Untrained CNN and coefficient-space regularization for joint reconstruction in photoacoustic tomography

Alfio Bozi; Anwesa Dey; Souvik Roy

cs.LGOct 16, 2026

Untrained CNN and coefficient-space regularization for joint reconstruction in photoacoustic tomography

Authors: Alfio Bozi, Anwesa Dey, Souvik Roy

Organizations: Institut für Mathematik, Universität Würzburg, Emil-Fischer-Strasse 30, 97074 Würzburg, Germany. · Department of Mathematics, The University of Utah, John A. Widtsoe Building 155 S 1400 E, Salt Lake City, 84112, Utah, USA. · Department of Mathematics, The University of Texas at Arlington, 411 S. Nedderman Drive, Arlington, 76019, Texas, USA.

Abstract

The joint reconstruction of the initial pressure and spatially varying sound speed in photoacoustic tomography from boundary measurements is considered. A regularized data-driven framework is proposed in which the unknown fields are represented in finite-dimensional wavelet spaces and fixed nonlinear features generated by an untrained convolutional neural network (CNN) are used to reconstruct the initial-pressure coefficients through Tikhonov-regularized least squares. A learned coefficient-space prior is subsequently used to infer the squared sound-speed coefficients. Numerical experiments demonstrate accurate reconstruction on held-out data and strong generalization to previously unseen phantom geometries. The frozen reconstruction map also remains stable under measurement noise not represented during training, with only minor changes in reconstruction accuracy for noise levels up to $10\%$ . In comparison with the supervised trained-CNN approach considered in this work, the untrained representation yields substantially smaller reconstruction errors while requiring approximately one-sixth of the computational time. These results demonstrate that fixed random convolutional features combined with regularized coefficient-space learning provide an efficient and robust approach to joint reconstruction in photoacoustic tomography.

Figures & tables

Reconstruction	Data set	Mean error	Median error	$R^{2}$
Pressure untrained CNN+ridge $f_{0}$	Training	0.19521	0.19410	0.95536
	Test	0.24616	0.24472	0.92470
Sound speed untrained CNN+ridge $c_{0}^{\rm rec}$	Training	0.00828	0.00814	0.80582
	Test	0.01013	0.00996	0.71181

Table 1: Reconstruction performance of the untrained CNN and ridge-regression framework on the training and held-out test data. The reported

R^{2}

values are computed globally over all samples and spatial grid points.

Noise level	Mean $E_{f}$	$R_{f}^{2}$	Mean $E_{c}$	$R_{c}^{2}$
$0\%$	0.246162	0.924696	0.010129	0.711812
$1\%$	0.246163	0.924696	0.010129	0.711806
$2\%$	0.246167	0.924693	0.010129	0.711814
$5\%$	0.246203	0.924673	0.010129	0.711788
$10\%$	0.246337	0.924591	0.010132	0.711642

Table 2: Stability of the frozen reconstruction map under additive Gaussian perturbations of the held-out boundary measurements. The nonzero-noise results are averaged over three independent noise realizations.

Method	Quantity	Train mean/median	Test mean/median	Test $R^{2}$
Untrained CNN + ridge	$f_{0}$	0.19521/0.19410	0.24616/0.24472	0.92470
	$c_{0}$	0.00828/0.00814	0.01013/0.00996	0.71181
Trained CNN + ridge	$f_{0}$	0.56853/0.56722	0.58519/0.58262	0.57635
	$c_{0}$	0.01361/0.01351	0.01424/0.01380	0.25921

Table 3: Comparison of the untrained and trained convolutional feature extractors. Relative reconstruction errors are reported as mean/median. Both approaches use the same multiscale ridge reconstruction and coefficient-space sound-speed prior.

Panel of Figure 1 — Figure 1: Reconstruction of the unseen disk phantom. The first column shows the true initial pressure and sound speed. The second and third columns show the reconstructions obtained using the proposed untrained CNN+ridge method from clean measurements and measurements contaminated by $10\%$ relative Gaussian noise, respectively. The fourth column shows the reconstruction obtained using the trained CNN+ridge method from clean measurements.

Panel of Figure 2 — Figure 2: Reconstruction of the unseen Gaussian phantom. The first column shows the true initial pressure and sound speed. The second and third columns show the reconstructions obtained using the proposed untrained CNN+ridge method from clean measurements and measurements contaminated by $10\%$ relative Gaussian noise, respectively. The fourth column shows the reconstruction obtained using the trained CNN+ridge method from clean measurements.

Panel of Figure 3 — Figure 3: Reconstruction of the unseen heart–lung phantom. The first column shows the true initial pressure and sound speed. The second and third columns show the reconstructions obtained using the proposed untrained CNN+ridge method from clean measurements and measurements contaminated by $10\%$ relative Gaussian noise, respectively. The fourth column shows the reconstruction obtained using the trained CNN+ridge method from clean measurements.

Phantom	Method/data	$E_{f}$	$R_{f}^{2}$	$E_{c}$	$R_{c}^{2}$
Disk	Untrained, clean	0.270509	0.919631	0.007996	0.899434
	Untrained, $10\%$ noise	0.270908	0.919393	0.008011	0.899053
	Trained, clean	0.623817	0.572596	0.017175	0.536073
Gaussian	Untrained, clean	0.038391	0.998336	0.001138	0.993573
	Untrained, $10\%$ noise	0.039248	0.998261	0.001152	0.993409
	Trained, clean	0.945115	-0.008256	0.014777	-0.084208

Table 4: Reconstruction performance on the three completely unseen phantom geometries. The untrained CNN+ridge method is evaluated using both clean measurements and measurements contaminated by

10\%

relative Gaussian noise. The noisy results are averages over three independent realizations. For comparison, the clean-measurement results obtained using the trained CNN+ridge method are also reported.

Photoacoustic Tomography

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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

Department of Mathematics, Indian Institute of Technology, Gandhinagar, Gujarat, India. · Department of Mathematics, University of Utah, Salt Lake City, Utah, UT, USA. · Department of Mathematics, The University of Texas at Arlington, Arlington, TX, USA.

Medical Imaging Gradient-Free Optimization Photoacoustic Tomography PDE Inverse Problems Inverse Problems

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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.

Abstract

Photoacoustic Tomography

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