arXiv subjects

Per Christian Hansen

Publications and source records attributed to Per Christian Hansen.

At least 19 recordsLinked to original sources

arXiv2026

On spectral properties and fast initial convergence of the Kaczmarz method

The Kaczmarz method is successfully used for solving discretizations of linear inverse problems, especially in computed tomography where it is known as ART. Practitioners often observe and appreciate its fast convergence in the first few iterations, leading to the same favorable semi-convergence that we observe for simultaneous iterative reconstruction methods. While the latter methods have symmetric and positive definite iteration operators that facilitate their analysis, the operator in Kaczmarz's method is nonsymmetric and it has been an open question so far to understand this fast initial convergence. We perform a spectral analysis of Kaczmarz's method that gives new insight into its (often fast) initial behavior. We also carry out a statistical analysis of how the data noise enters the iteration vectors, which sheds new light on the semi-convergence. Our results are illustrated with several numerical examples.

math.NA↗

arXiv2024

CUQIpy: I. Computational uncertainty quantification for inverse problems in Python

This paper introduces CUQIpy, a versatile open-source Python package for computational uncertainty quantification (UQ) in inverse problems, presented as Part I of a two-part series. CUQIpy employs a Bayesian framework, integrating prior knowledge with observed data to produce posterior probability distributions that characterize the uncertainty in computed solutions to inverse problems. The package offers a high-level modeling framework with concise syntax, allowing users to easily specify their inverse problems, prior information, and statistical assumptions. CUQIpy supports a range of efficient sampling strategies and is designed to handle large-scale problems. Notably, the automatic sampler selection feature analyzes the problem structure and chooses a suitable sampler without user intervention, streamlining the process. With a selection of probability distributions, test problems, computational methods, and visualization tools, CUQIpy serves as a powerful, flexible, and adaptable tool for UQ in a wide selection of inverse problems. Part II of the series focuses on the use of CUQIpy for UQ in inverse problems with partial differential equations (PDEs).

math.NA↗

arXiv2024

CUQIpy: II. Computational uncertainty quantification for PDE-based inverse problems in Python

Inverse problems, particularly those governed by Partial Differential Equations (PDEs), are prevalent in various scientific and engineering applications, and uncertainty quantification (UQ) of solutions to these problems is essential for informed decision-making. This second part of a two-paper series builds upon the foundation set by the first part, which introduced CUQIpy, a Python software package for computational UQ in inverse problems using a Bayesian framework. In this paper, we extend CUQIpy's capabilities to solve PDE-based Bayesian inverse problems through a general framework that allows the integration of PDEs in CUQIpy, whether expressed natively or using third-party libraries such as FEniCS. CUQIpy offers concise syntax that closely matches mathematical expressions, streamlining the modeling process and enhancing the user experience. The versatility and applicability of CUQIpy to PDE-based Bayesian inverse problems are demonstrated on examples covering parabolic, elliptic and hyperbolic PDEs. This includes problems involving the heat and Poisson equations and application case studies in electrical impedance tomography and photo-acoustic tomography, showcasing the software's efficiency, consistency, and intuitive interface. This comprehensive approach to UQ in PDE-based inverse problems provides accessibility for non-experts and advanced features for experts.

math.NA↗

arXiv2024

Inferring Object Boundaries and their Roughness with Uncertainty Quantification

This work describes a Bayesian framework for reconstructing the boundaries that represent targeted features in an image, as well as the regularity (i.e., roughness vs. smoothness) of these boundaries.This regularity often carries crucial information in many inverse problem applications, e.g., for identifying malignant tissues in medical imaging. We represent the boundary as a radial function and characterize the regularity of this function by means of its fractional differentiability. We propose a hierarchical Bayesian formulation which, simultaneously, estimates the function and its regularity, and in addition we quantify the uncertainties in the estimates. Numerical results suggest that the proposed method is a reliable approach for estimating and characterizing object boundaries in imaging applications, as illustrated with examples from X-ray CT and image inpainting. We also show that our method is robust under various noise types, noise levels, and incomplete data.

math.NA↗

arXiv2022

Uncertainty Quantification of Inclusion Boundaries in the Context of X-ray Tomography

In this work, we describe a Bayesian framework for reconstructing the boundaries of piecewise smooth regions in the X-ray computed tomography (CT) problem in an infinite-dimensional setting. In addition to the reconstruction, we are also able to quantify the uncertainty of the predicted boundaries. Our approach is goal oriented, meaning that we directly detect the discontinuities from the data, instead of reconstructing the entire image. This drastically reduces the dimension of the problem, which makes the application of Markov Chain Monte Carlo (MCMC) methods feasible. We show that our method provides an excellent platform for challenging X-ray CT scenarios (e.g., in case of noisy data, limited angle, or sparse angle imaging). We investigate the performance and accuracy of our method on synthetic data as well as on real-world data. The numerical results indicate that our method provides an accurate method in detecting boundaries of piecewise smooth regions and quantifies the uncertainty in the prediction.

math.NA↗

arXiv2022

Horseshoe priors for edge-preserving linear Bayesian inversion

In many large-scale inverse problems, such as computed tomography and image deblurring, characterization of sharp edges in the solution is desired. Within the Bayesian approach to inverse problems, edge-preservation is often achieved using Markov random field priors based on heavy-tailed distributions. Another strategy, popular in statistics, is the application of hierarchical shrinkage priors. An advantage of this formulation lies in expressing the prior as a conditionally Gaussian distribution depending of global and local hyperparameters which are endowed with heavy-tailed hyperpriors. In this work, we revisit the shrinkage horseshoe prior and introduce its formulation for edge-preserving settings. We discuss a sampling framework based on the Gibbs sampler to solve the resulting hierarchical formulation of the Bayesian inverse problem. In particular, one of the conditional distributions is high-dimensional Gaussian, and the rest are derived in closed form by using a scale mixture representation of the heavy-tailed hyperpriors. Applications from imaging science show that our computational procedure is able to compute sharp edge-preserving posterior point estimates with reduced uncertainty.

stat.CO↗

arXiv2022

Iterative image reconstruction for CT with unmatched projection matrices using the generalized minimal residual algorithm

The generalized minimal residual (GMRES) algorithm is applied to image reconstruction using linear computed tomography (CT) models. The GMRES algorithm iteratively solves square, non-symmetric linear systems and it has practical application to CT when using unmatched back-projector/projector pairs and when applying preconditioning. The GMRES algorithm is demonstrated on a 3D CT image reconstruction problem where it is seen that use of unmatched projection matrices does not prevent convergence, while using an unmatched pair in the related conjugate gradients for least-squares (CGLS) algorithm leads to divergent iteration. Implementation of preconditioning using GMRES is also demonstrated.

physics.med-ph↗

arXiv2022

GMRES Methods for Tomographic Reconstruction with an Unmatched Back Projector

Unmatched pairs of forward and back projectors are common in X-ray CT computations for large-scale problems; they are caused by the need for fast algorithms that best utilize the computer hardware, and it is an interesting and challenging task to develop fast and easy-to-use algorithms for these cases. Our approach is to use preconditioned GMRES, in the form of the AB- and BA-GMRES algorithms, to handle the unmatched normal equations associated with an unmatched pair. These algorithms are simple to implement, they rely only on computations with the available forward and back projectors, and they do not require the tuning of any algorithm parameters. We show that these algorithms are equivalent to well-known LSQR and LSMR algorithms in the case of a matched projector. Our numerical experiments demonstrate that AB- and BA-GMRES exhibit a desired semi-convergence behavior that is comparable with LSQR/LSMR and that standard stopping rules work well. Hence, AB- and BA-GMRES are suited for large-scale CT reconstruction problems with noisy data and unmatched projector pairs.

math.NA↗

arXiv2021

Stopping Rules for Algebraic Iterative Reconstruction Methods in Computed Tomography

Algebraic models for the reconstruction problem in X-ray computed tomography (CT) provide a flexible framework that applies to many measurement geometries. For large-scale problems we need to use iterative solvers, and we need stopping rules for these methods that terminate the iterations when we have computed a satisfactory reconstruction that balances the reconstruction error and the influence of noise from the measurements. Many such stopping rules are developed in the inverse problems communities, but they have not attained much attention in the CT world. The goal of this paper is to describe and illustrate four stopping rules that are relevant for CT reconstructions.

math.NA↗

arXiv2021

A hybrid Gibbs sampler for edge-preserving tomographic reconstruction with uncertain view angles

In computed tomography, data consist of measurements of the attenuation of X-rays passing through an object. The goal is to reconstruct the linear attenuation coefficient of the object's interior. For each position of the X-ray source, characterized by its angle with respect to a fixed coordinate system, one measures a set of data referred to as a view. A common assumption is that these view angles are known, but in some applications they are known with imprecision. We propose a framework to solve a Bayesian inverse problem that jointly estimates the view angles and an image of the object's attenuation coefficient. We also include a few hyperparameters that characterize the likelihood and the priors. Our approach is based on a Gibbs sampler where the associated conditional densities are simulated using different sampling schemes - hence the term hybrid. In particular, the conditional distribution associated with the reconstruction is nonlinear in the image pixels, non-Gaussian and high-dimensional. We approach this distribution by constructing a Laplace approximation that represents the target conditional locally at each Gibbs iteration. This enables sampling of the attenuation coefficients in an efficient manner using iterative reconstruction algorithms. The numerical results show that our algorithm is able to jointly identify the image and the view angles, while also providing uncertainty estimates of both. We demonstrate our method with 2D X-ray computed tomography problems using fan beam configurations.

stat.CO↗

arXiv2021

A twin error gauge for Kaczmarz's iterations

We propose two new algebraic reconstruction techniques based on Kaczmarz's method that produce a regularized solution to noisy tomography problems. Tomography problems exhibit semi-convergence when iterative methods are employed, and the aim is therefore to stop near the semi-convergence point. Our approach is based on an error gauge that is constructed by pairing standard down-sweep Kaczmarz's method with its up-sweep version; we stop the iterations when this error gauge is minimal. The reconstructions of the new methods differ from standard Kaczmarz iterates in that our final result is the average of the stopped up- and down-sweeps. Even when Kaczmarz's method is supplied with an oracle that provides the exact error -- and is thereby able to stop at the best possible iterate -- our methods have a lower two-norm error in the vast majority of our test cases. In terms of computational cost, our methods are a little cheaper than standard Kaczmarz equipped with a statistical stopping rule.

math.NA↗

arXiv2019

Fixing Nonconvergence of Algebraic Iterative Reconstruction with an Unmatched Backprojector

We consider algebraic iterative reconstruction methods with applications in image reconstruction. In particular, we are concerned with methods based on an unmatched projector/backprojector pair; i.e., the backprojector is not the exact adjoint or transpose of the forward projector. Such situations are common in large-scale computed tomography, and we consider the common situation where the method does not converge due to the nonsymmetry of the iteration matrix. We propose a modified algorithm that incorporates a small shift parameter, and we give the conditions that guarantee convergence of this method to a fixed point of a slightly perturbed problem. We also give perturbation bounds for this fixed point. Moreover, we discuss how to use Krylov subspace methods to efficiently estimate the leftmost eigenvalue of a certain matrix to select a proper shift parameter. The modified algorithm is illustrated with test problems from computed tomography.

math.NA↗

arXiv2017

IR Tools: A MATLAB Package of Iterative Regularization Methods and Large-Scale Test Problems

This paper describes a new MATLAB software package of iterative regularization methods and test problems for large-scale linear inverse problems. The software package, called IR Tools, serves two related purposes: we provide implementations of a range of iterative solvers, including several recently proposed methods that are not available elsewhere, and we provide a set of large-scale test problems in the form of discretizations of 2D linear inverse problems. The solvers include iterative regularization methods where the regularization is due to the semi-convergence of the iterations, Tikhonov-type formulations where the regularization is explicitly formulated in the form of a regularization term, and methods that can impose bound constraints on the computed solutions. All the iterative methods are implemented in a very flexible fashion that allows the problem's coefficient matrix to be available as a (sparse) matrix, a function handle, or an object. The most basic call to all of the various iterative methods requires only this matrix and the right hand side vector; if the method uses any special stopping criteria, regularization parameters, etc., then default values are set automatically by the code. Moreover, through the use of an optional input structure, the user can also have full control of any of the algorithm parameters. The test problems represent realistic large-scale problems found in image reconstruction and several other applications. Numerical examples illustrate the various algorithms and test problems available in this package.

math.NA↗

arXiv2015

A Tensor-Based Dictionary Learning Approach to Tomographic Image Reconstruction

We consider tomographic reconstruction using priors in the form of a dictionary learned from training images. The reconstruction has two stages: first we construct a tensor dictionary prior from our training data, and then we pose the reconstruction problem in terms of recovering the expansion coefficients in that dictionary. Our approach differs from past approaches in that a) we use a third-order tensor representation for our images and b) we recast the reconstruction problem using the tensor formulation. The dictionary learning problem is presented as a non-negative tensor factorization problem with sparsity constraints. The reconstruction problem is formulated in a convex optimization framework by looking for a solution with a sparse representation in the tensor dictionary. Numerical results show that our tensor formulation leads to very sparse representations of both the training images and the reconstructions due to the ability of representing repeated features compactly in the dictionary.

cs.CV↗

arXiv2015

Noise Robustness of a Combined Phase Retrieval and Reconstruction Method for Phase-Contrast Tomography

Classical reconstruction methods for phase-contrast tomography consist of two stages: phase retrieval and tomographic reconstruction. A novel algebraic method combining the two was suggested by Kostenko et al. (Opt. Express, 21, 12185, 2013) and preliminary results demonstrating improved reconstruction compared to a two-stage method given. Using simulated free-space propagation experiments with a single sample-detector distance, we thoroughly compare the novel method with the two-stage method to address limitations of the preliminary results. We demonstrate that the novel method is substantially more robust towards noise; our simulations point to a possible reduction in counting times by an order of magnitude.

math.NA↗

arXiv2015

Tomographic Image Reconstruction using Training images

We describe and examine an algorithm for tomographic image reconstruction where prior knowledge about the solution is available in the form of training images. We first construct a nonnegative dictionary based on prototype elements from the training images; this problem is formulated as a regularized non-negative matrix factorization. Incorporating the dictionary as a prior in a convex reconstruction problem, we then find an approximate solution with a sparse representation in the dictionary. The dictionary is applied to non-overlapping patches of the image, which reduces the computational complexity compared to other algorithms. Computational experiments clarify the choice and interplay of the model parameters and the regularization parameters, and we show that in few-projection low-dose settings our algorithm is competitive with total variation regularization and tends to include more texture and more correct edges.

cs.CV↗

arXiv2014

"Plug-and-Play" Edge-Preserving Regularization

In many inverse problems it is essential to use regularization methods that preserve edges in the reconstructions, and many reconstruction models have been developed for this task, such as the Total Variation (TV) approach. The associated algorithms are complex and require a good knowledge of large-scale optimization algorithms, and they involve certain tolerances that the user must choose. We present a simpler approach that relies only on standard computational building blocks in matrix computations, such as orthogonal transformations, preconditioned iterative solvers, Kronecker products, and the discrete cosine transform -- hence the term "plug-and-play." We do not attempt to improve on TV reconstructions, but rather provide an easy-to-use approach to computing reconstructions with similar properties.

math.NA↗

arXiv2013

Generalized Row-Action Methods for Tomographic Imaging

Row-action methods play an important role in tomographic image reconstruction. Many such methods can be viewed as incremental gradient methods for minimizing a sum of a large number of convex functions, and despite their relatively poor global rate of convergence, these methods often exhibit fast initial convergence which is desirable in applications where a low-accuracy solution is acceptable. In this paper, we propose relaxed variants of a class of incremental proximal gradient methods, and these variants generalize many existing row-action methods for tomographic imaging. Moreover, they allow us to derive new incremental algorithms for tomographic imaging that incorporate different types of prior information via regularization. We demonstrate the efficacy of the approach with some numerical examples.

math.OC↗