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Initial condition recovery in nonlinear damped viscous photoacoustic tomography using a convolutional neural network-guided gradient-free optimization framework

Madhu Gupta, Anwesa Dey, Prapti Tala, Souvik Roy

arXiv:2610.01015v1math.OCcs.LG

TL;DR

Recovering initial pressure from PAT boundary data is complicated by ill-posedness and acoustic attenuation. The paper analyzes a nonlinear damped wave model and combines CNN initialization with SQH optimization; numerical experiments report improved reconstruction quality, contrast, and resolution over standalone time-reversal and CNN reconstructions.

  • Problem

    Recovering initial pressure from PAT boundary measurements is challenging because the inverse problem is ill-posed and reconstructions can be noise-sensitive and low-resolution.

    Source: §1 · p. 2

  • Method

    The paper analyzes a nonlinear damped viscoelastic wave model and combines a CNN-generated initial guess with SQH optimization for reconstruction.

    Source: §1 · pp. 3–4

  • Results

    In two-dimensional experiments, the hybrid framework produced improved contrast, resolution, and reconstruction accuracy compared with standalone time-reversal and CNN-based reconstructions.

    Source: §6 · p. 37

  • Takeaways & Limitations

    The reported hybrid reconstruction combines data-driven initialization with SQH enforcement of the governing PDE dynamics in the tested attenuating and nonlinear setting.

    Source: §6 · p. 37

    The inverse-problem analysis assumes the initial pressure f is compactly supported in Ω.

    Source: §3 · p. 18

Abstract

from arXiv · show

Photoacoustic tomography (PAT) is a hybrid imaging modality that combines high optical contrast with high ultrasonic resolution for biomedical imaging applications. In this work, we investigate the inverse problem of recovering the initial pressure distribution from boundary measurements in the presence of nonlinear acoustic propagation and viscous attenuation effects. To model these phenomena more accurately, we consider a nonlinear damped viscoelastic wave equation incorporating spatially varying sound speed, temporal attenuation, and nonlinear propagation mechanisms. We first establish the well-posedness of the corresponding forward problem using a Galerkin approximation combined with energy estimates and a fixed-point argument. For the inverse problem, we derive existence, uniqueness, and local uniqueness results under suitable assumptions through a harmonic extension reduction, spectral Laplace transform techniques, and observability estimates. To numerically reconstruct the initial pressure field, we develop a hybrid reconstruction framework that combines a convolutional neural network (CNN) with a gradient-free optimization strategy based on the sequential quadratic Hamiltonian (SQH) method derived from Pontryagin's maximum principle. The CNN is used to generate an informative initial guess, while the SQH framework enforces the governing PDE dynamics during the reconstruction process. Numerical experiments demonstrate that the proposed hybrid strategy significantly improves reconstruction quality, contrast, and robustness compared to standalone time-reversal and CNN-based approaches.

1 Introduction

The introduction frames PAT reconstruction as recovering initial pressure from boundary measurements while accounting for acoustic effects that simpler models may omit. It presents mathematical analysis of the nonlinear damped model and a CNN-initialized SQH reconstruction framework.

1 Introduction

  • The inverse problem is highly ill-posed, making reconstructions sensitive to noise and typically low in spatial resolution.Source: §1 · p. 2
  • PAT seeks to recover the initial pressure distribution from boundary measurements, but attenuation and nonlinear propagation complicate the acoustic model.The initial pressure is proportional to absorbed optical energy; ignoring attenuation and nonlinear mechanisms can degrade image quality.Source: §1 · p. 2
  • The paper analyzes the nonlinear damped viscoelastic wave equation and establishes forward well-posedness plus inverse existence, uniqueness, stability, and local uniqueness under suitable assumptions.The analysis uses Galerkin approximation, energy estimates, a fixed-point framework, harmonic extension, spectral Laplace transforms, and observability estimates.Source: §1 · p. 3
  • Time-reversal reconstructions in attenuating media can lose substantial amplitude, while CNN-based methods lack explicit physical-law enforcement and may produce nonphysical artifacts.Source: §1 · p. 4
  • The hybrid strategy initializes SQH with a combination of time-reversal reconstruction and CNN output, then uses SQH optimization for the inverse problem.The CNN maps boundary measurements to initial pressure and supplies structural priors for subsequent optimization.Source: §1 · p. 4

2 Analysis of the nonlinear damped wave model

The analysis establishes existence and uniqueness for the nonlinear damped wave model under stated assumptions. It also develops the linearized problem and proves continuous dependence on initial and boundary data.

2 Analysis of the nonlinear damped wave model

  • Under Assumptions (4), the nonlinear damped wave equation has a unique solution in X_T.The proof constructs Galerkin approximations, obtains uniform energy bounds, and uses a fixed-point argument.Source: §2 · pp. 5, 7, 10, 12
  • For each fixed v in the prescribed ball, the linearized equation has a unique solution in X_T under condition (5).The Galerkin weak formulation yields a finite-dimensional Volterra integro-differential system with a unique local solution.Source: §2 · pp. 7, 10
  • Solutions depend continuously on the initial and boundary data when the compared solutions are sufficiently regular.Source: §2 · p. 12

3 Theory of the inverse problem

The inverse theory addresses recovery of initial pressure from boundary measurements and establishes uniqueness and existence results under stated assumptions. Its analysis uses harmonic extension, spectral Laplace techniques, and observability estimates.

3 Theory of the inverse problem

  • The paper establishes local uniqueness for recovering the initial pressure from the forward map, using an observability estimate.Source: §3 · pp. 18–19
  • For compactly supported initial pressure and constant sound speed and damping, a linear inverse problem has a reconstruction for given boundary measurements.Source: §3 · p. 19
  • The linear reconstruction argument subtracts a harmonic extension of the boundary data, then uses Dirichlet eigenfunctions and Laplace transforms.The subtraction gives an auxiliary function with homogeneous Dirichlet boundary data.Source: §3 · pp. 20–21
  • For sufficiently small T, attenuation, and linear reconstruction norm, the nonlinear inverse problem has a solution near the linear reconstruction.The fixed-point argument gives a unique solution in a ball around that reconstruction.Source: §3 · pp. 21, 23

4 Numerical reconstruction framework

This section presents a constrained, regularized initial-condition reconstruction framework that combines a CNN-generated estimate from boundary measurements with SQH optimization derived from Pontryagin’s maximum principle. It also establishes existence and convergence properties for the framework, including finite termination and subsequential convergence of the algorithm.

4 Numerical reconstruction framework

  • The objective combines least-squares data fitting with L2–L1 regularization for sparsity, under pointwise bounds on the initial condition.Its regularization parameters are positive, and the admissible initial conditions lie in Uad.Source: §4 · p. 23
  • The objective is weakly lower semicontinuous, bounded below, and coercive on Uad, and an optimal initial-condition/state pair exists.The existence argument uses compactness of the admissible set and strong L2 convergence of a subsequence of states.Source: §4 · p. 24
  • PMP characterizes the optimal initial condition by pointwise Hamiltonian minimization, avoiding derivatives of the objective with respect to the initial condition and motivating SQH.The optimality condition holds almost everywhere in the domain.Source: §4 · pp. 24, 26
  • SQH iteratively minimizes an augmented Hamiltonian pointwise, adaptively increasing its penalty parameter when the objective decrease is insufficient and decreasing it otherwise.The quadratic penalty limits updates from deviating excessively from the previous initial-condition iterate; state and adjoint remain fixed during each spatial sweep.Source: §4 · pp. 26–27
  • The SQH penalty sequence is bounded; with ε = C + η at every iteration and κ = 0, a subsequence converges to a pair satisfying the PMP criterion.The convergence statement concerns the states and adjoints in L2(0, T; L2(Ω)), and the limiting initial condition minimizes the Hamiltonian almost everywhere.Source: §4 · p. 29
  • For κ > 0, Algorithm 13 is well defined, stops in finitely many steps, and converges weakly along a subsequence to a local minimum satisfying the PMP optimality condition.The limiting control satisfies the Hamiltonian minimization condition almost everywhere.Source: §4 · p. 30

4.1 CNN-guided initial guess construction

  • The CNN processes four boundary-measurement channels independently before concatenation and shared encoding, anisotropic downsampling, and single-channel output.Each channel uses three 32-filter 3 × 3 convolutional layers; the shared encoder uses 64-filter 3 × 3 convolutions, and the output is produced by a 1 × 1 linear convolution.Source: §4.1 · p. 30

5 Numerical results

Across the tested phantoms, SQH reconstructions are reported to improve on time-reversal and CNN reconstructions in image quality; this advantage is also reported with space-time-varying damping and nonlinearity. The experiments compare qualitative reconstructions and MSE, PSNR, and SSIM.

5 Numerical results

  • The evaluation compares time reversal, CNN, and SQH using MSE, PSNR, and SSIM, alongside 2D reconstructions and 1D cross-sectional profiles.The CNN training dataset contains 750 samples of Gaussian, characteristic-function, and combined outputs.Source: §5 · p. 31, Figure 1 · p. 32, Figure 2 · p. 33
  • For the two-phantom case, SQH provides better resolution and contrast than time reversal and CNN.The test phantom combines a Gaussian and a disk; the reconstructions include 2D profiles and 1D cross-sections.Source: §5 · pp. 31–32, Figure 1 · p. 32
  • With three phantoms, time reversal has severe artifacts, while SQH clears artifacts and improves contrast and resolution over the CNN reconstruction.The CNN captures the disk discontinuity but also produces superficial structures despite not being trained on three-phantom cases.Source: §5 · pp. 32–33, Figure 2 · p. 33
  • For the heart-and-lung phantom, CNN sharp edges help SQH improve contrast and resolution despite superficial structures in the CNN reconstruction.The CNN training set contained disks and Gaussians, with at most two objects; the test combines ellipses and a disk.Source: §5 · pp. 33–34, Figure 3 · p. 34
  • SQH significantly outperforms time reversal and CNN on MSE and PSNR and achieves SSIM values close to 1 across the test cases, including space-time-varying coefficients.The authors describe these results as evidence of robustness and versatility; the supplied table passages give metric scopes but no cell values.Source: §5 · pp. 35–36, Table 3 · p. 35, Table 6 · p. 36
  • With space-time-varying damping and nonlinearity, SQH again clears artifacts and improves contrast and resolution for both three-phantom and heart-and-lung cases.The CNN remained trained on the original coefficient settings; for the final case it was also trained on disks and Gaussians with at most two objects.Source: §5 · pp. 35–36, Figure 4 · p. 35, Figure 5 · p. 36

6 Conclusion

The paper combines theoretical analysis of a nonlinear damped viscoelastic wave model with a CNN-initialized SQH reconstruction framework. In two-dimensional experiments, the hybrid method improves reconstruction accuracy, contrast, and resolution over standalone time-reversal and CNN reconstructions, particularly with attenuation and nonlinear effects.

6 Conclusion

  • The forward problem is well-posed, and the inverse problem has existence, uniqueness, and local-uniqueness results under suitable assumptions.The analysis uses Galerkin approximations, energy estimates, a fixed-point argument, harmonic extension methods, spectral Laplace transforms, and observability estimates.Source: §6 · p. 37
  • In two-dimensional experiments, the hybrid CNN–SQH framework improves contrast, resolution, and reconstruction accuracy over standalone time-reversal and CNN reconstructions, particularly with attenuation and nonlinear effects.The CNN supplies a data-driven initialization, while SQH enforces the PDE dynamics and improves physical consistency.Source: §6 · p. 37
  • The authors identify physics-informed and uncertainty-aware reconstruction for complex heterogeneous media as future work.They frame the current results as support for combining machine learning with structure-preserving optimization in nonlinear, ill-posed PAT inverse problems.Source: §6 · p. 37
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