSF. For these reasons, we use the same approach as Besag [7] and Bouman and Sauer [10] and choose an iterative relaxation method. In addition, we adopt an over relaxed scheme since such a choice has been shown to yield faster convergence [15] 11] The problem with local iterative methods is that, for any penalty function #, there is no closed form expression for the minimum of J ....

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C. Bouman and K. Sauer, "A generalized Gaussian image model for edge-preserving MAP estimation", IEEE Trans. Image Processing, vol. 2, no. 3, pp. 296--310, July 1993.

....numerical tools, such as quadratic programming methods. In the present paper, we restrict the choice to strictly convex penalty functions in order to ensure that is also strictly convex. As a consequence, admits no local minima. Moreover, the minimizer is unique and continuous w.r.t. the data [21]; this guarantees the well posedness of the regularized problem [22] Finally, many deterministic descent methods (such as gradient based methods and the IRLS algorithm [23] 24] will be ensured to converge toward if is continuously differentiable (10a) strictly convex (10b) infinite at ....

....above. In practical simulations (see Section V B 2) we have selected the hyperbolic potential in . C. Smooth Spectra 1) Complex Gibbs Markov Regularization: In the field of signal and image restoration, Gibbs Markov potential functions are often used as roughness penalty functions [11] 13] [21], 26] 27] 30] Adopting this approach in the case of spectral regularity, one might think of simply penalizing differences between complex coefficients, using (12) where because of the circularity constraint. In (12) the subscript stands for smooth. Then, provided that is convex and ....

C. A. Bouman and K. D. Sauer, "A generalized Gaussian image model for edge-preserving MAP estimation," IEEE Trans. Image Processing, vol. 2, pp. 296--310, July 1993.

.... could be written: where V is a hnction (called potential) of local groups of points c (cliques) In this work, we only took into account potential functions of the form: p 5 ; j neighbourn 2312 which restricts us to a class of MRFs called Gener alised Gauss Markov Random Fields [5], and horizontal and vertical neighbourhood (first order) For p near or equal 2, it severely penalises large differences between the values of the neighhours and is useful to reconstruct smooth objects, on the other hand, for p near to 1, it allows such differences; and could be used to ....

C. Bouman and t(. Sauer "A generalized Gaussian image model for edge-preserving MAP estimation," [EEE Trans. Image Processing, vol. 2, pp 296-310, (1993).

....an Lp norm rather than a quadratic norm. Then, the noise models and prior models take the following form: Pb(Y Ar; b) oc exp [blly Axllp] 12) px(r; A) oc exp [ kllMllq] 13) where M can be a difference matrix as used by BOUMAN SAUER and BESAG on the Generalized Gauss Markov Models [9], and L Markov models [10] Finally, with q 1 and M an identity matrix it leads to a L deconvolution algorithm in the context of seismic deconvolution [11] According to the scale transformation x kx and y ky, the models change in the following way: pb(ky Akr; b) oc exp [kPbl]y Arl]p] ....

....the following way: pb(ky Akr; b) oc exp [kPbl]y Arl]p] 14) px(kr; A) oc exp [kqAllMrllq] 15) If we set (bk, kk) kPb, kq;k) the two estimates are scale invariant. Moreover, if p = q, we can drop the scale k in the MAP criteria (eq. 11) which becomes scale invariant. This is done in [9] [11] but it makes the choice of the prior and the noise models mutually dependent. We can also remark that )q P is scale invariant and can be interpreted as a generalized SNR. Maximum Entropy methods: Maximum Entropy reconstruction methods have been extensively used in the last decade. A ....

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C. Bouman and K. Sauer, "A generalized Gaussian image model for edge-preserving MAP estimation," IEEE Transactions on Medical Imaging, vol. MI-2, no. 3, pp. 296310, 1993.

....great dimension of f, even though the matrix R is sparse. Many works have been done on choosing appropriate regularization functionals or equivalently appropriate prior probability laws for f to enforce some particular properties of the image such as smoothness, positivity or piecewise smoothness [2, 3, 20, 23]. Among these, one can mainly mention the types of functions fl(f) which enforce the binary assumption of the object (See Sauer et al. in this book) 1.2.2 Level set approach The second approach consists in modeling directly the closed contour of the object as the zero crossing (or any level set) ....

C. Bouman and K. Sauer, "A generalized Gaussian image model for edge-preserving MAP estimation," IEEE Transactions on Image Processing, vol. IP-2, pp. 296-310, July 1993.

....of its calculation is huge due to the great dimension of f. Many works have been done on choosing appropriate regularization functionals or equivalently appropriate prior probability laws for to enforce some special properties of the image such as smoothness, positivity or piecewise smoothness [5, 6, 7, 8, 9, 10]. Among these, one can mention mainly two types of functions for 2( Entropic laws: 2(f) fj) with (x) xP, xlogx,logx, Homogeneous Markovian laws: c) fj,fi) with (x,y) Ix ylp, Ix yllog,logcoshlx yl, j=l iAij See for example [6] for the entropic laws, 7, 5] for scale ....

.... [5, 6, 7, 8, 9, 10] Among these, one can mention mainly two types of functions for 2( Entropic laws: 2(f) fj) with (x) xP, xlogx,logx, Homogeneous Markovian laws: c) fj,fi) with (x,y) Ix ylp, Ix yllog,logcoshlx yl, j=l iAij See for example [6] for the entropic laws, [7, 5] for scale invariant markovian laws and [8, 9, 10] for other specific choices. In the second approach, one starts by giving a parametric model for the object and then tries to estimate these parameters using least squares (LS) or maximum likelihood (ML) methods. In general, in this approach one ....

C. Bouman and K. Saucr, "A generalized Gaussian image model for edge-preserving MAP estimation," IEEE Transactions on Image Processing, vol. IP-2, pp. 296 310, July 1993.

....be understood as Markovian models. For example the classical zeroth order Tikhonov regularization corresponds to a Gaussian prior model (H(x) IIxll 2) 6] One major challenge, is to model an image with smooth areas and discontinuity between the areas. It leads to convex energy functions (e.g. [9]) or, by modeling implicit or explicit line processes, to nonconvex energy function (e.g. 10] 11] 121) The choice of such a function is not the goal of this paper and is in itself a wide research area. Following the work of Bouman and Saner [9] we chose the Generalized Gauss Markov model, ....

....It leads to convex energy functions (e.g. 9] or, by modeling implicit or explicit line processes, to nonconvex energy function (e.g. 10] 11] 121) The choice of such a function is not the goal of this paper and is in itself a wide research area. Following the work of Bouman and Saner [9], we chose the Generalized Gauss Markov model, with the potential function: Vc(zC) Ixj xil p, xj, 28 i c, 1 p 2. For p near or equal 2, it severely penalizes large differences between the values of the neighbors and is useful to reconstruct smoothly varying images. On the other hand, ....

C. Bouman and K. Sauer, "A generalized Gaussian image model for edge-preserving MAP estimation", IEEE Transactions on Image Processing, vol. IP-2, no. 3, pp. 296-310, July 1993.

.... images with a spiky appearance over a homogeneous background [13] 27] Gaussian MRF s give rise to linear estimators, but the basic homogeneous Gaussian MRF s are well known to allow noise cancellation only at the expense of oversmoothing the object [35] 40] Generalized Gaussian (GG) MRF s [7] preserve edges better while maintaining convex energies. In the same way, other useful MRF s weight the differences between neighbors using a Huber function or a log cosh function [20] A discrepancy measure on neighbors was introduced in [32] in order to model correlations in positive objects. ....

....Data with 10 dB SNR: real and imaginary parts. a) b) Fig. 5. Reconstruction using a Gaussian MRF from the data shown in Fig. 4: a) 20 dB SNR, fl = 0:08. b) 10 dB SNR, fl =0:14. where is convex. Standard descent algorithms have been used for the minimization. Generalized Gaussians (GG s) [7] are defined by , where controls smoothing. The prior is a Gaussian MRF when and it cancels noise at the expense of over smoothing the edges (Fig. 5) Subsequently, GG s with were introduced, and the 20 dB SNR and 10 dB SNR data were processed with and , respectively, Fig. 6) In the first case, ....

C. Bouman and K. Sauer, "A generalized Gaussian image model for edge-preserving map estimation," IEEE Trans. Image Processing, vol. 2, pp. 296--310, July 1993.

.... in the literature are methods based on explicit modeling of edges and other boundary like features (see, for example, 132, 234] approaches that use non Gaussian models in order to better capture the heavy tail nature of imagery (for example the generalized Gaussian models studied in depth in [41]) and an array of procedures using wavelet transforms (e.g. 104, 301, 57, 2, 68, 330, 80, 192, 193, 261, 59, 281, 58, 333] For this latter set of methods the general idea is to exploit the localization properties of wavelets to allow much easier and more transparent adaptive processing in ....

....but not their dependence involves a simple modification to the model described previously. In particular, we still use (6) with A(s) 0, but we use a non Gaussian, distribution for w(s) One possibility is a heavy tailed distribution from the class of so called generalized Gaussian distributions [41, 250], of the form Kexp x a , with 0 a 2. An alternate class of choices for the distributions of w(s)aremixture distributions [218, 80] For example, one of the simpler models introduced in [80] consists of modeling wavelet coe#cients as independent with distributions consisting of finite ....

C. A. Bouman and K. Sauer. A generalized Gaussian image model for edge-preserving MAP estimation. IEEE Trans. on Image Processing, 2(3):296--310, July 1993.

....We accomplished this by incorporating an image model in the reconstruction process that models our a priori knowledge regarding the unknown fields D and a . Markov random fields (MRF) have been extensively used in image processing applications. We model D as a generalized Gaussian MRF (GGMRF) [13] with an energy function of the form u fs;rg2N b s Gammar fi fi D s Gamma D r fi fi ; 8) where N is the set of all neighboring pixel pairs. The popular choice of p = 2 in the signal processing literature yields a quadratic cost function, which tends to excessively penalize ....

C. A. Bouman and K. Sauer. A generalized Gaussian image model for edge-preserving MAP estimation. IEEE Trans. Image Processing, 2:296-- 310, 1993.

....We accomplish this by employing an image model in the reconstruction process that incorporates our a priori knowledge regarding the unknown fields D and a . Markov random fields (MRF) have been extensively used in image processing applications. We model D as a generalized Gaussian MRF (GGMRF) [19] with an energy function of the form V fs;rg2N b s Gammar fi fi D s Gamma D r fi fi ; 10) where N is the set of all neighboring pixel pairs, b is the weight assigned to the specific neighbors, and oe D is the scale parameter of the model. The popular choice of p = 2 in the ....

C. A. Bouman and K. Sauer, "A generalized Gaussian image model for edgepreserving MAP estimation," IEEE Trans. on Image Proc., 2, pp. 296--310, 1993.

....problems, adaptive wavelet filtering techniques [9, 10, 11, 12] and estimators based on hidden Markov models (HMT) 13, 14] outperform simpler Wiener filtering techniques. The use of hidden Markov tree models has also been beneficial in image classification [15, 16] Markov random field models [17, 18] have been used successfully in some applications. The potential advantages of using a particular model can be validated by an improved performance in a specific application, as in the above papers, or by a direct characterization of the discrepancy between this model and a simpler one. This ....

C. Bouman and K. Sauer, "A generalized Gaussian image model for edge-preserving MAP estimation," IEEE Trans. on Image Processing, vol. 2, pp. 296--310, July 1993.

....competitive learning, rival penalizedcompetitive learning, fuzzy c means clustering and K means clustering. Each time after we finish the clustering, we use the clustering results to perform classification and compare the classification results with those given from Maximum a posterior (MAP) [27]. The results are summarized as follows: Table 2: Comparison on the perfermance between FCC and several traditional clustering algorithms. Average Error Rate (in percentage) Fuzzy Competitive Clustering 9.70 Fuzzy c Means 15.48 K Means 13.29 Competitive Learning 16.07 Rival Penalized ....

C. Bouman and K. Sauer, "A generalized gaussian image model for edgepreserving map estimation," in IEEE Transactions on Image Processing,vol. 2, pp. 296--310, Jul. 1993.

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C. A. Bouman and K. Sauer, "A generalized Gaussian image model for edge-preserving MAP estimation," IEEE Trans. Image Process. 2, 296 --310 #1993#.

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C. A. Bouman and K. Sauer, "A generalized Gaussian image model for edge-preserving map estimation," IEEE Trans. Image Processing, vol. 2, pp. 296--310, July 1993.

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C. A. Bouman and K. Sauer, "A generalized Gaussian image model for edge-preserving MAP estimation, " IEEE Trans. on Image Process., vol. 2, no. 3, pp. 296--310, July 1993.

....MAP estimate of given the measurement vector is (8) where is the prior density for the image and maximization over enforces the required positivity constraint, i.e. that , as required for the physical problem. As in [12] we use the generalized Gaussian Markov random field (GGMRF) prior model [37] (9) where is a normalization hyperparameter and controls the degree of edge smoothness, with corresponding to the Gaussian case. This prior model enforces smoothness in the solution while preserving sharp edge transitions. We adaptively estimate during the reconstruction procedure. Initially, ....

C. A. Bouman and K. Sauer, "A generalized Gaussian image model for edge-preserving MAP estimation," IEEE Trans. Image Processing, vol. 2, pp. 296--310, July 1993.

....a symmetric function 13 that penalizes the differences in adjacent pixel values. Such a stabilizing functional results from the selection of a prior density p(x) corresponding to a Markov random field (MRF) 59] A wide variety of functionals ( have been suggested for this purpose [60] 61] [62]. Generally, these methods attempt to select these functionals so that large differences in pixel value are not excessively penalized, thereby allowing the accurate formation of sharp edge discontinuities. The stabilizing functional at scale q must be selected so that = S(x) 23) This ....

....dimension of the problem. Here we assume that x i x j = x j ) 2 , and we use the constant 2 to compensate for the reduction in the number of terms as the sampling grid is coarsened. In our experiments, we use the generalized Gaussian Markov random field (GGMRF) image prior model [62], 52] 45] 13] 14] given by p(x) z(p) exp 5 ; 25) where is a normalization parameter, 1 p 2 controls the degree of edge smoothness, and z(p) is a partition function. For the GGMRF prior, the stabilizing functional is given by S(x) 26) and the corresponding ....

C. A. Bouman and K. Sauer, "A generalized Gaussian image model for edge-preserving MAP estimation," IEEE Trans. on Image Processing, vol. 2, no. 3, pp. 296--310, July 1993.

....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, ....

....model uses no adaptation and only a single parameter vector at each scale such that # s = f(n) The two fixed resolution MAP reconstruction algorithms were based on a Gaussian Markov random field (GMRF) and a generalized Gaussian Markov random field (GGMRF) prior model respectively. The GGMRF[10] is an edge preserving, spatially homogeneous MRF that uses a non quadratic penalty term. For the results shown here, the generalized Gaussian parameter was set to p = 1.2. The algorithms used ICD optimization with a large, fixed number of iterations to insure complete convergence. The CBP ....

C. A. Bouman and K. Sauer. A generalized Gaussian image model for edge-preserving MAP estimation. IEEE Trans. on Image Processing, 2(3):296--310, July 1993.

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C. A. Bouman and K. Sauer, "A generalized Gaussian image model for edge-preserving MAP estimation," IEEE Trans. Image Process. 2, 296--310 (1993).

....and divides per iteration as EM. All the sequential algorithms typically include a root finding step at each pixel, which may raise cost somewhat for non Gaussian prior models. The a priori image model here consists of two choices of for the generalized Gaussian Markov random field (GGMRF) [22] prior model with the prior log density function of , where is the coefficient linking pixels and , is a scale (temperature) parameter, and is a parameter which controls the smoothness of the reconstruction. The parameter for all six cases and the dosage parameter for the transmission data were ....

C. A. Bouman and K. Sauer, "A generalized Gaussian image model for edge-preserving map estimation," IEEE Trans. Image Processing, vol. 2, pp. 296--310, July 1993.

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C. Bouman and K. Sauer, "A generalized Gaussian image model for edge-preserving MAP estimation," IEEE Trans. Image Processing, vol. 2, pp. 296--310, 1993.

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C. Bouman and K. Sauer, "A Generalized Gaussian Image Model for Edge-Preserving MAP Estimation," IEEE Trans. on Image Processing, vol. 2, pp. 296-310, 1993.

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C. Bouman and K. Sauer, "A Generalized Gaussian Image Model for Edge-Preserving MAP Estimation," IEEE Trans. Image Processing, vol. 2, no.3, July 1993. (a) (b) (c) (d)

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C. A. Bouman and K. D. Sauer, "A generalized Gaussian image model for edge-preserving MAP estimation," IEEE Trans. Image Processing, vol. 2, pp. 296--310, July 1993.

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C. Bouman and K. Sauer, "A generalized Gaussian image model for edge-preserving MAP estimation," IEEE Trans. Image Processing, vol. 2, pp. 296--310, July 1993.

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C. Bouman and K. Sauer, "A generalized Gaussian image model for edge-preserving MAP estimation," IEEE Trans. on Image Processing, vol. 2, pp. 296-- 310, 1993.

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C. A. Bouman and K. Sauer, "A generalized Gaussian image model for edge-preserving MAP estimation," IEEE Trans. on Image Processing, vol. 2, no. 3, pp. 296--310, July 1993.

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C. Bouman and K. Sauer, "A generalized gaussian image model for edge preserving map estimation," IEEE Trans. Signal Proc. 2, pp. 296--310, 1993.

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C. A. Bouman and K. Sauer, "A generalized Gaussian image model for edge-preserving MAP estimation," IEEE Trans. on Image Processing, vol. 2, no. 3, pp. 296--310, July 1993.

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C. Bouman and K. Sauer, \A Generalized Gaussian Image Model for EdgePreserving MAP Estimation", IEEE Trans. Image Processing. Vol. 2, No. 3, pp, 296-310, 1993. 78

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C. Bouman and K. Sauer. A Generalized Gaussian Image Model for Edge-Preserving MAP Estimation. IEEE Trans. Image Proc., 2(3):296--310, July 1993.

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C. Bouman and K. Sauer, "A generalized Gaussian image model for edge-preserving MAP estimation," IEEE Trans. Imag. Process. 2, 3 (1993) 296--310.

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C. Bouman, and K. Sauer, "A generalized Gaussian image model for edge-preserving MAP estimation," IEEE Trans. Image Processing, vol. 2, pp. 296-310, Jul. 1993.

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C. A. Bouman and K. D. Sauer, "A generalized Gaussian image model for edge-preserving MAr estimation," IEEE Trans. Image Processing IP-2, pp. 296-310, July 1993.

No context found.

C. Bouman and K. Saner, "A generalized Gaussian image model for edge-preserving MAP estima- tion", IEEE Trans. Image Processing, vol. IP-2, no. 3, pp. 296-310, 1993.

No context found.

C. Bouman and K. Sauer, "A generalized Gaussian image model for edge-preserving MAP estimation," IEEE Transactions on Image Processing IP-2, pp. 296-310, July 1993.

No context found.

C. Bouman and K. Sauer, "A generalized Gaussian image model for edge-preserving MAr estimation," IEEE Trans- actions on Image Processing, vol. IP-2, pp. 296-310, July1993.

No context found.

C. A. Bouman and K. Sauer, #A generalized Gaussian image model for edge-preserving MAP estimation," IEEE Trans. on Image Processing 2#3#, pp. 296#310, 1993.

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