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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 ..."
Abstract

Cited by 181 (13 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...
Iterative Regularized LeastMean MixedNorm Image Restoration
 Opt. Eng
, 2002
"... We develop a regularized mixednorm image restoration algorithm to deal with various types of noise. A mixednorm functional is introduced, which combines the least mean square (LMS) and the least mean fourth (LMF) functionals, as well as a smoothing functional. Two regularization parameters are int ..."
Abstract

Cited by 1 (0 self)
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We develop a regularized mixednorm image restoration algorithm to deal with various types of noise. A mixednorm functional is introduced, which combines the least mean square (LMS) and the least mean fourth (LMF) functionals, as well as a smoothing functional. Two regularization parameters are introduced: one to determine the relative importance of the LMS and LMF functionals, which is a function of the kurtosis, and another to determine the relative importance of the smoothing functional. The two parameters are chosen in such a way that the proposed functional is convex, so that a unique minimizer exists. An iterative algorithm is utilized for obtaining the solution, and its convergence is analyzed. The novelty of the proposed algorithm is that no knowledge of the noise distribution is required, and the relative contributions of the LMS, the LMF, and the smoothing functionals are adjusted based on the partially restored image. Experimental results demonstrate the effectiveness of the proposed algorithm. 2002 Society of PhotoOptical Instrumentation Engineers. [DOI: 10.1117/1.1503072] Subject terms: mixednorm functional; LMS; LMF; kurtosis; convex.
Asymptotic Behavior of RAestimates in Autoregressive 2D
, 2007
"... In this work we study the asymptotic behavior of a robust class of estimator of coe ¢ cients of AR2D process. We established the consistency and asymptotic normality of the RA estimator under precise conditions. This class of models has diverse applications in image modeling and statistical image p ..."
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In this work we study the asymptotic behavior of a robust class of estimator of coe ¢ cients of AR2D process. We established the consistency and asymptotic normality of the RA estimator under precise conditions. This class of models has diverse applications in image modeling and statistical image processing.
Robust Solutions To LeastSquares Problems With Uncertain Data
, 1997
"... We consider leastsquares problems where the coefficient matrices A, b are unknown but bounded. 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 interprete ..."
Abstract
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We consider leastsquares problems where the coefficient matrices A, b are unknown but bounded. 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.
Correspondence Adaptive Image Restoration Using a Generalized Gaussian Model for Unknown Noise
"... AbstracrA model adaptive method is proposed for restorating blurred and noise corrupted images. The generalized pGaussian family of probability density functions is used as the approximating parametric noise model. Distribution shape parameters are estimated from the image, and the resulting maxim ..."
Abstract
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AbstracrA model adaptive method is proposed for restorating blurred and noise corrupted images. The generalized pGaussian family of probability density functions is used as the approximating parametric noise model. Distribution shape parameters are estimated from the image, and the resulting maximum likelihood optimization problem is solved. An iterative algorithm for datadirected restoration is presented and analyzed. I.