Search

In this paper, we study a special instance of the split inverse problem (SIP), which is the split variational inclusion problem (SVIP). Three simple iterative methods for solving it are introduced and weak and strong convergence theorems are established under mild and standard assumptions. As an application, the problem of minimizing two proper, convex, and lower semi-continuous functions is considered. We compare and illustrate the efficiency and applicability of our schemes for several numerical experiments as well as an example in the field of compressed sensing. [ABSTRACT FROM AUTHOR]

In this paper, we present a novel class of iterative reconstruction methods for severely angular undersampled or/and limited-view tomographic problems with fan-beam scanning geometry. The proposed algorithms are based on a new analytical transform which generalizes Fourier-slice theorem to divergent-beam scanning geometries. Using a non-rigid coordinate transform, divergent rays can be reorganized into parallel ones. Therefore, one can employ a simpler parallel-beam projection model instead of more complicated divergent-beam geometries. Various existing iterative reconstruction techniques for divergent-beam geometries can be easily adapted to the proposed framework. The significant advantage of this formulation is the possibility of exploiting efficient Fourier-based recovery methods without rebinning of the projections. In case of highly sparse measurements (few-view data), rebinning methods are not suitable due to error-prone angular interpolation involved. In this work, three new methods based on the novel analytical framework for fan-beam geometry are presented: the Gerchberg-Papoulis algorithm, the Neumann decomposition method and its total variation regularized version. Presented numerical experiments demonstrate that the methods can be competitive in reconstructing from few-view noisy tomographic measurements. [ABSTRACT FROM AUTHOR]

In this paper we consider discrete inverse problems for which noise becomes negligible compared to data with increasing model norm. We introduce two novel definitions of regularization for characterizing inversion methods which provide approximations of ill-conditioned inverse operators consistent with such noisy data. In particular, these d efinitions, respectively, require that the reconstruction error computed from normalized data (p-asymptotic regularization) and the relative reconstruction error (p-relative regularization) go to zero as the model norm tends to infinity, 0 ≤ p < 1 being a parameter controlling the increase rate of the noise level. We investigate the relationship between these two definitions and we prove that t hey are all equivalent for positively homogeneous iterative algorithms with suitable stopping rules. This result has as a crucial consequence that such iterative algorithms realize regularization independently of the noise model. Then we give sufficient conditions for such methods to be p-asymptotic and p-relative regularizations in a discrete setting and we prove that the classical expectation maximization algorithm for Poisson data and the Landweber algorithm, if suitably stopped, are regu larization methods in this sense. We perform numerical simulations in the case of image deconvolution and computerized tomography to show that, in the presence of model-dependent noise, the reconst ructions provided by the above mentioned methods improve with increasing model norm as required by the p-asymptotic and p-relative regularization properties. More extensive studies on the p-asym ptotic and p-relative regularizations for Tikhonov-type methods will be the object of future work. [ABSTRACT FROM AUTHOR]

In this paper, the nonlinear iterative methods, which are different from the classical algorithms, to solve inverse problems are presented. Our methods by denoting some parameters and some properties of the algorithm in both noise and noiseless cases are studied. Finally, the convergence of the sequence generated by the algorithm without noise is discussed. [ABSTRACT FROM AUTHOR]

Bregman iterative algorithms have been extensively used forand total variation regularization problems, allowing to obtain simple, fast and effective algorithms. In this paper, three already-available algorithms for geosounding inversion are modified by including them in a Bregman iterative procedure. The resulting algorithms are easy to implement and do not require any optimization package. Modelling results are presented for synthetic and field data, observing better convergence properties than the original versions, avoiding the need of any continuation descent procedure. [ABSTRACT FROM AUTHOR]

The paper presents the iterative solutions of two coefficient inverse problems (CIPs) arising in frequency sounding and electrical prospecting. An iterative algorithm is constructed to obtain such solutions. Exploiting the Beilina–Klibanov approach to CIPs, this algorithm possesses the new iterative and refinement procedures. These features enhance significantly both the spatial and contrast resolutions of reconstructed coefficients. The computational effectiveness of the proposed numerical technique is demonstrated in computational experiments with two applied CIPs: electromagnetic or acoustic frequency sounding and electrical prospecting of layered media. The Slichter–Langer–Tikhonov formulation is exploited as a mathematical model of the latter. [ABSTRACT FROM PUBLISHER]