Learning a Filtered Backprojection Reconstruction Method for Photoacoustic Computed Tomography with Hemispherical Measurement Geometries

In certain three-dimensional (3D) applications of photoacoustic computed tomography (PACT), including \textit{in vivo} breast imaging, hemispherical measurement apertures that enclose the object within their convex hull are employed for data acquisition. Data acquired with such measurement geometries are referred to as \textit{half-scan} data, as only half of a complete spherical measurement aperture is employed. Although previous studies have demonstrated that half-scan data can uniquely and stably reconstruct the sought-after object, no closed-form reconstruction formula for use with half-scan data has been reported. To address this, a semi-analytic reconstruction method in the form of filtered backprojection (FBP), referred to as the half-scan FBP method, is developed in this work. Because the explicit form of the filtering operation in the half-scan FBP method is not currently known, a learning-based method is proposed to approximate it. The proposed method is systematically investigated by use of virtual imaging studies of 3D breast PACT that employ ensembles of numerical breast phantoms and a physics-based model of the data acquisition process. The method is subsequently applied to experimental data acquired in an \textit{in vivo} breast PACT study. The results confirm that the half-scan FBP method can accurately reconstruct 3D images from half-scan data. Importantly, because the sought-after inverse mapping is well-posed, the reconstruction method remains accurate even when applied to data that differ considerably from those employed to learn the filtering operation.

Physics and Deep Learning in Computational Wave Imaging

Computational wave imaging (CWI) extracts hidden structure and physical properties of a volume of material by analyzing wave signals that traverse that volume. Applications include seismic exploration of the Earth's subsurface, acoustic imaging and non-destructive testing in material science, and ultrasound computed tomography in medicine. Current approaches for solving CWI problems can be divided into two categories: those rooted in traditional physics, and those based on deep learning. Physics-based methods stand out for their ability to provide high-resolution and quantitatively accurate estimates of acoustic properties within the medium. However, they can be computationally intensive and are susceptible to ill-posedness and nonconvexity typical of CWI problems. Machine learning-based computational methods have recently emerged, offering a different perspective to address these challenges. Diverse scientific communities have independently pursued the integration of deep learning in CWI. This review delves into how contemporary scientific machine-learning (ML) techniques, and deep neural networks in particular, have been harnessed to tackle CWI problems. We present a structured framework that consolidates existing research spanning multiple domains, including computational imaging, wave physics, and data science. This study concludes with important lessons learned from existing ML-based methods and identifies technical hurdles and emerging trends through a systematic analysis of the extensive literature on this topic.

Revisiting the joint estimation of initial pressure and speed-of-sound distributions in photoacoustic computed tomography with consideration of canonical object constraints

In photoacoustic computed tomography (PACT) the accurate estimation of the initial pressure (IP) distribution generally requires knowledge of the object's heterogeneous speed-of-sound (SOS) distribution. Although hybrid imagers that combine ultrasound tomography with PACT have been proposed, in many current applications of PACT the SOS distribution remains unknown. Joint reconstruction (JR) of the IP and SOS distributions from PACT measurement data alone can address this issue. However, this joint estimation problem is ill-posed and corresponds to a non-convex optimization problem. While certain regularization strategies have been deployed, stabilizing the JR problem to yield accurate estimates of the IP and SOS distributions has remained an open challenge. To address this, the presented numerical studies explore the effectiveness of easy to implement canonical object constraints for stabilizing the JR problem. The considered constraints include support, bound, and total variation constraints, which are incorporated into an optimization-based method for JR. Computer-simulation studies that employ anatomically realistic numerical breast phantoms are conducted to evaluate the impact of these object constraints on JR accuracy. Additionally, the impact of certain data inconsistencies, such as caused by measurement noise and physics modeling mismatches, on the effectiveness of the object constraints is investigated. The results demonstrate, for the first time, that the incorporation of canonical object constraints in an optimization-based image reconstruction method holds significant potential for mitigating the ill-posed nature of the PACT JR problem.

ProxNF: Neural Field Proximal Training for High-Resolution 4D Dynamic Image Reconstruction

Accurate spatiotemporal image reconstruction methods are needed for a wide range of biomedical research areas but face challenges due to data incompleteness and computational burden. Data incompleteness arises from the undersampling often required to increase frame rates and reduce acquisition times, while computational burden emerges due to the memory footprint of high-resolution images with three spatial dimensions and extended time horizons. Neural fields, an emerging class of neural networks that act as continuous representations of spatiotemporal objects, have previously been introduced to solve these dynamic imaging problems by reframing image reconstruction to a problem of estimating network parameters. Neural fields can address the twin challenges of data incompleteness and computational burden by exploiting underlying redundancies in these spatiotemporal objects. This work proposes ProxNF, a novel neural field training approach for spatiotemporal image reconstruction leveraging proximal splitting methods to separate computations involving the imaging operator from updates of the network parameter. Specifically, ProxNF evaluates the (subsampled) gradient of the data-fidelity term in the image domain and uses a fully supervised learning approach to update the neural field parameters. By reducing the memory footprint and the computational cost of evaluating the imaging operator, the proposed ProxNF approach allows for reconstructing large, high-resolution spatiotemporal images. This method is demonstrated in two numerical studies involving virtual dynamic contrast-enhanced photoacoustic computed tomography imaging of an anatomically realistic dynamic numerical mouse phantom and a two-compartment model of tumor perfusion.

Technical Note: An Efficient Implementation of the Spherical Radon Transform with Cylindrical Apertures

The spherical Radon transform (SRT) is an integral transform that maps a function to its integrals over concentric spherical shells centered at specified sensor locations. It has several imaging applications, including synthetic aperture radar and photoacoustic computed tomography. However, computation of the SRT can be expensive. Efficient implementation of SRT on general purpose graphic processing units (GPGPUs) often utilizes non-matched implementation of the adjoint operator, leading to inconsistent gradients in optimization-based image reconstruction methods. This work details an efficient implementation of the SRT and its adjoint for the case of a cylindrical measurement aperture. Exploiting symmetry of the cylindrical geometry, the SRT can then be expressed as the composition of two circular Radon transforms (CRT). Utilizing this formulation then allows for an efficient implementation of the SRT as a discrete-to-discrete operator utilizing sparse matrix representation.

Investigating the Use of Traveltime and Reflection Tomography for Deep Learning-Based Sound-Speed Estimation in Ultrasound Computed Tomography

Ultrasound computed tomography (USCT) is actively being developed to quantify acoustic tissue properties such as the speed-of-sound (SOS). Although full-waveform inversion (FWI) is an effective method for accurate SOS reconstruction, it can be computationally challenging for large-scale problems. Deep learning-based image-to-image learned reconstruction (IILR) methods are being investigated as scalable and computationally efficient alternatives. This study investigates the impact of the chosen input modalities on IILR methods for high-resolution SOS reconstruction in USCT. The selected modalities are traveltime tomography (TT) and reflection tomography (RT), which produce a low-resolution SOS map and a reflectivity map, respectively. These modalities have been chosen for their lower computational cost relative to FWI and their capacity to provide complementary information: TT offers a direct -- while low resolution -- SOS measure, while RT reveals tissue boundary information. Systematic analyses were facilitated by employing a stylized USCT imaging system with anatomically realistic numerical breast phantoms. Within this testbed, a supervised convolutional neural network (CNN) was trained to map dual-channel (TT and RT images) to a high-resolution SOS map. Moreover, the CNN was fine-tuned using a weighted reconstruction loss that prioritized tumor regions to address tumor underrepresentation in the training dataset. To understand the benefits of employing dual-channel inputs, single-input CNNs were trained separately using inputs from each modality alone (TT or RT). The methods were assessed quantitatively using normalized root mean squared error and structural similarity index measure for reconstruction accuracy and receiver operating characteristic analysis to assess signal detection-based performance measures.

Spatiotemporal Image Reconstruction to Enable High-Frame Rate Dynamic Photoacoustic Tomography with Rotating-Gantry Volumetric Imagers

Significance: Dynamic photoacoustic computed tomography (PACT) is a valuable technique for monitoring physiological processes. However, current dynamic PACT techniques are often limited to 2D spatial imaging. While volumetric PACT imagers are commercially available, these systems typically employ a rotating gantry in which the tomographic data are sequentially acquired. Because the object varies during the data-acquisition process, the sequential data-acquisition poses challenges to image reconstruction associated with data incompleteness. The proposed method is highly significant in that it will address these challenges and enable volumetric dynamic PACT imaging with existing imagers. Aim: The aim of this study is to develop a spatiotemporal image reconstruction (STIR) method for dynamic PACT that can be applied to commercially available volumetric PACT imagers that employ a sequential scanning strategy. The proposed method aims to overcome the challenges caused by the limited number of tomographic measurements acquired per frame. Approach: A low-rank matrix estimation-based STIR method (LRME-STIR) is proposed to enable dynamic volumetric PACT. The LRME-STIR method leverages the spatiotemporal redundancies to accurately reconstruct a 4D spatiotemporal image. Results: The numerical studies substantiate the LRME-STIR method's efficacy in reconstructing 4D dynamic images from measurements acquired with a rotating gantry. The experimental study demonstrates the method's ability to faithfully recover the flow of a contrast agent at a frame rate of 0.1 s even when only a single tomographic measurement per frame is available. Conclusions: The LRME-STIR method offers a promising solution to the challenges faced by enabling 4D dynamic imaging using commercially available volumetric imagers. By enabling accurate 4D reconstruction, this method has the potential to advance preclinical research.

Learned Full Waveform Inversion Incorporating Task Information for Ultrasound Computed Tomography

Ultrasound computed tomography (USCT) is an emerging imaging modality that holds great promise for breast imaging. Full-waveform inversion (FWI)-based image reconstruction methods incorporate accurate wave physics to produce high spatial resolution quantitative images of speed of sound or other acoustic properties of the breast tissues from USCT measurement data. However, the high computational cost of FWI reconstruction represents a significant burden for its widespread application in a clinical setting. The research reported here investigates the use of a convolutional neural network (CNN) to learn a mapping from USCT waveform data to speed of sound estimates. The CNN was trained using a supervised approach with a task-informed loss function aiming at preserving features of the image that are relevant to the detection of lesions. A large set of anatomically and physiologically realistic numerical breast phantoms (NBPs) and corresponding simulated USCT measurements was employed during training. Once trained, the CNN can perform real-time FWI image reconstruction from USCT waveform data. The performance of the proposed method was assessed and compared against FWI using a hold-out sample of 41 NBPs and corresponding USCT data. Accuracy was measured using relative mean square error (RMSE), structural self-similarity index measure (SSIM), and lesion detection performance (DICE score). This numerical experiment demonstrates that a supervised learning model can achieve accuracy comparable to FWI in terms of RMSE and SSIM, and better performance in terms of task performance, while significantly reducing computational time.

Ideal Observer Computation by Use of Markov-Chain Monte Carlo with Generative Adversarial Networks

Medical imaging systems are often evaluated and optimized via objective, or task-specific, measures of image quality (IQ) that quantify the performance of an observer on a specific clinically-relevant task. The performance of the Bayesian Ideal Observer (IO) sets an upper limit among all observers, numerical or human, and has been advocated for use as a figure-of-merit (FOM) for evaluating and optimizing medical imaging systems. However, the IO test statistic corresponds to the likelihood ratio that is intractable to compute in the majority of cases. A sampling-based method that employs Markov-Chain Monte Carlo (MCMC) techniques was previously proposed to estimate the IO performance. However, current applications of MCMC methods for IO approximation have been limited to a small number of situations where the considered distribution of to-be-imaged objects can be described by a relatively simple stochastic object model (SOM). As such, there remains an important need to extend the domain of applicability of MCMC methods to address a large variety of scenarios where IO-based assessments are needed but the associated SOMs have not been available. In this study, a novel MCMC method that employs a generative adversarial network (GAN)-based SOM, referred to as MCMC-GAN, is described and evaluated. The MCMC-GAN method was quantitatively validated by use of test-cases for which reference solutions were available. The results demonstrate that the MCMC-GAN method can extend the domain of applicability of MCMC methods for conducting IO analyses of medical imaging systems.

Derivative-Informed Neural Operator: An Efficient Framework for High-Dimensional Parametric Derivative Learning

Neural operators have gained significant attention recently due to their ability to approximate high-dimensional parametric maps between function spaces. At present, only parametric function approximation has been addressed in the neural operator literature. In this work we investigate incorporating parametric derivative information in neural operator training; this information can improve function approximations, additionally it can be used to improve the approximation of the derivative with respect to the parameter, which is often the key to scalable solution of high-dimensional outer-loop problems (e.g. Bayesian inverse problems). Parametric Jacobian information is formally intractable to incorporate due to its high-dimensionality, to address this concern we propose strategies based on reduced SVD, randomized sketching and the use of reduced basis surrogates. All of these strategies only require only $O(r)$ Jacobian actions to construct sample Jacobian data, and allow us to reduce the linear algebra and memory costs associated with the Jacobian training from the product of the input and output dimensions down to $O(r^2)$, where $r$ is the dimensionality associated with the dimension reduction technique. Numerical results for parametric PDE problems demonstrate that the addition of derivative information to the training problem can significantly improve the parametric map approximation, particularly given few data. When Jacobian actions are inexpensive compared to the parametric map, this information can be economically substituted for parametric map data. Additionally we show that Jacobian error approximations improve significantly with the introduction of Jacobian training data. This result opens the door to the use of derivative-informed neural operators (DINOs) in outer-loop algorithms where they can amortize the additional training data cost via repeated evaluations.