Discrete imaging models for three-dimensional optoacoustic tomography using radially symmetric expansion functions

S-Wave Accelerates Optimization-based Photoacoustic Image Reconstruction in vivo

OBJECTIVE

Photoacoustic imaging has undergone rapid development in recent years. To simulate photoacoustic imaging on a computer, the most popular MATLAB toolbox currently used for the forward projection process is k-Wave. However, k-Wave suffers from significant computation time. Here we propose a straightforward simulation approach based on superposed Wave (s-Wave) to accelerate photoacoustic simulation.

METHODS

In this study, we consider the initial pressure distribution as a collection of individual pixels. By obtaining standard sensor data from a single pixel beforehand, we can easily manipulate the phase and amplitude of the sensor data for specific pixels using loop and multiplication operators. The effectiveness of this approach is validated through an optimization-based reconstruction algorithm.

RESULTS

The results reveal significantly reduced computation time compared with k-Wave. Particularly in a sparse 3-D configuration, s-Wave exhibits a speed improvement >2000 times compared with k-Wave. In terms of optimization-based image reconstruction, in vivo imaging results reveal that using the s-Wave method yields images highly similar to those obtained using k-Wave, while reducing the reconstruction time by approximately 50 times.

CONCLUSION

Proposed here is an accelerated optimization-based algorithm for photoacoustic image reconstruction, using the fast s-Wave forward projection simulation. Our method achieves substantial time savings, particularly in sparse system configurations. Future work will focus on further optimizing the algorithm and expanding its applicability to a broader range of photoacoustic imaging scenarios.

Model-Based 3-D X-Ray Induced Acoustic Computerized Tomography

X-ray-induced acoustic (XA) computerized tomography (XACT) is an evolving imaging technique that aims to reconstruct the X-ray energy deposition from XA measurements. Main challenges in XACT are the poor signal-to-noise ratio and limited field-of-view, which cause artifacts in the images. We demonstrate the efficacy of model-based (MB) algorithms for three-dimensional XACT and compare with the traditional algorithms. The MB algorithm is based on iterative, matrix-free approach for regularized-least-squares minimization corresponding to XACT. The matrix-free-LSQR (MF-LSQR) and the non-iterative model-backprojection (MBP) reconstructions were evaluated and compared with universal backprojection (UBP), time-reversal (TR) and fast-Fourier transform (FFT)-based reconstructions for numerical and experimental XACT datasets. The results demonstrate the capability of MF-LSQR algorithm to reduce noisy artifacts thus yielding better reconstructions. MBP and MF-LSQR algorithms perform particularly well with the experimental XACT dataset, where noise in signals significantly affects the reconstruction of the target in UBP and FFT-based reconstructions. The TR reconstruction for experimental XACT are comparable to MF-LSQR, but takes thrice as much time and filters the frequency components greater than maximum frequency supported by the grid, resulting loss of resolution. The MB algorithms are able to overcome the challenges in XACT and hence are vital for the clinical translation of XACT.

Learned regularization for image reconstruction in sparse-view photoacoustic tomography

Constrained data acquisitions, such as sparse view measurements, are sometimes used in photoacoustic computed tomography (PACT) to accelerate data acquisition. However, it is challenging to reconstruct high-quality images under such scenarios. Iterative image reconstruction with regularization is a typical choice to solve this problem but it suffers from image artifacts. In this paper, we present a learned regularization method to suppress image artifacts in model-based iterative reconstruction in sparse view PACT. A lightweight dual-path network is designed to learn regularization features from both the data and the image domains. The network is trained and tested on both simulation and in vivo datasets and compared with other methods such as Tikhonov regularization, total variation regularization, and a U-Net based post-processing approach. Results show that although the learned regularization network possesses a size of only 0.15% of a U-Net, it outperforms other methods and converges after as few as five iterations, which takes less than one-third of the time of conventional methods. Moreover, the proposed reconstruction method incorporates the physical model of photoacoustic imaging and explores structural information from training datasets. The integration of deep learning with a physical model can potentially achieve improved imaging performance in practice.

Computing a projection operator onto the null space of a linear imaging operator: tutorial

Many imaging systems can be approximately described by a linear operator that maps an object property to a collection of discrete measurements. However, even in the absence of measurement noise, such operators are generally "blind" to certain components of the object, and hence information is lost in the imaging process. Mathematically, this is explained by the fact that the imaging operator can possess a null space. All objects in the null space, by definition, are mapped to a collection of identically zero measurements and are hence invisible to the imaging system. As such, characterizing the null space of an imaging operator is of fundamental importance when comparing and/or designing imaging systems. A characterization of the null space can also facilitate the design of regularization strategies for image reconstruction methods. Characterizing the null space via an associated projection operator is, in general, a computationally demanding task. In this tutorial, computational procedures for establishing projection operators that map an object to the null space of a discrete-to-discrete imaging operator are surveyed. A new machine-learning-based approach that employs a linear autoencoder is also presented. The procedures are demonstrated by use of biomedical imaging examples, and their computational complexities and memory requirements are compared.

Discretization of Learned NETT Regularization for Solving Inverse Problems

Deep learning based reconstruction methods deliver outstanding results for solving inverse problems and are therefore becoming increasingly important. A recently invented class of learning-based reconstruction methods is the so-called NETT (for Network Tikhonov Regularization), which contains a trained neural network as regularizer in generalized Tikhonov regularization. The existing analysis of NETT considers fixed operators and fixed regularizers and analyzes the convergence as the noise level in the data approaches zero. In this paper, we extend the frameworks and analysis considerably to reflect various practical aspects and take into account discretization of the data space, the solution space, the forward operator and the neural network defining the regularizer. We show the asymptotic convergence of the discretized NETT approach for decreasing noise levels and discretization errors. Additionally, we derive convergence rates and present numerical results for a limited data problem in photoacoustic tomography.

Model-Based Reconstruction of Large Three-Dimensional Optoacoustic Datasets

Iterative model-based algorithms are known to enable more accurate and quantitative optoacoustic (photoacoustic) tomographic reconstructions than standard back-projection methods. However, three-dimensional (3D) model-based inversion is often hampered by high computational complexity and memory overhead. Parallel implementations on a graphics processing unit (GPU) have been shown to efficiently reduce the memory requirements by on-the-fly calculation of the actions of the optoacoustic model matrix, but the high complexity still makes these approaches impractical for large 3D optoacoustic datasets. Herein, we show that the computational complexity of 3D model-based iterative inversion can be significantly reduced by splitting the model matrix into two parts: one maximally sparse matrix containing only one entry per voxel-transducer pair and a second matrix corresponding to cyclic convolution. We further suggest reconstructing the images by multiplying the transpose of the model matrix calculated in this manner with the acquired signals, which is equivalent to using a very large regularization parameter in the iterative inversion method. The performance of these two approaches is compared to that of standard back-projection and a recently introduced GPU-based model-based method using datasets from in vivo experiments. The reconstruction time was accelerated by approximately an order of magnitude with the new iterative method, while multiplication with the transpose of the matrix is shown to be as fast as standard back-projection.

A survey of computational frameworks for solving the acoustic inverse problem in three-dimensional photoacoustic computed tomography

Photoacoustic computed tomography (PACT), also known as optoacoustic tomography, is an emerging imaging technique that holds great promise for biomedical imaging. PACT is a hybrid imaging method that can exploit the strong endogenous contrast of optical methods along with the high spatial resolution of ultrasound methods. In its canonical form that is addressed in this article, PACT seeks to estimate the photoacoustically-induced initial pressure distribution within the object. Image reconstruction methods are employed to solve the acoustic inverse problem associated with the image formation process. When an idealized imaging scenario is considered, analytic solutions to the PACT inverse problem are available; however, in practice, numerous challenges exist that are more readily addressed within an optimization-based, or iterative, image reconstruction framework. In this article, the PACT image reconstruction problem is reviewed within the context of modern optimization-based image reconstruction methodologies. Imaging models that relate the measured photoacoustic wavefields to the sought-after object function are described in their continuous and discrete forms. The basic principles of optimization-based image reconstruction from discrete PACT measurement data are presented, which includes a review of methods for modeling the PACT measurement system response and other important physical factors. Non-conventional formulations of the PACT image reconstruction problem, in which acoustic parameters of the medium are concurrently estimated along with the PACT image, are also introduced and reviewed.

Optoacoustic image reconstruction: the full inverse problem with variable bases

Optoacoustic imaging was for a long time concerned with the reconstruction of energy density or optical properties. In this work, we present the full inverse problem with respect to optical absorption and diffusion as well as speed of sound and mass density. The inverse problem is solved by an iterative gradient-based optimization procedure. Since the ill-conditioning increases with the number of sought parameters, we propose two approaches to improve the conditioning. The first approach is based on the reduction of the size of the basis for the parameter spaces, that evolves according to the particular characteristics of the solution, while maintaining the flexibility of element-wise parameter selection. The second approach is a material identification technique that incorporates prior knowledge of expected material types and uses the acoustical gradients to identify materials uniquely. We present numerical studies to illustrate the properties and functional principle of the proposed methods. Significant convergence speed-ups are gained by the two approaches countering ill-conditioning. Additionally, we show results for the reconstruction of a mouse brain from in vivo measurements.

Image-guided filtering for improving photoacoustic tomographic image reconstruction

Several algorithms exist to solve the photoacoustic image reconstruction problem depending on the expected reconstructed image features. These reconstruction algorithms promote typically one feature, such as being smooth or sharp, in the output image. Combining these features using a guided filtering approach was attempted in this work, which requires an input and guiding image. This approach act as a postprocessing step to improve commonly used Tikhonov or total variational regularization method. The result obtained from linear backprojection was used as a guiding image to improve these results. Using both numerical and experimental phantom cases, it was shown that the proposed guided filtering approach was able to improve (as high as 11.23 dB) the signal-to-noise ratio of the reconstructed images with the added advantage being computationally efficient. This approach was compared with state-of-the-art basis pursuit deconvolution as well as standard denoising methods and shown to outperform them.

Mitigation of artifacts due to isolated acoustic heterogeneities in photoacoustic computed tomography using a variable data truncation-based reconstruction method

Photoacoustic computed tomography (PACT) is an emerging computed imaging modality that exploits optical contrast and ultrasonic detection principles to form images of the absorbed optical energy density within tissue. If the object possesses spatially variant acoustic properties that are unaccounted for by the reconstruction method, the estimated image can contain distortions. While reconstruction methods have recently been developed to compensate for this effect, they generally require the object’s acoustic properties to be known a priori. To circumvent the need for detailed information regarding an object’s acoustic properties, we previously proposed a half-time reconstruction method for PACT. A half-time reconstruction method estimates the PACT image from a data set that has been temporally truncated to exclude the data components that have been strongly aberrated. However, this method can be improved upon when the approximate sizes and locations of isolated heterogeneous structures, such as bones or gas pockets, are known. To address this, we investigate PACT reconstruction methods that are based on a variable data truncation (VDT) approach. The VDT approach represents a generalization of the half-time approach, in which the degree of temporal truncation for each measurement is determined by the distance between the corresponding ultrasonic transducer location and the nearest known bone or gas void location. Computer-simulated and experimental data are employed to demonstrate the effectiveness of the approach in mitigating artifacts due to acoustic heterogeneities.

Impact of nonstationary optical illumination on image reconstruction in optoacoustic tomography

Optoacoustic tomography (OAT), also known as photoacoustic tomography, is a rapidly emerging hybrid imaging technique that possesses great potential for a wide range of biomedical imaging applications. In OAT, a laser is employed to illuminate the tissue of interest and acoustic signals are produced via the photoacoustic effect. From these data, an estimate of the distribution of the absorbed optical energy density within the tissue is reconstructed, referred to as the object function. This quantity is defined, in part, by the distribution of light fluence within the tissue that is established by the laser source. When performing three-dimensional imaging of large objects, such as a female human breast, it can be difficult to achieve a relatively uniform coverage of light fluence within the volume of interest when the position of the laser source is fixed. To circumvent this, researchers have proposed illumination schemes in which the relative position of the laser source and ultrasound probe is fixed, and both are rotated together to acquire a tomographic dataset. A problem with this rotating-illumination scheme is that the tomographic data are inconsistent; namely, the acoustic data recorded at each tomographic view angle (i.e., probe position) are produced by a distinct object function. In this work, the impact of this data inconsistency on image reconstruction accuracy is investigated systematically. This is accomplished by use of computer-simulation studies and application of mathematical results from the theory of microlocal analysis. These studies specify the set of image discontinuities that can be stably reconstructed with a nonstationary optical illumination setup. The study also includes a comparison of the ability of iterative and analytic image reconstruction methods to mitigate artifacts attributable to the data inconsistency.

Tutorial on photoacoustic tomography

Photoacoustic tomography (PAT) has become one of the fastest growing fields in biomedical optics. Unlike pure optical imaging, such as confocal microscopy and two-photon microscopy, PAT employs acoustic detection to image optical absorption contrast with high-resolution deep into scattering tissue. So far, PAT has been widely used for multiscale anatomical, functional, and molecular imaging of biological tissues. We focus on PAT’s basic principles, major implementations, imaging contrasts, and recent applications.

A Constrained Variable Projection Reconstruction Method for Photoacoustic Computed Tomography Without Accurate Knowledge of Transducer Responses

Photoacoustic computed tomography (PACT) is an emerging computed imaging modality that exploits optical contrast and ultrasonic detection principles to form images of the absorbed optical energy density within tissue. When the imaging system employs conventional piezoelectric ultrasonic transducers, the ideal photoacoustic (PA) signals are degraded by the transducers' acousto-electric impulse responses (EIRs) during the measurement process. If unaccounted for, this can degrade the accuracy of the reconstructed image. In principle, the effect of the EIRs on the measured PA signals can be ameliorated via deconvolution; images can be reconstructed subsequently by application of a reconstruction method that assumes an idealized EIR. Alternatively, the effect of the EIR can be incorporated into an imaging model and implicitly compensated for during reconstruction. In either case, the efficacy of the correction can be limited by errors in the assumed EIRs. In this work, a joint optimization approach to PACT image reconstruction is proposed for mitigating errors in reconstructed images that are caused by use of an inaccurate EIR. The method exploits the bi-linear nature of the imaging model and seeks to refine the measured EIR during the process of reconstructing the sought-after absorbed optical energy density. Computer-simulation and experimental studies are conducted to investigate the numerical properties of the method and demonstrate its value for mitigating image distortions and enhancing the visibility of fine structures.

Pre- and postmortem imaging of transplanted cells

Therapeutic interventions based on the transplantation of stem and progenitor cells have garnered increasing interest. This interest is fueled by successful preclinical studies for indications in many diseases, including the cardiovascular, central nervous, and musculoskeletal system. Further progress in this field is contingent upon access to techniques that facilitate an unambiguous identification and characterization of grafted cells. Such methods are invaluable for optimization of cell delivery, improvement of cell survival, and assessment of the functional integration of grafted cells. Following is a focused overview of the currently available cell detection and tracking methodologies that covers the entire spectrum from pre- to postmortem cell identification.

Waveform inversion with source encoding for breast sound speed reconstruction in ultrasound computed tomography

Ultrasound computed tomography (USCT) holds great promise for improving the detection and management of breast cancer. Because they are based on the acoustic wave equation, waveform inversion-based reconstruction methods can produce images that possess improved spatial resolution properties over those produced by ray-based methods. However, waveform inversion methods are computationally demanding and have not been applied widely in USCT breast imaging. In this work, source encoding concepts are employed to develop an accelerated USCT reconstruction method that circumvents the large computational burden of conventional waveform inversion methods. This method, referred to as the waveform inversion with source encoding (WISE) method, encodes the measurement data using a random encoding vector and determines an estimate of the sound speed distribution by solving a stochastic optimization problem by use of a stochastic gradient descent algorithm. Both computer simulation and experimental phantom studies are conducted to demonstrate the use of the WISE method. The results suggest that the WISE method maintains the high spatial resolution of waveform inversion methods while significantly reducing the computational burden.