Implicit learning to determine variable sound speed and the reconstruction operator in photoacoustic tomography

TL;DR

This work addresses the problem of jointly recovering the spatially varying sound speed $c(\boldsymbol{x})$ and the reconstruction operator $\mathcal{D}_{c}^{-1}$ in photoacoustic tomography from boundary Dirichlet and Neumann data, without ground-truth internal pressures. It introduces an implicit learning framework consisting of a sound-speed network $\tilde{c}$, a reconstruction network $\mathcal{R}$, and a differentiable wave forward operator $\mathcal{W}_{\tilde{c}}$, trained to enforce boundary-data consistency via a loss $\mathcal{L}(\mathscr{T}) = \mathcal{L}_{A}(\mathscr{T}) + \mathcal{L}_{B}(\mathscr{T})$ and a TV regularizer. The method is tested on synthetic Shepp–Logan phantoms with two speed profiles and noise, showing accurate estimation of $c$ and reconstruction of $f$ under both clean and noisy data, and demonstrating robustness to unknown speed and data limitations. This approach reduces dependence on labeled targets in PAT, enabling boundary-data-driven inversion under variable sound speed and advancing data-driven inverse problems in medical imaging.

Abstract

Photoacoustic tomography (PAT) is a hybrid medical imaging technique that offer high contrast and a high spatial resolution. One challenging mathematical problem associated with PAT is reconstructing the initial pressure of the wave equation from data collected at the specific surface where the detectors are positioned. The study addresses this problem when PAT is modeled by a wave equation with unknown sound speed $c$, which is a function of spatial variables, and under the assumption that both the Dirichlet and Neumann boundary values on the detector surface are measured. In practical, we introduce a novel implicit learning framework to simultaneously estimate the unknown $c$ and the reconstruction operator using only Dirichlet and Neumann boundary measurement data. The experimental results confirm the success of our proposed framework, demonstrating its ability to accurately estimate variable sound speed and the reconstruction operator in PAT.

Figures (8)

• Figure 1: Proposed framework
• Figure 2: Reconstruction network $\mathcal{R}$. The Conv(a, b) layer is a 2D convolution layer with a kernel size of (a, b), and a stride of (1, 1) while padding is employed to ensure that the output is the same size as the input. The AvgPool(a, b) layer is a 2D average pooling layer with a kernel size of (a, b). The linear layer is a fully connected layer that transforms an input with dimension of $64\times64$ to an output with the same dimensions. The Deconv(a, b) layer is a 2D transposed convolution layer with a kernel size of (a, b) and a stride of (2, 2). Each layer contains no bias.
• Figure 3: Estimated results for Type 1 and Type 2 sound speed profiles. The first, second, and third columns represent the ground truth, the estimated speed for noise-free data, and the estimated speed for noisy data, respectively.
• Figure 4: In the left image, the ground truth is shown with three lines labeled Line 1, Line 2, and Line 3. The right images present cross-sectional views corresponding to these lines. The blue lines represent the ground truth, while the red dashed lines represent the estimated speed.
• Figure 5: Reconstruction results: (a) ground truth (GT); (b) and (c) output of the reconstruction network for noise-free data for Type 1 and Type 2 sound speed profiles; (d) and (e) output of the reconstruction network for noisy data for Type 1 and Type 2 sound speed profiles.

Theorems & Definitions (2)

• Theorem A
• Theorem B