Results 1  10
of
18
Regularization of Wavelets Approximations
, 1999
"... this paper, weintroduce nonlinear regularized wavelet estimators for estimating nonparametric regression functions when sampling points are not uniformly spaced. The approach can apply readily to many other statistical contexts. Various new penalty functions are proposed. The hardthresholding and s ..."
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Cited by 85 (7 self)
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this paper, weintroduce nonlinear regularized wavelet estimators for estimating nonparametric regression functions when sampling points are not uniformly spaced. The approach can apply readily to many other statistical contexts. Various new penalty functions are proposed. The hardthresholding and softthresholding estimators of Donoho and Johnstone (1994) are specic members of nonlinear regularized wavelet estimators. They correspond to the lower and upper bound of a class of the penalized leastsquares estimators. Necessary conditions for penalty functions are given for regularized estimators to possess thresholding properties. Oracle inequalities and universal thresholding parameters are obtained for a large class of penalty functions. The sampling properties of nonlinear regularized wavelet estimators are established, and are shown to be adaptively minimax. To eciently solve penalized leastsquares problems, Nonlinear Regularized Sobolev Interpolators (NRSI) are proposed as initial estimators, which are shown to have good sampling properties. The NRSI is further ameliorated by Regularized OneStep Estimators (ROSE), which are the onestep estimators of the penalized leastsquares problems using the NRSI as initial estimators. Two other approaches, the graduated nonconvexity algorithm and wavelet networks, are also introduced to handle penalized leastsquares problems. The newly introduced approaches are also illustrated by a few numerical examples. ####### ########## ## ########## ########### ## ############# ## ####### ######################### ##### ######## ##### ## ####### ######## ### ## ########## ########## ## ########### ########## ## ########### ### ######## ## ########## ### ### ####### ########## ## #### ##### ##### ########### ######### ######### ## ###...
Exact distribution of edgepreserving MAP estimators for linear signal models with Gaussian measurement noise
 IEEE Transactions on image processing
, 2000
"... Abstract—We derive the exact statistical distribution of maximum a posteriori (MAP) estimators having edgepreserving nonGaussian priors. Such estimators have been widely advocated for image restoration and reconstruction problems. Previous investigations of these image recovery methods have been p ..."
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Cited by 11 (2 self)
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Abstract—We derive the exact statistical distribution of maximum a posteriori (MAP) estimators having edgepreserving nonGaussian priors. Such estimators have been widely advocated for image restoration and reconstruction problems. Previous investigations of these image recovery methods have been primarily empirical; the distribution we derive enables theoretical analysis. The signal model is linear with Gaussian measurement noise. We assume that the energy function of the prior distribution is chosen to ensure a unimodal posterior distribution (for which convexity of the energy function is sufficient), and that the energy function satisfies a uniform Lipschitz regularity condition. The regularity conditions are sufficiently general to encompass popular priors such as the generalized Gaussian Markov random field prior and the Huber prior, even though those priors are not everywhere twice continuously differentiable. Index Terms—Bayesian methods, image reconstruction, image restoration. I.
Regularized Estimation of Mixed Spectra Using a Circular GibbsMarkov Model
 IEEE Trans. Signal Processing
, 2001
"... Formulated as a linear inverse problem, spectral estimation is particularly underdetermined when only short data sets are available. Regularization by penalization is an appealing nonparametric approach to solve such illposed problems. Following Sacchi et al., we first address line spectra reco ..."
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Cited by 9 (3 self)
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Formulated as a linear inverse problem, spectral estimation is particularly underdetermined when only short data sets are available. Regularization by penalization is an appealing nonparametric approach to solve such illposed problems. Following Sacchi et al., we first address line spectra recovering in this framework. Then, we extend the methodology to situations of increasing difficulty: the case of smooth spectra and the case of mixed spectra, i.e., peaks embedded in smooth spectral contributions.
Deblurring of Color Images Corrupted by Impulsive Noise
, 2006
"... We consider the problem of restoring a multichannel image corrupted by blur and impulsive noise (e.g. saltandpepper noise). Using the variational framework, we consider the L¹ fidelity term and several possible regularizers. In particular, we use generalizations of the MumfordShah functional to ..."
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Cited by 6 (1 self)
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We consider the problem of restoring a multichannel image corrupted by blur and impulsive noise (e.g. saltandpepper noise). Using the variational framework, we consider the L¹ fidelity term and several possible regularizers. In particular, we use generalizations of the MumfordShah functional to color images and Γconvergence approximations to unify deblurring and denoising. Experimental comparisons show that the MumfordShah stabilizer yields better results with respect to Beltrami and Total Variation regularizers. Color edge detection is a beneficial byproduct of our methods.
Joint NDT image restoration and segmentation using Gauss–Markov– Potts prior models and variational bayesian computation
 IEEE Transactions on Image Processing
, 2010
"... In this paper, we propose a method to simultaneously restore and to segment piecewise homogenous images degraded by a known point spread function (PSF) and additive noise. For this purpose, we propose a family of nonhomogeneous GaussMarkov fields with Potts region labels model for images to be use ..."
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Cited by 4 (1 self)
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In this paper, we propose a method to simultaneously restore and to segment piecewise homogenous images degraded by a known point spread function (PSF) and additive noise. For this purpose, we propose a family of nonhomogeneous GaussMarkov fields with Potts region labels model for images to be used in a Bayesian estimation framework. The joint posterior law of all the unknowns (the unknown image, its segmentation (hidden variable) and all the hyperparameters) is approximated by a separable probability law via the variational Bayes technique. This approximation gives the possibility to obtain practically implemented joint restoration and segmentation algorithm. We will present some preliminary results and comparison with a MCMC Gibbs sampling based algorithm. We may note that the prior models proposed in this work are particularly appropriate for the images of the scenes or objects that are composed of a finite set of homogeneous materials. This is the case of many images obtained in nondestructive testing (NDT) applications.
HIERARCHICAL MARKOVIAN MODELS FOR 3D COMPUTED TOMOGRAPHY IN NON DESTRUCTIVE TESTING APPLICATIONS
"... So as to detect and characterize potential defects in pipes, inspections are carried out with the help of nondestructive examination techniques (NDE) including Xor radiography. Should a defect be detected, one can be asked to prove the component still stands the mechanical constraints. In these ca ..."
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Cited by 3 (2 self)
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So as to detect and characterize potential defects in pipes, inspections are carried out with the help of nondestructive examination techniques (NDE) including Xor radiography. Should a defect be detected, one can be asked to prove the component still stands the mechanical constraints. In these cases of expertise, the use of a 3D reconstruction processing technique can be very useful. One characteristic of such applications is that, in general the number and angles of projections are very limited and the data are very noisy, so the problem is severely ill posed. Hopefully, in these applications we know a priori the number and the types of materials in the object under the study and this is a great piece of prior information. In this work, we first propose a particular hierarchical MarkovPotts a priori model which takes into account for the specificity of the Non Destructive Technique (NDT) Computed Tomography (CT). Then, we give details of a Bayesian estimation computation based on MCMC and EM techniques. Finally, we show the performances of the proposed 3D CT reconstruction method with a very limited number and angles of projections and very low signal to noise ratio simulating from simulating data. These data have been obtained from very simple defects (cubic form) with acquisition conditions that are supposed to be representatives of real inspection in power plants. 1.
A Bayesian Approach To Shape Reconstruction Of A Compact Object From A Few Number Of Projections
 in Maximum Entropy and Bayesian Methods, (MaxEnt96, Aug. 96, BergenDal, South Africa
, 1996
"... Image reconstruction in X ray tomography consists in determining an object from its projections. In many applications such as non destructive testing, we look for an image who has a constant value inside a region (default) and another constant value outside that region (homogeneous region surroundin ..."
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Cited by 3 (2 self)
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Image reconstruction in X ray tomography consists in determining an object from its projections. In many applications such as non destructive testing, we look for an image who has a constant value inside a region (default) and another constant value outside that region (homogeneous region surrounding the default). The image reconstruction problem becomes then the determination of the shape of that region. In this work we model the object (the default region) as a polygonal disc and propose a new method for the estimation of the coordinates of its vertices directly from a very limited number of its projections.
Unsupervised Frequency Tracking Beyond the Nyquist Frequency Using Markov Chains
 IEEE Trans. Signal Processing
, 2002
"... This paper deals with the estimation of a sequence of frequencies from a corresponding sequence of signals. This problem arises in fields such as Doppler imaging, where its specificity is twofold. First, only short noisy data records are available (typically four sample long), and experimental c ..."
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Cited by 3 (1 self)
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This paper deals with the estimation of a sequence of frequencies from a corresponding sequence of signals. This problem arises in fields such as Doppler imaging, where its specificity is twofold. First, only short noisy data records are available (typically four sample long), and experimental constraints may cause spectral aliasing so that measurements provide unreliable, ambiguous information. Second, the frequency sequence is smooth. Here, this information is accounted for by a Markov model, and application of the Bayes rule yields the a posteriori density. The maximum a posteriori is computed by a combination of Viterbi and descent procedures. One of the major features of the method is that it is entirely unsupervised. Adjusting the hyperparameters that balance databased and priorbased information is done automatically by maximum likelihood (ML) using an expectationmaximization (EM)based gradient algorithm. We compared the proposed estimate to a reference one and found that it performed better: Variance was greatly reduced, and tracking was correct, even beyond the Nyquist frequency.
Shape Reconstruction in XRay Tomography
"... Xray tomographic image reconstruction consists in determining an object function from its projections. In many ap plications such as non destructive testing, we look for a default region (air) in a homogeneous known background (metal). The image reconstruction problem becomes then the determinatio ..."
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Cited by 2 (1 self)
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Xray tomographic image reconstruction consists in determining an object function from its projections. In many ap plications such as non destructive testing, we look for a default region (air) in a homogeneous known background (metal). The image reconstruction problem becomes then the determination of the shape of the default region. Two approaches can bc used: modelling the image as a binary Markov random field and estimating the whole pixels of the image or modeling the shape of the default and estimating it directly from the projections. In this work wc model the default shape by a polygonal disc and propose a new method for estimating directly the coordinates of its vertices from a very limited number of its projections. The idea is not new, but in other competing methods, in general, the default shape is modelled by a small number of parameters (polygonal shapes with very small number of vertices, snakes and deformable templates) and these parameters are estimated either by least squares or by maximum likelihood methods. What wc propose is to model the shape of the default region by a polygon with a great number of vertices to bc able to model any shapes and to estimate directly its vertices coordinates from the projections by defining the solution as the minimizer of an appropriate rcgularizcd criterion which can also bc interpreted as a maximum a postcriori (MAP) estimate in a Bayesian estimation framework. To optimize this criterion we use either a simulated annealing or a special purpose deterministic algorithm based on iterated conditional modes (ICM). The simulated results are very encouraging specially when the number and the angles of projections arc very limited (5 projections limited in45 to 45 degrees). Some comparisons with classical methods are prov...