Ben T. Cox

Antonio Stanziola, Simon R. Arridge, Ben T. Cox et al.

We present an open-source differentiable acoustic simulator, j-Wave, which can solve time-varying and time-harmonic acoustic problems. It supports automatic differentiation, which is a program transformation technique that has many applications, especially in machine learning and scientific computing. j-Wave is composed of modular components that can be easily customized and reused. At the same time, it is compatible with some of the most popular machine learning libraries, such as JAX and TensorFlow. The accuracy of the simulation results for known configurations is evaluated against the widely used k-Wave toolbox and a cohort of acoustic simulation software. j-Wave is available from https://github.com/ucl-bug/jwave.

Antonio Stanziola, Simon Arridge, Ben T. Cox et al.

A new method for solving the wave equation is presented, called the learned Born series (LBS), which is derived from a convergent Born Series but its components are found through training. The LBS is shown to be significantly more accurate than the convergent Born series for the same number of iterations, in the presence of high contrast scatterers, while maintaining a comparable computational complexity. The LBS is able to generate a reasonable prediction of the global pressure field with a small number of iterations, and the errors decrease with the number of learned iterations.

Antonio Stanziola, Simon R. Arridge, Ben T. Cox et al.

Differentiable simulators are an emerging concept with applications in several fields, from reinforcement learning to optimal control. Their distinguishing feature is the ability to calculate analytic gradients with respect to the input parameters. Like neural networks, which are constructed by composing several building blocks called layers, a simulation often requires computing the output of an operator that can itself be decomposed into elementary units chained together. While each layer of a neural network represents a specific discrete operation, the same operator can have multiple representations, depending on the discretization employed and the research question that needs to be addressed. Here, we propose a simple design pattern to construct a library of differentiable operators and discretizations, by representing operators as mappings between families of continuous functions, parametrized by finite vectors. We demonstrate the approach on an acoustic optimization problem, where the Helmholtz equation is discretized using Fourier spectral methods, and differentiability is demonstrated using gradient descent to optimize the speed of sound of an acoustic lens. The proposed framework is open-sourced and available at \url{https://github.com/ucl-bug/jaxdf}

Janek Gröhl, Leonid Kunyansky, Jenni Poimala et al.

Quantitative comparison of the quality of photoacoustic image reconstruction algorithms remains a major challenge. No-reference image quality measures are often inadequate, but full-reference measures require access to an ideal reference image. While the ground truth is known in simulations, it is unknown in vivo, or in phantom studies, as the reference depends on both the phantom properties and the imaging system. We tackle this problem by using numerical digital twins of tissue-mimicking phantoms and the imaging system to perform a quantitative calibration to reduce the simulation gap. The contributions of this paper are two-fold: First, we use this digital-twin framework to compare multiple state-of-the-art reconstruction algorithms. Second, among these is a Fourier transform-based reconstruction algorithm for circular detection geometries, which we test on experimental data for the first time. Our results demonstrate the usefulness of digital phantom twins by enabling assessment of the accuracy of the numerical forward model and enabling comparison of image reconstruction schemes with full-reference image quality assessment. We show that the Fourier transform-based algorithm yields results comparable to those of iterative time reversal, but at a lower computational cost. All data and code are publicly available on Zenodo: https://doi.org/10.5281/zenodo.15388429.

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.

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.

Antonio Stanziola, Simon R. Arridge, Ben T. Cox et al.

Transcranial ultrasound therapy is increasingly used for the non-invasive treatment of brain disorders. However, conventional numerical wave solvers are currently too computationally expensive to be used online during treatments to predict the acoustic field passing through the skull (e.g., to account for subject-specific dose and targeting variations). As a step towards real-time predictions, in the current work, a fast iterative solver for the heterogeneous Helmholtz equation in 2D is developed using a fully-learned optimizer. The lightweight network architecture is based on a modified UNet that includes a learned hidden state. The network is trained using a physics-based loss function and a set of idealized sound speed distributions with fully unsupervised training (no knowledge of the true solution is required). The learned optimizer shows excellent performance on the test set, and is capable of generalization well outside the training examples, including to much larger computational domains, and more complex source and sound speed distributions, for example, those derived from x-ray computed tomography images of the skull.

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.