Marta M. Betcke

Tobías I. Liaudat, Matthijs Mars, Matthew A. Price et al.

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.

Matthijs Mars, Marta M. Betcke, Jason D. McEwen

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.

Bolin Pan, Marta M. Betcke

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.

Marta M. Betcke, Lisa Maria Kreusser, Davide Murari

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.

Matthijs Mars, Marta M. Betcke, Jason D. McEwen

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 good reconstruction quality and computation time, generalising well to other coverages and require little to no fine-tuning. Furthermore, they generalise well to more realistic radio observations and are able to reconstruct images with with a larger dynamic range than the training set.

Emilio McAllister Fognini, Marta M. Betcke, Ben T. Cox

The Fast Multipole Method (FMM) is an efficient numerical algorithm for computation of long-ranged forces in $N$-body problems within gravitational and electrostatic fields. This method utilizes multipole expansions of the Green's function inherent to the underlying dynamical systems. Despite its widespread application in physics and engineering, the integration of FMM with modern machine learning architectures remains underexplored. In this work, we propose a novel neural network architecture, the Neural FMM, that integrates the information flow of the FMM into a hierarchical machine learning framework for learning the Green's operator of an Elliptic PDE. Our Neural FMM architecture leverages a hierarchical computation flow of the FMM method to split up the local and far-field interactions and efficiently learn their respective representations.

Matthijs Mars, Tobías I. Liaudat, Jessica J. Whitney et al.

With the rise of large radio interferometric telescopes, particularly the SKA, there is a growing demand for computationally efficient image reconstruction techniques. Existing reconstruction methods, such as the CLEAN algorithm or proximal optimisation approaches, are iterative in nature, necessitating a large amount of compute. These methods either provide no uncertainty quantification or require large computational overhead to do so. Learned reconstruction methods have shown promise in providing efficient and high quality reconstruction. In this article we explore the use of generative neural networks that enable efficient approximate sampling of the posterior distribution for high quality reconstructions with uncertainty quantification. Our RI-GAN framework, builds on the regularised conditional generative adversarial network (rcGAN) framework by integrating a gradient U-Net (GU-Net) architecture - a hybrid reconstruction model that embeds the measurement operator directly into the network. This framework uses Wasserstein GANs to improve training stability in combination with regularisation terms that combat mode collapse, which are typical problems for conditional GANs. This approach takes as input the dirty image and the point spread function (PSF) of the observation and provides efficient, high-quality image reconstructions that are robust to varying visibility coverages, generalises to images with an increased dynamic range, and provides informative uncertainty quantification. Our methods provide a significant step toward computationally efficient, scalable, and uncertainty-aware imaging for next-generation radio telescopes.

Bolin Pan, Simon R. Arridge, Felix Lucka et al.

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.

Marta M. Betcke, Ben T. Cox, Nam Huynh et al.

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.

Simon R. Arridge, Marta M. Betcke, Ben T. Cox et al.

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.

Matthias J. Ehrhardt, Marta M. Betcke

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.