Elena Loli Piccolomini

The conjugate gradient regularization method in Computed Tomography problems

In this work we solve inverse problems coming from the area of Computed Tomography by means of regularization methods based on conjugate gradient iterations. We develop a stopping criterion which is efficient for the computation of a regularized solution for the least-squares normal equations. The stopping rule can be suitably applied also to the Tikhonov regularization method. We report computational experiments based on different physical models and with different degrees of noise. We compare the results obtained with those computed by other currently used methods such as Algebraic Reconstruction Techniques (ART) and Backprojection.

An improved Newton projection method for nonnegative deblurring of Poisson-corrupted images with Tikhonov regularization

In this paper a quasi-Newton projection method for image deblurring is presented. The image restoration problem is mathematically formulated as a nonnegatively constrained minimization problem where the objective function is the sum of the Kullback–Leibler divergence, used to express fidelity to the data in the presence of Poisson noise, and of a Tikhonov regularization term. The Hessian of the objective function is approximated so that the Newton system can be efficiently solved by using Fast Fourier Transforms. The numerical results show the potential of the proposed method both in terms of relative error reduction and computational efficiency.

NPTool: a Matlab software for nonnegative image restoration with Newton projection methods

Several image restoration applications require the solution of nonnegatively constrained minimization problems whose objective function is typically constituted by the sum of a data fit function and a regularization function. Newton projection methods are very attractive because of their fast convergence, but they need an efficient implementation to avoid time consuming iterations. In this paper we present NPTool, a set of Matlab functions implementing Newton projection methods for image denoising and deblurring applications. They are specifically thought for two different data fit functions, the Least Squares function and the Kullback–Leibler divergence, and two regularization functions, Tikhonov and Total Variation, giving the opportunity of solving a large variety of restoration problems. The package is easily extensible to other linear or nonlinear data fit and regularization functions. Some examples of its use are included in the package and shown in this paper.

A Total Variation-Based Reconstruction Method for Dynamic MRI

In recent years, total variation (TV) regularization has become a popular and powerful tool for image restoration and enhancement. In this work, we apply TV minimization to improve the quality of dynamic magnetic resonance images. Dynamic magnetic resonance imaging is an increasingly popular clinical technique used to monitor spatio-temporal changes in tissue structure. Fast data acquisition is necessary in order to capture the dynamic process. Most commonly, the requirement of high temporal resolution is fulfilled by sacrificing spatial resolution. Therefore, the numerical methods have to address the issue of images reconstruction from limited Fourier data. One of the most successful techniques for dynamic imaging applications is the reduced-encoded imaging by generalized-series reconstruction method of Liang and Lauterbur. However, even if this method utilizes a priori data for optimal image reconstruction, the produced dynamic images are degraded by truncation artifacts, most notably Gibbs ringing, due to the spatial low resolution of the data. We use a TV regularization strategy in order to reduce these truncation artifacts in the dynamic images. The resulting TV minimization problem is solved by the fixed point iteration method of Vogel and Oman. The results of test problems with simulated and real data are presented to illustrate the effectiveness of the proposed approach in reducing the truncation artifacts of the reconstructed images.

Constrained numerical optimization methods for blind deconvolution

This paper describes a nonlinear least squares framework to solve a separable nonlinear ill-posed inverse problem that arises in blind deconvolution. It is shown that with proper constraints and well chosen regularization parameters, it is possible to obtain an objective function that is fairly well behaved and the nonlinear minimization problem can be effectively solved by a Gauss–Newton method. Although uncertainties in the data and inaccuracies of linear solvers make it unlikely to obtain a smooth and convex objective function, it is shown that implicit filtering optimization methods can be used to avoid becoming trapped in local minima. Computational considerations, such as computing the Jacobian, are discussed, and numerical experiments are used to illustrate the behavior of the algorithms. Although the focus of the paper is on blind deconvolution, the general mathematical model addressed in this paper, and the approaches discussed to solve it, arise in many other applications.

A descent method for regularization of ill-posed problems

In this paper, we describe an iterative algorithm, called descent-TCG, based on truncated conjugate gradients iterations to compute Tikhonov regularized solutions of linear ill-posed problems. The sequence of approximate solutions and regularization parameters, computed by the algorithm, is shown to decrease the value of the Tikhonov functional. Suitable termination criteria are built-up to define an inner–outer iteration scheme that computes a regularized solution. Numerical experiments are performed to compare this algorithm with other well established regularization methods. We observe that the best descent-TCG results occur for highly noised data and we always get fairly reliable solutions, thus it prevents the dangerous error growth often appearing in other well established regularization methods. Finally, the descent-TCG method is computationally advantageous especially for large size problems.

Image Enhancement Variational Methods for Enabling Strong Cost Reduction in OLED-based Point-of-Care Immunofluorescent Diagnostic Systems

Immunofluorescence diagnostic systems cost is often dominated by high-sensitivity, low-noise CCD-based cameras that are used to acquire the fluorescence images. In this paper, we investigate the use of low-cost CMOS sensors in a point-of-care immunofluorescence diagnostic application for the detection and discrimination of 4 different serotypes of the Dengue virus in a set of human samples. A 2-phase postprocessing software pipeline is proposed, which consists in a first image-enhancement stage for resolution increasing and segmentation and a second diagnosis stage for the computation of the output concentrations. We present a novel variational coupled model for the joint super-resolution and segmentation stage and an automatic innovative image analysis for the diagnosis purpose. A specially designed forward backward-based numerical algorithm is introduced, and its convergence is proved under mild conditions. We present results on a cheap prototype CMOS camera compared with the results of a more expensive CCD device, for the detection of the Dengue virus with a low-cost OLED light source. The combination of the CMOS sensor and the developed postprocessing software allows to correctly identify the different Dengue serotype using an automatized procedure. The results demonstrate that our diagnostic imaging system enables camera cost reduction up to 99%, at an acceptable diagnostic accuracy, with respect to the reference CCD-based camera system. The correct detection and identification of the Dengue serotypes have been confirmed by standard diagnostic methods (RT-PCR and ELISA).

A total variation regularization strategy in dynamic MRI

Recently, the application of Magnetic Resonance Imaging to the study of dynamic processes has increased the search of ever-faster dynamic imaging methods. The Keyhole method is a popular technique for fast dynamic imaging that unfortunately produces images affected by artifacts. In this work, we propose a Total Variation regularization strategy for post-processing the dynamic Keyhole images. The strategy consists in the solution of an unconstrained minimization problem whose Euler–Lagrange equation is solved by a quasi-Newton method. A trace of the algorithm is given. Results of numerical experiments on simulated and real data are presented to illustrate the effectiveness of the proposed method. Our experimental results indicate that the method reduces the degrading artifacts and improves the quality of the final images.

A total-variation-based regularization strategy in magnetic resonance imaging

In this paper we present some variational functionals for the regularization of Magnetic Resonance (MR) images, usually corrupted by noise and artifacts. The mathematical problem has a Tikhonov-like formulation, where the regularization functional is a nonlinear variational functional. The problem is numerically solved as an optimization problem with a quasi-Newton algorithm. The algorithm has been applied to MR images corrupted by noise and to dynamic MR images corrupted by truncation artifacts due to limited resolution. The results on test problems obtained from simulated and real data are presented. The functionals actually reduce noise and artifacts, provided that a good regularizing parameter is used.

A descent method for computing the Tikhonov regularized solution of linear inverse problems

In this paper we describe an iterative algorithm, called Descent-TCG, based on truncated Conjugate Gradient iterations to compute Tikhonov regularized solutions of linear ill-posed problems. Suitable termination criteria are built-up to define an inner-outer iteration scheme for the computation of a regularized solution. Numerical experiments are performed to compare the algorithm with other well-established regularization methods. We observe that the best Descent-TCG results occur for highly noised data and we always get fairly reliable solutions, preventing the dangerous error growth often appearing in other well-established regularization methods. Finally, the Descent-TCG method is computationally advantageous especially for large size problems.