R. L. Stevenson, B. E. Schmitz and E. J. Delp, "Discontinuity Preserving Regularization of Inverse Visual Problems," IEEE Trans. Syst. Man Cybern., vol. 24, no. 3, Mar. 1994.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....of regularization, Table 1 reveals another useful classi cation in this context: while image driven models correspond to the class of quadratic regularizers [6] ow driven models belong to the more general class of nonquadratic convex regularizers. This latter class has been suggested in [11, 47, 52] for generalizing the well known quadratic regularization approaches used for early computational vision. 11 2.3 Spatio temporal regularizers All regularizers that we have discussed so far use only spatial smoothness constraints. Thus, it would be natural to impose some amount of (piecewise) ....

R.L. Stevenson, B.E. Schmitz, E.J. Delp, Discontinuity preserving regularization of inverse visual problems, IEEE Trans. Systems, Man and Cybernetics, Vol. 24, 455-469, 1994.

....of regularization, Table 1 reveals another useful classi cation in this context: while image driven models correspond to the class of quadratic regularizers [6] ow driven models belong to the more general class of non quadratic convex regularizers. This latter class has been suggested in [11, 45, 49] for generalizing the well known quadratic regularization approaches (cf. 6] used for early computational vision. 12 3.1 Assumptions In the following, we do not distinguish between the approaches (48) and (49) since with IR n , our results hold true for arbitrary n. Furthermore, we ....

R.L. Stevenson, B.E. Schmitz, E.J. Delp, Discontinuity preserving regularization of inverse visual problems, IEEE Trans. Systems, Man and Cybernetics, Vol. 24, 455-469, 1994.

....optimization [4] is not feasible for typical image sizes, and deterministic annealing procedures [5, 6] cannot guarantee to obtain a good local minimum. Therefore, the use of non quadratic but convex functionals has been advocated to simplify image smoothing from a computational viewpoint [2,8,9]. Furthermore, although being much simpler, convex functionals nevertheless provide reasonable approximations (cf. 10] to the prototypical but mathematically and computationally sophisticated variational smoothing approach of Mumford and Shah [1] Despite of e#cient digital implementations of ....

Stevenson, R.L., Schmitz, B.E. & Delp, E.J. (1994) Discontinuity preserving regularization of inverse problems. IEEE Trans. Systems, Man and Cyb. 24(3): 455 -- 469.

.... preserving smoothing has been well researched and there exist a number of successful models, such as the line process model [1] in the Markov random eld (MRF) framework and the weak string and membrane models [5] in the regularization framework; further studies can be found in [6] 7] 8] [9], 10] These models assume that the underlying surface has zero rst order derivatives and are suitable for preserving step edges but not for roof edges. Higher order derivatives have to be dealt with for roof edges, but such algorithms su er from instability [5] In this paper, a novel MRF ....

R. L. Stevenson, B. E. Schmitz, and E. J. Delp, \Discontinuity preserving regularization of inverse visual problems", IEEE Transactions on Systems, Man and Cybernetics, vol. 24, no. 3, pp. 455-469, March 1994.

....0 for any vector x as long as all the weighting coe#cients are positive. Hence Q is positive definite. 7.3 Statistical Spatial Approach: MAP Estimation The above technique tends to smear edges. Statistical techniques however have been successfully used in image processing for edge reconstruction [7, 73, 74, 75, 76]. The original image is modeled as a Markov random field (MRF) 6, 5, 7] and edges are reconstructed by maximum a posteriori (MAP) techniques. This is the approach adopted here. Each original frame X and its received version Y are modeled as discrete parameter random fields where each pixel is a ....

....first order derivatives at the i th j th pixel. #( is a cost function, # ascalingfactor,b (m) i,j weighting coe#cients, and the set of cliques [7] is C = i, j 1) i, j) i 1,j 1) i, j) i 1,j) i, j) i 1,j 1) i, j) Several cost functions have been proposed [73, 75]. A convex #( results in the minimization of a convex functional. The cost function used here is the one introduced by Huber for obtaining robust M estimates of location [79] Its advantage is that it is convex, does not heavily penalize edges, and is simpler to implement than most of 83the ....

R. L. Stevenson, B. E. Schmitz, and E. J. Delp, "Discontinuity preserving regularization of inverse visual problems," IEEE Transactions on Systems Man and Cybernetics, vol. 24, no. 3, pp. 455--469, March 1994. -

....observations do not fit to an assumed model or contain outliers that are anomalous data far away from the assumed error distribution. Thus, several conventional algorithms have been proposed by replacing each part of the MAP estimator by a robust estimator commonly used in robust statistics [12] [17]. On the other hand, least squares (LS) algorithms based on maximum likelihood (ML) were also employed in computer vision problems [18] While these algorithms are optimal for Gaussian noise, their performance is severely deteriorated by a few outliers, which is common in practical computer vision ....

....But this approach yields an oversmoothed solution at discontinuous regions. To alleviate this artifact, many approaches have been presented [4] 5] 6] On the other hand, MAP estimators using a Markov random field (MRF) with line process were proposed for various applications [12] [17]. Regularization converts ill posed or ill conditioned problems into well posed ones by constraining the solution with a priori assumption. The energy function J(Q, a) in Tikhonov s regularization is defined by [1] 6] JAy ii i y i Q,aa a q af = OE SD S Y r r r 2 where S and D ....

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# R.L. Stevenson, B.E. Schmitz, and E.J. Delp, "Discontinuity Preserving Regularization of Inverse Visual Problems," IEEE Trans. Systems, Man, and Cybernetics, vol. 24, no. 3, pp. 455-469, Mar. 1994.

....3 . In particular, positivity is an important convex constraint which can both improve the quality of reconstructions and significantly speed numerical convergence[21] Non Gaussian prior distributions are also important since they can substantially reduce noise while preserving edge detail[11, 22, 23, 24]. Our analysis starts with an approximate analysis of the optimization problem based on a Taylor expansion of the log likelihood function. This analysis is important for two 2 We choose to use the name ICD since it is most descriptive of the algorithmic approach. 3 We note that convex ....

....choice for a prior model is the Gaussian MRF, but the quadratic penalty extracted for the Gaussian often causes excessive smoothing of edges. Several prior models have been developed which include desirable edge preserving properties, and which maintain convexity in their log prior densities[11, 28, 23, 24]. Provided we choose one of these models, the MAP problem is also convex, and iterations converge to the unique global minimum solution. We use the Generalized Gaussian MRF (GGMRF) model proposed in [23] with the density function p # (#) 1 z exp 8 : # q X j,k #C b j k # j # k ....

R. Stevenson, B. Schmitz, and E. Delp, "Discontinuity Preserving Regularization of Inverse Visual Problems," IEEE Transactions on Systems, Man, and Cybernetics, vol. 24, no. 3, March, 1994.

....is important that inherit all properties of the continuously formulated approach. Property 3 stated above says that such approximations exist in a sense that will be made explicit in section 2.3. Meanwhile, convex but non quadratic functionals have been considered by other researchers, too [10, 11]. The present investigation is based on our variational model suggested in [12] and discusses several different aspects of the approach, apart from uniqueness of the minimizing solution. The abstract mathematical model involved enables the design of global optimization problems for different ....

R.L. Stevenson, B.E. Schmitz, and E.J. Delp. Discontinuity preserving regularization of inverse visual problems. IEEE Trans. Systems, Man and Cyb., 24(3):455-- 469, 1994.

....the contribution of the edges to the penalty term behave like the contributions of outliers to the regression terms, and in robust regression, the latter are similarly accounted for by nonlinear terms in place of the least squares terms. Motivated by this, Blauer Levine [1] and Stevenson et al. [21], also looked at the penalty term q(x) X j X i j H(x i Gamma x j ; ffi) 9) and at more elaborate version also incorporating local curvature information) where is H is the piecewise quadratic linear function H(d; ffi) d 2 if jdj ffi; 2ffijdj Gamma ffi 2 otherwise (10) ....

R.L. Stevenson, B.E. Schmitz and E.J. Delp, Discontinuity Preserving Regularization of Inverse Visual Problems, IEEE Trans. Systems, Man, Cybernetics 24 (1994), 455-469.

....approximation has to be used. Therefor, the resulting optimization problem of a nonconvex model often suffers from the uncertainty of the computed solution. Another disadvantage of nonconvex models is its instability. Their solutions may not depend continuously on the input data in some situations [9, 10]. A small change in the input might result in a drastic difference in the solutions (see Fig. 1) The phenomenon is also due to a hard decision making property of nonconvex models [4] As such, the solution often depends substantially on the method used to perform the minimization. In this sense, ....

....(influence functions) a) bounded smoothing and (b) convex potential functions. iii) ae 0 3fl (j) ae 2j jjj fl 2flsign(j) jjj fl ae 3fl (j) ae j 2 jjj fl 2fljjj Gamma fl 2 jjj fl : ae 3fl is Huber s error function in robust estimation [2] and has been examined in [5, 8, 10, 11]. Compared to other two potentials above, ae 3fl is the best in terms of computational complexity. Mathematically, it is sufficient to guarantee the convexity of ae fl (j) that its first derivative ae 0 fl (j) is monotonically increasing (criterion c) which ensures the ae 00 fl (j) to be ....

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R. Stevenson, B. E. Schmitz and E. J. Delp. Discontinuity preserving regularization of inverse visual problems. IEEE Transactions on Systems, Man, and Cybernetics, 24(3),March 1994.

....in intensity. The reason is that with the quadratic function, gradients are too much penalized. One solution to prevent the destruction of discontinuities but allows for isotropically smoothing uniform areas, is to change the above quadratic term. This point have been widely discussed [13, 53, 64, 66, 11, 8]. We refer to [26] for a review. The key idea is that for low gradients, isotropic smoothing is performed, and for high gradient, smoothing is only applied in the direction of the isophote and not across it. This condition can be mathematically formalized if we look at the Euler Lagrange Equation ....

R.L. Stevenson, B.E. Schmitz, and E.J. Delp. Discontinuity preserving regularization of inverse visual problems. IEEE Transactions on Systems, Man, and Cybernetics, 24(3):455469, March 1994.

....the contribution of the edges to the penalty term behave like the contributions of outliers to the regression terms, and in robust regression, the latter are similarly accounted for by nonlinear terms in place of the least squares terms. Motivated by this, Blauer Levine [1] and Stevenson et al. [21], also looked at the penalty term q(x) X j X i j H(x i Gamma x j ; ffi) 9) and at more elaborate versions also incorporating local curvature information) where is H is the piecewise quadratic linear function H(d; ffi) d 2 if jdj ffi; 2ffijdj Gamma ffi 2 otherwise (10) ....

R.L. Stevenson, B.E. Schmitz and E.J. Delp, Discontinuity Preserving Regularization of Inverse Visual Problems, IEEE Trans. Systems, Man, Cybernetics 24 (1994), 455-469.

....for typical image sizes like 512 Theta 512 pixels, say, and deterministic annealing procedures [2, 4] cannot guarantee to reach a good local minimum. Therefore, the use of non quadratic but convex functionals has been advocated to simplify image segmentation from a computational viewpoint [6, 7, 8]. In the present paper, we investigate algorithms to compute the unique function u minimizing the following convex functional [7] J(v) 1 2 Z Omega n (v Gamma g) 2 (jrvj) o dx ; 1) where g denotes a given image, t) ae 2 h t 2 ; 0 t c ae ; 2 l t 2 c (2t Gamma c ....

R.L. Stevenson, B.E. Schmitz, and E.J. Delp. Discontinuity preserving regularization of inverse problems. IEEE Trans. Systems, Man and Cyb., 24(3):455--469, 1994.

....in intensity. The reason is that with the quadratic function, gradients are too much penalized. One solution to prevent the destruction of discontinuities but allows for isotropically smoothing uniform areas, is to change the above quadratic term. This point have been widely discussed [23, 24, 3, 2]. We refer to [7] for a review. The key idea is that for low gradients, isotropic smoothing is performed, and for high gradient, smoothing is only applied in the direction of the isophote and not across it. This condition can be mathematically formalized if we look at the Euler Lagrange Equation ....

R. Stevenson, B. Schmitz, and E. Delp. Discontinuity preserving regularization of inverse visual problems. IEEE Transactions on Systems, Man, and Cybernetics, 24(3):455469, Mar. 1994.

....solution with respect to the data: A small change in the data may result in a drastic difference in the solution because of the nonconvexity. In recent years, there has been considerable interest in convex models for edge preserving smoothing with the edge preserving ability [6] 7] 8] 9] [10], 11] This class of models overcome the above mentioned disadvantages. The convexity guarantees the unique solution and its stability. Thus a local minimization technique suffices, making continuation or annealing unnecessary. An important issue to be addressed in edge preserving smoothing is ....

.... In Reference Potential Function Shulman and Herve [6] g 1 (j) 8 : j 2 jjj fl 2fljjj Gamma fl 2 jjj fl Green [7] g 2 (j) ln(cosh(j=fl) Lange [8] g 3 (j) 1 2 (jj=flj 1 1 jj=flj Gamma 1) Bouman and Sauer [9] g 4 (j) jjj p (1:0 p 2:0) Stevenson, Schmitz and Delp [10] g 5 (j) 8 : jjj p jjj fl (jjj ( p q fl p Gamma1 ) 1 q Gamma1 Gamma fl) q fl p Gamma ( p q fl p Gamma1 ) q q Gamma1 jjj fl p; q 1; q p Alternative 1 g 6 (j) j arctan(j=fl) Gamma fl 2 ln(1 (j=fl) 2 ) Alternative 2 g 7 (j) fljjj Gamma fl 2 ....

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R. L. Stevenson, B. E. Schmitz, and E. J. Delp, "Discontinuity preserving regularization of inverse visual problems", IEEE Transactions on Systems, Man and Cybernetics, vol. 24, pp. 455--469, March 1994.

....may result. Nonconvex energy models for edge preserving smoothing such as the wellknown line process model and alike have this problem in the minimization process. In recent years, there has been considerable interest in convex models for edge preserving smoothing with the edge preserving ability [12, 3, 5, 2, 13, 7]. This class of models overcome the above mentioned disadvantages. The convexity guarantees the unique solution and its stability. Thus local minimization techniques can be efficiently utilized, making continuation or annealing unnecessary. An important issue in edge preserving smoothing and ....

R. L. Stevenson, B. E. Schmitz, and E. J. Delp. "Discontinuity preserving regularization of inverse visual problems". IEEE Transactions on Systems, Man and Cybernetics, 24(3):455--469, March 1994. This article was processed using the L A T E X macro package with LLNCS style

....the partition function, is the MRF temperature parameter (inspired by the Gibbs distribution in thermodynamics) The summation is over the set of all cliques # with # computing local spatio temporal activity. The non linear spatial activity penalizing function # #x# is the Huber function [7], # #x## # x # #x## # #x## # #x# : 14) The clique structure determines the spatial and temporal interactions. Five clique types are divided into two classes: 1. Spatial activity is computed using finite difference approximations to second directional derivatives (vertical, ....

R. L. Stevenson, B. E. Schmitz, and E. J. Delp, "Discontinuity Preserving Regularization of Inverse Visual Problems," IEEE Transactions on Systems, Man and Cybernetics, vol. 24, no. 3, pp. 455--469, Mar. 1994.

....at the i th j th pixel. #( is a cost function, # a scaling factor, b (m) i,j weighting coe#cients, and the set of cliques [9] is C = i, j 1) i, j) i 1, j 1) i, j) i 1, j) i, j) i 1, j 1) i, j) 12) Several cost functions have been proposed [11, 12]. A convex #( results in the minimization of a convex functional. The cost function used here is the one introduced by Huber [13] It is defined to be # # (x) x 2 x # # # 2 2#( x #) x #. 13) Hence, X c#C V c (x) N1 1 X i=0 N2 1 X j=0 3 X m=0 b m i,j # # ( Dm ....

R. L. Stevenson, B. E. Schmitz, and E. J. Delp, "Discontinuity preserving regularization of inverse visual problems," IEEE Transactions on Systems Man and Cybernetics, vol. 24, no. 3, pp. 455--469, March 1994.

....the unknown high resolution image and the low resolution observations is not invertible, and thus a unique solution to the inverse problem cannot be computed. Regularization techniques include prior knowledge about the data in order to compute an approximate solution [1] 4] 7] 19] [20]. A Tikhonov regularization approach to image interpo1 lation was proposed by Karayiannis and Venetsanopoulos [4] in which a quadratic stabilizing functional added to a fidelity term for the constraints was defined. The resulting unconstrained optimization problem allowed for some noise within ....

.... 0:5z m 1;n 1 (16) These quantities approximate second order directional derivatives computed at z (k) m;n , with directions selected to account for horizontal, vertical, and diagonal edge orientations. The likelihood of edges in the data is controlled by the Huber edge penalty function [7] [ 20], ae ff (x) ae x 2 ; jxj ff; 2ffjxj Gamma ff 2 ; jxj ff; 17) where ff is a threshold parameter separating the quadratic and linear regions. A quadratic edge penalty, lim ff 1 ae ff (x) x 2 ; characterizes the Gauss Markov image model. Edges are severely penalized by the ....

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R. L. Stevenson, B. E. Schmitz, and E. J. Delp, "Discontinuity preserving regularization of inverse visual problems," IEEE Trans. Syst., Man, Cybern., vol. 24, no. 3, pp. 455--469, 1994.

....elusive. This paper proposes a robust multispectral image model, for use in Bayesian maximum a posteriori (MAP) estimation. A Gibbs prior over a Markov random field (MRF) containing spatial and spectral clique functions has been selected as the image model. The Huber edge penalty function [9] [ 22] is used to maintain spatial discontinuities within each channel, while spectral clique functions incorporate cross channel correlations. When this prior is used in conjunction with a Gaussian noise density, a convex optimization problem results for computing the MAP estimate. The paper will be ....

....extent results in moderately ill conditioned solutions. Spatially limited blurs, such as linear motion or out of focus camera blur, are severely ill conditioned [6] Regularization adds prior knowledge about the desired estimate to make the inverse problem well posed. Tikhonov regularization [22], 25] 26] is a deterministic technique which restricts the solution space, using a metric to distinguish between possible solutions. Stochastic regularization [25] 27] defines a probability space on the solution space, using a probability density to distinguish between solutions. In some ....

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R. L. Stevenson, B. E. Schmitz, and E. J. Delp, "Discontinuity preserving regularization of inverse visual problems," IEEE Trans. Syst. Man Cybern., vol. 24, no. 3, pp. 455--469, 1994.

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R. L. Stevenson, B. E. Schmitz and E. J. Delp, "Discontinuity Preserving Regularization of Inverse Visual Problems," IEEE Trans. Syst. Man Cybern., vol. 24, no. 3, Mar. 1994.

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R. L. Stevenson, B. E. Schimitz and E. J. Delp, "Discontinuity preserving regularization of inverse visual problems," IEEE Trans. on SMC, 24, 94, 455--469.

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R.L. Stevenson, B.E. Schmitz, and E.J. Delp. Discontinuity preserving regularization of inverse visual problems. IEEE Transactions on Systems, Man, and Cybernetics, 24(3):455469, March 1994.

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R. L. Stevenson, B. E. Schmitz, and E. J. Delp, Discontinuity preserving regularization of inverse visual problems, IEEE Transactions on Systems, Man and Cybernetics, 24 (1994), pp. 455--469.

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