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Problemi ai minimi quadrati separabili e loro applicazione alla blind deconvolution
No full textIn questo lavoro si analizzano problemi ai minimi quadrati separabili, in cui le variabili possono essere divise in un sottoinsieme lineare e uno non lineare. Questo tipo di problemi sono stati trattati in letteratura con due diversi approcci: il metodo di Proiezione delle Variabili e quello della Minimizzazione Alternata. L'idea su cui entrambi i metodi si basano è di sfruttare la separabilità delle variabili per modificare il problema originale ed ottenere una nuova formulazione, più semplice e di dimensioni minori della prima. E' presentato un dettagliato confronto tra i due metodi, sottolineando i rispettivi pro e contro e per quali applicazioni la scelta di un approccio è preferibile all'altro. Un punto cruciale per entrambi i metodi, che rappresenta la loro maggiore differenza, è nel calcolo della matrice Jacobiana.
In particolare, ci siamo concentrati su problemi ai minimi quadrati separabili vincolati, dando particolare attenzione al caso di vincoli di non-negatività sulla variabile lineare. Questa scelta è motivata dall'applicazione all'ambito dell'imaging, dove le componenti del vettore delle incognite sono pixel, che si riferiscono ad una quantità fisica quindi devo essere reali e non-negativi. Come contributo originale, si è estesa la Proiezione delle Variabili al caso non-negativo ed si è proposta una nuova formula per il calcolo della matrice Jacobiana.
Per validare i risultati teorici, è proposta una applicazione alla blind deconvolution. In questi problemi, sono fornite misure di radiazioni (microonde, raggi-X, ultrasuoni, ecc.), emesse o trasmesse da un oggetto incognito (tipicamente un segnale o un'immagine bidimensionale), insieme a parziali informazioni (o nessuna) sull'apparato che fornisce i dati. Si devono quindi risolvere il problema non banale della ricostruzione dell'oggetto e contemporaneamente del modello che descrive la formazione dei dati. Una strategia diffusa per affrontare la blind deconvolution consiste nel passare ad una formulazione ai minimi quadrati, che abbia per incognite sia l'oggetto che il modello dipendente da parametri da stimare. A causa della severa mal posizione, la presenza di rumore sui dati misurati rende inutilizzabile la strategia di semplice inversione dei dati, mentre solo un accurato studio dell'instabilità congiunta alla scelta di opportune tecniche di regolarizzazione può portare ad ottenere una soluzione fisicamente significativa.
Se l'oggetto è l'incognita lineare e i parametri da cui dipende il modello sonole variabili non lineari, la blind deconvolution può essere formulata come un problema separabile e risolta con i metodi di Proiezione delle Variabili e Minimizzazione Alternata. Per tali problemi di larga scala, l'operatore che descrive il modello possiede tipicamente speciali caratteristiche e strutture, da sfruttare per una efficiente implementazione dei metodi iterativi e delle tecniche di regolarizzazione.
Una seconda differenza cruciale tra i due metodi è nel tipo di regolarizzazione che ammettono. Infatti, nella Proiezioni delle Variabili è consentita solo la siddetta regolarizzazione diretta, che consiste nell'incorporare informazioni a priori sulla soluzione attesa. Invece la Minimizzazione Alternata ammette anche la regolarizzazione iterativa, basata sul costruire iterativamente una successione di vettori che converge alla soluzione e sul fermare il procedimento iterativo prima che il rumore introduca artefatti sulla ricostruzione.
Sono proposti numerosi esperimenti numerici nell'ambito dell'imaging astronomico, dove la formulazione "blind" della deconvoluzione arriva da una modellizzazione parametrica dell'operatore di blurring o, nel caso di dati di Fourier, da incertezza sulle frequenze spaziali corrispondenti ai dati. Tutti i test condotti mostrano che la presenza di vincoli di non-negatività sull'approccio separabile di blind deconvolution proposto forniscono miglioramenti nella qualità dei risultati ottenuti.In this work we analyze separable least squares problems, where the variables can be partitioned into a linear subset and a nonlinear one. This kind of problems have been addressed in literature with two approaches: the Variable Projection and the Alternating Optimization. The basic idea on which both the methods are based is to exploit the separability of the variables in order to modify the original problem and obtain a new formulation, easier to solve and of smaller dimension than the previous. A detailed comparison of the two approaches is given, pointing out their respective pros and cons and the application for which an approach is more convenient than the other. A crucial issue for both methods, representing the main difference between them, is in the computation of the Jacobian matrix related to the objective function.
In particular, we focus on a constrained formulation of separable least squares, giving special attention to the case of nonnegative constraints imposed on the linear variable. This choice is motivated by the application of the methods to the imaging framework, where the entries of the unknown are pixels contents refering to a physical quantity, that thus must be real and nonnegative. Iterative projection methods are applied to solve the problem.
As original contribution, we extend the Variable Projection method to this nonnegative case and a new formula for the computation of the Jacobian matrix is proposed.
In order to validate all the theoretical results, an application to blind deconvolution is given. In this problems, some measured radiation (microwaves, photons, X-rays, ultrasound, etc.) emitted or transmitted by an unknown target is available, together with only partial (or any) information on the apparatus providing the data. Reconstructions of both the target and the model connecting the data with the target itself have therefore to be addressed contemporaneously, which represents a particularly tough issue when dealing with restoration problems. A common strategy to address the solution of a blind deconvolution problem involves a least squares formulation, where the unknown are both the target and a parametric version of the model. Due to the sever ill-posedness characterizing the problem, the presence of noise in the measured data makes any naive inversion strategy useless, and only an accurate study of the instability together with a choice of an appropriate regularization technique can lead to a physically meaningful result.
By considering the target as the linear unknown and the parameters describing the model as the nonlinear variables, blind deconvolution can be formulated as a separable problem and addressed with the Variable Projection and the Alternating Optimization methods. For such large scale problems, the operator describing the parametric model typically has special features and structure, that can be exploited for an efficient implementation of the iterative methods and the regularization techniques.
Another crucial difference between the two methods lays in the regularization techniques they admit. Specifically, for Variable Projection only direct regularization is allowed, that consists on incorporating further a priori information about the expected solution. On the other hand, Alternating Optimization admits also iterative regularization, based on iteratively building up a sequence of arrays that converges to the solution and by stopping the iterative procedure before the noise introduces artefacts on the reconstructions.
Numerical experiments in the astronomical imaging framework are given, where the "blind" formulation comes from a parametric model of the blurring operator or, when dealing with Fourier data, uncertanties on the spatial frequencies.
All the tests conducted show that the presence of nonnegative constraints on the proposed separable blind deconvolution approach provides some improvements in the reconstructions
A convergent least-squares regularized blind deconvolution approach
No full textThe aim of this work is to present a new and efficient optimization method for the solution of blind deconvolution problems with data corrupted by Gaussian noise, which can be reformulated as a constrained minimization problem whose unknowns are the point spread function (PSF) of the acquisition system and the true image. The objective function we consider is the weighted sum of the least-squares fit-to-data discrepancy and possible regularization terms accounting for specific features to be preserved in both the image and the PSF. The solution of the corresponding minimization problem is addressed by means of a proximal alternating linearized minimization (PALM) algorithm, in which the updating procedure is made up of one step of a gradient projection method along the arc and the choice of the parameter identifying the steplength in the descent direction is performed
automatically by exploiting the optimality conditions of the problem. The resulting approach is a particular case of a general scheme whose convergence to stationary points of the constrained minimization problem has been recently proved. The effectiveness of the iterative method is validated in several numerical simulations in image reconstruction problems
Runge–Kutta-like scaling techniques for first-order methods in convex optimization
No full textIt is well known that there is a strong connection between time integration and convex optimization. In this work, inspired by the equivalence between the forward Euler scheme and the gradient descent method, we broaden our analysis to the family of Runge–Kutta methods and show that they enjoy a natural interpretation as first-order optimization algorithms. The strategies intrinsically suggested by Runge–Kutta methods are exploited in order to detail novel proposal for either scaling or preconditioning gradient-like approaches, whose convergence is ensured by the stability condition for Runge–Kutta schemes. The theoretical analysis is supported by the numerical experiments carried out on some test problems arising from suitable applications where the proposed techniques can be efficiently employed
Semi-blind deconvolution for Fourier-based image restoration
No full textIn this work we develop a new optimization algorithm for image reconstruction problems from Fourier data with uncertanties on the spatial
frequencies corresponding to the measured data. By considering such dependency on the frequencies as a further unknown, we obtain a so-called
semi-blind deconvolution. Both the image and the spatial frequencies are obtained as solutions of a reformulated constrained optimization problem,
approached by an alternating scheme. Numerical tests on simulated data, based on the imaging hardware of the NASA RHESSI satellite, show
that the proposed approach provides some improvements in the reconstruction
A new semi-blind deconvolution approach for Fourier-based image restoration: an application in astronomy
Full text linkThe 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 RHESSI satellite. Numerical experiments on simulated data show that, both in presence and in absence of statistical noise, the proposed approach provides some improvements in the reconstructions
Filter factor analysis of scaled gradient methods for linear least squares
No full textA typical way to compute a meaningful solution of a linear least squares problem involves the introduction of a filter factors array, whose aim is to avoid noise amplification due to the presence of small singular values. Beyond the classical direct regularization approaches, iterative gradient methods can be thought as filtering methods, due to their typical capability to recover the desired components of the true solution at the first iterations. For an iterative method, regularization is achieved by stopping the procedure before the noise introduces artifacts, making the iteration number playing the role of the regularization parameter. In this paper we want to investigate the filtering and regularizing effects of some first-order algorithms, showing in particular which benefits can be gained in recovering the filters of the true solution by means of a suitable scaling matrix
Going Beyond Counting First Authors in Author Co-citation Analysis
Full text linkThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
On the filtering effect of iterative regularization algorithms for discrete inverse problems
No full textMany real-world applications are addressed through a linear least-squares
problem formulation, whose solution is calculated by means of an iterative approach. A huge amount of studies has been carried out in the optimization field to provide the fastest methods for the reconstruction of the solution, involving choices of adaptive parameters and scaling matrices. However, in presence of an ill-conditioned model and real data, the need of a regularized solution instead of the least-squares one changed the point of view in favour of iterative algorithms able to combine a fast execution with a stable behaviour with respect to the restoration error. In this paper we analyze some classical and recent gradient approaches for the linear least-squares problem by looking at their way of filtering the singular values, showing in particular the effects of scaling matrices and non-negative constraints in recovering the correct filters of the solution. An original analysis of the filtering effect for the image deblurring problem with Gaussian noise on the data is also provided
An image reconstruction method from Fourier data with uncertainties on the spatial frequencies
Full text linkIn this work we develop a new optimization algorithm for image reconstruction problems from Fourier data with uncertainties on the spatial frequencies corresponding to the measured data. By considering such dependency on the frequencies as a further unknown, we obtain a so-called semi-blind deconvolution. Both the image and the spatial frequencies are obtained as solutions of a reformulated constrained optimization problem, approached by an alternating scheme. Numerical tests on simulated data, based on the imaging hardware of the NASA RHESSI satellite, show that the proposed approach provides some improvements in the reconstruction
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