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  1. Electrical Impedance Tomography (EIT) is a well-known imaging technique for detecting the electrical properties of an object in order to detect anomalies, such as conductive or resistive targets. More specifically, EIT has many applications in medical imaging for the detection and location of bodily tumors since it is an affordable and non-invasive method, which aims to recover the internal conductivity of a body using voltage measurements resulting from applying low frequency current at electrodes placed at its surface. Mathematically, the reconstruction of the internal conductivity is a severely ill-posed inverse problem and yields a poor quality image reconstruction. To remedy this difficulty, at least in part, we regularize and solve the nonlinear minimization problem by the aid of a Krylov subspace-type method for the linear sub problem during each iteration. In EIT, a tumor or general anomaly can be modeled as a piecewise constant perturbation of a smooth background, hence, we solve the regularized problem on a subspace of relatively small dimension by the Flexible Golub-Kahan process that provides solutions that have sparse representation. For comparison, we use a well-known modified Gauss–Newton algorithm as a benchmark. Using simulations, we demonstrate the effectiveness of the proposed method. The obtained reconstructions indicatemore »that the Krylov subspace method is better adapted to solve the ill-posed EIT problem and results in higher resolution images and faster convergence compared to reconstructions using the modified Gauss–Newton algorithm.« less
  2. Abstract Bregman-type iterative methods have received considerable attention in recent years due to their ease of implementation and the high quality of the computed solutions they deliver. However, these iterative methods may require a large number of iterations and this reduces their usefulness. This paper develops a computationally attractive linearized Bregman algorithm by projecting the problem to be solved into an appropriately chosen low-dimensional Krylov subspace. The projection reduces the computational effort required for each iteration. A variant of this solution method, in which nonnegativity of each computed iterate is imposed, also is described. Extensive numerical examples illustrate the performance of the proposed methods.