1,721,075 research outputs found
Regularization Methods for the Solution of Inverse Problems in Solar X-ray and Imaging Spectroscopy
No full textAstronomical practice often requires addressing remote sensing problems, whereby the radiation emitted by a source far in the sky and measured through ‘ad hoc’ observational techniques, contains very indirect information on the physical processat the basis of the emission. The main difficulties in this investigations rely on the poor quality of the measurements and on the ill-posedness of the mathematical model describing the relation between the measured data and the target functions. In the present paper we consider a set of problems in solar physics in the framework of the NASA Ramaty High Energy Solar Spectroscopic Imager (RHESSI) mission. The data analysis activity is essentially based on the regularization theory for ill-posed inverse problems and a review of the main regularization methods applied in this analysis is given. Furthermore, we describe the main results of these applications, in the case ofboth synthetic data and real observations recorded by RHESSI
A deconvolution algorithm for imaging problems from Fourier data
No full textIn this paper we address the problem of reconstructing a two-dimensional
image starting from the knowledge on nonuniform samples of its Fourier Transform. Such inverse problem has a natural semidiscrete formulation, that is analyzed together with its fully discrete counterpart. In particular, the image restoration problem in this case can be reformulated as the minimization of the data discrepancy under nonnegativity constraints, possibly with the addition of a further equality constraint on the total flux of the image. Moreover, we show that such problem is equivalent to a deconvolution in the image space, that represents a key property allowing the desing of a computationally efficient algorithm based on Fast Fourier Transforms to address its solution. Our proposal to compute a regularized solution in the discrete case involves a gradient projection method, with an adaptive choice for the steplength parameter that improves the convergence rate. A numerical experimentation on simulated data from the NASA RHESSI mission is also performed
Space-D: a software for nonnegative image deconvolution from sparse Fourier data
No full textThis code deals with image restoration problems where the data are nonuniform samples of the Fourier transform of the unknown object. 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
Inverse problems in machine learning: an application to brain activity interpretation
Full text linkIn a typical machine learning problem one has to build a model from a finite training set which is able to generalize the properties characterizing the examples of the training set to new examples. The model has to reflect as much as possible the set of trainingexamples but, especially in real-world problems in which the data are often corrupted by different sources of noise, it has to avoid a too strict dependence on the training examples themselves. Recent studies on the relationship between this kind of learning problem and the regularization theory for ill-posed inverse problems have given rise to new regularized learning algorithms. In this paper we recall some of these learning methods and we propose an accelerated version of the classical Landweber iterative scheme which results particularly efficient from thecomputational viewpoint. Finally, we compare the performances of these methods with the classical Support Vector Machines learning algorithm on a real-world experiment concerning brain activity interpretation through the analysis of functional magnetic resonance imaging data
A practical use of regularization for supervised learning with kernel methods
No full textIn several supervised learning applications, it happens that reconstruction methods have to be applied repeatedly before being able to achieve the final solution. In these situations, the availability of learning algorithms able to provide effective predictors in a very short time may lead to remarkable improvements in the overall computational requirement. Here we consider the kernel ridge regression problem and we look for predictors given by a linear combination of kernel functions plus a constant term, showing that an effective solution can be obtained very fastly by applying specific regularization algorithms directly to the linear system arising from the Empirical Risk Minimization problem
A novel gradient projection approach for Fourier-based image restoration
No full textThis work deals with the ill-posed inverse problem of reconstructing a two-dimensional image of an unknownobject starting from sparse and nonuniform measurements of its Fourier Transform. In particular, if we consider a prioriinformation about the target image (e.g., the nonnegativity of the pixels), this inverse problem can be reformulated as aconstrained optimization problem, in which the stationary points of the objective function can be viewed as the solutionsof a deconvolution problem with a suitable kernel. We propose a fast and effective gradient-projection iterative algorithmto provide regularized solutions of such a deconvolution problem by early stopping the iterations. Preliminary results on areal-world application in astronomy are presented
New convergence results for the inexact variable metric forward-backward method
No full textForward-backward methods are valid tools to solve a variety of optimization problems where the objective function is the sum of a smooth, possibly nonconvex term plus a convex, possibly nonsmooth function. The corresponding iteration is built on two main ingredients: the computation of the gradient of the smooth part and the evaluation of the proximity (or resolvent) operator associated to the convex term. One of the main difficulties, from both implementation and theoretical point of view, arises when the proximity operator is computed in an inexact way. The aim of this paper is to provide new convergence results about forward-backward methods with inexact computation of the proximity operator, under the assumption that the objective function satisfies the Kurdyka-Lojasiewicz property. In particular, we adopt an inexactness criterion which can be implemented in practice, while preserving the main theoretical properties of the proximity operator. The main result is the proof of the convergence of the iterates generated by the forward-backward algorithm in [1] to a stationary point. Convergence rate estimates are also provided. At the best of our knowledge, there exists no other inexact forward-backward algorithm with proved convergence in the nonconvex case and equipped with an explicit procedure to inexactly compute the proximity operator
Image Reconstruction from Nonuniform Fourier Data
No full textIn many scientific frameworks (e.g., radio and high energy astronomy, medical imaging) the data at one's disposal are encoded in the form of sparse and nonuniform samples of the desired unknown object's Fourier Transform. From the numerical point of view, reconstructing an image from sparse Fourier data is an ill-posed inverse problem in the sense of Hadamard, since there are infinite possible images which match the available Fourier samples. Moreover, the irregular distribution of such samples in the frequency space makes the use of any FFT-based reconstruction algorithm impossible, unless an interpolation and resampling (also known as gridding) procedure is previously applied to the original data. However, if the distribution of the Fourier samples in the frequency space is particularly irregular and/or the signal-to-noise ratio is poor, then the gridding step might either distort the information enclosed in the data or amplify the noise level on the re-sampled data with the result of artefacts formation and undesirable effects in the corresponding reconstructed image.This talk will deal with a different approach to the reconstruction of an image from a nonuniform sampling of its Fourier transform which acts straightly on the data without interpolation and re-sampling operations, exploiting in this way the real nature of the data themselves. In particular, we show that the minimization of the data discrepancy is equivalent to a deconvolution problem with a suitable kernel and we address its solution by means of a gradient projection method with an adaptive steplength parameter, chosen via an alternation of the two Barzilai–Borwein rules. Since the objective function involves a convolution operator, the algorithm can be effectively implemented exploiting the Fast Fourier Transform. The proposed algorithm is tested in a real-world problem, namely the restoration of X-ray images of the Sun during the solar flares by means of the datasets provided by the NASA RHESSI satellite
A new steplength selection for scaled gradient methods with application to image deblurring
No full textGradient methods are frequently used in large scale image deblurring problems since they avoid the onerous computation of the Hessian matrix of the objective function. Second order information is typically sought by a clever choice of the steplength parameter defining the descent direction, as in the case of the well-known Barzilai and Borwein rules. In a recent paper, a strategy for the steplength selection approximating the inverse of some eigenvalues of the Hessian matrix has been proposed for gradient methods applied to unconstrained minimization problems. In the quadratic case, this approach is based on a Lanczos process applied every m iterations to the matrix of the gradients computed in the previous m iterations, but the idea can be extended to a general objective function. In this paper we extend this rule to the case of scaled gradient projection methods applied
to constrained minimization problems, and we test the effectiveness of the proposed strategy in image deblurring problems in both the presence and the absence of an
explicit edge-preserving regularization term
- …
