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Rosemary Renaut

Publications and source records attributed to Rosemary Renaut.

11 recordsLinked to original sources
arXiv2026

Hybrid GKB Methods for X-Ray Tomography Problems with Unmatched Back Projector

X ray Computed Tomography (CT) is a widely used imaging modality in medical, industrial, and scientific applications. In practical CT implementations, forward, A, and back projection, B, operators are often constructed using different discretization schemes to improve computational efficiency on available software and hardware platforms. Consequently, the resulting projector pair is generally unmatched, meaning that the back projector is not the exact adjoint of the forward projector. This mismatch alters the mathematical properties of the reconstruction problem and can affect the convergence behavior of iterative reconstruction methods. Previous studies have proposed AB and BA Golub Kahan bidiagonalization (GKB) methods, as well as GMRES and hybrid GMRES methods, for solving CT reconstruction problems with unmatched projector pairs, demonstrating reduced semiconvergence effects compared with conventional approaches. In this work, we develop hybrid variants of the AB- and BA-GKB algorithms by incorporating regularization within the projection process. The singular value decomposition for the operator on the projected space is used to efficiently reconstruct the solution, and to automatically select the regularization parameter using either the L-curve or generalized cross-validation. Numerical experiments on several CT reconstruction problems demonstrate the effectiveness of the proposed hybrid methods in improving robustness against semiconvergence.

math.NA↗
arXiv2025

Fast $\ell_1$-Regularized EEG Source Localization Using Variable Projection

Electroencephalograms (EEG) are invaluable for treating neurological disorders, however, mapping EEG electrode readings to brain activity requires solving a challenging inverse problem. Due to the time series data, the use of $\ell_1$ regularization quickly becomes intractable for many solvers, and, despite the reconstruction advantages of $\ell_1$ regularization, $\ell_2$-based approaches such as sLORETA are used in practice. In this work, we formulate EEG source localization as a graphical generalized elastic net inverse problem and present a variable projected algorithm (VPAL) suitable for fast EEG source localization. We prove convergence of this solver for a broad class of separable convex, potentially non-smooth functions subject to linear constraints and include a modification of VPAL that reconstructs time points in sequence, suitable for real-time reconstruction. Our proposed methods are compared to state-of-the-art approaches including sLORETA and other methods for $\ell_1$-regularized inverse problems.

stat.ML↗
arXiv2024

Parameter Selection by GCV and a $χ^2$ test within Iterative Methods for $\ell_1$-regularized Inverse Problems

$\ell_1$ regularization is used to preserve edges or enforce sparsity in a solution to an inverse problem. We investigate the Split Bregman and the Majorization-Minimization iterative methods that turn this non-smooth minimization problem into a sequence of steps that include solving an $\ell_2$-regularized minimization problem. We consider selecting the regularization parameter in the inner generalized Tikhonov regularization problems that occur at each iteration in these $\ell_1$ iterative methods. The generalized cross validation and $χ^2$ degrees of freedom methods are extended to these inner problems. In particular, for the $χ^2$ method this includes extending the $χ^2$ result for problems in which the regularization operator has more rows than columns, and showing how to use the $A-$weighted generalized inverse to estimate prior information at each inner iteration. Numerical experiments for image deblurring problems demonstrate that it is more effective to select the regularization parameter automatically within the iterative schemes than to keep it fixed for all iterations. Moreover, an appropriate regularization parameter can be estimated in the early iterations and used fixed to convergence.

math.NA↗
arXiv2024

Split Bregman Isotropic and Anisotropic Image Deblurring with Kronecker Product Sum Approximations using Single Precision Enlarged-GKB or RSVD Algorithms to provide low rank truncated SVDs

We consider the solution of the $\ell_1$ regularized image deblurring problem using isotropic and anisotropic regularization implemented with the split Bregman algorithm. For large scale problems, we replace the system matrix $A$ using a Kronecker product approximation obtained via an approximate truncated singular value decomposition for the reordered matrix $\mathcal{R}(A)$. To obtain the approximate decomposition for $\mathcal{R}(A)$ we propose the enlarged Golub Kahan Bidiagonalization algorithm that proceeds by enlarging the Krylov subspace beyond either a given rank for the desired approximation, or uses an automatic stopping test that provides a suitable rank for the approximation. The resultant expansion is contrasted with the use of the truncated and the randomized singular value decompositions with the same number of terms. To further extend the scale of problem that can be considered we implement the determination of the approximation using single precision, while performing all steps for the regularization in standard double precision. The reported numerical tests demonstrate the effectiveness of applying the approximate single precision Kronecker product expansion for $A$, combined with either isotropic or anisotropic regularization implemented using the split Bregman algorithm, for the solution of image deblurring problems. As the size of the problem increases, our results demonstrate that the major costs are associated with determining the Kronecker product approximation, rather than with the cost of the regularization algorithm. Moreover, the enlarged Golub Kahan Bidiagonalization algorithm competes favorably with the randomized singular value decomposition for estimating the approximate singular value decomposition.

math.NA↗
arXiv2024

Efficient Decomposition-Based Algorithms for $\ell_1$-Regularized Inverse Problems with Column-Orthogonal and Kronecker Product Matrices

We consider an $\ell_1$-regularized inverse problem where both the forward and regularization operators have a Kronecker product structure. By leveraging this structure, a joint decomposition can be obtained using generalized singular value decompositions. We show how this joint decomposition can be effectively integrated into the Split Bregman and Majorization-Minimization methods to solve the $\ell_1$-regularized inverse problem. Furthermore, for cases involving column-orthogonal regularization matrices, we prove that the joint decomposition can be derived directly from the singular value decomposition of the system matrix. As a result, we show that framelet and wavelet operators are efficient for these decomposition-based algorithms in the context of $\ell_1$-regularized image deblurring problems.

math.NA↗
arXiv2022

The Variable Projected Augmented Lagrangian Method

Inference by means of mathematical modeling from a collection of observations remains a crucial tool for scientific discovery and is ubiquitous in application areas such as signal compression, imaging restoration, and supervised machine learning. The inference problems may be solved using variational formulations that provide theoretically proven methods and algorithms. With ever-increasing model complexities and growing data size, new specially designed methods are urgently needed to recover meaningful quantifies of interest. We consider the broad spectrum of linear inverse problems where the aim is to reconstruct quantities with a sparse representation on some vector space; often solved using the (generalized) least absolute shrinkage and selection operator (lasso). The associated optimization problems have received significant attention, in particular in the early 2000's, because of their connection to compressed sensing and the reconstruction of solutions with favorable sparsity properties using augmented Lagrangians, alternating directions and splitting methods. We provide a new perspective on the underlying l1 regularized inverse problem by exploring the generalized lasso problem through variable projection methods. We arrive at our proposed variable projected augmented Lagrangian (vpal) method. We analyze this method and provide an approach for automatic regularization parameter selection based on a degrees of freedom argument. Further, we provide numerical examples demonstrating the computational efficiency for various imaging problems.

math.NA↗
arXiv2020

Fast non-convex low-rank matrix decomposition for separation of potential field data using minimal memory

A fast non-convex low-rank matrix decomposition method for potential field data separation is proposed. The singular value decomposition of the large size trajectory matrix, which is also a block Hankel matrix, is obtained using a fast randomized singular value decomposition algorithm in which fast block Hankel matrix-vector multiplications are implemented with minimal memory storage. This fast block Hankel matrix randomized singular value decomposition algorithm is integrated into the \texttt{Altproj} algorithm, which is a standard non-convex method for solving the robust principal component analysis optimization problem. The improved algorithm avoids the construction of the trajectory matrix. Hence, gravity and magnetic data matrices of large size can be computed. Moreover, it is more efficient than the traditional low-rank matrix decomposition method, which is based on the use of an inexact augmented Lagrange multiplier algorithm. The presented algorithm is also robust and, hence, algorithm-dependent parameters are easily determined. The improved and traditional algorithms are contrasted for the separation of synthetic gravity and magnetic data matrices of different sizes. The presented results demonstrate that the improved algorithm is not only computationally more efficient but it is also more accurate. Moreover, it is possible to solve far larger problems. As an example, for the adopted computational environment, matrices of sizes larger than $205 \times 205$ generate "out of memory" exceptions with the traditional method, but a matrix of size $2001\times 2001$ can be calculated in $1062.29$s with the new algorithm. Finally, the improved method is applied to separate real gravity and magnetic data in the Tongling area, Anhui province, China. Areas which may exhibit mineralizations are inferred based on the separated anomalies.

physics.geo-ph↗
arXiv2015

Functional partial canonical correlation

A rigorous derivation is provided for canonical correlations and partial canonical correlations for certain Hilbert space indexed stochastic processes. The formulation relies on a key congruence mapping between the space spanned by a second order, $\mathcal{H}$-valued, process and a particular Hilbert function space deriving from the process' covariance operator. The main results are obtained via an application of methodology for constructing orthogonal direct sums from algebraic direct sums of closed subspaces.

math.ST↗
arXiv2014

Non-negatively constrained least squares and parameter choice by the residual periodogram for the inversion of electrochemical impedance spectroscopy

The inverse problem associated with electrochemical impedance spectroscopy requiring the solution of a Fredholm integral equation of the first kind is considered. If the underlying physical model is not clearly determined, the inverse problem needs to be solved using a regularized linear least squares problem that is obtained from the discretization of the integral equation. For this system, it is shown that the model error can be made negligible by a change of variables and by extending the effective range of quadrature. This change of variables serves as a right preconditioner that significantly improves the condition of the system. Still, to obtain feasible solutions the additional constraint of non-negativity is required. Simulations with artificial, but realistic, data demonstrate that the use of non-negatively constrained least squares with a smoothing norm provides higher quality solutions than those obtained without the non-negative constraint. Using higher-order smoothing norms also reduces the error in the solutions. The L-curve and residual periodogram parameter choice criteria, which are used for parameter choice with regularized linear least squares, are successfully adapted to be used for the non-negatively constrained Tikhonov least squares problem. Although these results have been verified within the context of the analysis of electrochemical impedance spectroscopy, there is no reason to suppose that they would not be relevant within the broader framework of solving Fredholm integral equations for other applications.

math.NA↗
arXiv2009

FDG-PET Parametric Imaging by Total Variation Minimization

Parametric imaging of the cerebral metabolic rate for glucose (CMRGlc) using [18F]-fluorodeoxyglucose positron emission tomography is considered. Traditional imaging is hindered due to low signal to noise ratios at individual voxels. We propose to minimize the total variation of the tracer uptake rates while requiring good fit of traditional Patlak equations. This minimization guarantees spatial homogeneity within brain regions and good distinction between brain regions. Brain phantom simulations demonstrate significant improvement in quality of images by the proposed method as compared to Patlak images with post-filtering using Gaussian or median filters.

q-bio.QM↗