Abstract not found

We propose an iterative method that solves constrained linear least-squares problems by formulating them as nonlinear systems of equations and applying the Newton scheme. The method reduces the size of the linear system to be solved at each iteration by considering only a subset of the unknown variables. Hence the linear system can be solved more efficiently. We prove that the method is locally quadratic convergent. Applications to image deblurring problems show that our method gives better restored images than those obtained by projecting or scaling the solution into the dynamic range.

Abstract: Current computed tomography (CT) scanners, including micro-CT scanners, utilize a point x-ray source. As we target higher and higher spatial resolutions, the reduced x-ray focal spot size limits the temporal and contrast resolutions achievable. To overcome this limitation, in this paper we propose to use a line-shaped x-ray source so that many more photons can be generated, given a data acquisition interval. In reference to the simultaneous algebraic reconstruction technique (SART) algorithm for image reconstruction from projection data generated by an x-ray point source, here we develop a generalized SART algorithm for image reconstruction from projection data generated by an x-ray line source. Our numerical simulation results demonstrate the feasibility of our novel line-source-based x-ray CT approach and the proposed generalized SART algorithm.

Abstract. Many questions in science and engineering give rise to ill-posed inverse problems whose solution is known to satisfy box constrains, such as nonnegativity. The solution of discretized versions of these problems is highly sensitive to perturbations in the data, discretization errors, and round-off errors introduced during the computations. It is therefore often beneficial to impose known constraints during the solution process. This paper paper describes a two-phase algorithm for the solution of large-scale box-constrained discrete ill-posed problems. The first phase applies a cascadic multilevel method and imposes the constraints on each level by orthogonal projection. The second phase improves the computed approximate solution on the finest level by an active set method. The latter allows several indices of the active set to be updated simultaneously. This reduces the computational effort significantly, when compared to standard active set methods that update one index at a time. Applications to image restoration are presented.

for material identification

least squares estimation for high-dimensional regression

least squares

Constrained least-squares regression problems, such as the Nonnegative Least Squares (NNLS) problem, where the variables are restricted to take only nonnegative values, often arise in applications. Motivated by the recent development of the fast Johnson-Lindestrauss transform, we present a fast random projection type approximation algorithm for the NNLS problem. Our algorithm employs a randomized Hadamard transform to construct a much smaller NNLS problem and solves this smaller problem using a standard NNLS solver. We prove that our approach finds a nonnegative solution vector that, with high probability, is close to the optimum nonnegative solution in a relative error approximation sense. We experimentally evaluate our approach on a large collection of term-document data and verify that it does offer considerable speedups without a significant loss in accuracy. Our analysis is based on a novel random projection type result that might be of independent interest. In particular, given a tall and thin matrix Φ ∈ R n×d (n ≫ d) and a vector y ∈ R d, we prove that the Euclidean length of Φy can be estimated very accurately by the Euclidean length of ˜ Φy, where ˜ Φ consists of a small subset of (appropriately rescaled) rows of Φ. 1