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This book presents recent mathematical methods in the area of inverse problems in imaging with a particular focus on the computational aspects and applications. The formulation of inverse problems in imaging requires accurate mathematical modeling in order to preserve the significant features of the image. The book describes computational methods to efficiently address these problems based on new optimization algorithms for smooth and nonsmooth convex minimization, on the use of structured (numerical) linear algebra, and on multilevel techniques. It also discusses various current and challenging applications in fields such as astronomy, microscopy, and biomedical imaging. The book is intended for researchers and advanced graduate students interested in inverse problems and imaging.



Recent Advances in Variable Metric First-Order Methods

Minimization problems often occur in modeling phenomena dealing with real-life applications that nowadays handle large-scale data and require real-time solutions. For these reasons, among all possible iterative schemes, first-order algorithms represent a powerful tool in solving such optimization problems since they admit a relatively simple implementation and avoid onerous computations during the iterations. On the other hand, a well known drawback of these methods is a possible poor convergence rate, especially showed when an high accurate solution is required. Consequently, the acceleration of first-order approaches is a very discussed field which has experienced several efforts from many researchers in the last decades. The possibility of considering a variable underlying metric changing at each iteration and aimed to catch local properties of the starting problem has been proved to be effective in speeding up first-order methods. In this work we deeply analyze a possible way to include a variable metric in first-order methods for the minimization of a functional which can be expressed as the sum of a differentiable term and a nondifferentiable one. Particularly, the strategy discussed can be realized by means of a suitable sequence of symmetric and positive definite matrices belonging to a compact set, together with an Armijo-like linesearch procedure to select the steplength along the descent direction ensuring a sufficient decrease of the objective function.
Silvia Bonettini, Federica Porta, Marco Prato, Simone Rebegoldi, Valeria Ruggiero, Luca Zanni

Structure Preserving Preconditioning for Frame-Based Image Deblurring

Regularizing preconditioners for accelerating the convergence of iterative regularization methods and improving their accuracy have been extensively investigated both in Hilbert and Banach spaces. For deconvolution problems, the classical approach defines preconditioners based on the circular convolution. On the other hand, for \(\ell _2\) regularization methods, it has been recently shown that a preconditioner preserving the structure of the convolution operator can be more effective. Such a preconditioner can improve both restoration quality and robustness of the method with respect to the choice of the regularization parameter when compared with the non-structured ones. In this paper we explore the use of structure preserving preconditioning for \(\ell _1\)-norm regularization in the wavelet domain in image deblurring. A recently proposed preconditioned variant of the linearized Bregman iteration is modified to preserve the structure of the coefficient matrix according to the imposed boundary conditions. The structured preconditioner is chosen as an approximation of a regularized inverse of the convolution matrix. Selected numerical experiments show that our preconditioning strategy improves the previous results obtained with circulant preconditioning providing restorations with lower ringing effects and sharper details.
Davide Bianchi, Alessandro Buccini, Marco Donatelli

Non-stationary Structure-Preserving Preconditioning for Image Restoration

Non-stationary regularizing preconditioners have recently been proposed for the acceleration of classical iterative methods for the solution of linear discrete ill-posed problems. This paper explores how these preconditioners can be combined with the flexible GMRES iterative method. A new structure-respecting strategy to construct a sequence of regularizing preconditioners is proposed. We show that flexible GMRES applied with these preconditioners is able to restore images that have been contaminated by strongly non-symmetric blur, while several other iterative methods fail to do this.
Pietro Dell’Acqua, Marco Donatelli, Lothar Reichel

Numerical Investigation of the Spectral Distribution of Toeplitz-Function Sequences

Solving Toeplitz-related systems has been of interest for their ubiquitous applications, particularly in image science and the numerical treatment of differential equations. Extensive study has been carried out for Toeplitz matrices \(T_n \in \mathbb {C}^{n \times n}\) as well as Toeplitz-function matrices \(h(T_n) \in \mathbb {C}^{n \times n}\), where h(z) is a certain function. Owing to its importance in developing effective preconditioning approaches, their spectral distribution associated with Lebesgue integrable generating functions f has been well investigated. While the spectral result concerning \(\{h(T_n) \}_n\) is largely known, such a study is not complete when considering \(\{Y_n h(T_n) \}_n \) with \(Y_n \in \mathbb {R}^{n \times n}\) being the anti-identity matrix. In this book chapter, we attempt to provide numerical evidence for showing that the eigenvalues of \(\{Y_n h(T_n)\}_n\) can be described by a spectral symbol which is precisely identified.
Sean Hon, Andy Wathen

The Hough Transform and the Impact of Chronic Leukemia on the Compact Bone Tissue from CT-Images Analysis

Computational analysis of X-ray Computed Tomography (CT) images allows the assessment of alteration of bone structure in adult patients with Advanced Chronic Lymphocytic Leukemia (ACLL), and may even offer a powerful tool to assess the development of the disease (prognostic potential). The crucial requirement for this kind of analysis is the application of a pattern recognition method able to accurately segment the intra-bone space in clinical CT images of the human skeleton. Our purpose is to show how this task can be accomplished by a procedure based on the use of the Hough transform technique for special families of algebraic curves. The dataset used for this study is composed of sixteen subjects including eight control subjects, one ACLL survivor, and seven ACLL victims. We apply the Hough transform approach to the set of CT images of appendicular bones for detecting the compact and trabecular bone contours by using ellipses, and we use the computed semi-axes values to infer information on bone alterations in the population affected by ACLL. The effectiveness of this method is proved against ground truth comparison. We show that features depending on the semi-axes values detect a statistically significant difference between the class of control subjects plus the ACLL survivor and the class of ACLL victims.
Anna Maria Massone, Cristina Campi, Francesco Fiz, Mauro Carlo Beltrametti

Multiple Image Deblurring with High Dynamic-Range Poisson Data

An interesting problem arising in astronomical imaging is the reconstruction of an image with high dynamic range, for example a set of bright point sources superimposed to smooth structures. A few methods have been proposed for dealing with this problem and their performance is not always satisfactory. In this paper we propose a solution based on the representation, already proposed elsewhere, of the image as the sum of a pointwise component and a smooth one, with different regularization for the two components. Our approach is in the framework of Poisson data and to this purpose we need efficient deconvolution methods. Therefore, first we briefly describe the application of the Scaled Gradient Projection (SGP) method to the case of different regularization schemes and subsequently we propose how to apply these methods to the case of multiple image deconvolution of high-dynamic range images, with specific reference to the Fizeau interferometer LBTI of the Large Binocular Telescope (LBT). The efficacy of the proposed methods is illustrated both on simulated images and on real images, observed with LBTI, of the Jovian moon Io. The software is available at http://​www.​oasis.​unimore.​it/​site/​home/​software.​html.
Marco Prato, Andrea La Camera, Carmelo Arcidiacono, Patrizia Boccacci, Mario Bertero

On the Segmentation of Astronomical Images via Level-Set Methods

Astronomical images are of crucial importance for astronomers since they contain a lot of information about celestial bodies that can not be directly accessible. Most of the information available for the analysis of these objects starts with sky explorations via telescopes and satellites. Unfortunately, the quality of astronomical images is usually very low with respect to other real images and this is due to technical and physical features related to their acquisition process. This increases the percentage of noise and makes more difficult to use directly standard segmentation methods on the original image. In this work we will describe how to process astronomical images in two steps: in the first step we improve the image quality by a rescaling of light intensity whereas in the second step we apply level-set methods to identify the objects. Several experiments will show the effectiveness of this procedure and the results obtained via various discretization techniques for level-set equations.
Silvia Tozza, Maurizio Falcone
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