A depth-dependent, transverse shift-invariant operator for fast iterative 3D photoacoustic tomography in planar geometry

Abstract

Iterative model-based image reconstruction in photoacoustic tomography (PAT) enables principled incorporation of detector physics, object-related priors, and complex acquisition strategies. However, for three-dimensional (3D) imaging scenario, the computational cost is often dominated by repeatedly solving wave equations. We propose a fast forward model for planar detection geometries that exploits transverse shift invariance. This symmetry enables to compute the full acoustic field from a 3D object, as a result of a set of 2D convolutions with depth-dependent impulse responses. This formulation yields a FFT-based forward operator and its corresponding discrete adjoint operator, making iterative reconstruction faster without calling partial differential equation (PDE) solvers at each iteration. We validate the model against commonly used PDE solver under matched discretization and boundary settings, and demonstrate accelerations of up to 2 orders of magnitude for iterative reconstructions from experimental all-optical photoacoustic datasets.

Figures (4)

• Figure 1: Depth-dependent transverse shift invariance in planar photoacoustic acquisition. (a) Transverse shift invariance: at fixed depth $z$, laterally translating a point absorber translates the measured wave-field by the same amount, implying convolution in $(x,y)$. (b) Depth dependence: increasing $z$ increases time-of-flight and changes the wavefront curvature on the sensor plane. These effects motivate the depth-indexed convolutional model \ref{['eq:conv_model_continuous']}--\ref{['eq:discrete_forward']}.
• Figure 2: Forward operators runtime as a function of grid size. Single forward operator runtime comparison between k-Wave implementation of pseudo-spectral numerical acoustic wave propagation and the proposed depth-dependent convolutional operator with $N_t=1000$ time steps, as a function of the 3D grid size.
• Figure 3: Image reconstructions of phantoms. Image reconstructions from experimental datasets, comparing direct baselines and iterative MB approach: (left) k-Wave TR, (middle) adjoint reconstruction under the proposed depth-dependent convolutional operator, (right) FISTA reconstruction with non-negativity and $\ell_1$ regularization ($\lambda = 2\times10^{-5}$, 15 iterations), using the proposed forward and adjoint operators. Two representative phantom samples are tested: a sparse collection of 10-20 µm black beads (top) and elongated structures formed by a 20 µm black wire, mimicking a vessel-like organisation (bottom). Scale bars: $200~\mu$m.
• Figure 4: Image reconstructions of in-vivo human forearm vasculature. Comparison of TR and iterative MB (FISTA with non-negativity and $\ell_1$ regularization) reconstructions of an in vivo human forearm. Overlay at the bottom highlights the higher noise level (green background) in TR reconstruction, while objects features appear in both reconstructions (white structures). Scale bars: 2 mm.