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Label-free quantitative phase imaging is a vital tool for optical microscopy and metrology applications. A hair-thin multimode fiber stands out as a very attractive platform for minimally invasive imaging. Here we propose and experimentally demonstrate a non-interferometric non-iterative approach for high-speed high-resolution label-free quantitative phase imaging via a multimode fiber, unlocking multiple applications in life science and bioimaging.

Ultrafast ultrasound imaging insonifies a medium with one or a combination of a few plane waves at different beam-steered angles instead of many focused waves. It can achieve much higher frame rates, but often at the cost of reduced image quality. Deep learning approaches have been proposed to mitigate this disadvantage, in particular for single plane wave imaging. Predominantly, image-to-image post-processing networks or fully learned data-to-image neural networks are used. Both construct their mapping purely data-driven and require expressive networks and large amounts of training data to perform well. In contrast, we consider data-to-image networks which incorporate a conventional image formation techniques as differentiable layers in the network architecture. This allows for end-to-end training with small amounts of training data. In this work, using f-k migration as an image formation layer is evaluated in-depth with experimental data. We acquired a data collection designed for benchmarking data-driven plane wave imaging approaches using a realistic breast mimicking phantom and an ultrasound calibration phantom. The evaluation considers global and local image similarity measures and contrast, resolution and lesion detectability analysis. The results show that the proposed network architecture is capable of improving the image quality of single plane wave images on all evaluation metrics. Furthermore, these image quality improvements can be achieved with surprisingly little amounts of training data., Comment: 8 pages, 9 figures, currently submitted to IEEE Transactions on Medical Imaging

Inverse problems are ubiquitous in science and engineering. Many of these are naturally formulated as a PDE-constrained optimization problem. These non-linear, large-scale, constrained optimization problems know many challenges, of which the inherent non-linearity of the problem is an important one. As an alternative to this physics-driven approach, data-driven methods have been proposed. These methods come with their own set of challenges, and it appears that, ideally, one would devise hybrid methods that combine the best of both worlds. In this paper, we propose one way of combining PDE-constrained optimization with recently proposed data-driven reduced-order models. Starting from an infinite-dimensional formulation of the inverse problem with discrete data, we propose a general framework for the analysis and discretisation of such problems. The proposed approach is based on a relaxed formulation of the PDE-constrained optimization problem, which reduces to a weighted non-linear least-squares problem. The weight matrix turns out to be the Gram matrix of solutions of the PDE, and it can be estimated directly from the measurements. We provide a number of representative case studies and numerical examples.

X-ray imaging can be efficiently used for high-throughput in-line inspection of industrial products. However, designing a system that satisfies industrial requirements and achieves high accuracy is a challenging problem. The effect of many system settings is application-specific and difficult to predict in advance. Consequently, the system is often configured using empirical rules and visual observations. The performance of the resulting system is characterized by extensive experimental testing. We propose to use computational methods to substitute real measurements with generated images corresponding to the same experimental settings. With this approach, it is possible to observe the influence of experimental settings on a large amount of data and to make a prediction of the system performance faster than with conventional methods. We argue that a high accuracy of the image generator may be unnecessary for an accurate performance prediction. We propose a quantitative methodology to characterize the quality of the generation model using POD curves. The proposed approach can be adapted to various applications and we demonstrate it on the poultry inspection problem. We show how a calibrated image generation model can be used to quantitatively evaluate the effect of the X-ray exposure time on the performance of the inspection system.

Measuring the dynamics and mechanical properties of muscles and joints is important to understand the (patho)physiology of muscles. However, acquiring dynamic time-resolved MRI data is challenging. We have previously developed Spectro-Dynamic MRI which allows the characterization of dynamical systems at a high spatial and temporal resolution directly from k-space data. This work presents an extended Spectro-Dynamic MRI framework that reconstructs 1) time-resolved MR images, 2) time-resolved motion fields, 3) dynamical parameters, and 4) an activation force, at a temporal resolution of 11 ms. An iterative algorithm solves a minimization problem containing four terms: a motion model relating the motion to the fully-sampled k-space data, a dynamical model describing the expected type of dynamics, a data consistency term describing the undersampling pattern, and finally a regularization term for the activation force. We acquired MRI data using a dynamic motion phantom programmed to move like an actively driven linear elastic system, from which all dynamic variables could be accurately reconstructed, regardless of the sampling pattern. The proposed method performed better than a two-step approach, where time-resolved images were first reconstructed from the undersampled data without any information about the motion, followed by a motion estimation step., Comment: 11 pages, 7 figures, 5 supplementary figures, 1 supplementary video. The video can be viewed by downloading the source file under "Other Formats"

A large percentage of apples are affected by internal disorders after long-term storage, which makes them unacceptable in the supply chain. CT imaging is a promising technique for in-line detection of these disorders. Therefore, it is crucial to understand how different disorders affect the image features that can be observed in CT scans. This paper presents a workflow for creating datasets of image pairs of photographs of apple slices and their corresponding CT slices. By having CT and photographic images of the same part of the apple, the complementary information in both images can be used to study the processes underlying internal disorders and how internal disorders can be measured in CT images. The workflow includes data acquisition, image segmentation, image registration, and validation methods. The image registration method aligns all available slices of an apple within a single optimization problem, assuming that the slices are parallel. This method outperformed optimizing the alignment separately for each slice. The workflow was applied to create a dataset of 1347 slice photographs and their corresponding CT slices. The dataset was acquired from 107 'Kanzi' apples that had been stored in controlled atmosphere (CA) storage for 8 months. In this dataset, the distance between annotations in the slice photograph and the matching CT slice was, on average, $1.47 \pm 0.40$ mm. Our workflow allows collecting large datasets of accurately aligned photo-CT image pairs, which can help distinguish internal disorders with a similar appearance on CT. With slight modifications, a similar workflow can be applied to other fruits or MRI instead of CT scans., Comment: 20 pages, 9 figures 13-Dec-2023 revision: The plan to make the paper part-one of a two-part series was cancelled. Therefore the title of this paper and the title in the reference to the part-two paper (Wood et al., 2023) were changed

Optical measurements often exhibit mixed Poisson-Gaussian noise statistics, which hampers image quality, particularly under low signal-to-noise ratio (SNR) conditions. Computational imaging falls short in such situations when solely Poissonian noise statistics are assumed. In response to this challenge, we define a loss function that explicitly incorporates this mixed noise nature. By using maximum-likelihood estimation, we devise a practical method to account for camera readout noise in gradient-based ptychography optimization. Our results, based on both experimental and numerical data, demonstrate that this approach outperforms the conventional one, enabling enhanced image reconstruction quality under challenging noise conditions through a straightforward methodological adjustment., Comment: Contains main and supplementary documents

In X-ray Computed Tomography (CT), projections from many angles are acquired and used for 3D reconstruction. To make CT suitable for in-line quality control, reducing the number of angles while maintaining reconstruction quality is necessary. Sparse-angle tomography is a popular approach for obtaining 3D reconstructions from limited data. To optimize its performance, one can adapt scan angles sequentially to select the most informative angles for each scanned object. Mathematically, this corresponds to solving and optimal experimental design (OED) problem. OED problems are high-dimensional, non-convex, bi-level optimization problems that cannot be solved online, i.e., during the scan. To address these challenges, we pose the OED problem as a partially observable Markov decision process in a Bayesian framework, and solve it through deep reinforcement learning. The approach learns efficient non-greedy policies to solve a given class of OED problems through extensive offline training rather than solving a given OED problem directly via numerical optimization. As such, the trained policy can successfully find the most informative scan angles online. We use a policy training method based on the Actor-Critic approach and evaluate its performance on 2D tomography with synthetic data.

Recent research in computational imaging largely focuses on developing machine learning (ML) techniques for image reconstruction, which requires large-scale training datasets consisting of measurement data and ground-truth images. However, suitable experimental datasets for X-ray Computed Tomography (CT) are scarce, and methods are often developed and evaluated only on simulated data. We fill this gap by providing the community with a versatile, open 2D fan-beam CT dataset suitable for developing ML techniques for a range of image reconstruction tasks. To acquire it, we designed a sophisticated, semi-automatic scan procedure that utilizes a highly-flexible laboratory X-ray CT setup. A diverse mix of samples with high natural variability in shape and density was scanned slice-by-slice (5000 slices in total) with high angular and spatial resolution and three different beam characteristics: A high-fidelity, a low-dose and a beam-hardening-inflicted mode. In addition, 750 out-of-distribution slices were scanned with sample and beam variations to accommodate robustness and segmentation tasks. We provide raw projection data, reference reconstructions and segmentations based on an open-source data processing pipeline.

Background: X-ray imaging is widely used for the non-destructive detection of defects in industrial products on a conveyor belt. In-line detection requires highly accurate, robust, and fast algorithms. Deep Convolutional Neural Networks (DCNNs) satisfy these requirements when a large amount of labeled data is available. To overcome the challenge of collecting these data, different methods of X-ray image generation are considered. Objective: Depending on the desired degree of similarity to real data, different physical effects should either be simulated or can be ignored. X-ray scattering is known to be computationally expensive to simulate, and this effect can greatly affect the accuracy of a generated X-ray image. We aim to quantitatively evaluate the effect of scattering on defect detection. Methods: Monte-Carlo simulation is used to generate X-ray scattering distribution. DCNNs are trained on the data with and without scattering and applied to the same test datasets. Probability of Detection (POD) curves are computed to compare their performance, characterized by the size of the smallest detectable defect. Results: We apply the methodology to a model problem of defect detection in cylinders. When trained on data without scattering, DCNNs reliably detect defects larger than 1.3 mm, and using data with scattering improves performance by less than 5%. If the analysis is performed on the cases with large scattering-to-primary ratio ($1 < SPR < 5$), the difference in performance could reach 15% (approx. 0.4 mm). Conclusion: Excluding the scattering signal from the training data has the largest effect on the smallest detectable defects, and the difference decreases for larger defects. The scattering-to-primary ratio has a significant effect on detection performance and the required accuracy of data generation., Comment: This paper appears in: Journal of X-Ray Science and Technology, vol. 32, no. 4, pp. 1099-1119, 2024. Print ISSN: 0895-3996 Online ISSN: 1095-9114 Digital Object Identifier: https://doi.org/10.3233/XST-230389

In this paper, we extend the ROM-based approach for inverse scattering with Neumann boundary conditions, introduced by Druskin at. al. (Inverse Problems 37, 2021), to the 1D Schr{\"o}dinger equation with impedance (Robin) boundary conditions. We also propose a novel data-assimilation (DA) inversion method based on the ROM approach, thereby avoiding the need for a Lanczos-orthogonalization (LO) step. Furthermore, we present a detailed numerical study and comparison of the accuracy and stability of the DA and LO methods.

We present a novel approach to transcranial ultrasound computed tomography that utilizes normalizing flows to improve the speed of imaging and provide Bayesian uncertainty quantification. Our method combines physics-informed methods and data-driven methods to accelerate the reconstruction of the final image. We make use of a physics-informed summary statistic to incorporate the known ultrasound physics with the goal of compressing large incoming observations. This compression enables efficient training of the normalizing flow and standardizes the size of the data regardless of imaging configurations. The combinations of these methods results in fast uncertainty-aware image reconstruction that generalizes to a variety of transducer configurations. We evaluate our approach with in silico experiments and demonstrate that it can significantly improve the imaging speed while quantifying uncertainty. We validate the quality of our image reconstructions by comparing against the traditional physics-only method and also verify that our provided uncertainty is calibrated with the error., Comment: Accepted into PMLR Medical Imaging with Deep Learning (MIDL) 2023

Motion artifacts often spoil the radiological interpretation of MR images, and in the most severe cases the scan needs be repeated, with additional costs for the provider. We discuss the application of a novel 3D retrospective rigid motion correction and reconstruction scheme for MRI, which leverages multiple scans contained in a MR session. Typically, in a multi-contrast MR session, motion does not equally affect all the scans, and some motion-free scans are generally available, so that we can exploit their anatomic similarity. The uncorrupted scan is used as a reference in a generalized rigid-motion registration problem to remove the motion artifacts affecting the corrupted scans. We discuss the potential of the proposed algorithm with a prospective in-vivo study and clinical 3D brain protocols. This framework can be easily incorporated into the existing clinical practice with no disruption to the conventional workflow.

Detection of unwanted (`foreign') objects within products is a common procedure in many branches of industry for maintaining production quality. X-ray imaging is a fast, non-invasive and widely applicable method for foreign object detection. Deep learning has recently emerged as a powerful approach for recognizing patterns in radiographs (i.e., X-ray images), enabling automated X-ray based foreign object detection. However, these methods require a large number of training examples and manual annotation of these examples is a subjective and laborious task. In this work, we propose a Computed Tomography (CT) based method for producing training data for supervised learning of foreign object detection, with minimal labour requirements. In our approach, a few representative objects are CT scanned and reconstructed in 3D. The radiographs that have been acquired as part of the CT-scan data serve as input for the machine learning method. High-quality ground truth locations of the foreign objects are obtained through accurate 3D reconstructions and segmentations. Using these segmented volumes, corresponding 2D segmentations are obtained by creating virtual projections. We outline the benefits of objectively and reproducibly generating training data in this way compared to conventional radiograph annotation. In addition, we show how the accuracy depends on the number of objects used for the CT reconstructions. The results show that in this workflow generally only a relatively small number of representative objects (i.e., fewer than 10) are needed to achieve adequate detection performance in an industrial setting. Moreover, for real experimental data we show that the workflow leads to higher foreign object detection accuracies than with standard radiograph annotation., Comment: This paper is under consideration at Expert Systems with Applications. 22 pages, 15 figures

Advances in multi-spectral detectors are causing a paradigm shift in X-ray Computed Tomography (CT). Spectral information acquired from these detectors can be used to extract volumetric material composition maps of the object of interest. If the materials and their spectral responses are known a priori, the image reconstruction step is rather straightforward. If they are not known, however, the maps as well as the responses need to be estimated jointly. A conventional workflow in spectral CT involves performing volume reconstruction followed by material decomposition, or vice versa. However, these methods inherently suffer from the ill-posedness of the joint reconstruction problem. To resolve this issue, we propose 'A Dictionary-based Joint reconstruction and Unmixing method for Spectral Tomography' (ADJUST). Our formulation relies on forming a dictionary of spectral signatures of materials common in CT and prior knowledge of the number of materials present in an object. In particular, we decompose the spectral volume linearly in terms of spatial material maps, a spectral dictionary, and the indicator of materials for the dictionary elements. We propose a memory-efficient accelerated alternating proximal gradient method to find an approximate solution to the resulting bi-convex problem. From numerical demonstrations on several synthetic phantoms, we observe that ADJUST performs exceedingly well compared to other state-of-the-art methods. Additionally, we address the robustness of ADJUST against limited and noisy measurement patterns. The demonstration of the proposed approach on a spectral micro-CT dataset shows its potential for real-world applications. Code is available at https://github.com/mzeegers/ADJUST., Comment: This paper is under consideration at Inverse Problems with minor revisions. 33 pages, 24 figures

We propose a variation on wavefield reconstruction inversion for seismic inversion, which takes advantage of randomized linear algebra as a way to overcome the typical limitations of conventional inversion techniques. Consequently, we can aim both to robustness towards convergence stagnation and large-sized 3D applications. The central idea hinges on approximating the optimal slack variables involved in wavefield reconstruction inversion via a low-rank stochastic approximation of the wave-equation error covariance. As a result, we obtain a family of inversion methods parameterized by a given model covariance (suited for the problem at hand) and the rank of the related stochastic approximation sketch. The challenges and advantages of our proposal are demonstrated with some numerical experiments.

In plane-wave imaging, multiple unfocused ultrasound waves are transmitted into a medium of interest from different angles and an image is formed from the recorded reflections. The number of plane waves used leads to a trade-off between frame-rate and image quality, with single-plane-wave (SPW) imaging being the fastest possible modality with the worst image quality. Recently, deep learning methods have been proposed to improve ultrasound imaging. One approach is to use image-to-image networks that work on the formed image and another is to directly learn a mapping from data to an image. Both approaches utilize purely data-driven models and require deep, expressive network architectures, combined with large numbers of training samples to obtain good results. Here, we propose a data-to-image architecture that incorporates a wave-physics-based image formation algorithm in-between deep convolutional neural networks. To achieve this, we implement the Fourier (FK) migration method as network layers and train the whole network end-to-end. We compare our proposed data-to-image network with an image-to-image network in simulated data experiments, mimicking a medical ultrasound application. Experiments show that it is possible to obtain high-quality SPW images, almost similar to an image formed using 75 plane waves over an angular range of $\pm$16$^\circ$. This illustrates the great potential of combining deep neural networks with physics-based image formation algorithms for SPW imaging.

Ultrasonic imaging is being used to obtain information about the acoustic properties of a medium by emitting waves into it and recording their interaction using ultrasonic transducer arrays. The Delay-And-Sum (DAS) algorithm forms images using the main path on which reflected signals travel back to the transducers. In some applications, different insonification paths can be considered, for instance by placing the transducers at different locations or if strong reflectors inside the medium are known a-priori. These different modes give rise to multiple DAS images reflecting different geometric information about the scatterers and the challenge is to either fuse them into one image or to directly extract higher-level information regarding the materials of the medium, e.g., a segmentation map. Traditional image fusion techniques typically use ad-hoc combinations of pre-defined image transforms, pooling operations and thresholding. In this work, we propose a deep neural network (DNN) architecture that directly maps all available data to a segmentation map while explicitly incorporating the DAS image formation for the different insonification paths as network layers. This enables information flow between data pre-processing and image post-processing DNNs, trained end-to-end. We compare our proposed method to a traditional image fusion technique using simulated data experiments, mimicking a non-destructive testing application with four image modes, i.e., two transducer locations and two internal reflection boundaries. Using our approach, it is possible to obtain much more accurate segmentation of defects.

X-ray imaging is a widely used technique for non-destructive inspection of agricultural food products. One application of X-ray imaging is the autonomous, in-line detection of foreign objects in food samples. Examples of such inclusions are bone fragments in meat products, plastic and metal debris in fish, fruit infestations. This article presents a processing methodology for unsupervised foreign object detection based on dual-energy X-ray absorptiometry (DEXA). A foreign object is defined as a fragment of material with different X-ray attenuation properties than those belonging to the food product. A novel thickness correction model is introduced as a pre-processing technique for DEXA data. The aim of the model is to homogenize regions in the image that belong to the food product and enhance contrast where the foreign object is present. In this way, the segmentation of the foreign object is more robust to noise and lack of contrast. The proposed methodology was applied to a dataset of 488 samples of meat products. The samples were acquired from a conveyor belt in a food processing factory. Approximately 60\% of the samples contain foreign objects of different types and sizes, while the rest of the samples are void of foreign objects. The results show that samples without foreign objects are correctly identified in 97% of cases, the overall accuracy of foreign object detection reaches 95%., Comment: This work has been submitted to the IEEE for possible publication. Copyright may be transferred without notice, after which this version may no longer be accessible

We introduce single-shot X-ray tomography that aims to estimate the target image from a single cone-beam projection measurement. This linear inverse problem is extremely under-determined since the measurements are far fewer than the number of unknowns. Moreover, it is more challenging than conventional tomography where a sufficiently large number of projection angles forms the measurements, allowing for a simple inversion process. However, single-shot tomography becomes less severe if the target image is only composed of known shapes. Hence, the shape prior transforms a linear ill-posed image estimation problem to a non-linear problem of estimating the roto-translations of the shapes. In this paper, we circumvent the non-linearity by using a dictionary of possible roto-translations of the shapes. We propose a convex program CoShaRP to recover the dictionary-coefficients successfully. CoShaRP relies on simplex-type constraint and can be solved quickly using a primal-dual algorithm. The numerical experiments show that CoShaRP recovers shapes stably from moderately noisy measurements., Comment: Paper is currently under consideration for Pattern Recognition Letters

In many ultrasonic imaging systems, data acquisition and image formation are performed on separate computing devices. Data transmission is becoming a bottleneck, thus, efficient data compression is essential. Compression rates can be improved by considering the fact that many image formation methods rely on approximations of wave-matter interactions, and only use the corresponding part of the data. Tailored data compression could exploit this, but extracting the useful part of the data efficiently is not always trivial. In this work, we tackle this problem using deep neural networks, optimized to preserve the image quality of a particular image formation method. The Delay-And-Sum (DAS) algorithm is examined which is used in reflectivity-based ultrasonic imaging. We propose a novel encoder-decoder architecture with vector quantization and formulate image formation as a network layer for end-to-end training. Experiments demonstrate that our proposed data compression tailored for a specific image formation method obtains significantly better results as opposed to compression agnostic to subsequent imaging. We maintain high image quality at much higher compression rates than the theoretical lossless compression rate derived from the rank of the linear imaging operator. This demonstrates the great potential of deep ultrasonic data compression tailored for a specific image formation method., Comment: IEEE International Ultrasonics Symposium 2020

Ultrasonic imaging algorithms used in many clinical and industrial applications consist of three steps: A data pre-processing, an image formation and an image post-processing step. For efficiency, image formation often relies on an approximation of the underlying wave physics. A prominent example is the Delay-And-Sum (DAS) algorithm used in reflectivity-based ultrasonic imaging. Recently, deep neural networks (DNNs) are being used for the data pre-processing and the image post-processing steps separately. In this work, we propose a novel deep learning architecture that integrates all three steps to enable end-to-end training. We examine turning the DAS image formation method into a network layer that connects data pre-processing layers with image post-processing layers that perform segmentation. We demonstrate that this integrated approach clearly outperforms sequential approaches that are trained separately. While network training and evaluation is performed only on simulated data, we also showcase the potential of our approach on real data from a non-destructive testing scenario., Comment: IEEE International Ultrasonics Symposium 2020

We consider regularized least-squares problems of the form $\min_{x} \frac{1}{2}\Vert Ax - b\Vert_2^2 + \mathcal{R}(Lx)$. Recently, Zheng et al., 2019, proposed an algorithm called Sparse Relaxed Regularized Regression (SR3) that employs a splitting strategy by introducing an auxiliary variable $y$ and solves $\min_{x,y} \frac{1}{2}\Vert Ax - b\Vert_2^2 + \frac{\kappa}{2}\Vert Lx - y\Vert_2^2 + \mathcal{R}(x)$. By minimizing out the variable $x$ we obtain an equivalent system $\min_{y} \frac{1}{2} \Vert F_{\kappa}y - g_{\kappa}\Vert_2^2+\mathcal{R}(y)$. In our work we view the SR3 method as a way to approximately solve the regularized problem. We analyze the conditioning of the relaxed problem in general and give an expression for the SVD of $F_{\kappa}$ as a function of $\kappa$. Furthermore, we relate the Pareto curve of the original problem to the relaxed problem and we quantify the error incurred by relaxation in terms of $\kappa$. Finally, we propose an efficient iterative method for solving the relaxed problem with inexact inner iterations. Numerical examples illustrate the approach., Comment: 25 pages, 14 figures, submitted to SIAM Journal for Scientific Computing special issue Sixteenth Copper Mountain Conference on Iterative Methods

Full waveform inversion (FWI) aims at estimating subsurface medium properties from measured seismic data. It is usually cast as a non-linear least-squares problem that incorporates uncertainties in the measurements. In exploration seismology, extended formulations of FWI that allow for uncertaties in the physics have been proposed. Even when the physics is modelled accurately, these extensions have been shown to be beneficial because they reduce the non-lineary of the resulting data-fitting problem. In this note, I derive an alternative (but equivalent) formulation of extended full waveform inversion. This re-formulation takes the form of a conventional FWI formulation that includes a medium-dependent weight on the residuals. I discuss the implications of this re-formulation and illustrate its properties with a simple numerical example.

Binary tomography is concerned with the recovery of binary images from a few of their projections (i.e., sums of the pixel values along various directions). To reconstruct an image from noisy projection data, one can pose it as a constrained least-squares problem. As the constraints are non-convex, many approaches for solving it rely on either relaxing the constraints or heuristics. In this paper we propose a novel convex formulation, based on the Lagrange dual of the constrained least-squares problem. The resulting problem is a generalized LASSO problem which can be solved efficiently. It is a relaxation in the sense that it can only be guaranteed to give a feasible solution; not necessarily the optimal one. In exhaustive experiments on small images (2x2, 3x3, 4x4) we find, however, that if the problem has a unique solution, our dual approach finds it. In case of multiple solutions, our approach finds the commonalities between the solutions. Further experiments on realistic numerical phantoms and an experiment on X-ray dataset show that our method compares favourably to Total Variation and DART.

In tomographic reconstruction, the goal is to reconstruct an unknown object from a collection of line integrals. Given a complete sampling of such line integrals for various angles and directions, explicit inverse formulas exist to reconstruct the object. Given noisy and incomplete measurements, the inverse problem is typically solved through a regularized least-squares approach. A challenge for both approaches is that in practice the exact directions and offsets of the x-rays are only known approximately due to, e.g. calibration errors. Such errors lead to artifacts in the reconstructed image. In the case of sufficient sampling and geometrically simple misalignment, the measurements can be corrected by exploiting so-called consistency conditions. In other cases, such conditions may not apply and we have to solve an additional inverse problem to retrieve the angles and shifts. In this paper we propose a general algorithmic framework for retrieving these parameters in conjunction with an algebraic reconstruction technique. The proposed approach is illustrated by numerical examples for both simulated data and an electron tomography dataset.

This paper introduces a parametric level-set method for tomographic reconstruction of partially discrete images. Such images consist of a continuously varying background and an anomaly with a constant (known) grey-value. We represent the geometry of the anomaly using a level-set function, which we represent using radial basis functions. We pose the reconstruction problem as a bi-level optimization problem in terms of the background and coefficients for the level-set function. To constrain the background reconstruction we impose smoothness through Tikhonov regularization. The bi-level optimization problem is solved in an alternating fashion; in each iteration we first reconstruct the background and consequently update the level-set function. We test our method on numerical phantoms and show that we can successfully reconstruct the geometry of the anomaly, even from limited data. On these phantoms, our method outperforms Total Variation reconstruction, DART and P-DART., Comment: Paper submitted to 20th International Conference on Discrete Geometry for Computer Imagery

Seismic full-waveform inversion tries to estimate subsurface medium parameters from seismic data. Areas with subsurface salt bodies are of particular interest because they often have hydrocarbon reservoirs on their sides or underneath. Accurate reconstruction of their geometry is a challenge for current techniques. This paper presents a parametric level-set method for the reconstruction of salt-bodies in seismic full-waveform inversion. We split the subsurface model in two parts: a background velocity model and the salt body with known velocity but undetermined shape. The salt geometry is represented by a level-set function that evolves during the inversion. We choose radial basis functions to represent the level-set function, leading to an optimization problem with a modest number of parameters. A common problem with level-set methods is to fine tune the width of the level-set boundary for optimal sensitivity. We propose a robust algorithm that dynamically adapts the width of the level-set boundary to ensure faster convergence. Tests on a suite of idealized salt geometries show that the proposed method is stable against a modest amount of noise. We also extend the method to joint inversion of both the background velocity model and the salt-geometry.

We propose an extended full-waveform inversion formulation that includes general convex constraints on the model. Though the full problem is highly nonconvex, the overarching optimization scheme arrives at geologically plausible results by solving a sequence of relaxed and warm-started constrained convex subproblems. The combination of box, total-variation, and successively relaxed asymmetric total-variation constraints allows us to steer free from parasitic local minima while keeping the estimated physical parameters laterally continuous and in a physically realistic range. For accurate starting models, numerical experiments carried out on the challenging 2004 BP velocity benchmark demonstrate that bound and total-variation constraints improve the inversion result significantly by removing inversion artifacts, related to source encoding, and by clearly improved delineation of top, bottom, and flanks of a high-velocity high-contrast salt inclusion. The experiments also show that for poor starting models these two constraints by themselves are insufficient to detect the bottom of high-velocity inclusions such as salt. Inclusion of the one-sided asymmetric total-variation constraint overcomes this issue by discouraging velocity lows to buildup during the early stages of the inversion. To the author's knowledge the presented algorithm is the first to successfully remove the imprint of local minima caused by poor starting models and band-width limited finite aperture data., Comment: 25 pages, 15 figures

Common computational problems, such as parameter estimation in dynamic models and PDE constrained optimization, require data fitting over a set of auxiliary parameters subject to physical constraints over an underlying state. Naive quadratically penalized formulations, commonly used in practice, suffer from inherent ill-conditioning. We show that surprisingly the partial minimization technique regularizes the problem, making it well-conditioned. This viewpoint sheds new light on variable projection techniques, as well as the penalty method for PDE constrained optimization, and motivates robust extensions. In addition, we outline an inexact analysis, showing that the partial minimization subproblem can be solved very loosely in each iteration. We illustrate the theory and algorithms on boundary control, optimal transport, and parameter estimation for robust dynamic inference., Comment: 8 pages, 9 figures

Variable projection solves structured optimization problems by completely minimizing over a subset of the variables while iterating over the remaining variables. Over the last 30 years, the technique has been widely used, with empirical and theoretical results demonstrating both greater efficacy and greater stability compared to competing approaches. Classic examples have exploited closed-form projections and smoothness of the objective function. We extend the approach to problems that include non-smooth terms, and where the projection subproblems can only be solved inexactly by iterative methods. We propose an inexact adaptive algonrithm for solving such problems and analyze its computational complexity. Finally, we show how the theory can be used to design methods for selected problems occurring frequently in machine-learning and inverse problems.

Many inverse and parameter estimation problems can be written as PDE-constrained optimization problems. The goal, then, is to infer the parameters, typically coefficients of the PDE, from partial measurements of the solutions of the PDE for several right-hand-sides. Such PDE-constrained problems can be solved by finding a stationary point of the Lagrangian, which entails simultaneously updating the paramaters and the (adjoint) state variables. For large-scale problems, such an all-at-once approach is not feasible as it requires storing all the state variables. In this case one usually resorts to a reduced approach where the constraints are explicitly eliminated (at each iteration) by solving the PDEs. These two approaches, and variations thereof, are the main workhorses for solving PDE-constrained optimization problems arising from inverse problems. In this paper, we present an alternative method that aims to combine the advantages of both approaches. Our method is based on a quadratic penalty formulation of the constrained optimization problem. By eliminating the state variable, we develop an efficient algorithm that has roughly the same computational complexity as the conventional reduced approach while exploiting a larger search space. Numerical results show that this method indeed reduces some of the non-linearity of the problem and is less sensitive the initial iterate.

We consider an important class of signal processing problems where the signal of interest is known to be sparse, and can be recovered from data given auxiliary information about how the data was generated. For example, a sparse Green's function may be recovered from seismic experimental data using sparsity optimization when the source signature is known. Unfortunately, in practice this information is often missing, and must be recovered from data along with the signal using deconvolution techniques. In this paper, we present a novel methodology to simultaneously solve for the sparse signal and auxiliary parameters using a recently proposed variable projection technique. Our main contribution is to combine variable projection with sparsity promoting optimization, obtaining an efficient algorithm for large-scale sparse deconvolution problems. We demonstrate the algorithm on a seismic imaging example., Comment: 5 pages, 4 figures

The Helmholtz equation arises in many applications, such as seismic and medical imaging. These application are characterized by the need to propagate many wavelengths through an inhomogeneous medium. The typical size of the problems in 3D applications precludes the use of direct factorization to solve the equation and hence iterative methods are used in practice. For higher wavenumbers, the system becomes increasingly indefinite and thus good preconditioners need to be constructed. In this note we consider an accelerated Kazcmarz method (CGMN) and present an expression for the resulting iteration matrix. This iteration matrix can be used to analyze the convergence of the CGMN method. In particular, we present a Fourier analysis for the method applied to the 1D Helmholtz equation. This analysis suggests an optimal choice of the relaxation parameter. Finally, we present some numerical experiments.

Many inverse problems include nuisance parameters which, while not of direct interest, are required to recover primary parameters. Structure present in these problems allows efficient optimization strategies - a well known example is variable projection, where nonlinear least squares problems which are linear in some parameters can be very efficiently optimized. In this paper, we extend the idea of projecting out a subset over the variables to a broad class of maximum likelihood (ML) and maximum a posteriori likelihood (MAP) problems with nuisance parameters, such as variance or degrees of freedom. As a result, we are able to incorporate nuisance parameter estimation into large-scale constrained and unconstrained inverse problem formulations. We apply the approach to a variety of problems, including estimation of unknown variance parameters in the Gaussian model, degree of freedom (d.o.f.) parameter estimation in the context of robust inverse problems, automatic calibration, and optimal experimental design. Using numerical examples, we demonstrate improvement in recovery of primary parameters for several large- scale inverse problems. The proposed approach is compatible with a wide variety of algorithms and formulations, and its implementation requires only minor modifications to existing algorithms., Comment: 16 pages, 5 figures