The estimation of parameters in a linear model is considered under the hypothesis that the noise, with finite second order statistics, can be represented by random coefficients in a given deterministic basis. An extended underdetermined design matrix is then formed, and the estimator of the extended parameters with minimum l(1) norm is computed. It is proved that, if the noise variance is larger than a threshold, which depends on the unknown parameters and on the extended design matrix, then the proposed estimator of the original parameters dominates the least-squares estimator, in the sense of the mean square error. A small simulation illustrates its behavior. Moreover it is shown experimentally that it can be convenient, even if the design matrix is not known but only an estimate can be used. Furthermore the noise basis can eventually be used to introduce some prior information in the estimation process. These points are illustrated in a simulation by using the proposed estimator for solving a difficult inverse ill-posed problem, related to the complex moments of an atomic complex measure. (C) 2016 Elsevier Inc. All rights reserved.

A common problem, arising in many different applied contexts, consists in estimating the number of exponentially damped sinusoids whose weighted sum best fits a finite set of noisy data and in estimating their parameters. Many different methods exist to this purpose. The best of them are based on approximate Maximum Likelihood estimators, assuming to know the number of damped sinusoids, which can then be estimated by an order selection procedure. As the problem can be severely ill posed, a stochastic perturbation method is proposed which provides better results than Maximum Likelihood based methods when the signal-to-noise ratio is low. The method depends on some hyperparameters which turn out to be essentially independent of the application. Therefore they can be fixed once and for all, giving rise to a black box method.

Universality properties of the distribution of the generalized eigenvalues of a pencil of random Hankel matrices, arising in the solution of the exponential interpolation problem of a complex discrete stationary process, are proved under the assumption that every finite set of random variables of the process have a multivariate spherical distribution. An integral representation of the condensed density of the generalized eigenvalues is also derived. The asymptotic behavior of this function turns out to depend only on stationarity and not on the specific distribution of the process.

An atomic random complex measure defined on the unit disk with normally distributed moments is considered. An approximation to the distribution of the zeros of its Cauchy transform is computed. Implications of this result for solving several moment problems are discussed.

It is shown that the density of the ratio of two random variables with the same variance and joint Gaussian density satisfies a nonstationary diffusion equation. Implications of this result for adaptive kernel density estimation of the condensed density of the generalized eigenvalues of a random matrix pencil useful for solving the exponential analysis problem are discussed.

Pencils of matrices whose elements have a joint noncentral Gaussian distribution with nonidentical covariance are considered. An approximation to the distribution of the squared modulus of their determinant is computed which allows to get a closed form approximation of the condensed density of the generalized eigenvalues of the pencils. Implications of this result for solving several moments problems are discussed and some numerical examples are provided.

Bartlett's decomposition provides the distributional properties of the elements of the Cholesky factor of $A=G^TG$ where the elements of $G$ are i.i.d. standard Gaussian random variables. In this paper the most general case where the elements of $G$ have a joint multivariate Gaussian density is considered.

Many real life problems can he reduced to the solution of a complex exponentials approximation problem which is usually ill-posed. Recently a new transform for solving this problem, formulated as a specific moments problem in the plane, has been proposed in a theoretical framework. In this work some computational issues are addressed to make this new tool useful in practice. An algorithm is developed and used to solve a Nuclear Magnetic Resonance spectrometry problem, two time series interpolation and extrapolation problems and a shape from moments problem.

Many real life problems can be represented by an ordered sequence of digital images. At a given pixel a specific time course is observed which is morphologically related to the time courses at neighbor pixels. Useful information can be usually extracted from a set of such observations if we are able to classify pixels in groups, according to some features of interest for the final user. Moreover parameters with a physical meaning can be extracted from the time courses. In a continuous setting we can formalize the problem by assuming to observe a noisy version of a positive real function defined on a bounded set T ×\Omega ‚ R×R2, parameterized by a vector of unknown functions defined on R2 with discontinuities along regular curves in \Omega which separate regions with different features. Suitable regularity conditions on the parameters are also assumed in order to take into account the physical constraints. The problem consists in estimating the parameter functions, segmenting \Omega in subsets with regular boundaries and assigning to each subset a label according to the values that the parameters assume on the subset. A global model is proposed which allows to address all of the above subproblems in the same framework. A variational approach is then adopted to compute the solution and an algorithm has been developed. Some numerical results obtained by using the proposed method to solve a dynamic Magnetic Resonance imaging problem are reported.

The problem of estimating a complex measure made up by a linear combination of Dirac distributions centered on points of the complex plane from a finite number of its complex moments affected by additive i.i.d. Gaussian noise is considered. A random measure is defined whose expectation approximates the unknown measure under suitable conditions. An estimator of the approximating measure is then proposed as well as a new discrete transform of the noisy moments that allows computing in estimate of the Unknown measure. A small simulation study is also performed to experimentally check, the goodness of the approximations.

A contrast enhanced dynamic Magnetic Resonance clinical exam produces a set of images displaying over time the concentration of a contrast agent in the blood stream of an organ. The portion of tissue represented by each pixel can be classified as normal, benign or malignant tumoral, according to the qualitative behavior of the contrast agent uptake associated to it. These responses can be considered as the noisy output of a pharmacokinetic distributed model whose parameters have an intrinsic diagnostic importance. Fundamental MR imaging characteristics force a compromise between the noise level and the spatial and temporal resolution of the dynamic sequence. This makes the identification of the pharmacokinetic parameters and the classification problem difficult especially if short computation time is required by physicians. In this paper, a fast method is proposed to solve simultaneously the parameter identification and the classification problems. The complexity of the algorithm is $O(N\cdot n_p)$ flops where $N$ is the number of pixels and $n_p$ is the number of pharmacokinetic parameters per pixel. A family of functions for the parameters and the classification labels is defined. Each function is the weighted sum, with unknown weights, of a coherence-to-data term, several terms which enforce a roughness penalty on the model parameters, a term measuring the distance between the parameters in each pixel and the expected parameters for each class and a term which enforces a roughness penalty on the classification labels. A constrained optimization problem is solved to choose a member of the family, i.e. to estimate the unknown weights, and to minimize it in order to jointly estimate the parameters and the classification labels. A tuning procedure have been also devised, which makes the algorithm fully automated. The performances of the method are illustrated on real data sets.

In Fricke-agarose gels, an accurate determination of the spatial dose distribution is hindered by the diffusion of ferric ions. In this work, a model was developed to describe the diffusion process within gel samples of finite length and, thus, permit the reconstruction of the initial spatial distribution of the ferric ions. The temporal evolution of the ion concentration as a function of the initial concentration is derived by solving Fick's second law of diffusion in two dimensions with boundary reflections. The model was applied to magnetic resonance imaging data acquired at high spatial resolution (0.3 mm) and was found to describe accurately the observed diffusion effects.

In the application of Pad\'{e} methods to signal processing a basic problem is to take into account the effect of measurement noise on the computed approximants. Qualitative deterministic noise models have been proposed which are consistent with experimental results. In this paper the Pad\'{e} approximants to the $Z$-transform of a complex Gaussian discrete white noise process are considered. Properties of the condensed density of the Pad\'{e} poles such as circular symmetry, asymptotic concentration on the unit circle and independence on the noise variance are proved. An analytic model of the condensed density of the Pad\'{e} poles for all orders of the approximants is also computed. Some Montecarlo simulations are provided.

A new fast iterative algorithm is proposed to denoise images affected by Gaussian additive noise, by solving a constrained optimization problem. A family of functionals parameterized by a few hyperparameters already proposed in the literature to solve denoising problems is considered. The algorithm estimates jointly the image and the hyperparameters, therefore providing an automatic method. Each member of the family is made up of a coherence with the data term and a term enforcing a roughness penalty and preserving jump discontinuities. Assuming we know the noise variance, an adequacy constraint is also considered. The algorithm computes a member of the family and a minimizer of it which satisfies the constraint. A convergence proof is provided. We then consider a heuristic version of the algorithm which gives restorations of comparable quality whose computational complexity is a linear function of the pixels number. Experimental results on synthetic and real data are presented. Moreover, numerical comparisons with several fast denoising methods are provided.

H-1 magnetic resonance studies on MCF-7 and HeLa cells were undertaken to reveal differences in lipid and lipid metabolite signals during the growth in culture. High intensity mobile lipid (ML) signals were found during the first days in culture, while afterwards the same signals declined and started increasing again at confluence and at late confluence. At the same time, signals from the lipid metabolite phosphocholine decreased in intensity while signals from glycerophosphocholine in MCF-7 and from choline in HeLa increased as cells approached confluence. Spectral parameters from actively proliferating and non-proliferating cells were used to classify cells with respect to the proliferative conditions by means of a multivariate statistical analysis. Furthermore, it was shown that polyunsaturation of mobile lipid chains was lower in the confluent group with respect to the actively proliferating cells. The examination of spectra from suspensions of MCF-7 spheroids with diameter smaller than 500 mum suggests that cells in spheroids are in condition of lipid metabolism similar to that of confluent cultured cells.

Dynamic magnetic resonance imaging with contrast agent is a very promising technique for mammography. A temporal sequence of magnetic resonance images of the same slice are acquired following the injection of a contrast agent in the blood stream. The image intensity depends on the local concentration of the contrast agent so that tissue perfusion can be studied using the image sequence. A new statistical method of analyzing such sequences is presented. The method is developed within the Bayesian framework. A specific statistical model is used to take into account image degradation. In addition, a suitable Markov random field allows us to model some relevant ``a priori'' information on the quantities to be estimated. Inference is based on simulations from the posterior distribution obtained by means of Markov chain algorithms. The issue of hyper-parameter estimation is also addressed. Image classification is also performed by means of a new Bayesian method. Some results obtained from sequences of dynamic magnetic resonance images of human breasts will be illustrated.