We consider the high-resolution imaging problem of 3-dimensional (3D) point source image recovery from 2-dimensional data using a method based on point spread function (PSF) engineering. The method involves a new technique, recently proposed by Prasad, based on the use of a rotating PSF with a single lobe to obtain depth from defocus. The amount of rotation of the PSF encodes the depth position of the point source. Applications include high-resolution single molecule localization microscopy as well as the problem addressed in this paper on localization of space debris using a space-based telescope. The localization problem is discretized on a cubical lattice where the coordinates of nonzero entries represent the 3D locations and the values of these entries the fluxes of the point sources. Finding the locations and fluxes of the point sources is a large-scale sparse 3D inverse problem. A new non-convex regularization method with a data-fitting term based on Kullback--Leibler (KL) divergence is proposed for 3D localization for the Poisson noise model. In addition, we propose a new scheme of estimation of the source fluxes from the KL data-fitting term. Numerical experiments illustrate the efficiency and stability of the algorithms that are trained on a random subset of image data before being applied to other images. Our 3D localization algorithms can readily be applied to other kinds of depth-encoding PSFs as well.

MRI phase contrast imaging methods that assemble slice-wise acquisitions into volumes can contain interslice phase discontinuities (IPDs) over the course of the scan from sources, including unavoidable physiological activity. In magnetic resonance elastography (MRE), this can alter wavelength and tissue stiffness estimates, invalidating the analysis. We first model this behavior as jitter along the z-axis of the phase of 3D complex-valued wave volumes. A two-step image processing pipeline is then proposed that removes IPDs. First, constant slicewise phase shift is removed with a novel, non-convex dejittering algorithm. Then, regional physiological noise artifacts are removed with novel filtering of 3D wavelet coefficients. Calibration of two pipeline coefficients, the dejitter parameter$\alpha $and the wavelet band high-pass coefficient$\omega _{c}$, was first performed on a finite-element method brain phantom. A comparative investigation was then performed, on a cohort of 48 brain acquisitions, of four approaches to IPDs: 1) the proposed method; 2) a “control” condition of neglect of IPDs; 3) an anisotropic wavelet-based method; and 4) a method of in-plane (2D) processing. The present method showed medians of$\lvert {G}^{*} \rvert = \textsf {1873}$Pa for a multifrequency wave inversion centered at 40 Hz which was within 6% of methods 3) and 4), while neglect produced$\lvert {G}^{*} \rvert $estimates a mean of 17% lower. The proposed method reduced the value range of the cohort against methods 3) and 4) by 29% and 31%, respectively. Such reduction in variance enhances the ability of brain MRE to predict subtler physiological changes. Our theoretical approach further enables more powerful applications of fundamental findings in noise and denoising to MRE.

Nonlocal patch-based methods, in particular the Bayesian approach of Lebrun, Buades, and Morel [SIAM J. Imaging Sci., 6 (2013), pp. 1665--1688], are considered to be state-of-the-art methods for denoising (color) images corrupted by white Gaussian noise of moderate variance. This paper is the first attempt to generalize this technique to manifold-valued images. Such images, for example, images with phase or directional entries or with values in the manifold of symmetric positive definite matrices, are frequently encountered in real-world applications. Generalizing the normal law to manifolds is not canonical, and different attempts have been considered. Here, we focus on a straightforward intrinsic model and discuss the relation to other approaches for specific manifolds. We reinterpret the Bayesian approach of Lebrun, Buades, and Morel [SIAM J. Imaging Sci., 6 (2013), pp. 1665--1688] in terms of minimum mean squared error estimation, which motivates our definition of a corresponding estimator on the manifold. With this estimator at hand we present a nonlocal patch-based method for the restoration of manifold-valued images. Various proof-of-concept examples demonstrate the potential of the proposed algorithm.

Color image enhancement is a complex and challenging task in digital imaging with abundant applications. Preserving the hue of the input image is crucial in a wide range of situations. We propose simple image enhancement algorithms which conserve the hue and preserve the range (gamut) of the R, G, B channels in an optimal way. In our setup, the intensity input image is transformed into a target intensity image whose histogram matches a specified, well-behaved histogram. We derive a new color assignment methodology where the resulting enhanced image fits the target intensity image. We analyse the obtained algorithms in terms of chromaticity improvement and compare them with the unique and quite popular histogram based hue and range preserving algorithm of Naik and Murthy. Numerical tests confirm our theoretical results and show that our algorithms perform much better than the Naik-Murthy algorithm. In spite of their simplicity, they compete with well-established alternative methods for images where hue-preservation is desired.

This paper provides a fast algorithm to order in a meaningful, strict way the integer gray values in digital (quantized) images. It can be used in any exact histogram specification-based application. Our algorithm relies on the ordering procedure based on the specialized variational approach. This variational method was shown to be superior to all other state-of-the art ordering algorithms in terms of faithful total strict ordering but not in speed. Indeed, the relevant functionals are in general difficult to minimize because their gradient is nearly flat over vast regions. In this paper, we propose a simple and fast fixed point algorithm to minimize these functionals. The fast convergence of our algorithm results from known analytical properties of the model. Our algorithm is equivalent to an iterative nonlinear filtering. Furthermore, we show that a particular form of the variational model gives rise to much faster convergence than other alternative forms. We demonstrate that only a few iterations of this filter yield almost the same pixel ordering as the minimizer. Thus, we apply only few iteration steps to obtain images, whose pixels can be ordered in a strict and faithful way. Numerical experiments confirm that our algorithm outperforms by far its main competitors.

We have an $\sf{M}\times\sf{N}$ real-valued arbitrary matrix $A$ (e.g., a dictionary) with $\sf{M}<\sf{N}$ and data $d$ describing the sought-after object with the help of $A$. This work provides an in-depth analysis of the (local and global) minimizers of an objective function ${\mathcal{F}}_d$ combining a quadratic data-fidelity term and an $\ell_0$ penalty applied to each entry of the sought-after solution, weighted by a regularization parameter $\beta>0$. For several decades, this objective has attracted a ceaseless effort to conceive algorithms approaching a good minimizer. Our theoretical contributions, summarized below, shed new light on the existing algorithms and can help in the conception of innovative numerical schemes. Solving the normal equation associated with any $\sf{M}$-row submatrix of $A$ is equivalent to computing a local minimizer $\hat u$ of ${\mathcal{F}}_d$. (Local) minimizers $\hat u$ of ${\mathcal{F}}_d$ are strict if and only if the submatrix, composed of those columns of $A$ whose indices form the support of $\hat u$, has full column rank. An outcome is that strict local minimizers of ${\mathcal{F}}_d$ are easily computed without knowing the value of $\beta$. Each strict local minimizer is linear in data. It is proved that ${\mathcal{F}}_d$ has global minimizers and that they are always strict. They are studied in more detail under the (standard) assumption that rank$(A)=\sf{M}<\sf{N}$. The global minimizers with $\sf{M}$-length support are seen to be impractical. Given $d$, critical values $\beta_{\sf{K}}$ for any ${\sf{K}}\leqslant\sf{M}-1$ are exhibited such that if $\beta>\beta_{\sf{K}}$, all global minimizers of ${\mathcal{F}}_d$ are ${\sf{K}}$-sparse. An assumption on $A$ is adopted and proved to fail only on a closed negligible subset. Then for all data $d$ beyond a closed negligible subset, the objective ${\mathcal{F}}_d$ for $\beta>\beta_{\sf{K}}$, ${\sf{K}}\leqslant\sf{M}-1$, has a unique global minimizer, and this minimizer is ${\sf{K}}$-sparse. Instructive small-size ($5\times 10$) numerical illustrations confirm the main theoretical results.

Nonconvex nonsmooth regularization has advantages over convex regularization for restoring images with neat edges. However, its practical interest used to be limited by the difficulty of the computational stage which requires a nonconvex nonsmooth minimization. In this paper, we deal with nonconvex nonsmooth minimization methods for image restoration and reconstruction. Our theoretical results show that the solution of the nonconvex nonsmooth minimization problem is composed of constant regions surrounded by closed contours and neat edges. The main goal of this paper is to develop fast minimization algorithms to solve the nonconvex nonsmooth minimization problem. Our experimental results show that the effectiveness and efficiency of the proposed algorithms.

We consider the restoration of piecewise constant images where the number of the regions and their values are not fixed in advance, with a good difference of piecewise constant values between neighboring regions, from noisy data obtained at the output of a linear operator (e.g., a blurring kernel or a Radon transform). Thus we also address the generic problem of unsupervised segmentation in the context of linear inverse problems. The segmentation and the restoration tasks are solved jointly by minimizing an objective function (an energy) composed of a quadratic data-fidelity term and a nonsmooth nonconvex regularization term. The pertinence of such an energy is ensured by the analytical properties of its minimizers. However, its practical interest used to be limited by the difficulty of the computational stage which requires a nonsmooth nonconvex minimization. Indeed, the existing methods are unsatisfactory since they (implicitly or explicitly) involve a smooth approximation of the regularization term and often get stuck in shallow local minima. The goal of this paper is to design a method that efficiently handles the nonsmooth nonconvex minimization. More precisely, we propose a continuation method where one tracks the minimizers along a sequence of approximate nonsmooth energies $\{J_\eps\}$, the first of which being strictly convex and the last one the original energy to minimize. Knowing the importance of the nonsmoothness of the regularization term for the segmentation task, each $J_\eps$ is nonsmooth and is expressed as the sum of an $\ell_1$ regularization term and a smooth nonconvex function. Furthermore, the local minimization of each $J_{\eps}$ is reformulated as the minimization of a smooth function subject to a set of linear constraints. The latter problem is solved by the modified primal-dual interior point method, which guarantees the descent direction at each step. Experimental results are presented and show the effectiveness and the efficiency of the proposed method. Comparison with simulated annealing methods further shows the advantage of our method.

A popular way to restore images comprising edges is to minimize a cost function combining a quadratic data-fidelity term and an edge-preserving (possibly nonconvex) regularizalion term. Mainly because of the latter term, the calculation of the solution is slow and cumbersome. Half-quadratic (HQ) minimization (multiplicative form) was pioneered by Geman and Reynolds (1992) in order to alleviate the computational task in the context of image reconstruction with nonconvex regularization. By promoting the idea of locally homogeneous image models with a continuous-valued line process, they reformulated the optimization problem in terms of an augmented cost function which is quadratic with respect to the image and separable with respect to the line process, hence the name "half quadratic." Since then, a large amount of papers were dedicated to HQ minimization and important results--including edge-preservation along with convex regularization and convergence-have been obtained. In this paper, we show that HQ minimization (multiplicative form) is equivalent to the most simple and basic method where the gradient of the cost function is linearized at each iteration step. In fact, both methods give exactly the same iterations. Furthermore, connections of HQ minimization with other methods, such as the quasi-Newton method and the generalized Weiszfeld's method, are straightforward.

Restoring binary images is a problem which arises in various application fields. In our paper, this problem is considered in a variational framework: the sought-after solution minimizes an energy. Energies defined over the set of the binary images are inevitably nonconvex and there are no general methods to calculate the global minimum, while local minimizers are very often of limited interest. In this paper we define the restored image as the global minimizer of the total-variation (TV) energy functional constrained to the collection of all binary-valued images. We solve this constrained non-convex optimization problem by deriving another functional which is convex and whose (unconstrained) minimum is proven to be reached for the global minimizer of the binary constrained TV functional. Practical issues are discussed and a numerical example is provided.

Recently, a two-phase scheme for removing salt-and-pepper impulse noise has been proposed [R.H. Chan et al], In the first phase, an adaptive median filter is used to identify pixels which are likely to be contaminated by noise (noise candidates). In the second phase, the image is restored by minimizing a specialized regularization functional that applies only to those selected noise candidates. As an extension of this work, we propose an efficient method to accomplish the second phase. The speed of our method can be double as that of the method proposed in [R.H. Chan et al] for images contaminated by 30% salt-and-pepper noise and is faster for higher noise level.

We consider the restoration of discrete signals and images using least-squares with non-convex regularization. Our goal is to understand how the restoration of edges at the (local) minimizers of the cost function is determined by the shape of the regularization term. This question is of paramount importance for a relevant choice of regularization term. We show that the differences between neighboring pixels in homogeneous regions are smaller than a small threshold while they are larger than a large threshold at edges: we can say that the former are shrunk while the latter are enhanced. This entails a natural classification of differences as smooth or edges. Furthermore, if the original signal or image is a scaled characteristic function of a subset, we show that the global minimizer is smooth everywhere if the contrast is low, whereas edges are correctly recovered at higher (finite) contrast. Explicit expressions are derived for the truncated quadratic and the "0-1" regularization function. It is seen that restoration using non-convex regularization is fundamentally different from edge-preserving convex regularization. Our theoretical results are illustrated using a numerical experiment.

This paper proposes a two-phase scheme for removing salt-and-pepper impulse noise. In the first phase, an adaptive median filter is used to identify pixels which are likely to be contaminated by noise (noise candidates). In the second phase, the image is restored using a specialized regularization method that applies only to those selected noise candidates. In terms of edge preservation and noise suppression, our restored images show a significant improvement compared to those restored by using just nonlinear filters or regularization methods only. Our scheme can remove salt-and-pepper-noise with a noise level as high as 90%.

This work proposes a two-stage iterative method for removing random-valued impulse noise. In the first phase, we use the adaptive center-weighted median filter to identify pixels which are likely to be corrupted by noise (noise candidates). In the second phase, these noise candidates are restored using a detail-preserving regularization method which allows edges and noise-free pixels to be preserved. These two phases are applied alternatively. Simulation results indicate that the proposed method is significantly better than those using just nonlinear filters or regularization only.

We consider the reconstruction of MRI images by minimizing regularized cost-functions. To accelerate the computation of the estimate, two forms of half-quadratic regularization, multiplicative and additive, are often used. In Nikolova and Ng (2002), we have compared both theoretically and experimentally the efficiency of these two forms using one-dimensional signals. The goal of this paper is to compare experimentally the efficiency of these two forms using MRI image reconstruction. We find that using the additive form is more computationally effective than using the multiplicative form.

In this paper, we consider images corrupted by impulsive noise (outliers). A large variety of methods are based on order statistic filters. Other methods use state-conditioned filters. We propose a different approach, consisting of two stages. First, outliers are detected based on the minimizer of a cost-function composed of an l/sub 1/ data-fidelity and an l/sub 2/ regularization term. The computation of this minimizer is speed and the detection of outliers reliable. Then, only outliers are removed and replaced by the median of the nearest neighboring regular (uncorrupted) data samples. This method is justified by some recent theoretical result. The numerical experiments show that our method is very efficient in a broad range of situations, including highly corrupted images.

We consider the reconstruction of images by minimizing regularized cost-functions. To accelerate the computation of the estimate, two forms of half-quadratic regularization, multiplicative and additive, are often used. The goal of this paper is to compare both theoretically and experimentally the efficiency of these two forms. We provide a theoretical and experimental analysis of the speed of convergence that they allow. We show that the multiplicative form gives rise to a better rate of convergence.

We present a theoretical study of the recovery of images x from noisy data y by minimizing a regularized cost-function F(x, y) = /spl Psi/(x, y) + /spl alpha//spl Phi/(x), where /spl Psi/ is a data-fidelity term, /spl Phi/ is a smooth regularization term and a>0 is a parameter. Generally, /spl Psi/ is a smooth function; only a few papers make an exception. Nonsmooth data-fidelity terms are avoided in image processing. In spite of this, our ambition is to catch essential features exhibited by the local minimizers of F in relation with the smoothness of /spl Psi/. We focus on /spl Psi/(x,y)=/spl Sigma//sub i//spl psi/(a/sub i//sup T/x-y/sub i/) where a/sub i//sup T/ are linear operators and /spl psi/ is C/sup m/-smooth on R{0}. We show that if /spl psi/'(0/sup -/)

We focus on image deconvolution and image reconstruction problems where a sought image is recovered from degraded observed data. The solution is defined to be the minimizer of an objective function combining a data-fidelity term and an edge-preserving, convex regularization term. Our objective is to speed up the calculation of the solution in a wide range of situations. To this end, we propose a method applying pertinent preconditioning to an adapted half-quadratic equivalent form of the objective function. The optimal solution is then found using an alternating minimization (AM) scheme. We focus specifically on Huber regularization. We exhibit the possibility of getting very fast calculations while preserving the edges in the solution. Preliminary numerical results are reported to illustrate the effectiveness of our method.

A block algorithm is presented to solve the joint blind channel identification and blind symbol estimation problem. It is based on a deterministic maximum likelihood (DML) method. A partial prior on the symbols is incorporated into the DML criterion in order to improve the estimation accuracy. We propose a test which permits to circumvent the local minima problem and which is pertinent for a large class of criteria. The structure of the block algorithm is well-suited for deriving recursive and adaptive versions. We prove that, in the noiseless case, the obtained recursive algorithm converges only towards the global minimum. Numerical results show that the prior on the symbols improves the accuracy of the estimators and brings robustness to the lack of channel diversity. At the same time, this method introduces fewer local minima than the use of a full prior.

Bayesian maximum a posteriori estimation (MAP) is a very popular way to recover unknown signals and images by using jointly observed data and priors formulated as a probability law. In a variational context, a MAP estimate minimizes an objective function where the priors are seen as a regularization or diffusion term. Independently of such interpretations, MAP estimates are implicit functions of the data and of the functions expressing the priors. This point of view enabled the author to exhibit analytical relations between prior functions and the features of the relevant estimates. These results entail important consequences and questions which are the subject of this paper. Namely, they reveal an essential gap between prior models and the way these are effectively involved in a MAP estimate. Hence the question about the rationale of MAP estimation. At the same time, they give precious indications about the hyperparameters and suggest how to construct estimators which indeed respect the priors.

We present an original method to reconstruct 3D magnetic resonance images of high resolution in the 3 directions of space from two anisotropic volumes. The resolution of each volume is degraded in a different direction. The reconstruction method is based on an optimization technique, the constraints being fidelity to the acquired data on the one hand, smoothness and edge preservation on the other hand. The interest of such a method is to significantly decrease the acquisition time of MR images, without degrading the spatial resolution.

We present a theoretical study of the reconstruction of locally homogeneous images from noisy data, by minimizing regularized objective functions. The latters are composed of a smooth data-fidelity term and a regularizer where a potential function (PF) is applied to each difference between neighbouring pixels. We confine our attention to the reconstruction of images which contain either weakly homogeneous regions (where the differences between neighbours are close to zero, but are non-zero) or strongly homogeneous regions (where these differences are zero). This distinction is suggested by the essentially different visual effect produced by the regions of these two types. We show that the recovery of regions of either type is determined by the smoothness at zero of the PF. So, a smooth at zero PF leads to minimizers-image estimates-which are only weakly homogeneous. In contrast, a nonsmooth at zero PF yields image estimates which contain large strongly homogeneous regions. In this case, the data domain is composed of volumes where the data yield minimizers are images which have the same strongly homogeneous regions. These theoretical results are eloquently illustrated by a numerical example.

This paper is concerned with the reconstruction of images (or signals) from incomplete, noisy data, obtained at the output of an observation system. The solution is defined in maximum a posteriori (MAP) sense and it appears as the global minimum of an energy function joining a convex data-fidelity term and a Markovian prior energy. The sought images are composed of nearly homogeneous zones separated by edges and the prior term accounts for this knowledge. This term combines general nonconvex potential functions (PFs) which are applied to the differences between neighboring pixels. The resultant MAP energy generally exhibits numerous local minima. Calculating its local minimum, placed in the vicinity of the maximum likelihood estimate, is inexpensive but the resultant estimate is usually disappointing. Optimization using simulated annealing is practical only in restricted situations. Several deterministic suboptimal techniques approach the global minimum of special MAP energies, employed in the field of image denoising, at a reasonable numerical cost. The latter techniques are not directly applicable to general observation systems, nor to general Markovian prior energies. This work is devoted to the generalization of one of them, the graduated nonconvexity (GNC) algorithm, in order to calculate nearly-optimal MAP solutions in a wide range of situations. In fact, GNC provides a solution by tracking a set of minima along a sequence of approximate energies, starting from a convex energy and progressing toward the original energy. In this paper, we develop a common method to derive efficient GNC-algorithms for the minimization of MAP energies which arise in the context of any observation system giving rise to a convex data-fidelity term and of Markov random field (MRF) energies involving any nonconvex and/or nonsmooth PFs. As a side-result, we propose how to construct pertinent initializations which allow us to obtain meaningful solutions using local minimization of these MAP energies. Two numerical experiments-an image deblurring and an emission tomography reconstruction-illustrate the performance of the proposed technique.

Reconstructing binary images is a problem which arises in various application fields. In this paper, this problem is considered in a regularization framework: the sought solution minimizes an objective function (a criterion). Criteria defined over the set of the binary images are nonconvex and there are no general methods permitting one to calculate the global minimum, while approximate solutions are very often of limited interest. On the other hand, general-purpose reconstruction techniques, based on convex criteria, yield continuous-valued smooth estimates which are far from binarity. We propose two methods which are based on convex criteria and introduce binarity only partially. More precisely, we construct objective functions whose minimizers are continuous-valued but have a quasi-binary shape. In other words, these estimates are composed essentially of binary-valued pixels while nonbinary pixels are rare. According to our approach, the construction of these objective functions is based on analytical considerations. The resultant methods are stable and numerically attractive.

We propose a method for the reconstruction of signals and images observed partially through a linear operator with a large support (e.g., a Fourier transform on a sparse set). This inverse problem is ill-posed and we resolve it by incorporating the prior information that the reconstructed objects are composed of smooth regions separated by sharp transitions. This feature is modeled by a piecewise Gaussian (PG) Markov random field (MRF), known also as the weak-string in one dimension and the weak-membrane in two dimensions. The reconstruction is defined as the maximum a posteriori estimate. The prerequisite for the use of such a prior is the success of the optimization stage. The posterior energy corresponding to a PG MRF is generally multimodal and its minimization is particularly problematic. In this context, general forms of simulated annealing rapidly become intractable when the observation operator extends over a large support. In this paper, global optimization is dealt with by extending the graduated nonconvexity (GNC) algorithm to ill-posed linear inverse problems. GNC has been pioneered by Blake and Zisserman in the field of image segmentation. The resulting algorithm is mathematically suboptimal but it is seen to be very efficient in practice. We show that the original GNC does not correctly apply to ill-posed problems. Our extension is based on a proper theoretical analysis, which provides further insight into the GNC. The performance of the proposed algorithm is corroborated by a synthetic example in the area of diffraction tomography.

We characterise the features of a regularised ML estimate, or equivalently a MAP estimate, of an image (or a signal) in relation with the form of the regularisation. The unknown image (signal) is observed through a linear operator and the data are corrupted by white Gaussian noise. Its reconstruction is regularised by the energy of a first-order Markov random field where the contributions of the transitions between adjacent neighbours are weighted using general potential functions (PFs). We exhibit the relationship between several features of the estimate and the form of the PF. Points of interest are the edge recovery, the stability of the estimator, the estimation of locally constant regions, the bias over large transitions, the resolution. The exposed theoretical considerations are corroborated by numerical simulations.

We propose a method for the parameter selection for a Bayesian reconstruction of 1D or 2D signals, constituted by locally homogeneous regions, from incomplete and noisy projection data. A piecewise Gaussian Markov model (PG MM), defined as a sum of truncated quadratic potential functions, is used to regularise the reconstruction, which is otherwise ill-posed. This model is called the weak string in 1D and the weak membrane in 2D. The posterior energy is highly non-convex and the MAP estimator is piecewise continuous; the model parameters play a particularly decisive role. The resolution of the reconstruction-the finest recoverable features-is determined jointly by the parameters and the observation model. On the other hand, we propose a method for the determination of the parameters in order to reach, or at least to approach as closely as possible, a desired resolution. This model needs the evaluation of several posterior edge detection thresholds.

We propose a method for the reconstruction of an image, only partially observed through a linear integral operator. As such an inverse problem is ill-posed, prior information must be introduced. We consider the case of a compound Markov random field with a non-interacting line process. In order to maximise the posterior likelihood function, we propose an extension of the graduated non convexity principle pioneered by Blake and Zisserman (1987) which allows its use for ill-posed linear inverse problems. We discuss the role of the observation scale and some aspects of the implemented algorithm. Finally, we present an application of the method to a diffraction tomography imaging problem.>

In computed imaging an object which is not directly observable has to be imaged from some deficient projection data; the authors suppose that the image-data relation is linearised. Recovering the image is an ill-posed inverse problem and requires the use of prior information. The currently used quadratic regularisation is not satisfactory. The authors propose a class of reconstruction algorithms permitting the introduction of various scene features as priors for an MAP estimation, by means of Markov random fields with implicit line processes. The resulting optimisation problem, numerically not feasible by stochastic algorithms, is solved in a sub-optimal but very efficient manner with graduated non-convexity algorithms. The proposed technique is directly applicable to any ill-posed inverse problem whose log-likelihood is convex.>