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Super-resolution fluorescence microscopy overcomes blurring arising from light diffraction, allowing the reconstruction of fine scale details in biological structures. Standard methods come at the expense of long acquisition time and/or harmful effects on the biological sample, which makes the problem quite challenging for the imaging of body cells. A promising new avenue is the exploitation of molecules fluctuations, allowing live-cell imaging with good spatio-temporal resolution through common microscopes and conventional fluorescent dyes. Several numerical algorithms have been developed in the literature and used for fluctuant time series. These techniques are developed within the discrete setting, namely the super-resolved image is defined on a finer grid than the observed images. On the contrary, gridless optimisation does not rely on a fine grid and is rather an optimisation of Dirac measures in number, amplitudes and positions. In this work, we present a gridless problem accounting for the independence of fluctuations.

On-grid based direction-of-arrival (DOA) estimation methods rely on the resolution of a difficult group-sparse optimization problem that involves the ℓ2,0pseudo-norm. In this work, we show that an exact relaxation of this problem can be obtained by replacing the ℓ2,0term with a group minimax concave penalty with suitable parameters. This relaxation is more amenable to non-convex optimization algorithms as it is continuous and admits less local (not global) minimizers than the initial ℓ2,0-regularized criteria. We then show on numerical simulations that the minimization of the proposed relaxation with an iteratively reweighted ℓ2,0algorithm leads to an improved performance over traditional approaches.

Lemma 4.4 in [E. Soubies, L. Blanc-Féraud and G. Aubert, SIAM J. Imaging Sci., 8 (2015), pp. 1607--1639] is wrong for local minimizers of the continuous exact $\ell_0$ (CEL0) functional. The argument used to conclude the proof of this lemma is not sufficient in the case of local minimizers. In this note, we supply a revision of this lemma where new results are established for local minimizers. Theorem 4.8 in that paper remains unchanged but the proof has to be rewritten according to the new version of the lemma. Finally, some remarks of this paper are also rewritten using the corrected lemma.

Within the framework of the $\ell_0$ regularized least squares problem, we focus, in this paper, on nonconvex continuous penalties approximating the $\ell_0$-norm. Such penalties are known to better promote sparsity than the $\ell_1$ convex relaxation. Based on some results in one dimension and in the case of orthogonal matrices, we propose the continuous exact $\ell_0$ penalty (CEL0) leading to a tight continuous relaxation of the $\ell_2-\ell_0$ problem. The global minimizers of the CEL0 functional contain the global minimizers of $\ell_2 - \ell_0$, and from each global minimizer of CEL0 one can easily identify a global minimizer of $\ell_2 - \ell_0$. We also demonstrate that from each local minimizer of the CEL0 functional, a local minimizer of $\ell_2 - \ell_0$ is easy to obtain. Moreover, some strict local minimizers of the initial functional are eliminated with the proposed tight relaxation. Then solving the initial $\ell_2 - \ell_0$ problem is equivalent, in a sense, to solving it by replacing the $\ell_0$-norm with the CEL0 which provides better properties for the objective function in terms of minimization, such as the continuity and the convexity with respect to each direction of the standard $\mathbb{R}^N$ basis, although the problem remains nonconvex. Finally, recent nonsmooth nonconvex algorithms are used to address this relaxed problem within a macro algorithm ensuring the convergence to a critical point of the relaxed functional which is also a (local) optimum of the initial problem.

Super-resolution microscopy techniques allow to overstep the diffraction limit of conventional optics. Theses techniques are very promising since they give access to the visualisation of finer structures which is of fundamental importance in biology. In this paper we deal with Multiple-Angle Total Internal Reflection Microscopy (MA-TIRFM) which allows to reconstruct 3D sub-cellular structures of a single layer of ~ 300 nm behind the glass coverslip with a high axial resolution. The 3D volume reconstruction from a set of 2D measurements is an ill-posed inverse problem and a regularization is essential. Our aim in this work is to propose a new reconstruction method for sparse structures robust to Poisson noise and background fluorescence. The sparse property of the solution can be seen as a regularization using the `£° norm'. In order to solve this combinatorial problem, we propose a new algorithm based on smoothed `£° norm' allowing minimizing a non convex energy, composed of the Kullback-Leibler divergence data term and the £° regularization term, in a Graduated Non Convexity framework.

We are interested in blind restoration of optical space variant blurred Poissonian images. For example, blur variation is due to refractive index mismatch in three-dimensional fluorescence microscopy or due to atmospheric turbulence in astrophysical images. In this work, the space variant point spread function (PSF) is approximated by a convex combination of a set of space invariant blurring functions. The latter is jointly estimated with the image by optimizing a given criterion including $l_1$ and $l_2$ norms for regularizing the image and the PSFs. We prove, in the continuous setting, the existence of a solution to this optimization problem. We then propose an alternating optimization algorithm based on a scaled gradient projection method. We show the efficiency of the proposed method on simulated and real optical images.

In this paper we are interested in regularizing hyperparameter estimation by maximum likelihood in inverse problems with wavelet regularization. One parameter per subband will be estimated by gradient ascent algorithm. We have to face with two main difficulties: i) sampling the a posteriori image distribution to compute the gradient; ii) choosing a suited step-size to ensure good convergence properties. We first show that introducing an auxiliary variable makes the sampling feasible using classical Metropolis-Hastings algorithm and Gibbs sampler. Secondly, we propose an adaptive step-size selection and a line-search strategy to improve the gradient-based method. Good performances of the proposed approach are demonstrated on both synthetic and real data.

We are interested in blind image restoration in confocal laser scanning microscopy (CLSM). Two challenging problems in this imaging system are considered: First, spherical aberrations due to refractive index mismatch leads to a depth variant (DV) blur. Second, low illumination leads to a signal dependent Poisson noise. In addition, the DV point spread function (PSF) is unknown, which increases the complexity of the problem considered. Our goal is to remove in a blind framework both the DV blur and the Poisson noise from CLSM images. Using an approximation of the DV PSF, we define in a Bayesian framework a criterion to be jointly minimized w.r.t. the specimen function and the PSF. We then adopt an alternate minimization scheme for the optimization problem. For each elementary minimization, we use the recently proposed scaled gradient projection (SGP) algorithm that has shown a fast convergence rate. Results are shown on simulated and real CLSM images.

Like many other imaging techniques, 3D fluorescence microscopy suffers from degradations that are basically varying with the depth of the point source. This is due to the light refraction phenomenon. In this article, we focus on modeling and removing depth variant blur in such a system. In particular, we study some of the existing space-variant blur approximations and consider an efficient approximation where the space variant blur function is a linear combination of a set of space-invariant ones. We then focus on restoring space-variant blurred images using such a model. For that, we fit a domain decomposition-based minimization approach to the deconvolution problem with a space variant blur model. We thus obtain a fast restoration algorithm where the image estimation is performed in a parallel way on different sub-images.

This paper contains errata corrige on the paper A formal $\Gamma$-convergence approach for the detection of points in $2$-D images [D. Graziani, L. Blanc-Féraud, and G. Aubert, SIAM J. Imaging Sci., 3 (2010), pp. 578--594].

Deblurring noisy Poisson images has recently been a subject of an increasing amount of works in many areas such as astronomy and biological imaging. In this paper, we focus on confocal microscopy, which is a very popular technique for 3-D imaging of biological living specimens that gives images with a very good resolution (several hundreds of nanometers), although degraded by both blur and Poisson noise. Deconvolution methods have been proposed to reduce these degradations, and in this paper, we focus on techniques that promote the introduction of an explicit prior on the solution. One difficulty of these techniques is to set the value of the parameter, which weights the tradeoff between the data term and the regularizing term. Only few works have been devoted to the research of an automatic selection of this regularizing parameter when considering Poisson noise; therefore, it is often set manually such that it gives the best visual results. We present here two recent methods to estimate this regularizing parameter, and we first propose an improvement of these estimators, which takes advantage of confocal images. Following these estimators, we secondly propose to express the problem of the deconvolution of Poisson noisy images as the minimization of a new constrained problem. The proposed constrained formulation is well suited to this application domain since it is directly expressed using the antilog likelihood of the Poisson distribution and therefore does not require any approximation. We show how to solve the unconstrained and constrained problems using the recent alternating-direction technique, and we present results on synthetic and real data using well-known priors, such as total variation and wavelet transforms. Among these wavelet transforms, we specially focus on the dual-tree complex wavelet transform and on the dictionary composed of curvelets and an undecimated wavelet transform.

We propose a new variational method to restore point-like and curve-like singularities in 2-D images. As points and open curves are fine structures, they are difficult to restore by means of first order derivative operators computed in the noisy image. In this paper we propose to use the Laplacian operator of the observed intensity, since it be comes singular at points and curves. Then we propose to restore these singularities by introducing suitable regularization involving the M-norm of the Laplacian operator. Results are shown on synthetic an real data.

Deblurring noisy Poisson images has recently been subject of an increasingly amount of works in many areas such as astronomy or biological imaging. Several methods have promoted explicit prior on the solution to regularize the ill-posed inverse problem and to improve the quality of the image. In each of these methods, a regularizing parameter is introduced to control the weight of the prior. Unfortunately, this regularizing parameter has to be manually set such that it gives the best qualitative results. To tackle this issue, we present in this paper two constrained formulations for the Poisson deconvolution problem, derived from recent advances in regularizing parameter estimation for Poisson noise. We first show how to improve the accuracy of these estimators and how to link these estimators to constrained formulations. We then propose an algorithm to solve the resulting optimization problems and detail how to per form the projections on the constraints. Results on real and synthetic data are presented.

We analyze the illumination invariance of the level lines of an image. We show that if the scene surface has Lambertian reflectance and the light is directed, then a necessary and sufficient condition for the level lines to be illumination invariant is that the three-dimensional scene be developable and that its albedo satisfy some geometrical constraints. We then show that the level lines are “almost” invariant for piecewise developable surfaces. Such surfaces fit most of the urban structures. This allows us to devise a fast and simple algorithm that detects changes between pairs of remotely sensed images of urban areas, independently of the lighting conditions. We show the effectiveness of the algorithm both on synthetic OpenGL scenes and real QuickBird images. The synthetic results illustrate the theory developed in this paper. The two real QuickBird images show that the proposed change detection algorithm is discriminant. For easy scenes it achieves a rate of 85% detected changes for 10% false positives, while it reaches a rate of 75% detected changes for 25% false positives on demanding scenes.

The goal of this paper is to develop an algorithm for extracting point features from sequences of aerial infrared images. We propose an efficient method for the detection of threats in a sequence of infrared images by looking for a trajectory which optimizes a regularized criterion. The regularity is introduced by a new concept of total curvature which eliminates too oscillating trajectories while allowing those with ponctual changes of direction. In practice the research of an optimal trajectory is performed with an algorithm of dynamical programming type. Experimental results are also presented.

We propose a new variational model to locate points in 2-dimensional biological images. To this purpose we introduce a suitable functional whose minimizers are given by the points we want to detect. In order to provide numerical experiments we replace this energy with a sequence of more treatable functionals by means of the notion of $\Gamma$-convergence.

Many problems in remote sensing can be modeled as the minimization of the sum of a data term and a prior term. We propose to use a new complex wavelet based prior and an efficient scheme to solve these problems. We show some results on a problem of image reconstruction with noise, irregular sampling and blur. We also show a comparison between two widely used priors in image processing: sparsity and regularity priors.

We propose a new method to solve a problem of image restoration with many different aspects: reconstruction from irregular samples, deconvolution and denoising. The model we propose is robust to different kind of noises, in particular, impulse and Gaussian noise. We compare our results to the ones obtained in [1] and show that our problem presents some advantages particularly in satellite imaging. At last, we conclude on a discussion about resolution schemes for variational problems' minimization and propose some faster resolution shemes for our problem and the one in [1].

Many compression algorithms consist of quantizing the coefficients of an image in a linear basis. This introduces compression noise that often look like ringing. Recently some authors proposed variational methods to reduce those artifacts. They consists of minimizing a regularizing functional in the set of antecedents of the compressed image. In this paper we propose a fast algorithm to solve that problem. Our experiments lead us to the conclusion that these algorithms effectively reduce oscillations but also reduce contrasts locally. To handle that problem, we propose a fast contrast enhancement procedure. Experiments on a large dataset suggest that this procedure effectively improves the image quality at low bitrates.

We propose an algorithm that equalizes the contrast of grayscale image pairs to simplify the task of change detection. To ensure robustness of the detection under different illumination conditions, some authors recently proposed algorithms that compare the level lines of the images. We show - using ideas from the ldquoshape from shadingrdquo community - that under directed light, a necessary condition for the level lines to be illumination invariant is that the underlying surfaces be developable. The surfaces of cities can be modeled as piece-wise smooth developable surfaces, and it is therefore sensible to make use of the level lines for change detection. Our algorithm is robust and efficient both on synthetic OpenGL scenes and natural Quickbird images.

Confocal laser scanning microscopy is a powerful and increasingly popular technique for 3D imaging of biological specimens. However, the acquired images are degraded by blur from out-of-focus light and Poisson noise due to photon-limited detection. Several deconvolution and/or denoising methods have been proposed to reduce these degradations. Here, we propose a wavelet denoising method, which turns out to be very effective for 3D confocal images. To obtain a translation and rotation invariant algorithm, we have developed the 3D complex wavelet transform introduced by N. Kingsbury. These wavelets allow, moreover, a better directional selectivity of the wavelet coefficients. We show on simulated and real biological data the good performance of this algorithm.

The deconvolution of blurred and noisy satellite images is an ill-posed inverse problem, which can be regularized within a Bayesian context by using an a priori model of the reconstructed solution. Since real satellite data show spatially variant characteristics, we propose here to use an inhomogeneous model. We use the maximum likelihood estimator (MLE) to estimate its parameters and we show that the MLE computed on the corrupted image is not suitable for image deconvolution because it is not robust to noise. We then show that the estimation is correct only if it is made from the original image. Since this image is unknown, we need to compute an approximation of sufficiently good quality to provide useful estimation results. Such an approximation is provided by a wavelet-based deconvolution algorithm. Thus, a hybrid method is first used to estimate the space-variant parameters from this image and then to compute the regularized solution. The obtained results on high resolution satellite images simultaneously exhibit sharp edges, correctly restored textures, and a high SNR in homogeneous areas, since the proposed technique adapts to the local characteristics of the data.

A supervised classification model based on a variational approach is presented. This model is specifically devoted to textured images. We want to get a partition of an image, composed of texture regions separated by regular interfaces. Each kind of texture defines a class. We use a wavelet packet transform to analyze the textures, characterized by their energy distribution in each sub-band. In order to have an image segmentation according to the classes, we model the regions and their interfaces by level set functions. We define a functional on these level sets whose minimizers define the optimal classification according to textures. A system of coupled PDEs is deduced from the functional. By solving this system, each region evolves according to its wavelet coefficients and interacts with the neighbour region in order to obtain a partition with regular contours. Experiments are shown on synthetic and real images.

We present an anisotropic diffusion equation designed to restore interferometric images. It has two main purposes, the first is to preserve the structures and discontinuities formed by the fringes. The second is to incorporate the noise modeling which is specific to this kind of images. Besides we show that our model formalizes previous related work in interferometry filtering.

We present a supervised classification model based on a variational approach. This model is specifically devoted to textured images. We want to get a partition of an image, composed of texture regions separated by regular interfaces. Each kind of texture defines a class. We use a wavelet packet transform to analyze the textures, characterized by their energy distribution in each sub-band. In order to have an image segmentation according to the classes, we model the regions and their interfaces by level set functions. We define a functional on these level sets whose minimizers define the optimal classification according to texture. A system of coupled PDEs is deduced from the functional. By solving this system, each region evolves according to its wavelet coefficients and interacts with the neighbor regions in order to obtain a partition with regular contours. Experiments are shown on synthetic and real images.

In this paper we propose a new algorithm to estimate the parameters of the noise related to the sensor and the impulse response of the optical system, from a blurred and noisy satellite or aerial image. The noise is supposed to be white, Gaussian and stationary. The blurring kernel has a parametric form and is modeled in such a way as to take into account the physics of the system (the atmosphere, the optics and the sensor). The observed scene is described by a fractal model, taking into account the scale invariance properties of natural images. The estimation is performed automatically by maximizing a marginalized likelihood, which is achieved by a deterministic algorithm whose complexity is limited to O (N), where N is the number of pixels.

The deconvolution of blurred and noisy satellite images is an ill-posed inverse problem. Donoho (1994) has proposed to deconvolve the image without regularization and to denoise the result in a wavelet basis by thresholding the transformed coefficients. We have developed a new filtering method, consisting of using a complex wavelet packet basis. Herein, the thresholding functions associated to the proposed method are automatically estimated. The estimation is performed within a Bayesian framework, by modeling the subbands using generalized Gaussian distributions, and by applying the maximum a posteriori (MAP) estimator on each coefficient. Compared to real wavelet-packet-based algorithms, the proposed method is shift invariant, provides good directionality properties and remains of complexity O(N).

The deconvolution of blurred and noisy satellite images is an ill-posed inverse problem, which can be regularized within a Bayesian context by using an a priori model of the reconstructed solution. Since real satellite data show spatially variant characteristics, we propose to use an inhomogeneous model. We use the maximum likelihood estimator (MLE) to estimate its parameters. We demonstrate that the MLE computed on the corrupted image is not suitable for image deconvolution, because it is not robust to noise. Then we show that the estimation is correct only if it is made from the original image. As this image is unknown, we need to compute an approximation of sufficiently good quality to provide useful estimation results. Such an approximation is provided by a wavelet-based deconvolution algorithm. Thus, an hybrid method is first used to estimate the space-variant parameters from this image and second to compute the regularized solution. The obtained results on high resolution satellite images simultaneously exhibit sharp edges, correctly restored textures and a high SNR in homogeneous areas, since the proposed technique adapts to the local characteristics of the data.

We present a variational model devoted to image classification coupled with an edge-preserving regularization process. The discrete nature of classification (i.e., to attribute a label to each pixel) has led to the development of many probabilistic image classification models, but rarely to variational ones. In the last decade, the variational approach has proven its efficiency in the field of edge-preserving restoration. We add a classification capability which contributes to provide images composed of homogeneous regions with regularized boundaries, a region being defined as a set of pixels belonging to the same class. The soundness of our model is based on the works developed on the phase transition theory in mechanics. The proposed algorithm is fast, easy to implement, and efficient. We compare our results on both synthetic and satellite images with the ones obtained by a stochastic model using a Potts regularization.

Herein, we present a variational model devoted to image classification coupled with an edge-preserving regularization process. In the last decade, the variational approach has proven its efficiency in the field of edge-preserving restoration. In this paper, we add a classification capability which contributes to provide images compound of homogeneous regions with regularized boundaries. The soundness of this model is based on the works developed on the phase transition theory in mechanics. The proposed algorithm is fast, easy to implement and efficient. We compare our results on both synthetic and satellite images with the ones obtained by a stochastic model using a Potts regularization.

This paper deals with motion segmentation. We use an active contour method to segment the apparent motion computed from two successive frames of an image sequence. In order to obtain accurate segmentation, we combine both intensity and motion information. We examine two different ways for combination and we show results obtained on a real sequence.

This paper focuses on hyperparameter estimation of a variational model for image deconvolution. Using the generalized maximum likelihood (GML) estimator, the estimation problem is reduced to the ML estimation in the case of perfectly observed data. A method based on stochastic gradient is then developed for the estimation of both linear and nonlinear hyperparameters.

This paper deals with edge-preserving regularization for inverse problems in image processing. We first present a synthesis of the main results we have obtained in edge-preserving regularization by using a variational approach. We recall the model involving regularizing functions phi and we analyze the geometry-driven diffusion process of this model in the three-dimensional (3-D) case. Then a half-quadratic theorem is used to give a very simple reconstruction algorithm. After a critical analysis of this model, we propose another functional to minimize for edge-preserving reconstruction purposes. It results in solving two coupled partial differential equations (PDEs): one processes the intensity, the other the edges. We study the relationship with similar PDE systems in particular with the functional proposed by Ambrosio-Tortorelli in order to approach the Mumford-Shah functional developed in the segmentation application. Experimental results on synthetic and real images are presented.

This paper deals with segmentation and edge-preserving restoration. We use an active contour method to segment an image which may not be very noisy with regard to its edges. In order to both restore and segment a noisy image, we propose in this paper to derive a system of two partial differential equations, one for edge-preserving restoration (using edges given by segmentation), and the other for segmentation (using the restored image).

Many image processing problems are ill-posed and must be regularized. Usually, a roughness penalty is imposed on the solution. The difficulty is to avoid the smoothing of edges, which are very important attributes of the image. In this paper, we first give conditions for the design of such an edge-preserving regularization. Under these conditions, we show that it is possible to introduce an auxiliary variable whose role is twofold. First, it marks the discontinuities and ensures their preservation from smoothing. Second, it makes the criterion half-quadratic. The optimization is then easier. We propose a deterministic strategy, based on alternate minimizations on the image and the auxiliary variable. This leads to the definition of an original reconstruction algorithm, called ARTUR. Some theoretical properties of ARTUR are discussed. Experimental results illustrate the behavior of the algorithm. These results are shown in the field of 2D single photon emission tomography, but this method can be applied in a large number of applications in image processing.

This paper deals with edge-preserving regularization for image reconstruction. We use a non-quadratic regularization term involving a /spl phi/-function applied on the intensity gradient modulus. During the process, small gradients are smoothed while high gradients are preserved. In order to take into account the noise more specifically, we propose to use the explicit version of the regularization term involving the edge variable. We add a nonlinear constraint on this edge variable, in order to remove the noise. It allows edge enhancement while smoothing the noise, even in the case where the edge and noise generate the same high gradient modulus. We have previously proposed a model composed of two coupled partial differential equations (PDE) on the image intensity and image edges. In this paper, we show that, for a particular regularization intensity function, the two coupled PDE can be slightly modified in order to correspond to the Euler equations associated with the minimization of a global criterion. This new criterion contains a nonlinear regularization term on both the intensity and the edges. We use convergence towards Mumford and Shah (1989) functional to improve our results.

In emission computerized tomography, the use of realistic constraints such as edge-preserving smoothing lead to nonlinear regularisation. Charbonnier et al. (see IEEE Trans. on Image Processing, 1994) used the half-quadratic regularization in order to solve this problem. Applied together with a Gaussian likelihood, it formed the ARTUR algorithm. We propose a new regularized algorithm called MOISE which takes into account the Poisson nature of the statistical noise and uses this half-quadratic regularization. For that reason, MOISE differ from the MAP EM (maximum a posteriori expectation maximization) algorithm developed by P.J. Green (1990) which uses the one step late technique. We tested MOISE and compared it with ARTUR, on numerical simulation and real data. The results show that, despite the slowness of convergence, the half-quadratic regularization can be applied in the case of a Poisson statistic.

This paper deals with edge-preserving regularization. The definition of the regularizing functions and properties such as edge modeling or stability are studied in the variational approach (by minimizing of a criterion) and in the anisotropic diffusion approach (by solving a PDE). We propose sufficient conditions to define an edge-preserving regularizing function, and analyze comparatively several usual functions. We use the algorithm ARTUR based on the half-quadratic transform to solve the nonlinear equation. The image and the edge map are simultaneously estimated.

A new method for tomographic reconstruction using 2-D wavelet transform (WT) is proposed. Computed tomography involves huge matrices which make the reconstruction process costly. The WT can focus the useful information on a small number of entries. Then, the computation time can be significantly reduced by using special algorithms to exploit the sparsity of the matrices. Moreover, computing the solution in the WT domain scatters errors on several resolutions and thus, control the quality of the reconstruction. We also show that the 2-D WT is more efficient than the 1-D WT.>

Many image processing problems are ill-posed and must be regularized. Usually, a roughness penalty is imposed on the solution. The difficulty is to avoid the smoothing of edges, which are very important attributes of the image. The authors first give sufficient conditions for the design of such an edge-preserving regularization. Under these conditions, it is possible to introduce an auxiliary variable whose role is twofold. Firstly, it marks the discontinuities and ensures their preservation from smoothing. Secondly, it makes the criterion half-quadratic. The optimization is then easier. The authors propose a deterministic strategy, based on alternate minimizations on the image and the auxiliary variable. This yields two algorithms, ARTUR and LEGEND. The authors apply these algorithms to the problem of SPECT reconstruction.>

Many image processing problems are ill-posed and must be regularized. Usually, a roughness penalty is imposed on the solution. The difficulty is to avoid the smoothing of edges, which are very important attributes of the image. We first propose sufficient conditions for the design of such an edge-preserving regularization. Using the Legendre transform, it is then possible to introduce an auxiliary variable which role is twofold. Firstly, it marks the discontinuities and ensures their preservation from smoothing. Secondly, it makes the criterion half-quadratic. The optimization is then easier. We propose a deterministic algorithm, based on alternate minimizations over the image and the auxiliary variable. We apply this algorithm to the problem of SPECT reconstruction.

Recursive 2D Kalman filtering has been successfully used in image restoration. In order to reduce computational work and large amounts of storage, a fast algorithm has been introduced by Biemond. However this method is based on the DFT transform and ringing on the restored image results from boundary conditions. Furthermore, the Kalman gain factors may increase due to the zeros of the Fourier transform of the PSF periodicity is realized by means of a new image mirror. No more computational work is required since this new image is symmetric. Then the authors introduce adaptive Kalman filtering in order to avoid divergence of Kalman gain factors and to reduce the remaining ringing.>