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A Reciprocity-Law-Compliant Photoacoustic Forward-Adjoint Operator

Ashkan Javaherian

Abstract

We extend the forward-adjoint operator framework derived in our previous study to photoacoustic tomography (PAT). In that earlier work, the acoustic forward operator included a reception operator that maps, at each time step, the pressure wavefield in free space onto the boundary (receiver surface). It was shown that this reception operator serves as a left-inverse of an emission operator that maps the pressure restricted to the boundary (emitter surface) onto free space, perfectly complying with the reciprocity law of physics. In this study, we define the full PAT forward operator as a composite mapping composed of an acoustic forward operator equipped with a scaled variant of the previously proposed reception operator, and an operator describing the photoacoustic source. Singularities arising both in the reception step (due to the boundary restriction) and in the photoacoustic source (due to its instantaneous nature) are regularized using regularized Dirac delta distributions. The resulting PAT forward-adjoint operator pair satisfies an inner-product relation, which we verify through numerical experiments on a discretized domain. The effectiveness of the proposed operator pair is further demonstrated using an iterative minimization framework that yields both qualitatively and quantitatively accurate reconstructions of an initial pressure distribution from the corresponding Dirichlet-type boundary data.

A Reciprocity-Law-Compliant Photoacoustic Forward-Adjoint Operator

Abstract

We extend the forward-adjoint operator framework derived in our previous study to photoacoustic tomography (PAT). In that earlier work, the acoustic forward operator included a reception operator that maps, at each time step, the pressure wavefield in free space onto the boundary (receiver surface). It was shown that this reception operator serves as a left-inverse of an emission operator that maps the pressure restricted to the boundary (emitter surface) onto free space, perfectly complying with the reciprocity law of physics. In this study, we define the full PAT forward operator as a composite mapping composed of an acoustic forward operator equipped with a scaled variant of the previously proposed reception operator, and an operator describing the photoacoustic source. Singularities arising both in the reception step (due to the boundary restriction) and in the photoacoustic source (due to its instantaneous nature) are regularized using regularized Dirac delta distributions. The resulting PAT forward-adjoint operator pair satisfies an inner-product relation, which we verify through numerical experiments on a discretized domain. The effectiveness of the proposed operator pair is further demonstrated using an iterative minimization framework that yields both qualitatively and quantitatively accurate reconstructions of an initial pressure distribution from the corresponding Dirichlet-type boundary data.

Paper Structure

This paper contains 22 sections, 1 theorem, 33 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Existing Approaches and Our Contributions
  3. Existing Approaches
  4. Our Contributions
  5. The PAT Forward and Adjoint Operators
  6. PAT Forward Operator
  7. Minimization Problem
  8. PAT Adjoint Operator
  9. System of Three-Coupled First-Order Wave Equations
  10. Scaled Full Composite PAT Forward Operator
  11. Scaled Full Composite PAT Adjoint Operator
  12. Discretized Algorithm
  13. Discretized Scaled PAT Forward Operator
  14. Discretized Scaled PAT Adjoint Operator
  15. Numerical Results
  16. ...and 7 more sections

Key Result

Lemma 1

The action of the adjoint of the operator $\mathcal{Q}$, defined by Eq. eq:photoacoustic-op, on any test function $f^{s_{\mathcal{Q}}} \in C_0^\infty(\mathbb{R}^d \times \mathbb{R})$ is given by $\blacktriangleleft$$\blacktriangleleft$

Figures (3)

  • Figure 1: Experimental setup to quantify the consistency of the PAT forward--adjoint operator with respect to the inner product idenitity \ref{['eq:inner-product-test']} using on-grid line receivers. (a) Randomly generated initial pressure source distribution $p_0$ and the on-grid line receivers on the left, bottom, and right sides of the grid. (b) Randomly generated scaled Dirichlet-type boundary data $\frac{y^D}{a_R}$.
  • Figure 2: Experimental setup used to quantify the consistency of the PAT forward--adjoint operator with respect to the inner product identity \ref{['eq:inner-product-test']} using off-grid line receivers. (a) Randomly generated initial pressure distribution $p_0$ and the circular array of off-grid line receivers. (b) Randomly generated scaled Dirichlet-type boundary data $\frac{y^D}{a_R}$.
  • Figure 3: Binary mask of the sampled points contributing to the approximation of the Dirac delta distributions for all receiver nodes: (a) simulation of measurements, (b) iterative image reconstruction. The initial pressure source distribution $p_0$: (c) phantom used for the simulation of measurements (ground truth), (d) the final reconstructed image after 25 iterations. Algorithm convergence: (e) objective function values plotted against the iteration index; (f) relative error values plotted against the iteration index.

Theorems & Definitions (2)

  • Lemma 1
  • proof