Rosemary A Renaut

• Arizona State University

The singular value decomposition (SVD) of a reordering of a matrix A can be used to determine an efficient Kronecker product (KP) sum approximation to A. We present the use of an approximate truncated SVD (TSVD) to find the KP approximation, and contrast using a randomized singular value decomposition algorithm (RSVD), a new enlarged Golub Kahan Bi...

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 $\ell_1$...

We consider the solution of the ℓ 1 regularized image deblurring problem using isotropic and anisotropic regularization implemented with the split Bregman algorithm. We replace the system matrix A using a Kronecker product approximation obtained via an approximate truncated singular value decomposition for a reordering of the matrix A . To obtain t...

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 reord...

ℓ1 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 ℓ2-regularized minimization problem. We consider selecting the regul...

We consider an ℓ 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 Maj...

Within the iron metallogenic province of southeast Missouri, USA, there are several mines that contain not only economic iron resources, magnetite and/or hematite, but also contain rare earth elements, copper and gold. An area including three major deposits, Pea Ridge, Bourbon, and Kratz Spring, was selected for detailed modeling for the upper crus...

The Tell Atlas of Algeria has a huge potential for hydrothermal energy from over 240 thermal springs with temperatures up to 98◦ C in the Guelma area. The most exciting region is situated in the northeastern part which is known to have the hottest hydrothermal systems. In this work, we use a high-resolution gravity study to identify the location an...

The mixed L_p -norm, 0 ≤ p ≤ 2, stabilization algorithm is flexible for constructing a suite of subsurface models with either distinct, or a combination of, smooth, sparse, or blocky structures. This general purpose algorithm can be used for the inversion of data from regions with different subsurface characteristics. Model interpreta...

Complementary independent and joint focusing inversion investigations are applied on the magnetic and gravity data sets for two kimberlite pipes, BK54 and BK55, in Botswana. The magnetic data are high resolution and, clearly, indicates two anomalies in the survey area. Independent inversion of this magnetic data provides a focused image of the subs...

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 m...

A fast algorithm for the large-scale joint inversion of gravity and magnetic data is developed. The algorithm uses a nonlinear Gramian constraint to impose correlation between the density and susceptibility of the reconstructed models. The global objective function is formulated in the space of the weighted parameters, but the Gramian constraint is...

A fast algorithm for the large-scale joint inversion of gravity and magnetic data is developed. It uses a nonlinear Gramian constraint to impose correlation between density and susceptibility of reconstructed models. The global objective function is formulated in the space of the weighted parameters, but the Gramian constraint is implemented in the...

During the inversion of discrete linear systems, noise in data can be amplified and result in meaningless solutions. To combat this effect, characteristics of solutions that are considered desirable are mathematically implemented during inversion, which is a process called regularization. The influence of provided prior information is controlled by...

The purpose of this study was to estimate how much resident outcomes can improve with an increase in hours per resident day (HPRD) of registered nurses (RNs) staffing. Nursing home (NH) staff in Korea have serious problems with inappropriate nurse staffing standards and poor working conditions, which lead to poor quality of care for NH residents. T...

An efficient algorithm for the $\mathrm {L}_{ \mathrm {p}}$ -norm joint inversion of gravity and magnetic data using the cross-gradient constraint is presented. The presented framework incorporates stabilizers that use $\mathrm {L}_{ \mathrm {p}}$ -norms ( $0\leq \mathrm {p} \leq 2$ ) of the model parameters, and/or the gradient of the model p...

An efficient algorithm for the Lp-norm joint inversion of gravity and magnetic data using the cross-gradient constraint is presented. The presented framework incorporates stabilizers that use Lp-norms (0 � p � 2) of the model parameters, and/or the gradient of the model parameters. The formulation is developed from standard approaches for independe...

Fast computation of three-dimensional gravity and magnetic forward models is considered. When the measurement data is assumed to be obtained on a uniform grid which is staggered with respect to the discretization of the parameter volume, the resulting kernel sensitivity matrices exhibit block-Toeplitz Toeplitz-block (BTTB) structure. These matrices...

The focusing inversion of gravity and magnetic potential field data using the randomized singularvalue decomposition methodology is considered. This approach facilitates tackling the computational challenge that arises in the solution of the inversion problem that uses the standard and accurate approximation of the integral equation kernel. A compr...

Focusing inversion of potential field data for the recovery of sparse subsurface structures from surface measurement data on a uniform grid is discussed. For the uniform grid the model sensitivity matrices exhibit block Toeplitz Toeplitz block structure, by blocks for each depth layer of the subsurface. Then, through embedding in circulant matrices...

A generalized unifying approach for $L_{p}$-norm joint inversion of gravity and magnetic data using the cross-gradient constraint is presented. The presented framework incorporates stabilizers that use $L_{0}$, $L_{1}$, and $L_{2}$-norms of the model parameters, and/or the gradient of the model parameters. Furthermore, the formulation is developed...

Fast computation of three-dimensional gravity and magnetic forward models is considered. When the measurement data is assumed to be obtained on a uniform grid which is staggered with respect to the discretization of the parameter volume, the resulting kernel sensitivity matrices exhibit block-Toeplitz Toeplitz-block (BTTB) structure. These matrices...

We present a brief review of the widely-used and well-known stabilizers in the inversion of potential field data. These include stabilizers that use L 2 , L 1 , and L 0 norms of the model parameters, and the gradients of the model parameters. These stabilizers may all be realized in a common setting using two general forms with different weighting...

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 imp...

In gravitational inverse problems, the density and geometry of the subsurface body are usually determined. In contrast, recently, Bijani et al. (2015) developed a 3D inversion methodology based on graph theory, which delineates the skeleton of a subsurface body. An ensemble of similar point masses is used to model the subsurface homogenous body. Th...

The large-scale focusing inversion of gravity and magnetic potential field data using $L_1$-norm regularization is considered. The use of the randomized singular value decomposition methodology facilitates tackling the computational challenge that arises in the solution of these large-scale inverse problems. As such the powerful randomized singular...

The truncated singular value decomposition may be used to find the solution of linear discrete ill-posed problems in conjunction with Tikhonov regularization and requires the estimation of a regularization parameter that balances between the sizes of the fit to data function and the regularization term. The unbiased predictive risk estimator is one...

We present an open source MATLAB package, IGUG, for 3D inversion of gravity data. The algorithm implemented in this package is based on methodology that was introduced by Bijani et al. (2015). A homogeneous subsurface body is modeled by an ensemble of simple point masses. The model parameters are the Cartesian coordinates of the point masses and th...

The open source MATLAB package IGUG for 3D inversion of gravity data is presented. It is based on methodology that was introduced by Bijani et al (2015), in which a homogeneous subsurface body is modeled by an ensemble of simple point masses. The model parameters are the Cartesian coordinates of the point masses and their total mass. Associating th...

The truncated singular value decomposition (TSVD) may be used to find the solution of the linear discrete ill-posed problem $A x \approx b$ using Tikhonov regularization. Regularization parameter $\alpha^2$ balances between the sizes of the fit to data functional $\|A x - b\|_2^2$ and the regularization term $\| x\|_2^2$. Minimization of the unbias...

Tikhonov regularization for projected solutions of large-scale ill-posed problems is considered. Traditionally hybrid LSQR iterative methods are used to find the solution on a subspace that inherits the ill-conditioning of the original problem and regularization is imposed at the subspace level. Modern techniques employ a randomized singular value...

We present a fast algorithm for the total variation regularization of the $3$-D gravity inverse problem. Through imposition of the total variation regularization, subsurface structures presenting with sharp discontinuities are preserved better than when using a conventional minimum-structure inversion. The associated problem formulation for the reg...

We present a fast algorithm for the total variation regularization of the $3$-D gravity inverse problem. Through imposition of the total variation regularization, subsurface structures presenting with sharp discontinuities are preserved better than when using a conventional minimum-structure inversion. The associated problem formulation for the reg...

Sparse inversion of gravity data based on L1-norm regularization is discussed. An iteratively reweighted least squares algorithm is used to solve the problem. At each iteration the solution of a linear system of equations and the determination of a suitable regularization parameter are considered. The LSQR iteration is used to project the system of...

A fast algorithm for solving the under-determined 3-D linear gravity inverse problem based on the randomized singular value decomposition (RSVD) is developed. The algorithm combines an iteratively reweighted approach for $L_1$-norm regularization with the RSVD methodology in which the large scale linear system at each iteration is replaced with a m...

A fast algorithm for solving the under-determined 3-D linear gravity inverse problem based on the randomized singular value decomposition (RSVD) is developed. The algorithm combines an iteratively reweighted approach for $L_1$-norm regularization with the RSVD methodology in which the large scale linear system at each iteration is replaced with a m...

The recent paper of Ghalehnoee et al. Ghalehnoee et al. (2016), “Improving compact gravity inversion based on new weighting functions” discusses weighting functions for the compact inversion of gravity data. We studied the paper with great interest but deduced that the paper presents minor changes to already published methods. In the manuscript, th...

The solution, \(\varvec{x}\), of the linear system of equations \(A\varvec{x}\approx \varvec{b}\) arising from the discretization of an ill-posed integral equation \(g(s)=\int H(s,t) f(t) \,dt\) with a square integrable kernel H(s, t) is considered. The Tikhonov regularized solution \(\varvec{x}(\lambda )\) approximating the Galerkin coefficients o...

Tikhonov regularization for projected solutions of large-scale ill-posed problems is considered. The Golub-Kahan iterative bidiagonalization is used to project the problem onto a subspace and regularization then applied to find a subspace approximation to the full problem. Determination of the regularization parameter using the method of unbiased p...

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' covar...

The $\chi^2$ principle and the unbiased predictive risk estimator are used to determine optimal regularization parameters in the context of 3D focusing gravity inversion with the minimum support stabilizer. At each iteration of the focusing inversion the minimum support stabilizer is determined and then the fidelity term is updated using the standa...

The χ 2 > principle generalizes the Morozov discrepancy principle to the augmented residual of the Tikhonov regularized least squares problem. For weighting of the data fidelity by a known Gaussian noise distribution on the measured data, when the stabilizing, or regularization, term is considered to be weighted by unknown inverse covariance inform...

The solution of the linear system of equations $A\mathbf{x}\approx \mathbf{b}$ arising from the discretization of an ill-posed integral equation with a square integrable kernel is considered. The solution by means of Tikhonov regularization in which $\mathbf{x}$ is found to as the minimizer of $J(\mathbf{x})=\{ \|A \mathbf{x} -\mathbf{b}\|_2^2 + \l...

We investigate the use of Tikhonov regularization with the minimum support stabilizer for underdetermined 2-D inversion of gravity data. This stabilizer produces models with non-smooth properties which is useful for identifying geologic structures with sharp boundaries. A very important aspect of using Tikhonov regularization is the choice of the r...

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 discretizatio...

This document contains supplementary derivations and discussions not provided in the submitted paper with the same title. Additional results for the NCP and L-Curve comparisons with higher noise levels are given.

Determining parameters which describe the performance of a microbial fuel cell requires the solution of an inverse problem. Two formulations have been presented in the literature: a convolutional approach or a direct quadrature approach. A complete study and analysis of the direct quadrature method, which leads to two systems for the unknown signal...

The method of multisplitting (MS), implemented as a restricted additive Schwarz type algorithm, is extended for the solution of regularized least squares problems. The presented non-stationary version of the algorithm uses dynamic updating of the weights applied to the subdomains in reconstituting the global solution. Standard convergence results f...

We present a new method for estimating the edges in a piecewise smooth function from blurred and noisy Fourier data. The proposed method is constructed by combining the so called concentration factor edge detection method, which uses a finite number of Fourier coefficients to approximate the jump function of a piecewise smooth function, with compre...

A method to improve the signal-to-noise-ratio (SNR) of positron emission tomography (PET) scans is presented. A wavelet-based image decomposition technique decomposes an image into two parts, one which primarily contains the desired restored image and the other primarily the remaining unwanted portion of the image. Because the method is based on a...

This paper is concerned with estimating the solutions of numerically ill-posed least squares problems through Tikhonov regularization. Given a priori estimates on the covariance structure of errors in the measurement data b, and a suitable statistically-chosen , the Tikhonov regularized least squares functional J( ) = kAx bk2 Wb + 1= 2kD(x x0)k2 2,...

The expectation maximization algorithm is commonly used to reconstruct images obtained from positron emission tomography sinograms. For images with acceptable signal to noise ratios, iterations are terminated prior to convergence. A new quantitative and reproducible stopping rule is designed and validated on simulations using a Monte-Carlo generate...

This paper addresses the reconstruction of compactly supported functions from non-uniform samples of their Fourier transform. We briefly investigate the consequences of acquiring non-uniform spectral data. We summarize two often applied reconstruction methods, convolutional gridding and uniform re-sampling, and investigate the reconstruction accura...

Graphical analysis methods are widely used in positron emission tomography quantification because of their simplicity and model independence. But they may, particularly for reversible kinetics, lead to bias in the estimated parameters. The source of the bias is commonly attributed to noise in the data. Assuming a two-tissue compartmental model, we...

Linear least squares problems with box constraints are commonly solved to find model parameters within bounds based on physical considerations. Common algorithms include Bounded Variable Least Squares (BVLS) and the Matlab function lsqlin. Here, the goal is to find solutions to ill-posed inverse problems that lie within box constraints. To do this,...

We present a novel deconvolution approach to accurately restore piecewise smooth signals from blurred data. The first stage uses Higher Order Total Variation restorations to obtain an estimate of the location of jump discontinuities from the blurred data. In the second stage the estimated jump locations are used to determine the local orders of a V...

Graphical processing units introduce the capability for large scale computa-tion in the desk top environment. For the solution of linear systems of equa-tions, much effort has been devoted to efficient implementation of Krylov subspace-based solvers in high performance computing environments. Here the focus is to improve the computational efficienc...

We propose a new formulation of the Support Vector Ma-chine (SVM) for classifying genetic data. It is based on the development of ideas from the method of total least squares, in which assumed error in measured data are incorporated in the model design. For genetic data the number of features is always far greater than the sample size. Consequently...

Parametric imaging of the cerebral metabolic rate for glucose (CMRGlc) using [(18)F]-fluoro deoxyglucose 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 Pat...

In this paper we extend the method of inter-modality image registration using the maximization of normalized mutual information (NMI) for the registration of [18F]-2-fluoro-deoxy-D-glucose (FDG)-positron emission tomography (PET) with T1-weighted magnetic resonance (MR) volumes. We investigate the impact on the NMI maximization with respect to usin...

The preconditioned iteratively regularized Gauss–Newton algorithm for the minimization of general nonlinear functionals was introduced by Smirnova, Renaut and Khan (Inverse Problems 23: 1547–1563, 2007). In this paper, we establish theoretical convergence results for an extended stabilized family of Generalized Preconditioned Iterative methods whic...

Graphical analysis methods are widely used in positron emission tomography quantification because of their simplicity and model independence. But they may, particularly for reversible kinetics, lead to bias in the estimated parameters. The source of the bias is commonly attributed to noise in the data. Assuming a two-tissue compartmental model, we...

Logan's graphical analysis (LGA) is a widely-used approach for quantification of biochemical and physiological processes from Positron emission tomography (PET) image data. A well-noted problem associated with the LGA method is the bias in the estimated parameters. We recently systematically evaluated the bias associated with the linear model appro...

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...

We discuss the solution of numerically ill-posed overdetermined systems of equations using Tikhonov a-priori-based regularization. When the noise distribution on the measured data is available to appropriately weight the fidelity term, and the regularization is assumed to be weighted by inverse covariance information on the model parameters, the un...

Numerical methods for the solution of the structured data least squares problem with special application in digital filtering are investigated. While the minimum mean-square error, i.e. ordinary least squares formulation, solves the linear system of equations for the case of noise in the right hand side, data least squares is formulated for the pro...

We present an approach to automatically estimate an empirical source characterization of deep earthquakes recorded teleseismically and subsequently remove the source from the recordings by applying regularized deconvolution. A principle goal in this work is to effectively deblur the seismograms, resulting in more impulsive and narrower pulses, perm...

We previously developed a noninvasive technique for the quantification of fluorodeoxyglucose (FDG) positron emission tomography (PET) images using an image-derived input function obtained from a manually drawn carotid artery region. Here, we investigate the use of independent component analysis (ICA) for more objective identification of the carotid...

A new model of an input function for human [(18)F]-2-Deoxy-2-fluoro-d-glucose fluoro (FDG) positron emission tomography (PET) brain studies with bolus injection is presented. Input data for early time, roughly up to 0.6 min, were obtained noninvasively from the time-activity curve (TAC) measured from a carotid artery region of interest. Representat...

Theoretical convergence results for an iteratively regularized Gauss–Newton (IRGN) algorithm with a Tikhonov regularization term using a seminorm generated by a linear operator are established. The convergence theorem uses an a posteriori stopping rule and a modified source condition, without any restriction on the nonlinearity of the operator. The...

This paper presents a cell evolution analysis (CEA) scheme for bladder cancer cells. The proposed scheme consists of a cell migration analysis component for computing the overall migration rate of the cell cluster, and a cell proliferation analysis component based on counting the individually segmented cells within the cell cluster at different tim...

We present a method of signal restoration to improve the signal-to-noise ratio, sharpen seismic arrival onset, and act as an empirical source deconvolution of specific seismic arrivals. Observed time-series gi are modelled as a convolution of a simpler time-series fi, and an invariant point spread function (PSF) h that attempts to account for the e...

A reliable, semi-automated method for estimation of a minimally-invasive image-derived input function is validated for human [ 18 F]-fluoro deoxyglucose (FDG) positron emission tomography (PET) studies. Two time windows can be recognized in the time activity curve measured from a carotid artery region of interest (CA-ROI). During the first short wi...

A novel parallel method for determining an approximate total least squares (TLS) solution is introduced. Based on domain decomposition, the global TLS problem is partitioned into several dependent TLS subproblems. A convergent algorithm using the parallel variable distribution technique (SIAM J. Optim. 1994; 4:815–832) is presented. Numerical resul...

A new approach for cell migration analysis is presented from an automated image segmentation and ROI-compression. ROI-compression is accomplished through classical edge detection technique, followed by image opening to help identify the cell-occupied image region. That region is then compressed for better archival and analysis, to more efficiently...

This paper considers the issue of parameter estimation for biomedical applications using nonuniformly sampled data. The generalized linear least squares (GLLS) algorithm, first introduced by Feng and Ho (1993), is used in the medical imaging community for removal of bias when the data defining the model are correlated. GLLS provides an efficient it...

This article introduces an automated method for the computation of changes in brain volume from sequential magnetic resonance images (MRIs) using an iterative principal component analysis (IPCA) and demonstrates its ability to characterize whole-brain atrophy rates in patients with Alzheimer's disease (AD). The IPCA considers the voxel intensity pa...

Fast algorithms, based on the unsymmetric look-ahead Lanczos and the Arnoldi process, are developed for the estimation of the functional Φ(ƒ)=uTƒ(A) v for fixed u, v and A, where A∈n×n is a large-scale unsymmetric matrix. Numerical results are presented which validate the comparable accuracy of both approaches. Although the Arnoldi process reaches...

Error-contaminated systems Ax ≈ b, for which A is ill-conditioned, are considered. Such systems may,be solved using Tikhonov-like regularized total least squares (RTLS) methods. Golub, Hansen, and O’Leary [SIAM J. Matrix Anal. Appl., 21 (1999), pp. 185–194] presented a parameter-dependent direct algorithm for the solution of the augmented,Lagrange...

The Gegenbauer image reconstruction method, previously shown to improve the quality of magnetic resonance images, is utilized in this study as a segmentation preprocessing step. It is demonstrated that, for all simulated and real magnetic resonance images used in this study, the Gegenbauer reconstruction method improves the accuracy of segmentation...