E. Loli Piccolomini

An efficient method for nonnegatively constrained Total Variation-based denoising of medical images corrupted by Poisson noise

Medical images obtained with emission processes are corrupted by noise of Poisson type. In the paper the denoising problem is modeled in a Bayesian statistical setting by a nonnegatively constrained minimization problem, where the objective function is constituted by a data fitting term, the Kullback-Leibler divergence, plus a regularization term, the Total Variation function, weighted by a regularization parameter. Aim of the paper is to propose an efficient numerical method for the solution of the constrained problem. The method is a Newton projection method, where the inner system is solved by the Conjugate Gradient method, preconditioned and implemented in an efficient way for this specific application. The numerical results on simulated and real medical images prove the effectiveness of the method, both for the accuracy and the computational cost.

Scaling techniques for gradient projection-type methods in astronomical image deblurring

The aim of this paper is to present a computational study on scaling techniques in gradient projection-type (GP-type) methods for deblurring of astronomical images corrupted by Poisson noise. In this case, the imaging problem is formulated as a non-negatively constrained minimization problem in which the objective function is the sum of a fit-to-data term, the Kullback–Leibler divergence, and a Tikhonov regularization term. The considered GP-type methods are formulated by a common iteration formula, where the scaling matrix and the step-length parameter characterize the different algorithms. Within this formulation, both first-order and Newton-like methods are analysed, with particular attention to those implementation features and behaviours relevant for the image restoration problem. The numerical experiments show that suited scaling strategies can enable the GP methods to quickly approximate accurate reconstructions and then are useful for designing effective image deblurring algorithms.

An automatic regularization parameter selection algorithm in the total variation model for image deblurring

Image restoration is an inverse problem that has been widely studied in recent years. The total variation based model by Rudin-Osher-Fatemi (1992) is one of the most effective and well known due to its ability to preserve sharp features in restoration. This paper addresses an important and yet outstanding issue for this model in selection of an optimal regularization parameter, for the case of image deblurring. We propose to compute the optimal regularization parameter along with the restored image in the same variational setting, by considering a Karush Kuhn Tucker (KKT) system. Through establishing analytically the monotonicity result, we can compute this parameter by an iterative algorithm for the KKT system. Such an approach corresponds to solving an equation using discrepancy principle, rather than using discrepancy principle only as a stopping criterion. Numerical experiments show that the algorithm is efficient and effective for image deblurring problems and yet is competitive.

A projected Newton-CG method for nonnegative astronomical image deblurring

Astronomical images are usually assumed to be corrupted by a space-invariant Point Spread Function and Poisson noise. In this paper we propose an original projected inexact Newton method for the solution of the constrained nonnegative minimization problem arising from image deblurring. The problem is ill-posed and the objective function must be regularized. The inner system is inexactly solved by few Conjugate Gradient iterations. The convergence of the method is proved and its efficiency is tested on simulated astronomical blurred images. The results show that the method produces good reconstructed images at low computational cost.

An Algorithm for Image Denoising with Automatic Noise Estimate

In this paper, we propose a general algorithm for image denoising when no a priori information on the noise is available. The image denoising problem is formulated as an inequality constrained minimization problem where the objective is a general convex regularization functional and the right-hand side of the constraint depends on the noise norm and is not known. The proposed method is an iterative procedure which, at each iteration, automatically computes both an approximation of the noise norm and an approximate solution of the minimization problem. Experimental results demonstrate the effectiveness of the proposed automatic denoising procedure.

Parallel image restoration on parallel and distributed computers

Abstract In this paper we present a parallel application for image restoration. The problem is relevant in some application fields, such as medicine or astronomy, and has a very high computational complexity so that it is difficult to solve it on scalar computers. The algorithm is based on data parallelism, that is realized with an adaptive decomposition of the image spatial domain for a class of degradation functions. We discuss the implementation on a cluster of six workstations connected through Ethernet network and on a Cray T3E with 128 processors. The results obtained with different images and different number of tasks show good scalability and speed-up.

A B-spline parametric model for high resolution dynamic magnetic resonance imaging

In this paper we propose a numerical method for the reconstruction of dynamic magnetic resonance images from limited data in the Fourier space. In some medical dynamic applications, temporal sequences of magnetic resonance data of the same slice of the body are acquired. In order to hasten the acquisition process, only the low frequencies are acquired for all the data sets, with the exception of the first one which is completely acquired. Then, it is not possible to use a Fourier technique to obtain high resolution images from limited data. In the proposed method, the images of the sequence are represented via a B-spline parametric model using a priori information from the first image. The use of B-splines is due to their computational efficiency and their good behaviour in image fitting and representation. The use of this parametric model allows to obtain high resolution images of good quality and zoom regions of interest without loosing important details. Mathematical and numerical aspects of the method are investigated and an outline of the algorithm is given. Reconstructions from simulated and experimental data are presented and analyzed.

Quasi-Newton projection methods and the discrepancy principle in image restoration

Abstract In this work, the problem of the restoration of images corrupted by space invariant blur and noise is considered. This problem is ill-posed and regularization is required. The image restoration problem is formulated as a nonnegatively constrained minimization problem whose objective function depends on the statistical properties of the noise corrupting the observed image. The cases of Gaussian and Poisson noise are both considered. A Newton-like projection method with early stopping of the iterates is proposed as an iterative regularization method in order to determine a nonnegative approximation to the original image. A suitable approximation of the Hessian of the objective function is proposed for a fast solution of the Newton system. The results of the numerical experiments show the effectiveness of the method in computing a good solution in few iterations, when compared with some methods recently proposed as best performing.

A fast Total Variation-based iterative algorithm for digital breast tomosynthesis image reconstruction

In this work, we propose a fast iterative algorithm for the reconstruction of digital breast tomosynthesis images. The algorithm solves a regularization problem, expressed as the minimization of the sum of a least-squares term and a weighted smoothed version of the Total Variation regularization function. We use a Fixed Point method for the solution of the minimization problem, requiring the solution of a linear system at each iteration, whose coefficient matrix is a positive definite approximation of the Hessian of the objective function. We propose an efficient implementation of the algorithm, where the linear system is solved by a truncated Conjugate Gradient method. We compare the Fixed Point implementation with a fast first order method such as the Scaled Gradient Projection method, that does not require any linear system solution. Numerical experiments on a breast phantom widely used in tomographic simulations show that both the methods recover microcalcifications very fast while the Fixed Point is more efficient in detecting masses, when more time is available for the algorithm execution.

An iterative algorithm for large size least-squares constrained regularization problems

Abstract In this paper we propose an iterative algorithm to solve large size linear inverse ill posed problems. The regularization problem is formulated as a constrained optimization problem. The dual Lagrangian problem is iteratively solved to compute an approximate solution. Before starting the iterations, the algorithm computes the necessary smoothing parameters and the error tolerances from the data. The numerical experiments performed on test problems show that the algorithm gives good results both in terms of precision and computational efficiency.