arXiv subjects

Anne Gelb

Publications and source records attributed to Anne Gelb.

At least 37 records · Page 2Linked to original sources

arXiv2022

Improving numerical accuracy for the viscous-plastic formulation of sea ice

Accurate modeling of sea ice dynamics is critical for predicting environmental variables and is important in applications such as navigating ice breaker ships. Research for both modeling and simulating sea ice dynamics is ongoing, with the most widely accepted model based on the viscous-plastic (VP) formulation introduced by Hibler in 1979. Due to its highly nonlinear features, this model is intrinsically challenging for computational solvers. In particular, sea ice simulations often significantly differ from satellite observations. This study therefore focuses on improving the numerical accuracy of the VP sea ice model. Since the poor convergence observed in existing numerical simulations stems from the nonlinear nature of the VP formulation, this investigation proposes using the celebrated weighted essentially non-oscillatory (WENO) scheme -- as opposed to the frequently employed centered difference (CD) scheme -- for the spatial derivatives in the VP sea ice model. We then proceed to numerically demonstrate that WENO yields higher-order convergence for smooth solutions, and that furthermore it is able to resolve the discontinuities in the sharp features of sea ice covers -- something that is not possible using CD methods. Finally, our proposed framework integrates a potential function method that utilizes the phase field method to naturally incorporates the physical restrictions of ice thickness and ice concentration in transport equations, resulting in a modified transport equations which includes additional forcing terms. Our method does not require post-processing, thereby avoiding the possible introduction of discontinuities and corresponding negative impact on the solution behavior. Numerical experiments are provided to demonstrate the efficacy of our new methodology.

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arXiv2022

Sequential image recovery from noisy and under-sampled Fourier data

A new algorithm is developed to jointly recover a temporal sequence of images from noisy and under-sampled Fourier data. Specifically, we consider the case where each data set is missing vital information that prevents its (individual) accurate recovery. Our new method is designed to restore the missing information in each individual image by "borrowing" it from the other images in the sequence. As a result, {\em all} of the individual reconstructions yield improved accuracy. The use of high resolution Fourier edge detection methods is essential to our algorithm. In particular, edge information is obtained directly from the Fourier data which leads to an accurate coupling term between data sets. Moreover, data loss is largely avoided as coarse reconstructions are not required to process inter- and intra-image information. Numerical examples are provided to demonstrate the accuracy, efficiency and robustness of our new method.

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arXiv2021

Empirical Bayesian Inference using Joint Sparsity

This paper develops a new empirical Bayesian inference algorithm for solving a linear inverse problem given multiple measurement vectors (MMV) of under-sampled and noisy observable data. Specifically, by exploiting the joint sparsity across the multiple measurements in the sparse domain of the underlying signal or image, we construct a new support informed sparsity promoting prior. Several applications can be modeled using this framework, and as a prototypical example we consider reconstructing an image from synthetic aperture radar (SAR) observations using nearby azimuth angles. Our numerical experiments demonstrate that using this new prior not only improves accuracy of the recovery, but also reduces the uncertainty in the posterior when compared to standard sparsity producing priors.

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arXiv2021

Stabilizing Radial Basis Function Methods for Conservation Laws Using Weakly Enforced Boundary Conditions

It is well understood that boundary conditions (BCs) may cause global radial basis function (RBF) methods to become unstable for hyperbolic conservation laws (CLs). Here we investigate this phenomenon and identify the strong enforcement of BCs as the mechanism triggering such stability issues. Based on this observation we propose a technique to weakly enforce BCs in RBF methods. In the case of hyperbolic CLs, this is achieved by carefully building RBF methods from the weak form of the CL, rather than the typically enforced strong form. Furthermore, we demonstrate that global RBF methods may violate conservation, yielding physically unreasonable solutions when the approximation does not take into account these considerations. Numerical experiments validate our theoretical results.

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arXiv2020

Estimation and uncertainty quantification for piecewise smooth signal recovery

This paper presents a sparse Bayesian learning (SBL) algorithm for linear inverse problems with a high order total variation (HOTV) sparsity prior. For the problem of sparse signal recovery, SBL often produces more accurate estimates than maximum a posteriori estimates, including those that rely on l1 regularization. Moreover, rather than a single signal estimate, SBL yields a full posterior density estimate which can be used for uncertainty quantification. However, SBL is only immediately applicable to problems having a direct sparsity prior, or to those that can be formed via synthesis. This paper demonstrates how a problem with an HOTV sparsity prior can be formulated via synthesis, and then develops a corresponding Bayesian learning method. This expands the class of problems available to Bayesian learning to include, e.g., inverse problems dealing with the recovery of piecewise smooth functions or signals from data. Numerical examples are provided to demonstrate how this new technique is effectively employed.

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arXiv2020

Synthetic Aperture Radar Image Formation with Uncertainty Quantification

Synthetic aperture radar (SAR) is a day or night any-weather imaging modality that is an important tool in remote sensing. Most existing SAR image formation methods result in a maximum a posteriori image which approximates the reflectivity of an unknown ground scene. This single image provides no quantification of the certainty with which the features in the estimate should be trusted. In addition, finding the mode is generally not the best way to interrogate a posterior. This paper addresses these issues by introducing a sampling framework to SAR image formation. A hierarchical Bayesian model is constructed using conjugate priors that directly incorporate coherent imaging and the problematic speckle phenomenon which is known to degrade image quality. Samples of the resulting posterior as well as parameters governing speckle and noise are obtained using a Gibbs sampler. These samples may then be used to compute estimates, and also to derive other statistics like variance which aid in uncertainty quantification. The latter information is particularly important in SAR, where ground truth images even for synthetically-created examples are typically unknown. An example result using real-world data shows that the sampling-based approach introduced here to SAR image formation provides parameter-free estimates with improved contrast and significantly reduced speckle, as well as unprecedented uncertainty quantification information.

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arXiv2019

Thresholded Non-Uniform Fourier Frame-Based Reconstruction for Stripmap SAR

Fourier domain methods are fast algorithms for SAR imaging. They typically involve an interpolation in the frequency domain to re-grid non-uniform data so inverse fast Fourier transforms can be performed. In this paper, we apply a frame reconstruction algorithm, extending the non-uniform fast Fourier transform, to stripmap SAR data. Further, we present an improved thresholded frame reconstruction algorithm for robust performance and improved computational efficiency. We demonstrate compelling results on real stripmap SAR data.

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arXiv2019

Accelerated Variance Based Joint Sparsity Recovery of Images from Fourier Data

Several problems in imaging acquire multiple measurement vectors (MMVs) of Fourier samples for the same underlying scene. Image recovery techniques from MMVs aim to exploit the joint sparsity across the measurements in the sparse domain. This is typically accomplished by extending the use of $\ell_1$ regularization of the sparse domain in the single measurement vector (SMV) case to using $\ell_{2,1}$ regularization so that the "jointness" can be accounted for. Although effective, the approach is inherently coupled and therefore computationally inefficient. The method also does not consider current approaches in the SMV case that use spatially varying weighted $\ell_1$ regularization term. The recently introduced variance based joint sparsity (VBJS) recovery method uses the variance across the measurements in the sparse domain to produce a weighted MMV method that is more accurate and more efficient than the standard $\ell_{2,1}$ approach. The efficiency is due to the decoupling of the measurement vectors, with the increased accuracy resulting from the spatially varying weight. This paper introduces the accelerated VBJS which reduces computational cost even further by eliminating the requirement to first approximate the underlying image in order to construct the weights. Eliminating this preprocessing step moreover reduces the amount of information lost from the data, so that our method is more accurate. Numerical examples provided in the paper verify these benefits.

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arXiv2019

High Order Edge Sensors with $\ell^1$ Regularization for Enhanced Discontinuous Galerkin Methods

This paper investigates the use of $\ell^1$ regularization for solving hyperbolic conservation laws based on high order discontinuous Galerkin (DG) approximations. We first use the polynomial annihilation method to construct a high order edge sensor which enables us to flag troubled elements. The DG approximation is enhanced in these troubled regions by activating $\ell^1$ regularization to promote sparsity in the corresponding jump function of the numerical solution. The resulting $\ell^1$ optimization problem is efficiently implemented using the alternating direction method of multipliers. By enacting $\ell^1$ regularization only in troubled cells, our method remains accurate and efficient, as no additional regularization or expensive iterative procedures are needed in smooth regions. We present results for the inviscid Burgers' equation as well as a nonlinear system of conservation laws using a nodal collocation-type DG method as a solver.

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arXiv2019

Total Variation Bayesian Learning via Synthesis

This paper presents a sparse Bayesian learning algorithm for inverse problems in signal and image processing with a total variation (TV) sparsity prior. Because of the prior used, and the fact that the prior parameters are estimated directly from the data, sparse Bayesian learning often produces more accurate results than the typical maximum a posteriori Bayesian estimates for sparse signal recovery. It also provides a full posterior distribution. However, sparse Bayesian learning is only available to problems with a direct sparsity prior or those formed via synthesis. This paper demonstrates how a problem with a TV sparsity prior can be formulated in a synthesis approach. We then develop a method that combines this synthesis-based TV with the sparse Bayesian learning algorithm and provide numerical examples to demonstrate how our new technique is effectively employed.

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arXiv2019

Image reconstruction enhancement via masked regularization

Image reconstruction based on an edge-sparsity assumption has become popular in recent years. Many methods of this type are capable of reconstructing nearly perfect edge-sparse images using limited data. In this paper, we present a method to improve the accuracy of a suboptimal image resulting from an edge-sparsity image reconstruction method when compressed sensing or empirical data requirements are not met. The method begins with an edge detection from an initial edge-sparsity based reconstruction. From this edge map, a mask matrix is created which allows us to regularize exclusively in regions away from edges. By accounting for the spatial distribution of the sparsity, our method preserves edge information and and furthermore enhances suboptimal reconstructions to be nearly perfect from fewer data than needed by the initial method. We present results for two phantom images using a variety of initial reconstruction methods.

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arXiv2019

Edge-masked CT image reconstruction from limited data

This paper presents an iterative inversion algorithm for computed tomography image reconstruction that performs well in terms of accuracy and speed using limited data. The computational method combines an image domain technique and statistical reconstruction by using an initial filtered back projection reconstruction to create a binary edge mask, which is then used in an l2-regularized reconstruction. Both theoretical and empirical results are offered to support the algorithm. While in this paper a simple forward model is used and physical edges are used as the sparse feature, the proposed method is flexible and can accommodate any forward model and sparsifying transform.

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arXiv2018

Detecting edges from non-uniform Fourier data via sparse Bayesian learning

In recent investigations, the problem of detecting edges given non-uniform Fourier data was reformulated as a sparse signal recovery problem with an l1-regularized least squares cost function. This result can also be derived by employing a Bayesian formulation. Specifically, reconstruction of an edge map using l1 regularization corresponds to a so-called type-I (maximum a posteriori) Bayesian estimate. In this paper, we use the Bayesian framework to design an improved algorithm for detecting edges from non-uniform Fourier data. In particular, we employ what is known as type-II Bayesian estimation, specifically a method called sparse Bayesian learning. We also show that our new edge detection method can be used to improve downstream processes that rely on accurate edge information like image reconstruction, especially with regards to compressed sensing techniques.

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arXiv2018

Edge-adaptive l2 regularization image reconstruction from non-uniform Fourier data

Total variation regularization based on the l1 norm is ubiquitous in image reconstruction. However, the resulting reconstructions are not always as sparse in the edge domain as desired. Iteratively reweighted methods provide some improvement in accuracy, but at the cost of extended runtime. In this paper we examine these methods for the case of data acquired as non-uniform Fourier samples. We then develop a non-iterative weighted regularization method that uses a pre-processing edge detection to find exactly where the sparsity should be in the edge domain. We show that its performance in terms of both accuracy and speed has the potential to outperform reweighted TV regularization methods.

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arXiv2017

Using Frame Theoretic Convolutional Gridding for Robust Synthetic Aperture Sonar Imaging

Recent progress in synthetic aperture sonar (SAS) technology and processing has led to significant advances in underwater imaging, outperforming previously common approaches in both accuracy and efficiency. There are, however, inherent limitations to current SAS reconstruction methodology. In particular, popular and efficient Fourier domain SAS methods require a 2D interpolation which is often ill conditioned and inaccurate, inevitably reducing robustness with regard to speckle and inaccurate sound-speed estimation. To overcome these issues, we propose using the frame theoretic convolution gridding (FTCG) algorithm to handle the non-uniform Fourier data. FTCG extends upon non-uniform fast Fourier transform (NUFFT) algorithms by casting the NUFFT as an approximation problem given Fourier frame data. The FTCG has been show to yield improved accuracy at little more computational cost. Using simulated data, we outline how the FTCG can be used to enhance current SAS processing.

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arXiv2017

Recovering Fine Details from Under-Resolved Electron Tomography Data using HOTV Regularization

Over the last decade or so, reconstruction methods using $\ell_1$ regularization, often categorized as compressed sensing (CS) algorithms, have significantly improved the capabilities of high fidelity imaging in electron tomography. The most popular $\ell_1$ regularization approach within electron tomography has been total variation (TV) regularization. In addition to reducing unwanted noise, TV regularization encourages a piecewise constant solution with sparse boundary regions. In this paper we propose an alternative $\ell_1$ regularization approach for electron tomography based on higher order total variation (HOTV). Like TV, the HOTV approach promotes solutions with sparse boundary regions. In smooth regions however, the solution is not limited to piecewise constant behavior. We demonstrate that this allows for more accurate reconstruction of a broader class of images -- even those for which TV was designed for -- particularly when dealing with pragmatic tomographic sampling patterns and very fine image features. We develop results for an electron tomography data set as well as a phantom example, and we also make comparisons with discrete tomography approaches.

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arXiv2015

A Two-dimensional Inverse Frame Operator Approximation Technique

The ability to efficiently and accurately construct an inverse frame operator is critical for establishing the utility of numerical frame approximations. Recently, the admissible frame method was developed to approximate inverse frame operators for one-dimensional problems. Using the admissible frame approach, it is possible to project the corresponding frame data onto a more suitable (admissible) frame, even when the sampling frame is only weakly localized. As a result, a target function may be approximated as a finite frame expansion with its asymptotic convergence solely dependent on its smoothness. In this investigation, we seek to expand the admissible frame approach to two dimensions, which requires some additional constraints. We prove that the admissible frame technique converges in two dimensions and then demonstrate its usefulness with some numerical experiments that use sampling patterns inspired by applications that sample data non-uniformly in the Fourier domain.

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arXiv2014

A frame theoretic approach to the Non-Uniform Fast Fourier Transform

Nonuniform Fourier data are routinely collected in applications such as magnetic resonance imaging, synthetic aperture radar, and synthetic imaging in radio astronomy. To acquire a fast reconstruction that does not require an online inverse process, the non-uniform fast Fourier transform (NFFT), also called convolutional gridding, is frequently employed. While various investigations have led to improvements in accuracy, efficiency, and robustness of the NFFT, not much attention has been paid to the fundamental analysis of the scheme, and in particular its convergence properties. This paper analyzes the convergence of the NFFT by casting it as a Fourier frame approximation. In so doing, we are able to design parameters for the method that satisfy conditions for numerical convergence. Our so called frame theoretic convolutional gridding algorithm can also be applied to detect features (such as edges) from non-uniform Fourier samples of piecewise smooth functions.

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