The inversion of two-dimensional NMR data is an ill-posed problem related to the numerical computation of the inverse Laplace transform. In this paper we present the 2DUPEN algorithm that extends the Uniform Penalty (UPEN) algorithm [Borgia, Brown, Fantazzini, Journal of Magnetic Resonance, 1998] to two-dimensional data. The UPEN algorithm, defined for the inversion of one-dimensional NMR relaxation data, uses Tikhonov-like regularization and optionally non-negativity constraints in order to implement locally adapted regularization. In this paper, we analyze the regularization properties of this approach. Moreover, we extend the one-dimensional UPEN algorithm to the two-dimensional case and present an efficient implementation based on the Newton Projection method. Without any a-priori information on the noise norm, 2DUPEN automatically computes the locally adapted regularization parameters and the distribution of the unknown NMR parameters by using variable smoothing. Results of numerical experiments on simulated and real data are presented in order to illustrate the potential of the proposed method in reconstructing peaks and flat regions with the same accuracy.
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
The Generalized Minimal RESidual (GMRES) method and conjugate gradient method applied to the normal equations (CGNR) are popular iterative schemes for the solution of large linear systems of equations. GMRES requires the matrix of the linear system to be square and nonsingular, while CGNR also can be applied to overdetermined or underdetermined linear systems of equations. When equipped with a suitable stopping rule, both GMRES and CGNR are regularization methods for the solution of linear ill-posed problems. Many linear ill-posed problems that arise in the sciences and engineering have nonnegative solutions. This paper describes iterative schemes, based on the GMRES or CGNR methods, for the computation of non-negative solutions of linear ill-posed problems. The computations with these schemes are terminated as soon as a non-negative approximate solution which satisfies the discrepancy principle has been found. Several computed examples illustrate that the schemes of this paper are able to compute non-negative approximate solutions of higher quality with less computational effort than several available numerical methods.
Three patients affected by carcinoma cuniculatum involving the nail apparatus are reported. The toes were affected in 2 cases, the thumb in 1 case. In the first patient the tumour developed in the subungual area and resulted in loss of the toe-nail. In the second patient the tumour originated on the dorsum of the toe and subsequently involved the proximal nail fold. In the third patient the tumour developed in the nail bed resulting in loss of the lateral part of the nail plate. The pathology showed in all cases invaginating strands of well-differentiated keratinocytes, some of which had central crypts containing keratinous debris. Radical excision of the tumour required disarticulation of the digit in 2 cases, whereas Mohs micrographic surgery was performed in the third case.
In many science and engineering applications, the discretization of linear ill-posed problems gives rise to large ill-conditioned linear systems with the righthand side degraded by noise. The solution of such linear systems requires the solution of minimization problems with one quadratic constraint, depending on an estimate of the variance of the noise. This strategy is known as regularization. In this work, we propose a modification of the Lagrange method for the solution of the noise constrained regularization problem. We present the numerical results of test problems, image restoration and medical imaging denoising. Our results indicate that the proposed Lagrange method is effective and efficient in computing good regularized solutions of ill-conditioned linear systems and in computing the corresponding Lagrange multipliers. Moreover, our numerical experiments show that the Lagrange method is computationally convenient. Therefore, the Lagrange method is a promising approach for dealing with ill-posed problems.
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