A method for the numerical inversion of the Laplace transform of a continuous positive function $f(t)$ is proposed. Random matrices distributed according to a Gibbs law whose energy $V(x)$ is a function of $f(t)$ are considered as well as random polynomials orthogonal with respect to $w(x)=e^{-V(x)}$. The equation relating $w(x)$ to the reproducing kernel and to the condensed density of the roots of the random orthogonal polynomials is exploited. Basic results from the theories of orthogonal polynomials, random matrices and random polynomials are revisited in order to provide a unified and almost self--contained context. The qualitative behavior of the solutions provided by the proposed method is illustrated by numerical examples and discussed by using logarithmic potentials with external fields that give insight into the asymptotic behavior of the condensed density when the number of data points goes to infinity.

This paper is concerned with improving the performance of certain Markov chain algorithms for Monte Carlo simulation. We propose a new algorithm for simulating from multivariate Gaussian densities. This algorithm combines ideas from coupled Markov chain methods and from an existing algorithm based only on over-relaxation. The rate of convergence of the proposed and existing algorithms can be measured in terms of the square of the spectral radius of certain matrices. We present examples in which the proposed algorithm converges faster than the existing algorithm and the Gibbs sampler. We also derive an expression for the asymptotic variance of any linear combination of the variables simulated by the proposed algorithm. We outline how the proposed algorithm can be extended to non-Gaussian densities.

We study general over-relaxation Markov Chain Monte Carlo samplers for multivariate Gaussian densities. We provide conditions for convergence based on the spectral radius of the transition matrix and on detailed balance. We illustrate these algorithms using an image analysis example.

In this paper we study several different methods both deterministic and stochastic to solve the Nuclear Magnetic Resonance (NMR) relaxometry problem. This problem is strongly related to finding a non-negative function given a finite number of values of its Laplace transform embedded in noise. Some of the methods considered here are new. We also propose a procedure which exploits and combines the main features of these methods. To show the performances of this procedure, some results of applying it to synthetic data are finally reported.

An inverse diffusion problem that appears in Magnetic Resonance dosimetry is studied. The problem is shown to be equivalent to a deconvolution problem with a known kernel. To cope with the singularity of the kernel, nonlinear regularization functionals are considered which can provide regular solutions, reproduce steep gradients and impose positivity constraints. A fast deterministic algorithm for solving the involved non-convex minimization problem is used. Accurate restorations on real 256×256 images are obtained by the algorithm in a few minutes on a 266-MHz PC that allow to precisely quantitate the relative absorbed dose.

The problem of reconstructing a piecewise constant function from a finite number of its Fourier coefficients perturbed by noise is considered. A reconstruction method, based on the computation of the Padè approximants to the Z-transform of the sequence of the noisy Fourier coefficients is proposed. The method is based on the remark that the distribution of the poles of the Padè approximants shows, asymptotically, clusters in the complex plane which allow the identification of the discontinuities of the function. It turns out that the Z-transform is a multiple-valued function and the location of the clusters corresponds to the branch points of such a function. By using this property of the Padè poles, a very effective reconstruction method can be developed. Some numerical experiments are presented to show the feasibility of the method.

The authors present a novel method for processing T1-weighted images acquired with Inversion-Recovery (IR) sequence. The method, developed within the Bayesian framework, takes into account a priori knowledge about the spatial regularity of the parameters to be estimated. Inference is drawn by means of Markov Chains Monte Carlo algorithms. The method has been applied to the processing of IR images from irradiated Fricke-agarose gels, proposed in the past as relative dosimeter to verify radiotherapeutic treatment planning systems. Comparison with results obtained from a standard approach shows that signal-to noise ratio (SNR) is strongly enhanced when the estimation of the longitudinal relaxation rate (R1) is performed with the newly proposed statistical approach. Furthermore, the method allows the use of more complex models of the signal. Finally, an appreciable reduction of total acquisition time can be obtained due to the possibility of using a reduced number of images. The method can also be applied to T1 mapping of other systems.

The noisy trigonometric moment problem for a finite linear combination of box functions is considered, and a research program, possibly leading to a superresolving method, is outlined and some initial steps are performed. The method is based on the remark that the poles of the Padè approximant to the Z-transform of the noiseless moments show, asymptotically, a regular pattern in the complex plane. The pattern can be described by a set of arcs, connecting points on the unit circle, and a pole density function defined on the arcs. When a moderate noise affects the moments, more arcs are needed to describe the pole pattern, but the noiseless pattern is slightly deformed, still allowing its identification. When this identification is possible, a very effective noise filter and moment extrapolator should be easily constructed. In this paper only some preliminary steps of the above research program are performed. Specifically, the case of one box function is considered. A method for computing the pole patterns, based on the solution of a singular integral equation of Cauchy type, is developed. The method is general enough to be used also for several box functions. Some numerical results, showing the feasibility of the program, are discussed.

In this paper, the modal analysis model, made up by a linear combination of complex exponential functions, is considered. Padè approximants to the Z-transform of a noisy sample are then considered, and the asymptotic locus of their poles is studied. It turns out that this locus is strongly related to the complex exponentials of the model. By exploiting these properties, powerful methods for estimating the model parameters can be devised, which have both denoising and super-resolution capabilities.