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This paper considers one of the fundamental parallel-in-time methods for the solution of ordinary differential equations, Parareal, and extends it by adopting a neural network as a coarse propagator. We provide a theoretical analysis of the convergence properties of the proposed algorithm and show its effectiveness for several examples, including Lorenz and Burgers' equations. In our numerical simulations, we further specialize the underpinning neural architecture to Random Projection Neural Networks (RPNNs), a 2-layer neural network where the first layer weights are drawn at random rather than optimized. This restriction substantially increases the efficiency of fitting RPNN's weights in comparison to a standard feedforward network without negatively impacting the accuracy, as demonstrated in the SIR system example.

With the next generation of interferometric telescopes, such as the Square Kilometre Array (SKA), the need for highly computationally efficient reconstruction techniques is particularly acute. The challenge in designing learned, data-driven reconstruction techniques for radio interferometry is that they need to be agnostic to the varying visibility coverages of the telescope, since these are different for each observation. Because of this, learned post-processing or learned unrolled iterative reconstruction methods must typically be retrained for each specific observation, amounting to a large computational overhead. In this work we develop learned post-processing and unrolled iterative methods for varying visibility coverages, proposing training strategies to make these methods agnostic to variations in visibility coverage with minimal to no fine-tuning. Learned post-processing techniques are heavily dependent on the prior information encoded in training data and generalise poorly to other visibility coverages. In contrast, unrolled iterative methods, which include the telescope measurement operator inside the network, achieve state-of-the-art reconstruction quality and computation time, generalising well to other coverages and require little to no fine-tuning. Furthermore, they generalise well to realistic radio observations and are able to reconstruct the high dynamic range of these images.

Next-generation radio interferometers like the Square Kilometer Array have the potential to unlock scientific discoveries thanks to their unprecedented angular resolution and sensitivity. One key to unlocking their potential resides in handling the deluge and complexity of incoming data. This challenge requires building radio interferometric imaging methods that can cope with the massive data sizes and provide high-quality image reconstructions with uncertainty quantification (UQ). This work proposes a method coined QuantifAI to address UQ in radio-interferometric imaging with data-driven (learned) priors for high-dimensional settings. Our model, rooted in the Bayesian framework, uses a physically motivated model for the likelihood. The model exploits a data-driven convex prior, which can encode complex information learned implicitly from simulations and guarantee the log-concavity of the posterior. We leverage probability concentration phenomena of high-dimensional log-concave posteriors that let us obtain information about the posterior, avoiding MCMC sampling techniques. We rely on convex optimisation methods to compute the MAP estimation, which is known to be faster and better scale with dimension than MCMC sampling strategies. Our method allows us to compute local credible intervals, i.e., Bayesian error bars, and perform hypothesis testing of structure on the reconstructed image. In addition, we propose a novel blazing-fast method to compute pixel-wise uncertainties at different scales. We demonstrate our method by reconstructing radio-interferometric images in a simulated setting and carrying out fast and scalable UQ, which we validate with MCMC sampling. Our method shows an improved image quality and more meaningful uncertainties than the benchmark method based on a sparsity-promoting prior. QuantifAI's source code: https://github.com/astro-informatics/QuantifAI., Comment: 30 pages, 14 figures, 10 tables, code available at https://github.com/astro-informatics/QuantifAI

The Segmented Planar Imaging Detector for Electro-Optical Reconnaissance (SPIDER) is an optical interferometric imaging device that aims to offer an alternative to the large space telescope designs of today with reduced size, weight and power consumption. This is achieved through interferometric imaging. State-of-the-art methods for reconstructing images from interferometric measurements adopt proximal optimization techniques, which are computationally expensive and require handcrafted priors. In this work we present two data-driven approaches for reconstructing images from measurements made by the SPIDER instrument. These approaches use deep learning to learn prior information from training data, increasing the reconstruction quality, and significantly reducing the computation time required to recover images by orders of magnitude. Reconstruction time is reduced to ${\sim} 10$ milliseconds, opening up the possibility of real-time imaging with SPIDER for the first time. Furthermore, we show that these methods can also be applied in domains where training data is scarce, such as astronomical imaging, by leveraging transfer learning from domains where plenty of training data are available., Comment: 21 pages, 14 figures

In photoacoustic tomography (PAT) with flat sensor, we routinely encounter two types of limited data. The first is due to using a finite sensor and is especially perceptible if the region of interest is large relative to the sensor or located farther away from the sensor. In this paper, we focus on the second type caused by a varying sensitivity of the sensor to the incoming wavefront direction which can be modelled as binary i.e. by a cone of sensitivity. Such visibility conditions result, in the Fourier domain, in a restriction of both the image and the data to a bow-tie, akin to the one corresponding to the range of the forward operator. The visible wavefrontsets in image and data domains, are related by the wavefront direction mapping. We adapt the wedge restricted Curvelet decomposition, we previously proposed for the representation of the full PAT data, to separate the visible and invisible wavefronts in the image. We optimally combine fast approximate operators with tailored deep neural network architectures into efficient learned reconstruction methods which perform reconstruction of the visible coefficients and the invisible coefficients are learned from a training set of similar data., Comment: Submitted to SIAM Journal on Imaging Sciences 2nd Review

Curvelet frame is of special significance for photoacoustic tomography (PAT) due to its sparsifying and microlocalisation properties. We derive a one-to-one map between wavefront directions in image and data spaces in PAT which suggests near equivalence between the recovery of the initial pressure and PAT data from compressed/subsampled measurements when assuming sparsity in Curvelet frame. As the latter is computationally more tractable, investigation to which extent this equivalence holds conducted in this paper is of immediate practical significance. To this end we formulate and compare DR, a two step approach based on the recovery of the complete volume of the photoacoustic data from the subsampled data followed by the acoustic inversion, and p0R, a one step approach where the photoacoustic image (the initial pressure, p0) is directly recovered from the subsampled data. Effective representation of the photoacoustic data requires basis defined on the range of the photoacoustic forward operator. To this end we propose a novel wedge-restriction of Curvelet transform which enables us to construct such basis. Both recovery problems are formulated in a variational framework. As the Curvelet frame is heavily overdetermined, we use reweighted l1 norm penalties to enhance the sparsity of the solution. The data reconstruction problem DR is a standard compressed sensing recovery problem, which we solve using an ADMMtype algorithm, SALSA. Subsequently, the initial pressure is recovered using time reversal as implemented in the k-Wave Toolbox. The p0 reconstruction problem, p0R, aims to recover the photoacoustic image directly via FISTA, or ADMM when in addition including a non-negativity constraint. We compare and discuss the relative merits of the two approaches and illustrate them on 2D simulated and 3D real data in a fair and rigorous manner., Comment: 06 August 2021 (Accepted Version)

The majority of the solvers for the acoustic problem in Photoacoustic Tomography (PAT) rely on full solution of the wave equation which makes them less suitable for real-time and dynamic applications where only partial data is available. This is in contrast to other tomographic modalities, e.g. X-ray tomography, where partial data implies partial cost for the application of the forward and adjoint operators. In this work we present a novel solver for the forward and adjoint wave equations for the acoustic problem in PAT. We term the proposed solver Hamilton-Green as it approximates the fundamental solution to the respective wave equation along the trajectories of the Hamiltonian system resulting from the high frequency asymptotics for the wave equation. This approach is fast and scalable in the sense that it allows computing the solution for each sensor independently at a fraction of the cost of the full wave solution. The theoretical foundations of our approach are rooted in results available in seismics and ocean acoustics. We present results for 2D numerical phantom with heterogeneous sound speed which we evaluate against a full wave solution obtained with a pseudospectral method implemented in k-Wave toolbox., Comment: 35 pages, 15 figures

We present a framework for accelerated iterative reconstructions using a fast and approximate forward model that is based on k-space methods for photoacoustic tomography. The approximate model introduces aliasing artefacts in the gradient information for the iterative reconstruction, but these artefacts are highly structured and we can train a CNN that can use the approximate information to perform an iterative reconstruction. We show feasibility of the method for human in-vivo measurements in a limited-view geometry. The proposed method is able to produce superior results to total variation reconstructions with a speed-up of 32 times.

A crucial limitation of current high-resolution 3D photoacoustic tomography (PAT) devices that employ sequential scanning is their long acquisition time. In previous work, we demonstrated how to use compressed sensing techniques to improve upon this: images with good spatial resolution and contrast can be obtained from suitably sub-sampled PAT data acquired by novel acoustic scanning systems if sparsity-constrained image reconstruction techniques such as total variation regularization are used. Now, we show how a further increase of image quality can be achieved for imaging dynamic processes in living tissue (4D PAT). The key idea is to exploit the additional temporal redundancy of the data by coupling the previously used spatial image reconstruction models with sparsity-constrained motion estimation models. While simulated data from a two-dimensional numerical phantom will be used to illustrate the main properties of this recently developed joint-image-reconstruction-and-motion-estimation framework, measured data from a dynamic experimental phantom will also be used to demonstrate their potential for challenging, large-scale, real-world, three-dimensional scenarios. The latter only becomes feasible if a carefully designed combination of tailored optimization schemes is employed, which we describe and examine in more detail.

We propose an extension of a special form of gradient descent -- in the literature known as linearised Bregman iteration -- to a larger class of non-convex functions. We replace the classical (squared) two norm metric in the gradient descent setting with a generalised Bregman distance, based on a proper, convex and lower semi-continuous function. The algorithm's global convergence is proven for functions that satisfy the Kurdyka-\L ojasiewicz property. Examples illustrate that features of different scale are being introduced throughout the iteration, transitioning from coarse to fine. This coarse-to-fine approach with respect to scale allows to recover solutions of non-convex optimisation problems that are superior to those obtained with conventional gradient descent, or even projected and proximal gradient descent. The effectiveness of the linearised Bregman iteration in combination with early stopping is illustrated for the applications of parallel magnetic resonance imaging, blind deconvolution as well as image classification with neural networks., Comment: 29 pages, 6 figures, journal submission

Recent advances in deep learning for tomographic reconstructions have shown great potential to create accurate and high quality images with a considerable speed-up. In this work we present a deep neural network that is specifically designed to provide high resolution 3D images from restricted photoacoustic measurements. The network is designed to represent an iterative scheme and incorporates gradient information of the data fit to compensate for limited view artefacts. Due to the high complexity of the photoacoustic forward operator, we separate training and computation of the gradient information. A suitable prior for the desired image structures is learned as part of the training. The resulting network is trained and tested on a set of segmented vessels from lung CT scans and then applied to in-vivo photoacoustic measurement data.

We discuss a special form of gradient descent that in the literature has become known as the so-called linearised Bregman iteration. The idea is to replace the classical (squared) two norm metric in the gradient descent setting with a generalised Bregman distance, based on a more general proper, convex and lower semi-continuous functional. Gradient descent as well as the entropic mirror descent by Nemirovsky and Yudin are special cases, as is a specific form of non-linear Landweber iteration introduced by Bachmayr and Burger. We are going to analyse the linearised Bregman iteration in a setting where the functional we want to minimise is neither necessarily Lipschitz-continuous (in the classical sense) nor necessarily convex, and establish a global convergence result under the additional assumption that the functional we wish to minimise satisfies the so-called Kurdyka-{\L}ojasiewicz property., Comment: Conference proceedings of '2016 Geometric Numerical Integration and its Applications Maths Conference at La Trobe University, Melbourne Australia', MI Lecture Notes series of Kyushu University, six pages, one figure, program code: https://doi.org/10.17863/CAM.6714

We present a method for the recovery of compressively sensed acoustic fields using patterned, instead of point-by-point, detection. From a limited number of such compressed measurements, we propose to reconstruct the field on the sensor plane in each time step independently assuming its sparsity in a Curvelet frame. A modification of the Curvelet frame is proposed to account for the smoothing effects of data acquisition and motivated by a frequency domain model for photoacoustic tomography. An ADMM type algorithm, SALSA, is used to recover the pointwise data in each individual time step from the patterned measurements. For photoacoustic applications, the photoacoustic image of the initial pressure is reconstructed using time reversal in ${\bf k}$-Wave Toolbox., Comment: submitted to IEEE Transactions on Image Processing

Current 3D photoacoustic tomography (PAT) systems offer either high image quality or high frame rates but are not able to deliver high spatial and temporal resolution simultaneously, which limits their ability to image dynamic processes in living tissue. A particular example is the planar Fabry-Perot (FP) scanner, which yields high-resolution images but takes several minutes to sequentially map the photoacoustic field on the sensor plane, point-by-point. However, as the spatio-temporal complexity of many absorbing tissue structures is rather low, the data recorded in such a conventional, regularly sampled fashion is often highly redundant. We demonstrate that combining variational image reconstruction methods using spatial sparsity constraints with the development of novel PAT acquisition systems capable of sub-sampling the acoustic wave field can dramatically increase the acquisition speed while maintaining a good spatial resolution: First, we describe and model two general spatial sub-sampling schemes. Then, we discuss how to implement them using the FP scanner and demonstrate the potential of these novel compressed sensing PAT devices through simulated data from a realistic numerical phantom and through measured data from a dynamic experimental phantom as well as from in-vivo experiments. Our results show that images with good spatial resolution and contrast can be obtained from highly sub-sampled PAT data if variational image reconstruction methods that describe the tissues structures with suitable sparsity-constraints are used. In particular, we examine the use of total variation regularization enhanced by Bregman iterations. These novel reconstruction strategies offer new opportunities to dramatically increase the acquisition speed of PAT scanners that employ point-by-point sequential scanning as well as reducing the channel count of parallelized schemes that use detector arrays., Comment: submitted to "Physics in Medicine and Biology"

Photoacoustic Tomography (PAT) is an emerging biomedical "imaging from coupled physics" technique, in which the image contrast is due to optical absorption, but the information is carried to the surface of the tissue as ultrasound pulses. Many algorithms and formulae for PAT image reconstruction have been proposed for the case when a complete data set is available. In many practical imaging scenarios, however, it is not possible to obtain the full data, or the data may be sub-sampled for faster data acquisition. In such cases, image reconstruction algorithms that can incorporate prior knowledge to ameliorate the loss of data are required. Hence, recently there has been an increased interest in using variational image reconstruction. A crucial ingredient for the application of these techniques is the adjoint of the PAT forward operator, which is described in this article from physical, theoretical and numerical perspectives. First, a simple mathematical derivation of the adjoint of the PAT forward operator in the continuous framework is presented. Then, an efficient numerical implementation of the adjoint using a k-space time domain wave propagation model is described and illustrated in the context of variational PAT image reconstruction, on both 2D and 3D examples including inhomogeneous sound speed. The principal advantage of this analytical adjoint over an algebraic adjoint (obtained by taking the direct adjoint of the particular numerical forward scheme used) is that it can be implemented using currently available fast wave propagation solvers., Comment: submitted to "Inverse Problems"

Magnetic resonance imaging (MRI) is a versatile imaging technique that allows different contrasts depending on the acquisition parameters. Many clinical imaging studies acquire MRI data for more than one of these contrasts---such as for instance T1 and T2 weighted images---which makes the overall scanning procedure very time consuming. As all of these images show the same underlying anatomy one can try to omit unnecessary measurements by taking the similarity into account during reconstruction. We will discuss two modifications of total variation---based on i) location and ii) direction---that take structural a priori knowledge into account and reduce to total variation in the degenerate case when no structural knowledge is available. We solve the resulting convex minimization problem with the alternating direction method of multipliers that separates the forward operator from the prior. For both priors the corresponding proximal operator can be implemented as an extension of the fast gradient projection method on the dual problem for total variation. We tested the priors on six data sets that are based on phantoms and real MRI images. In all test cases exploiting the structural information from the other contrast yields better results than separate reconstruction with total variation in terms of standard metrics like peak signal-to-noise ratio and structural similarity index. Furthermore, we found that exploiting the two dimensional directional information results in images with well defined edges, superior to those reconstructed solely using a priori information about the edge location., Comment: 18 pages, 16 figures

This article presents a method for solving large-scale linear inverse problems regular- ized with a nonlinear, edge-preserving penalty term such as the total variation or Perona-Malik. In the proposed scheme, the nonlinearity is handled with lagged diffusivity fixed point iteration which involves solving a large-scale linear least squares problem in each iteration. Because the convergence of Krylov methods for problems with discontinuities is notoriously slow, we propose to accelerate it by means of priorconditioning. Priorconditioning is a technique which embeds the information contained in the prior (expressed as a regularizer in Bayesian framework) directly into the forward operator and hence into the solution space. We derive a factorization-free priorconditioned LSQR algorithm, allowing implicit application of the preconditioner through efficient schemes such as multigrid. The resulting method is matrix-free i.e. the forward map can be defined through its action on a vector. We demonstrate the effectiveness of the proposed scheme on a three-dimensional problem in fluorescence diffuse optical tomography using algebraic multigrid preconditioner.