[Paper Review] Hamilton-Green solver for the forward and adjoint problems in photoacoustic tomography

Francesc Rul⋅lan, Marta M. Betcke|arXiv (Cornell University)|Oct 31, 2018

Photoacoustic and Ultrasonic Imaging69 references4 citations

TL;DR

This paper introduces the Hamilton-Green (HG) solver, a high-frequency asymptotic method for solving the forward and adjoint wave equations in photoacoustic tomography (PAT) by approximating Green's functions along Hamiltonian ray trajectories. The method enables efficient, sensor-independent computation with significant speedup over full-wave solvers, achieving accurate results even near caustics, as validated against k-Wave simulations in 2D heterogeneous media with sound speed variations.

ABSTRACT

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.

Motivation & Objective

• To address the computational inefficiency of full-wave solvers in PAT, especially for real-time and dynamic applications with partial or sparse data.
• To develop a scalable, sensor-independent solver for the forward and adjoint wave equations that avoids full wavefield computation.
• To enable accurate solution of the acoustic problem in PAT using high-frequency asymptotic approximations based on ray tracing and Hamiltonian dynamics.
• To handle challenging features such as caustics and focusing lenses in heterogeneous media, where standard ray methods fail due to amplitude blowup.
• To provide a numerically stable and efficient alternative to full-wave solvers for use in iterative reconstruction and variational methods.

Proposed method

• The Hamilton-Green solver approximates the fundamental solution of the wave equation by tracing rays along trajectories derived from the Hamiltonian system of the high-frequency asymptotic solution.
• It computes the Green's function using ray amplitudes modulated by the determinant of the Jacobian of the ray map, with a regularization to avoid singularities at caustics.
• The method uses a time-reversed source in the adjoint problem, where sensor data is backpropagated via rays to reconstruct the initial pressure distribution.
• The solver employs a smooth cutoff function to limit the time window of integration, ensuring compatibility with real-world measurement constraints.
• It handles caustics by setting ray amplitudes to zero when the ray tube determinant |q(t; x₀)| < ε, and uses the absolute value of q to compensate for coordinate system reversal.
• The approach is implemented in 2D with a fixed number of rays per sensor, allowing independent computation per sensor and enabling scalability.

Experimental results

Research questions

• RQ1 Can a high-frequency asymptotic method based on ray tracing achieve accurate and efficient solutions for the forward and adjoint wave equations in PAT, especially in heterogeneous media?
• RQ2 How does the Hamilton-Green solver perform in the presence of caustics, where ray amplitudes diverge and standard ray methods fail?
• RQ3 To what extent can the solver reduce computational cost compared to full-wave solvers like k-Wave while maintaining accuracy in limited-view and dynamic PAT scenarios?
• RQ4 How does the regularization of ray amplitudes at caustics affect the accuracy of the reconstructed pressure signal at the sensor?
• RQ5 Can the solver be effectively used in iterative reconstruction frameworks, given its efficient, sensor-wise computation and accurate adjoint mapping?

Key findings

• The Hamilton-Green solver achieves accurate forward and adjoint solutions in 2D domains with heterogeneous sound speed, including acoustic lenses that focus rays and create caustics.
• For initial pressures located before the caustic, the HG solver matches the k-Wave solution closely, with minimal error.
• The largest error occurs for initial pressures at the caustic, due to the regularization of ray amplitudes when |q(t; x₀)| < ε, which sets amplitudes to zero and disrupts phase continuity.
• After the caustic, the error is reduced but still present, as the solver does not fully account for the complex phase shift exp(−im(t)π/2) associated with caustic formation.
• The method successfully handles ray focusing and bending around the lens, with outer rays unaffected by the caustic, and maintains numerical stability through amplitude regularization.
• The solver demonstrates significant computational advantage over full-wave solvers, enabling sensor-wise, independent computation at a fraction of the cost.