Marco Prato

Deducing Electron Properties from Hard X-Ray Observations

X-radiation from energetic electrons is the prime diagnostic of flare-accelerated electrons. The observed X-ray flux (and polarization state) is fundamentally a convolution of the cross-section for the hard X-ray emission process(es) in question with the electron distribution function, which is in turn a function of energy, direction, spatial location and time. To address the problems of particle propagation and acceleration one needs to infer as much information as possible on this electron distribution function, through a deconvolution of this fundamental relationship. This review presents recent progress toward this goal using spectroscopic, imaging and polarization measurements, primarily from the Reuven Ramaty High Energy Solar Spectroscopic Imager (RHESSI). Previous conclusions regarding the energy, angular (pitch angle) and spatial distributions of energetic electrons in solar flares are critically reviewed. We discuss the role and the observational evidence of several radiation processes: free-free electron-ion, free-free electron-electron, free-bound electron-ion, photoelectric absorption and Compton backscatter (albedo), using both spectroscopic and imaging techniques. This unprecedented quality of data allows for the first time inference of the angular distributions of the X-ray-emitting electrons and improved model-independent inference of electron energy spectra and emission measures of thermal plasma. Moreover, imaging spectroscopy has revealed hitherto unknown details of solar flare morphology and detailed spectroscopy of coronal, footpoint and extended sources in flaring regions. Additional attempts to measure hard X-ray polarization were not sufficient to put constraints on the degree of anisotropy of electrons, but point to the importance of obtaining good quality polarization data in the future.

Efficient deconvolution methods for astronomical imaging: algorithms and IDL-GPU codes

Context. The Richardson-Lucy method is the most popular deconvolution method in astronomy because it preserves the number of counts and the non-negativity of the original object. Regularization is, in general, obtained by an early stopping of Richardson-Lucy iterations. In the case of point-wise objects such as binaries or open star clusters, iterations can be pushed to convergence. However, it is well-known that Richardson-Lucy is an inefficient method. In most cases and, in particular, for low noise levels, acceptable solutions are obtained at the cost of hundreds or thousands of iterations, thus several approaches to accelerating Richardson-Lucy have been proposed. They are mainly based on Richardson-Lucy being a scaled gradient method for the minimization of the Kullback-Leibler divergence, or Csiszar I-divergence, which represents the data-fidelity function in the case of Poisson noise. In this framework, a line search along the descent direction is considered for reducing the number of iterations. Aims. A general optimization method, referred to as the scaled gradient projection method, has been proposed for the constrained minimization of continuously differentiable convex functions. It is applicable to the non-negative minimization of the Kullback-Leibler divergence. If the scaling suggested by Richardson-Lucy is used in this method, then it provides a considerable increase in the efficiency of Richardson-Lucy. Therefore the aim of this paper is to apply the scaled gradient projection method to a number of imaging problems in astronomy such as single image deconvolution, multiple image deconvolution, and boundary effect correction. Methods. Deconvolution methods are proposed by applying the scaled gradient projection method to the minimization of the Kullback-Leibler divergence for the imaging problems mentioned above and the corresponding algorithms are derived and implemented in interactive data language. For all the algorithms, several stopping rules are introduced, including one based on a recently proposed discrepancy principle for Poisson data. To attempt to achieve a further increase in efficiency, we also consider an implementation on graphic processing units. Results. The proposed algorithms are tested on simulated images. The acceleration of scaled gradient projection methods achieved with respect to the corresponding Richardson-Lucy methods strongly depends on both the problem and the specific object to be reconstructed, and in our simulations the improvement achieved ranges from about a factor of 4 to more than 30. Moreover, significant accelerations of up to two orders of magnitude have been observed between the serial and parallel implementations of the algorithms. The codes are available upon request.

Anisotropic Bremsstrahlung Emission and the Form of Regularized Electron Flux Spectra in Solar Flares

The cross section for bremsstrahlung photon emission in solar flares is, in general, a function of the angle θ between the incoming electron and the outgoing photon directions. Thus the electron spectrum required to produce a given photon spectrum is a function of this angle, which is related to the position of the flare on the solar disk and the direction(s) of the precollision electrons relative to the local solar vertical. We compare mean electron flux spectra for the flare of 2002 August 21 using cross sections for parameterized ranges of the angle θ. Implications for the shape of the mean source electron spectrum and for the injected power in nonthermal electrons are discussed.

A convergent blind deconvolution method for post-adaptive-optics astronomical imaging

In this paper, we propose a blind deconvolution method which applies to data perturbed by Poisson noise. The objective function is a generalized Kullback–Leibler (KL) divergence, depending on both the unknown object and unknown point spread function (PSF), without the addition of regularization terms; constrained minimization, with suitable convex constraints on both unknowns, is considered. The problem is non-convex and we propose to solve it by means of an inexact alternating minimization method, whose global convergence to stationary points of the objective function has been recently proved in a general setting. The method is iterative and each iteration, also called outer iteration, consists of alternating an update of the object and the PSF by means of a fixed number of iterations, also called inner iterations, of the scaled gradient projection (SGP) method. Therefore, the method is similar to other proposed methods based on the Richardson–Lucy (RL) algorithm, with SGP replacing RL. The use of SGP has two advantages: first, it allows one to prove global convergence of the blind method; secondly, it allows the introduction of different constraints on the object and the PSF. The specific constraint on the PSF, besides non-negativity and normalization, is an upper bound derived from the so-called Strehl ratio (SR), which is the ratio between the peak value of an aberrated versus a perfect wavefront. Therefore, a typical application, but not a unique one, is to the imaging of modern telescopes equipped with adaptive optics systems for the partial correction of the aberrations due to atmospheric turbulence. In the paper, we describe in detail the algorithm and we recall the results leading to its convergence. Moreover, we illustrate its effectiveness by means of numerical experiments whose results indicate that the method, pushed to convergence, is very promising in the reconstruction of non-dense stellar clusters. The case of more complex astronomical targets is also considered, but in this case regularization by early stopping of the outer iterations is required. However, the proposed method, based on SGP, allows generalization to the case of differentiable regularization terms added to the KL divergence, even if this generalization is outside the scope of this paper.

Nonnegative image reconstruction from sparse Fourier data: a new deconvolution algorithm

This paper deals with image restoration problems where the data are nonuniform samples of the Fourier transform of the unknown object. We study the inverse problem in both semidiscrete and fully discrete formulations, and our analysis leads to an optimization problem involving the minimization of the data discrepancy under nonnegativity constraints. In particular, we show that such a problem is equivalent to a deconvolution problem in the image space. We propose a practical algorithm, based on the gradient projection method, to compute a regularized solution in the discrete case. The key point in our deconvolution-based approach is that the fast Fourier transform can be employed in the algorithm implementation without the need of preprocessing the data. A numerical experimentation on simulated and real data from the NASA RHESSI mission is also performed.

Variable Metric Inexact Line-Search-Based Methods for Nonsmooth Optimization

We develop a new proximal-gradient method for minimizing the sum of a differentiable, possibly nonconvex, function plus a convex, possibly nondifferentiable, function. The key features of the proposed method are the definition of a suitable descent direction, based on the proximal operator associated to the convex part of the objective function, and an Armijo-like rule to determine the stepsize along this direction ensuring the sufficient decrease of the objective function. In this frame, we especially address the possibility of adopting a metric which may change at each iteration and an inexact computation of the proximal point defining the descent direction. For the more general nonconvex case, we prove that all limit points of the iterates sequence are stationary, while for convex objective functions we prove the convergence of the whole sequence to a minimizer, under the assumption that a minimizer exists. In the latter case, assuming also that the gradient of the smooth part of the objective function...

Electron-Electron Bremsstrahlung Emission and the Inference of Electron Flux Spectra in Solar Flares

Although both electron-ion and electron-electron bremsstrahlung contribute to the hard X-ray emission from solar flares, the latter is normally ignored. Such an omission is not justified at electron (and photon) energies above ~300 keV, and inclusion of the additional electron-electron bremsstrahlung in general makes the electron spectrum required to produce a given hard X-ray spectrum steeper at high energies. Unlike electron-ion bremsstrahlung, electron-electron bremsstrahlung cannot produce photons of all energies up to the electron energy involved. The maximum possible photon energy depends on the angle between the direction of the emitting electron and the emitted photon, and this suggests a diagnostic for an upper cutoff energy and/or for the degree of beaming of the accelerated electrons. We analyze the large event of 2005 January 17 and show that the upward break around 400 keV in the observed hard X-ray spectrum is naturally accounted for by the inclusion of electron-electron bremsstrahlung. Indeed, the mean source electron spectrum recovered through a regularized inversion of the hard X-ray spectrum, using a cross section that includes both electron-ion and electron-electron terms, has a relatively constant spectral index δ over the range from electron kinetic energy E = 200 keV to E = 1 MeV. Such a spectrum is indicative of an acceleration mechanism without a characteristic energy or corresponding scale.

A regularization algorithm for decoding perceptual temporal profiles from fMRI data

In several biomedical fields, researchers are faced with regression problems that can be stated as Statistical Learning problems. One example is given by decoding brain states from functional magnetic resonance imaging (fMRI) data. Recently, it has been shown that the general Statistical Learning problem can be restated as a linear inverse problem. Hence, new algorithms were proposed to solve this inverse problem in the context of Reproducing Kernel Hilbert Spaces. In this paper, we detail one iterative learning algorithm belonging to this class, called ν-method, and test its effectiveness in a between-subjects regression framework. Specifically, our goal was to predict the perceived pain intensity based on fMRI signals, during an experimental model of acute prolonged noxious stimulation. We found that, using a linear kernel, the psychophysical time profile was well reconstructed, while pain intensity was in some cases significantly over/underestimated. No substantial differences in terms of accuracy were found between the proposed approach and one of the state-of-the-art learning methods, the Support Vector Machines. Nonetheless, adopting the ν-method yielded a significant reduction in computational time, an advantage that became more evident when a relevant feature selection procedure was implemented. The ν-method can be easily extended and included in typical approaches for binary or multiple classification problems, and therefore it seems well-suited to build effective brain activity estimators.

New convergence results for the scaled gradient projection method

The aim of this paper is to deepen the convergence analysis of the scaled gradient projection (SGP) method, proposed by Bonettini et al. in a recent paper for constrained smooth optimization. The main feature of SGP is the presence of a variable scaling matrix multiplying the gradient, which may change at each iteration. In the last few years, an extensive numerical experimentation showed that SGP equipped with a suitable choice of the scaling matrix is a very effective tool for solving large scale variational problems arising in image and signal processing. In spite of the very reliable numerical results observed, only a weak, though very general, convergence theorem is provided, establishing that any limit point of the sequence generated by SGP is stationary. Here, under the only assumption that the objective function is convex and that a solution exists, we prove that the sequence generated by SGP converges to a minimum point, if the scaling matrices sequence satisfies a simple and implementable condition. Moreover, assuming that the gradient of the objective function is Lipschitz continuous, we are also able to prove the O(1/k) convergence rate with respect to the objective function values. Finally, we present the results of a numerical experience on some relevant image restoration problems, showing that the proposed scaling matrix selection rule performs well also from the computational point of view.

A New Semiblind Deconvolution Approach for Fourier-Based Image Restoration: An Application in Astronomy

The aim of this paper is to develop a new optimization algorithm for the restoration of an image starting from samples of its Fourier transform, when only partial information about the data frequencies is provided. The corresponding constrained optimization problem is approached with a cyclic block alternating scheme, in which projected gradient methods are used to find a regularized solution. Our algorithm is then applied to the imaging of high-energy radiation emitted during a solar flare through the analysis of the photon counts collected by the NASA Reuven Ramaty High Energy Solar Spectroscopic Imager satellite. Numerical experiments on simulated data show that, in both the presence and absence of statistical noise, the proposed approach provides some improvements in the reconstructions.