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40
Robust Solutions To LeastSquares Problems With Uncertain Data
, 1997
"... . We consider leastsquares problems where the coefficient matrices A; b are unknownbutbounded. We minimize the worstcase residual error using (convex) secondorder cone programming, yielding an algorithm with complexity similar to one singular value decomposition of A. The method can be interpret ..."
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Cited by 144 (12 self)
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. We consider leastsquares problems where the coefficient matrices A; b are unknownbutbounded. We minimize the worstcase residual error using (convex) secondorder cone programming, yielding an algorithm with complexity similar to one singular value decomposition of A. The method can be interpreted as a Tikhonov regularization procedure, with the advantage that it provides an exact bound on the robustness of solution, and a rigorous way to compute the regularization parameter. When the perturbation has a known (e.g., Toeplitz) structure, the same problem can be solved in polynomialtime using semidefinite programming (SDP). We also consider the case when A; b are rational functions of an unknownbutbounded perturbation vector. We show how to minimize (via SDP) upper bounds on the optimal worstcase residual. We provide numerical examples, including one from robust identification and one from robust interpolation. Key Words. Leastsquares, uncertainty, robustness, secondorder cone...
Nonlinear Image Recovery with HalfQuadratic Regularization
, 1993
"... One popular method for the recovery of an ideal intensity image from corrupted or indirect measurements is regularization: minimize an objective function which enforces a roughness penalty in addition to coherence with the data. Linear estimates are relatively easy to compute but generally introduce ..."
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Cited by 130 (0 self)
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One popular method for the recovery of an ideal intensity image from corrupted or indirect measurements is regularization: minimize an objective function which enforces a roughness penalty in addition to coherence with the data. Linear estimates are relatively easy to compute but generally introduce systematic errors; for example, they are incapable of recovering discontinuities and other important image attributes. In contrast, nonlinear estimates are more accurate, but often far less accessible. This is particularly true when the objective function is nonconvex and the distribution of each data component depends on many image components through a linear operator with broad support. Our approach is based on an auxiliary array and an extended objective function in which the original variables appear quadratically and the auxiliary variables are decoupled. Minimizing over the auxiliary array alone yields the original function, so the original image estimate can be obtained by joint min...
VectorValued Image Regularization with PDEs: A Common Framework for Different Applications
 IEEE Transactions on Pattern Analysis and Machine Intelligence
, 2003
"... We address the problem of vectorvalued image regularization with variational methods and PDE's. From the study of existing formalisms, we propose a unifying framework based on a very local interpretation of the regularization processes. The resulting equations are then specialized into new regulari ..."
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Cited by 114 (7 self)
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We address the problem of vectorvalued image regularization with variational methods and PDE's. From the study of existing formalisms, we propose a unifying framework based on a very local interpretation of the regularization processes. The resulting equations are then specialized into new regularization PDE's and corresponding numerical schemes that respect the local geometry of vectorvalued images. They are finally applied on a wide variety of image processing problems, including color image restoration, inpainting, magnification and flow visualization.
Motion Picture Restoration
, 1993
"... This dissertation presents algorithms for restoring some of the major corruptions observed in archived film or video material. The two principal problems of impulsive distortion (Dirt and Sparkle or Blotches) and noise degradation are considered. There is also an algorithm for suppressing the inter ..."
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Cited by 51 (9 self)
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This dissertation presents algorithms for restoring some of the major corruptions observed in archived film or video material. The two principal problems of impulsive distortion (Dirt and Sparkle or Blotches) and noise degradation are considered. There is also an algorithm for suppressing the interline jitter common in images decoded from noisy video signals. In the case of noise reduction and Blotch removal the thesis considers image sequences to be three dimensional signals involving evolution of features in time and space. This is necessary if any process presented is to show an improvement over standard twodimensional techniques. It is important to recognize that consideration of image sequences must involve an appreciation of the problems incurred by the motion of objects in the scene. The most obvious implication is that due to motion, useful three dimensional processing does not necessarily proceed in a direction `orthogonal' to the image frames. Therefore, attention is giv...
ML parameter estimation for Markov random fields, with applications to Bayesian tomography
 IEEE Trans. on Image Processing
, 1998
"... Abstract 1 Markov random fields (MRF) have been widely used to model images in Bayesian frameworks for image reconstruction and restoration. Typically, these MRF models have parameters that allow the prior model to be adjusted for best performance. However, optimal estimation of these parameters (so ..."
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Cited by 48 (18 self)
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Abstract 1 Markov random fields (MRF) have been widely used to model images in Bayesian frameworks for image reconstruction and restoration. Typically, these MRF models have parameters that allow the prior model to be adjusted for best performance. However, optimal estimation of these parameters (sometimes referred to as hyperparameters) is difficult in practice for two reasons: 1) Direct parameter estimation for MRF’s is known to be mathematically and numerically challenging. 2) Parameters can not be directly estimated because the true image crosssection is unavailable. In this paper, we propose a computationally efficient scheme to address both these difficulties for a general class of MRF models, and we derive specific methods of parameter estimation for the MRF model known as a generalized Gaussian MRF (GGMRF). The first section of the paper derives methods of direct estimation of scale and shape parameters for a general continuously valued MRF. For the GGMRF case, we show that the ML estimate of the scale parameter, σ, has a simple closed form solution, and we present an efficient scheme for computing the ML estimate of the shape parameter, p, by an offline numerical computation of the dependence of the partition function on p.
Degraded Image Analysis: An Invariant Approach
 IEEE Trans. Pattern Analysis and Machine Intelligence
, 1998
"... Analysis and interpretation of an image which was acquired by a nonideal imaging system is the key problem in many application areas. The observed image is usually corrupted by blurring, spatial degradations, and random noise. Classical methods like blind deconvolution try to estimate the blur param ..."
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Cited by 38 (10 self)
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Analysis and interpretation of an image which was acquired by a nonideal imaging system is the key problem in many application areas. The observed image is usually corrupted by blurring, spatial degradations, and random noise. Classical methods like blind deconvolution try to estimate the blur parameters and to restore the image. In this paper, we propose an alternative approach. We derive the features for image representation which are invariant with respect to blur regardless of the degradation PSF provided that it is centrally symmetric. As we prove in the paper, there exist two classes of such features: the first one in the spatial domain and the second one in the frequency domain. We also derive socalled combined invariants, which are invariant to composite geometric and blur degradations. Knowing these features, we can recognize objects in the degraded scene without any restoration. Index TermsDegraded image, symmetric blur, blur invariants, image moments, combined invariant...
A Computational Algorithm for Minimizing Total Variation in Image Restoration
 IEEE Trans. Image Processing
, 1996
"... A reliable and efficient computational algorithm for restoring blurred and noisy images is proposed. The restoration process is based on the minimal total variation principle introduced by Rudin et al [1], [2], [3]. For discrete images, the proposed algorithm minimizes a piecewise linear l 1 functio ..."
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Cited by 33 (1 self)
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A reliable and efficient computational algorithm for restoring blurred and noisy images is proposed. The restoration process is based on the minimal total variation principle introduced by Rudin et al [1], [2], [3]. For discrete images, the proposed algorithm minimizes a piecewise linear l 1 function (a measure of total variation) subject to a single 2norm inequality constraint (a measure of data fit). The algorithm starts by finding a feasible point for the inequality constraint using a (partial) conjugate gradient method. This corresponds to a deblurring process. Noise and other artifacts are removed by a subsequent total variation minimization process. The use of the linear l 1 objective function for the total variation measurement leads to a simplier computational algorithm. Both the steepest descent and an affine scaling Newton method are considered for solving this constrained piecewise linear l 1 minimization problem. The resulting algorithm, when viewed as an image restoratio...
Image restoration subject to a total variation constraint
 IEEE Transactions on Image Processing
, 2004
"... Abstract—Total variation has proven to be a valuable concept in connection with the recovery of images featuring piecewise smooth components. So far, however, it has been used exclusively as an objective to be minimized under constraints. In this paper, we propose an alternative formulation in which ..."
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Cited by 26 (2 self)
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Abstract—Total variation has proven to be a valuable concept in connection with the recovery of images featuring piecewise smooth components. So far, however, it has been used exclusively as an objective to be minimized under constraints. In this paper, we propose an alternative formulation in which total variation is used as a constraint in a general convex programming framework. This approach places no limitation on the incorporation of additional constraints in the restoration process and the resulting optimization problem can be solved efficiently via blockiterative methods. Image denoising and deconvolution applications are demonstrated. I. PROBLEM STATEMENT THE CLASSICAL linear restoration problem is to find the original form of an image in a real Hilbert space from the observation of a degraded image where
Regularized Constrained Total LeastSquares Image Restoration
 IEEE Trans. Image Processing
, 1995
"... In this paper the problem of restoring an image distorted by a linear spaceinvariant (LSI) pointspread function (psf) which is not exactly known is formulated as the solution of a perturbed set of linear equations. The regularized constrained total leastsquares (RCTLS) method is used to solve thi ..."
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Cited by 24 (6 self)
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In this paper the problem of restoring an image distorted by a linear spaceinvariant (LSI) pointspread function (psf) which is not exactly known is formulated as the solution of a perturbed set of linear equations. The regularized constrained total leastsquares (RCTLS) method is used to solve this set of equations. Using the diagonalization properties of the discrete Fourier transform (DFT) for circulant matrices, the RCTLS estimate is computed in the DFT domain. This significantly reduces the computational cost of this approach and makes its implementation possible even for large images. An error analysis of the RCTLS estimate, based on the meansquarederror (MSE) criterion is performed to verify its superiority over the constrained total leastsquares (CTLS) estimate. Numerical experiments for different psf errors are performed to test the RCTLS estimator for this problem. Objective and visual comparisons are presented with the linear minimum meansquarederror (LMMSE) and the re...
A Multiple Input Image Restoration Approach
 Journal of Visual Communication and Image Representation
, 1990
"... this paper image restoration applications, where multiple distorted versions of the same original image are available, are considered. A general adaptive restoration algorithm is derived on the basis of a set theoretic regularization technique. The adaptivity of the algorithm is introduced in two wa ..."
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Cited by 17 (5 self)
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this paper image restoration applications, where multiple distorted versions of the same original image are available, are considered. A general adaptive restoration algorithm is derived on the basis of a set theoretic regularization technique. The adaptivity of the algorithm is introduced in two ways: (a) by a constraint ,operator which incorporates properties of the response of the hu man visual system into the restoration process and (b) by a weight matrix which assigns greater importance for the deconvolution process to areas of high spatial activity than to areas of low spatial activity. Different degrees of trust are assigned to the various distorted images depending on the amounts of noise. The proposed algorithm is general and can be used for any type of linear distortion and constraint operators. It can also be used to restore signals other than images. Experimental results obtained by an iterative implementation of the proposed algorithms are pre sented. c 1990 Academic Press, Inc