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Regularization Term (regularization + term)

Distribution by Scientific Domains

Mathematics and Statistics	50%

Selected Abstracts

Regularized, fast, and robust analytical Q-ball imaging

MAGNETIC RESONANCE IN MEDICINE, Issue 3 2007
Maxime Descoteaux
Abstract We propose a regularized, fast, and robust analytical solution for the Q-ball imaging (QBI) reconstruction of the orientation distribution function (ODF) together with its detailed validation and a discussion on its benefits over the state-of-the-art. Our analytical solution is achieved by modeling the raw high angular resolution diffusion imaging signal with a spherical harmonic basis that incorporates a regularization term based on the Laplace,Beltrami operator defined on the unit sphere. This leads to an elegant mathematical simplification of the Funk,Radon transform which approximates the ODF. We prove a new corollary of the Funk,Hecke theorem to obtain this simplification. Then, we show that the Laplace,Beltrami regularization is theoretically and practically better than Tikhonov regularization. At the cost of slightly reducing angular resolution, the Laplace,Beltrami regularization reduces ODF estimation errors and improves fiber detection while reducing angular error in the ODF maxima detected. Finally, a careful quantitative validation is performed against ground truth from synthetic data and against real data from a biological phantom and a human brain dataset. We show that our technique is also able to recover known fiber crossings in the human brain and provides the practical advantage of being up to 15 times faster than original numerical QBI method. Magn Reson Med 58:497,510, 2007. © 2007 Wiley-Liss, Inc. [source]

Improving k - t SENSE by adaptive regularization

MAGNETIC RESONANCE IN MEDICINE, Issue 5 2007
Dan Xu
Abstract The recently proposed method known as k - t sensitivity encoding (SENSE) has emerged as an effective means of improving imaging speed for several dynamic imaging applications. However, k - t SENSE uses temporally averaged data as a regularization term for image reconstruction. This may not only compromise temporal resolution, it may also make some of the temporal frequency components irrecoverable. To address that issue, we present a new method called spatiotemporal domain-based unaliasing employing sensitivity encoding and adaptive regularization (SPEAR). Specifically, SPEAR provides an improvement over k - t SENSE by generating adaptive regularization images. It also uses a variable-density (VD), sequentially interleaved k - t space sampling pattern with reference frames for data acquisition. Simulations based on experimental data were performed to compare SPEAR, k - t SENSE, and several other related methods, and the results showed that SPEAR can provide higher temporal resolution with significantly reduced image artifacts. Ungated 3D cardiac imaging experiments were also carried out to test the effectiveness of SPEAR, and real-time 3D short-axis images of the human heart were produced at 5.5 frames/s temporal resolution and 2.4 × 1.2 × 8 mm3 spatial resolution with eight slices. Magn Reson Med 57:918,930, 2007. © 2007 Wiley-Liss, Inc. [source]

Analysis of block matrix preconditioners for elliptic optimal control problems

NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 4 2007
T. P. Mathew
Abstract In this paper, we describe and analyse several block matrix iterative algorithms for solving a saddle point linear system arising from the discretization of a linear-quadratic elliptic control problem with Neumann boundary conditions. To ensure that the problem is well posed, a regularization term with a parameter , is included. The first algorithm reduces the saddle point system to a symmetric positive definite Schur complement system for the control variable and employs conjugate gradient (CG) acceleration, however, double iteration is required (except in special cases). A preconditioner yielding a rate of convergence independent of the mesh size h is described for , , R2 or R3, and a preconditioner independent of h and , when , , R2. Next, two algorithms avoiding double iteration are described using an augmented Lagrangian formulation. One of these algorithms solves the augmented saddle point system employing MINRES acceleration, while the other solves a symmetric positive definite reformulation of the augmented saddle point system employing CG acceleration. For both algorithms, a symmetric positive definite preconditioner is described yielding a rate of convergence independent of h. In addition to the above algorithms, two heuristic algorithms are described, one a projected CG algorithm, and the other an indefinite block matrix preconditioner employing GMRES acceleration. Rigorous convergence results, however, are not known for the heuristic algorithms. Copyright © 2007 John Wiley & Sons, Ltd. [source]

Analysis of the bounded variation and the G regularization for nonlinear inverse problems

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 9 2010
I. Cimrák
Abstract We analyze the energy method for inverse problems. We study the unconstrained minimization of the energy functional consisting of a least-square fidelity term and two other regularization terms being the seminorm in the BV space and the norm in the G space. We consider a coercive (non)linear operator modelling the forward problem. We establish the uniqueness and stability results for the minimization problems. The stability is studied with respect to the perturbations in the data, in the operator, as well as in the regularization parameters. We settle convergence results for the general minimization schemes. Copyright © 2009 John Wiley & Sons, Ltd. [source]