Tristan van Leeuwen

Maximilian B. Kiss, Sophia B. Coban, K. Joost Batenburg et al.

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.

Aleksandr Y. Aravkin, Tristan van Leeuwen

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.

Aleksandr Aravkin, Michael P. Friedlander, Tristan van Leeuwen

We consider a class of inverse problems where it is possible to aggregate the results of multiple experiments. This class includes problems where the forward model is the solution operator to linear ODEs or PDEs. The tremendous size of such problems motivates dimensionality reduction techniques based on randomly mixing experiments. These techniques break down, however, when robust data-fitting formulations are used, which are essential in cases of missing data, unusually large errors, and systematic features in the data unexplained by the forward model. We survey robust methods within a statistical framework, and propose a semistochastic optimization approach that allows dimensionality reduction. The efficacy of the methods are demonstrated for a large-scale seismic inverse problem using the robust Student's t-distribution, where a useful synthetic velocity model is recovered in the extreme scenario of 60% data missing at random. The semistochastic approach achieves this recovery using 20% of the effort required by a direct robust approach.

Rafael Orozco, Mathias Louboutin, Ali Siahkoohi et al.

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.

Tristan van Leeuwen, Simon Maretzke, K. Joost Batenburg

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.

Ajinkya Kadu, Tristan van Leeuwen, K. Joost Batenburg

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.

Tianyuan Wang, Felix Lucka, Tristan van Leeuwen

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 an 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.

Tristan van Leeuwen

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.

Wangduo Xie, Richard Schoonhoven, Tristan van Leeuwen et al.

Computed tomography (CT) reconstruction plays a crucial role in industrial nondestructive testing and medical diagnosis. Sparse view CT reconstruction aims to reconstruct high-quality CT images while only using a small number of projections, which helps to improve the detection speed of industrial assembly lines and is also meaningful for reducing radiation in medical scenarios. Sparse CT reconstruction methods based on implicit neural representations (INRs) have recently shown promising performance, but still produce artifacts because of the difficulty of obtaining useful prior information. In this work, we incorporate a powerful prior: the total number of material categories of objects. To utilize the prior, we design AC-IND, a self-supervised method based on Attenuation Coefficient Estimation and Implicit Neural Distribution. Specifically, our method first transforms the traditional INR from scalar mapping to probability distribution mapping. Then we design a compact attenuation coefficient estimator initialized with values from a rough reconstruction and fast segmentation. Finally, our algorithm finishes the CT reconstruction by jointly optimizing the estimator and the generated distribution. Through experiments, we find that our method not only outperforms the comparative methods in sparse CT reconstruction but also can automatically generate semantic segmentation maps.

Mathé T. Zeegers, Ajinkya Kadu, Tristan van Leeuwen et al.

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.

Ryan A. L. Schoop, Gijs Hendriks, Tristan van Leeuwen et al.

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.

Selin Aslan, Tristan van Leeuwen, Allard Mosk et al.

In phase retrieval and similar inverse problems, the stability of solutions across different noise levels is crucial for applications. One approach to promote it is using signal priors in a form of a generative model as a regularization, at the expense of introducing a bias in the reconstruction. In this paper, we explore and compare the reconstruction properties of classical and generative inverse problem formulations. We propose a new unified reconstruction approach that mitigates overfitting to the generative model for varying noise levels.

Tristan van Leeuwen, Christoph Brune, Marcello Carioni

We formulate the inverse problem in a Bayesian framework and aim to train a generative model that allows us to simulate (i.e., sample from the likelihood) and do inference (i.e., sample from the posterior). We review the use of triangular normalizing flows for conditional sampling in this context and show how to combine two such triangular maps (an upper and a lower one) in to one invertible mapping that can be used for simulation and inference. We work out several useful properties of this invertible generative model and propose a possible training loss for training the map directly. We illustrate the workings of this new approach to conditional generative modeling numerically on a few stylized examples.

Tianyuan Wang, Felix Lucka, Daniël M. Pelt et al.

In industrial X-ray Computed Tomography (CT), the need for rapid in-line inspection is critical. Sparse-angle tomography plays a significant role in this by reducing the required number of projections, thereby accelerating processing and conserving resources. Most existing methods aim to balance reconstruction quality and scanning time, typically relying on fixed scan durations. Adaptive adjustment of the number of angles is essential; for instance, more angles may be required for objects with complex geometries or noisier projections. The concept of optimal stopping, which dynamically adjusts this balance according to varying industrial needs, remains overlooked. Building on our previous work, we integrate optimal stopping into sequential Optimal Experimental Design (sOED) and Reinforcement Learning (RL). We propose a novel method for computing the policy gradient within the Actor-Critic framework, enabling the development of adaptive policies for informative angle selection and scan termination. Additionally, we investigate the gap between simulation and real-world applications in the context of the developed learning-based method. Our trained model, developed using synthetic data, demonstrates reliable performance when applied to experimental X-ray CT data. This approach enhances the flexibility of CT operations and expands the applicability of sparse-angle tomography in industrial settings.

Ezgi Demircan-Tureyen, Felix Lucka, Tristan van Leeuwen

Recent works demonstrate the effectiveness of diffusion models as unsupervised solvers for inverse imaging problems. Sparse-view computed tomography (CT) has greatly benefited from these advancements, achieving improved generalization without reliance on measurement parameters. However, this comes at the cost of potential hallucinations, especially when handling out-of-distribution (OOD) data. To ensure reliability, it is essential to study OOD detection for CT reconstruction across both clinical and industrial applications. This need further extends to enabling the OOD detector to function effectively as an anomaly inspection tool. In this paper, we explore the use of a diffusion model, trained to capture the target distribution for CT reconstruction, as an in-distribution prior. Building on recent research, we employ the model to reconstruct partially diffused input images and assess OOD-ness through multiple reconstruction errors. Adapting this approach for sparse-view CT requires redefining the notions of ``input'' and ``reconstruction error''. Here, we use filtered backprojection (FBP) reconstructions as input and investigate various definitions of reconstruction error. Our proof-of-concept experiments on the MNIST dataset highlight both successes and failures, demonstrating the potential and limitations of integrating such an OOD detector into a CT reconstruction system. Our findings suggest that effective OOD detection can be achieved by comparing measurements with forward-projected reconstructions, provided that reconstructions from noisy FBP inputs are conditioned on the measurements. However, conditioning can sometimes lead the OOD detector to inadvertently reconstruct OOD images well. To counter this, we introduce a weighting approach that improves robustness against highly informative OOD measurements, albeit with a trade-off in performance in certain cases.

Vladyslav Andriiashen, Robert van Liere, Tristan van Leeuwen et al.

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.

Mathé T. Zeegers, Tristan van Leeuwen, Daniël M. Pelt et al.

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.

Georgios Pilikos, Chris L. de Korte, Tristan van Leeuwen et al.

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.

Georgios Pilikos, Lars Horchens, Tristan van Leeuwen et al.

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.

Vladyslav Andriiashen, Robert van Liere, Tristan van Leeuwen et al.

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%.

Ajinkya Kadu, Tristan van Leeuwen, K. Joost Batenburg

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.

Georgios Pilikos, Lars Horchens, Kees Joost Batenburg et al.

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.

Georgios Pilikos, Lars Horchens, Kees Joost Batenburg et al.

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.

Tristan van Leeuwen, Aleksandr Aravkin

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.