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Utilising a learned forward operator in the inverse problem of photoacoustic tomography

Karoliina Puronhaara, Teemu Sahlström, Andreas Hauptmann, Tanja Tarvainen

Abstract

We study the use of a learned forward operator in the inverse problem of photoacoustic tomography. The Fourier neural operator to approximate the photoacoustic wave propagation is used. Further, the inverse problem is solved using a gradient-based approach with automatic differentiation. The methodology is evaluated using numerical simulations, and the results are compared to a conventional approach, where the forward operator is approximated using the pseudospectral $k$-space method. The results show that the learned forward operator can be used to approximate the photoacoustic wave propagation with good accuracy, and that it can be utilised as a computationally efficient forward operator in solving the inverse problem of photoacoustic tomography.

Utilising a learned forward operator in the inverse problem of photoacoustic tomography

Abstract

We study the use of a learned forward operator in the inverse problem of photoacoustic tomography. The Fourier neural operator to approximate the photoacoustic wave propagation is used. Further, the inverse problem is solved using a gradient-based approach with automatic differentiation. The methodology is evaluated using numerical simulations, and the results are compared to a conventional approach, where the forward operator is approximated using the pseudospectral -space method. The results show that the learned forward operator can be used to approximate the photoacoustic wave propagation with good accuracy, and that it can be utilised as a computationally efficient forward operator in solving the inverse problem of photoacoustic tomography.
Paper Structure (12 sections, 16 equations, 7 figures, 3 tables)

This paper contains 12 sections, 16 equations, 7 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Photoacoustic tomography
  3. Utilising the Fourier neural operator in photoacoustic tomography
  4. Fourier neural operator
  5. Inverse problem with a learned forward operator
  6. Simulations
  7. Data simulation
  8. Fourier neural operator architecture
  9. Forward simulation
  10. Inverse problem
  11. Convergence and computation time
  12. Discussion and conclusions

Figures (7)

  • Figure 1: Neural network architecture of the FNO. The numbers in the parentheses indicate the tensor dimensions. One complete Fourier layer is indicated using a dashed box.
  • Figure 2: Simulated photoacoustic wave-field for a vessel phantom with an initial pressure distribution $p_0$ (first column) using the pseudospectral $k$-space method (k-Wave, first row) and the FNO (second row), and their difference (third row). Location used for illustration of photoacoustic data in Figure \ref{['fig:timePlot']} is indicated with a red dot in the first column.
  • Figure 3: Simulated photoacoustic wave-field for the Shepp-Logan phantom with an initial pressure distribution $p_0$ (first column) using the pseudospectral $k$-space method (k-Wave, first row) and the FNO (second row), and their difference (third row). Location used for illustration of photoacoustic data in Figure \ref{['fig:timePlot']} is indicated with a red dot in the first column.
  • Figure 4: Photoacoustic data simulated using the pseudospectral $k$-space method (k-Wave, black line) and the FNO (red dashed line) for the vessel (left image) and Shepp-Logan (right image) on a boundary pixel illustrated in Figures \ref{['fig:FNOforwardVessel']} and \ref{['fig:FNOforwardShepp']}.
  • Figure 5: The MAP estimates computed from a vessel phantom data in different sensor geometries: full-view (FV, first row), two-side limited view (LV2, second row) and one-side limited view (LV1, third row). Columns from left to right: the true initial pressure distribution $p_0$ (first column), MAP estimates calculated using the reference method (second column), MAP estimates calculated using the FNO and automatic differentation (third column), the difference between the true and the estimated initial pressure for the reference method (fourth column), and the difference between true and estimated initial pressure for the FNO approach (fifth column). Sensor locations are marked with the red line in the second column.
  • ...and 2 more figures