Computational Inverse Problems Research Articles

An accelerated inexact Newton regularization scheme with a learned feature-selection rule for non-linear inverse problems

With computational inverse problems, it is desirable to develop an efficient inversion algorithm to find a solution from measurement data through a mathematical model connecting the unknown solution and measurable quantity based on the first principles. However, most of mathematical models represent only a few aspects of the physical quantity of interest, and some of them are even incomplete in the sense that one measurement corresponds to many solutions satisfying the forward model. In this paper, in light of the recently developed iNETT method in (2023 Inverse Problems 39 055002), we propose a novel iterative regularization method for efficiently solving non-linear ill-posed inverse problems with potentially non-injective forward mappings and (locally) non-stable inversion mappings. Our approach integrates the inexact Newton iteration, the non-stationary iterated Tikhonov regularization, the two-point gradient acceleration method, and the structure-free feature-selection rule. The main difficulty in the regularization technique is how to design an appropriate regularization penalty, capturing the key feature of the unknown solution. To overcome this difficulty, we replace the traditional regularization penalty with a deep neural network, which is structure-free and can identify the correct solution in a huge null space. A comprehensive convergence analysis of the proposed algorithm is performed under standard assumptions of regularization theory. Numerical experiments with comparisons with other state-of-the-art methods for two model problems are presented to show the efficiency of the proposed approach.

A parallel sampling algorithm for some nonlinear inverse problems

AbstractWe derive a parallel sampling algorithm for computational inverse problems that present an unknown linear forcing term and a vector of nonlinear parameters to be recovered. It is assumed that the data are noisy and that the linear part of the problem is ill-posed. The vector of nonlinear parameters ${m} $ is modeled as a random variable. A dilation parameter $\alpha $ is used to scale the regularity of the linear unknown and is also modeled as a random variable. A posterior probability distribution for $({m}, \alpha )$ is derived following an approach related to the maximum likelihood (ML) regularization parameter selection (Galatsanos, N. P. & Katsaggelos, A. K. (1992). Methods for choosing the regularization parameter and estimating the noise variance in image restoration and their relation. IEEE Trans. Image Process., 1, 322–336). A major difference in our approach is that, unlike in Galatsanos, N. P. & Katsaggelos, A. K. (1992, Methods for choosing the regularization parameter and estimating the noise variance in image restoration and their relation. IEEE Trans. Image Process., 1, 322–336), we do not limit ourselves to the maximum likelihood value of $\alpha $. We then derive a parallel sampling algorithm where we alternate computing proposals in parallel and combining proposals to accept or reject them as in Calderhead, B. (2014, A general construction for parallelizing metropolis- hastings algorithms. Proc. Natl Acad Sci, 111, 17408–17413). This algorithm is well suited to problems where proposals are expensive to compute. We then apply it to an inverse problem in seismology. We show how our results compare favorably to those obtained from the ML, the Generalized Cross Validation and the Constrained Least Squares algorithms.

Self-calibrated 3D differential phase contrast microscopy with optimized illumination.

3D phase imaging recovers an object's volumetric refractive index from intensity and/or holographic measurements. Partially coherent methods, such as illumination-based differential phase contrast (DPC), are particularly simple to implement in a commercial brightfield microscope. 3D DPC acquires images at multiple focus positions and with different illumination source patterns in order to reconstruct 3D refractive index. Here, we present a practical extension of the 3D DPC method that does not require a precise motion stage for scanning the focus and uses optimized illumination patterns for improved performance. The user scans the focus by hand, using the microscope's focus knob, and the algorithm self-calibrates the axial position to solve for the 3D refractive index of the sample through a computational inverse problem. We further show that the illumination patterns can be optimized by an end-to-end learning procedure. Combining these two, we demonstrate improved 3D DPC with a commercial microscope whose only hardware modification is LED array illumination.

Computational Inverse Problems for Partial Differential Equations

Inverse problems in partial differential equations (PDEs) consist in reconstructing some part of a PDE such as a coefficient, a boundary condition, an initial condition, the shape of a domain, or a singularity from partial knowledge of solutions to the PDE. This has numerous applications in nondestructive testing, medical imaging, seismology, and optical imaging. Whereas classically mostly boundary or far field data of solutions to deterministic PDEs were considered, more recently also statistical properties of solutions to random PDEs have been studied. The study of numerical reconstruction methods of inverse problems in PDEs is at the interface of numerical analysis, PDE theory, functional analysis, statistics, optimization, and differential geometry. This workshop has mainly addressed five related topics of current interest: model reduction, control-based techniques in inverse problems, imaging with correlation data of waves, fractional diffusion, and model-based approaches using machine learning.

Robotics, Vision and Control: Fundamental Algorithms in MATLAB, Second Edition [Bookshelf

The second edition of this book covers an impressively broad array of important topics in robotics and computer vision. The book’s presentation style is approachable and engaging, not dense nor intimidating, and the material should be accessible to most upper-level, undergraduate engineering and computer science students. Each chapter combines foundational material, historical vignettes, colorful figures, descriptions of current technology, and software examples to help the reader quickly grasp the context, main ideas, and applications; this approach makes the book an excellent reference source. A nice feature is that, for many sections, it is possible to flip directly to that section and grasp the key ideas without having to read all of the material that comes before it. Due to the breadth of the material covered, this book can be used as a textbook for multiple courses in robotics and computer vision. The text sacrifices some mathematical details and derivations in favor of an example-driven, learn-by-doing treatment. Therefore, the instructor may wish to supplement the book with other sources that go into more detail on the fundamentals. This book is an introduction to both computational inverse problems and uncertainty quantification (UQ) for inverse problems. The text discusses more advanced material on Bayesian methods and UQ, including Markov chain Monte Carlo sampling methods for UQ in inverse problems. Each chapter contains the Matlab code, which implements the algorithms and generates the figures as well as a large number of exercises. This material is intended for graduate students, researchers, and applied scientists. It is appropriate for use with courses on computational inverse problems, Bayesian methods for inverse problems, and UQ methods for inverse problems.

Computational Inverse Problems for Partial Differential Equations

The problem of determining unknown quantities in a PDE from measurements of (part of) the solution to this PDE arises in a wide range of applications in science, technology, medicine, and finance. The unknown quantity may e.g. be a coefficient, an initial or a boundary condition, a source term, or the shape of a boundary. The identification of such quantities is often computationally challenging and requires profound knowledge of the analytical properties of the underlying PDE as well as numerical techniques. The focus of this workshop was on applications in phase retrieval, imaging with waves in random media, and seismology of the Earth and the Sun, a further emphasis was put on stochastic aspects in the context of uncertainty quantification and parameter identification in stochastic differential equations. Many open problems and mathematical challenges in application fields were addressed, and intensive discussions provided an insight into the high potential of joining deep knowledge in numerical analysis, partial differential equations, and regularization, but also in mathematical statistics, homogenization, optimization, differential geometry, numerical linear algebra, and variational analysis to tackle these challenges.

Iterative updating of model error for Bayesian inversion

In computational inverse problems, it is common that a detailed and accurate forward model is approximated by a computationally less challenging substitute. The model reduction may be necessary to meet constraints in computing time when optimization algorithms are used to find a single estimate, or to speed up Markov chain Monte Carlo (MCMC) calculations in the Bayesian framework. The use of an approximate model introduces a discrepancy, or modeling error, that may have a detrimental effect on the solution of the ill-posed inverse problem, or it may severely distort the estimate of the posterior distribution. In the Bayesian paradigm, the modeling error can be considered as a random variable, and by using an estimate of the probability distribution of the unknown, one may estimate the probability distribution of the modeling error and incorporate it into the inversion. We introduce an algorithm which iterates this idea to update the distribution of the model error, leading to a sequence of posterior distributions that are demonstrated empirically to capture the underlying truth with increasing accuracy. Since the algorithm is not based on rejections, it requires only limited full model evaluations.We show analytically that, in the linear Gaussian case, the algorithm converges geometrically fast with respect to the number of iterations when the data is finite dimensional. For more general models, we introduce particle approximations of the iteratively generated sequence of distributions; we also prove that each element of the sequence converges in the large particle limit under a simplifying assumption. We show numerically that, as in the linear case, rapid convergence occurs with respect to the number of iterations. Additionally, we show through computed examples that point estimates obtained from this iterative algorithm are superior to those obtained by neglecting the model error.

Combined arrival-time imaging and time reversal for scatterer identification

The computational inverse problem of identifying a scatterer in a time-dependent wave field is considered. The wave speed of the background medium and the wave source are assumed to be known. Wave measurements, possibly noisy, are given at chosen discrete points in space (sensor locations) and time. The goal is to find scatterer parameters such as location, size and shape. The computational solution procedure consists of two steps. In the first step, a standard fast Arrival-Time Imaging (ATI) algorithm is employed. This results in a rough image which provides possible regions for the location of the scatterer. In the second step an optimization scheme based on Time Reversal (TR) is used to determine the location, size and shape of the scatterer. The preliminary ATI step has the effect of reducing considerably the search space for the TR optimization. Also, proposed here is an improved definition of the objective function used for the optimization, which tends to eliminate spurious solutions. Numerical experiments, based on a finite element discretization in space and an explicit Newmark time-stepping, show the identification capability of the proposed scheme, for a model problem involving the linear scalar wave equation in a bounded domain. Two types of scatterers are considered: a crack with a known orientation, whose location and size are sought, and a rectangular scatterer whose location, aspect ratio and size are sought. The performance of the scheme in the presence of measurement noise is also demonstrated.

Accurate Solution of Bayesian Inverse Uncertainty Quantification Problems Combining Reduced Basis Methods and Reduction Error Models

Computational inverse problems related to partial differential equations (PDEs) often contain nuisance parameters that cannot be effectively identified but still need to be considered as part of the problem. The objective of this work is to show how to take advantage of a reduced order framework to speed up Bayesian inversion on the identifiable parameters of the system, while marginalizing away the (potentially large number of) nuisance parameters. The key ingredients are twofold. On the one hand, we rely on a reduced basis (RB) method, equipped with computable a posteriori error bounds, to speed up the solution of the forward problem. On the other hand, we develop suitable reduction error models (REMs) to quantify in an inexpensive way the error between the full-order and the reduced-order approximation of the forward problem, in order to gauge the effect of this error on the posterior distribution of the identifiable parameters. Numerical results dealing with inverse problems governed by elliptic PDEs in the case of both scalar parameters and parametric fields highlight the combined role played by RB accuracy and REM effectivity.

EVALUATING THE IMPORTANCE OF MITOTIC ASYMMETRY IN CYTON-BASED MODELS FOR FSE-BASED FLOW CYTOMETRY DATA

We carry out computational inverse problem investigations to de- termine the importance of allowing for unequal label allocation to daughter cells during mitosis. Parameter estimations are performed using previously de- veloped label-structured partial differential equation models that utilize cytons to account for variability in times between cell divisions as well as times until cell death. These models are augmented to allow for asymmetric cell divisions through inclusion of an additional model parameter. We employ well-known model comparison tests to analyze CFSE-based flow cytometry data with re- spect to the importance of asymmetric division during proliferation of human CD4+ and CD8+ T cells.

Left and right preconditioning for electrical impedance tomography with structural information

A common problem in computational inverse problems is to find an efficient way of solving linear or nonlinear least-squares problems. For large-scale problems, iterative solvers are the method of choice for solving the associated linear systems, and for nonlinear problems, an additional effective local linearization method is required. In this paper, we discuss an efficient preconditioning scheme for Krylov subspace methods, based on the Bayesian analysis of the inverse problem. The model problem to which we apply this methodology is electrical impedance tomography (EIT) augmented with prior information coming from a complementary modality, such as x-ray imaging. The particular geometry considered here models the x-ray-guided EIT for breast imaging. The interest in applying EIT concurrently with x-ray breast imaging arises from the experimental observation that the impedivity spectra of certain types of malignant and benign tissues differ significantly from each other, thus offering a possibility of diagnosis without more invasive tissue sampling. After setting up the EIT inverse problem within a Bayesian framework, we present an inner and outer iteration scheme for computing a maximum a posteriori estimate. The prior covariance provides a right preconditioner and the modeling error covariance provides a left preconditioner for the iterative method used to solve the linear least-squares problem at each outer iteration of the optimization problem. Moreover, the stopping criterion for the inner iterations is coupled with the progress of the solution of the outer iteration. Besides the preconditioning scheme, the computational efficiency relies on a very efficient method to compute the Jacobian, obtained by carefully organizing the forward computation. Computed examples illustrate the robustness and computational efficiency of the proposed algorithm.

Viscoelastic response of cylindrical vessels surrounded by gelatin and excited using impulsive ultrasound radiation force.

The objective of this study was to assess the potential for noninvasively identifying viscoelastic material properties in arterial vessels using ultrasound radiation force and modern computational inverse problem techniques. A rubber tube embedded in gelatin was excited at different points along the axis using impulsive ultrasound radiation force. The resulting mechanical waves were measured using a laser vibrometer and pulse‐echo ultrasound. The inverse problem was cast as an optimization problem, in which the discrepancy between the measured dynamic response and the computed finite element model response was minimized with respect to viscoelastic material parameters. The model was very sensitive to the tube’s viscoelastic parameters and relatively insensitive to those of the gelatin. The viscoelastic properties of the tube obtained from the inverse problem were compared to the results obtained from a dynamic mechanical analyzer (DMA). For the overlapping range of frequencies (130–190 Hz), the results of...

Theme Editor's Introduction: Computational Inverse Problems in Medicine

Inverse methods are the opposite of prediction. They're a little like imitating the great detective Sherlock Holmes except that the math is more complicated. Basically, you carefully study the evidence of the scene and from it try to infer who was there and what happened. To do this you might simply observe, but more likely, you will poke or tap at the scene--send various signals through it, for instance, and take sensitive detector readings of what happens to them.Then, fitting the evidence to your knowledge of how the world works and your common sense, you figure out what must have occurred, even though you were not an eyewitness. You essentially throw out all scenarios that are impossible (mathematicians call this constraining the problem), and whatever is left, however improbable, must be the truth. The difficulty comes when the evidence is so inconclusive or sparse or smooth that almost anything could have happened--or at least a very wide range of things--and still produced the same result. Mathematicians call this ill-posed.Introducing this first of various theme sections that IEEE CS&E will carry from time to time, Chris Johnson explains some of the mathematical and computational techniques that have been devised to solve inverse problems, which are closely related to remote sensing, imaging, and the whole business of parameter estimation in science and engineering. He touches on the vast generality of the subject, and some of the specific uses it is put to, especially in medicine. Though the theme articles focus on a particular branch of inverse problem-solving, a look at the diversity of the references Johnson cites gives a small taste of the interdisciplinary usefulness of the technique as a whole, and underscores an underlying purpose of this publication: that CSE practitioners can learn much from each other across discipline boundaries.