A discrepancy principle for Poisson data: uniqueness of the solution for 2D and 3D data

This paper is concerned with the uniqueness of the solution of a nonlinear equation, named discrepancy equation. For the restoration problem of data corrupted by Poisson noise, we have to minimize an objective function that combines a data-fidelity function, given by the generalized Kullback–Leibler divergence, and a regularization penalty function. Bertero et al. recently proposed to use the solution of the discrepancy equation as a convenient value for the regularization parameter. Furthermore they devised suitable conditions to assure the uniqueness of this solution for several regularization functions in 1D denoising and deblurring problems. The aim of this paper is to generalize this uniqueness result to 2D and 3D problems for several penalty functions, such as an edge preserving functional, a simple case of the class of Markov Random Field (MRF) regularization functionals and the classical Tikhonov regularization

Inexact Bregman iteration with an application to Poisson data reconstruction

This work deals with the solution of image restoration problems by an iterative regularization method based on the Bregman iteration. Any iteration of this scheme requires to exactly compute the minimizer of a function. However, in some image reconstruction applications, it is either impossible or extremely expensive to obtain exact solutions of these subproblems. In this paper, we propose an inexact version of the iterative procedure, where the inexactness in the inner subproblem solution is controlled by a criterion that preserves the convergence of the Bregman iteration and its features in image restoration problems. In particular, the method allows to obtain accurate reconstructions also when only an overestimation of the regularization parameter is known. The introduction of the inexactness in the iterative scheme allows to address image reconstruction problems from data corrupted by Poisson noise, exploiting the recent advances about specialized algorithms for the numerical minimization of the generalized Kullback–Leibler divergence combined with a regularization term. The results of several numerical experiments enable to evaluat

A variable metric forward--backward method with extrapolation

Forward-backward methods are a very useful tool for the minimization of a functional given by the sum of a differentiable term and a nondifferentiable one and their investigation has experienced several efforts from many researchers in the last decade. In this paper we focus on the convex case and, inspired by recent approaches for accelerating first-order iterative schemes, we develop a scaled inertial forward-backward algorithm which is based on a metric changing at each iteration and on a suitable extrapolation step. Unlike standard forward-backward methods with extrapolation, our scheme is able to handle functions whose domain is not the entire space. Both {an ${\mathcal O}(1/k^2)$ convergence rate estimate on the objective function values and the convergence of the sequence of the iterates} are proved. Numerical experiments on several {test problems arising from image processing, compressed sensing and statistical inference} show the {effectiveness} of the proposed method in comparison to well performing {state-of-the-art} algorithms

Spatial genetic structure in the clonal marine angiosperm <i>Cymodocea nodosa</i>: the influence of dispersal potential, mating system and species interactions

In the present thesis, the factors influencing population’s genetic structure in the clonal marine angiosperm Cymodocea nodosa have been investigated. C. nodosa is a dioecious seagrass, exhibiting both vegetative propagation and sexual reproduction. Seed dispersal is expected to be extremely restricted. I selected seven microsatellite loci through genomic library screening to investigate the relative effects of sexual and clonal reproduction on the genetic diversity and structure in a Cymodocea nodosa population from the Gulf of Naples (Italy). High clonal diversity and genet density were found. Autocorrelation analyses confirmed the expectations of very restricted seed dispersal (observed dispersal range 1-21m) in this species. The effect of mating system on genetic structure were investigated comparing the clonal architectures of the dioecious Cymodocea nodosa and monoecious Zostera noltii. An intermingled configuration of genets has been found in the dioecious Cymodocea nodosa while a clumped distribution of clones in the hermaphroditic Zostera noltii has been observed. I hypothesise that the possibility of reduction in the seed-set would drive genet distribution. On a phylogeographic spatial scale, the existence of population differentiation and of genetic disjunction within the Mediterranean Sea was investigated in Cymodocea nodosa. Populations displayed a wide variability in clonal diversity. A Bayesian analysis revealed that “supra-population” panmictic units are present in the Mediterranean basin. Genetic substructure from a phylogeographic tree coincided with major geographical boundaries within the Mediterranean basin. In general, in Cymodocea nodosa, seed dispersal is poor at the within-population level, but long-range dispersal events can occur, allowing high levels of gene flow at a phylogeographic scale. The observed “guerrilla” clonal architecture allows to reduce the effect of genetic identity on the genetic structure of the population, but it is also advantageous by allowing pollen availability and therefore a sufficient seed-set in this dioecious species

Recent and Ancient Signature of Balancing Selection around the S-Locus in Arabidopsis halleri and A. lyrata

Balancing selection can maintain different alleles over long evolutionary times. Beyond this direct effect on the molecular targets of selection, balancing selection is also expected to increase neutral polymorphism in linked genome regions, in inverse proportion to their genetic map distances from the selected sites. The genes controlling plant self-incompatibility are subject to one of the strongest forms of balancing selection, and they show clear signatures of balancing selection. The genome region containing those genes (the S-locus) is generally described as nonrecombining, and the physical size of the region with low recombination has recently been established in a few species. However, the size of the region showing the indirect footprints of selection due to linkage to the S-locus is only roughly known. Here, we improved estimates of this region by surveying synonymous polymorphism and estimating recombination rates at 12 flanking region loci at known physical distances from the S-locus region boundary, in two closely related self-incompatible plants Arabidopsis halleri and A. lyrata. In addition to studying more loci than previous studies and using known physical distances, we simulated an explicit demographic scenario for the divergence between the two species, to evaluate the extent of the genomic region whose diversity departs significantly from neutral expectations. At the closest flanking loci, we detected signatures of both recent and ancient indirect effects of selection on the S-locus flanking genes, finding ancestral polymorphisms shared by both species, as well as an excess of derived mutations private to either species. However, these effects are detected only in a physically small region, suggesting that recombination in the flanking regions is sufficient to quickly break up linkage disequilibrium with the S-locus. Our approach may be useful for distinguishing cases of ancient versus recently evolved balancing selection in other systems

Steplength selection in gradient projection methods for box-constrained quadratic programs

The role of the steplength selection strategies in gradient methods has been widely in- vestigated in the last decades. Starting from the work of Barzilai and Borwein (1988), many efficient steplength rules have been designed, that contributed to make the gradient approaches an effective tool for the large-scale optimization problems arising in important real-world applications. Most of these steplength rules have been thought in unconstrained optimization, with the aim of exploiting some second-order information for achieving a fast annihilation of the gradient of the objective function. However, these rules are successfully used also within gradient projection methods for constrained optimization, though, to our knowledge, a detailed analysis of the effects of the constraints on the steplength selections is still not available. In this work we investigate how the presence of the box constraints affects the spectral properties of the Barzilai\u2013Borwein rules in quadratic programming problems. The proposed analysis suggests the introduction of new steplength selection strategies specifically designed for taking account of the active constraints at each iteration. The results of a set of numerical experiments show the effectiveness of the new rules with respect to other state of the art steplength selections and their potential usefulness also in case of box-constrained non-quadratic optimization problems

Multiparameter quantum estimation of noisy phase shifts

Phase estimation is the most investigated protocol in quantum metrology, but its performance is affected by the presence of noise, also in the form of imperfect state preparation. Here we discuss how to address this scenario by using a multiparameter approach, in which noise is associated to a parameter to be measured at the same time as the phase. We present an experiment using two-photon states, and apply our setup to investigating optical activity of fructose solutions. Finally, we illustrate the scaling laws of the attainable precisions with the number of photons in the probe state