Self-Supervised Angular Deblurring in Photoacoustic Reconstruction via Noisier2Inverse

Abstract: Photoacoustic tomography (PAT) is an emerging imaging modality that combines the complementary strengths of optical contrast and ultrasonic resolution. A central task is image reconstruction, where measured acoustic signals are used to recover the initial pressure distribution. For ideal point-like or line-like detectors, several efficient and fast reconstruction algorithms exist, including Fourier methods, filtered backprojection, and time reversal. However, when applied to data acquired with finite-size detectors, these methods yield systematically blurred images. Although sharper images can be obtained by compensating for finite-detector effects, supervised learning approaches typically require ground-truth images that may not be available in practice. We propose a self-supervised reconstruction method based on Noisier2Inverse that addresses finite-size detector effects without requiring ground-truth data. Our approach operates directly on noisy measurements and learns to recover high-quality PAT images in a ground-truth-free manner. Its key components are: (i) PAT-specific modeling that recasts the problem as angular deblurring; (ii) a Noisier2Inverse formulation in the polar domain that leverages the known angular point-spread function; and (iii) a novel, statistically grounded early-stopping rule. In experiments, the proposed method consistently outperforms alternative approaches that do not use supervised data and achieves performance close to supervised benchmarks, while remaining practical for real acquisitions with finite-size detectors.

• The paper reformulates finite-size-detector photoacoustic reconstruction as angular deconvolution in polar coordinates, enabling targeted correction of aperture-induced blur under rotationally symmetric acquisition.
• The method combines Noisier2Inverse, which handles correlated post-inversion noise, with a U-Net and direct inference on noisy data, eliminating the need for clean training images.
• An Earth Mover’s Distance early-stopping rule selects effective models without ground truth; on simulated FIVES data, it reached 28.48 dB versus 26.25 dB for SSLTV and 26.62 dB for DIP under representative conditions.

Photoacoustic tomography (PAT) with finite-size detectors suffers from a systematic angular blurring that degrades spatial resolution. The paper by Haltmeier, Gruber, and Hwang (2604.15681) addresses this degradation with a self-supervised learning method that requires no ground-truth images. The approach combines three elements: a PAT-specific reformulation of reconstruction as an angular deconvolution in polar coordinates, a Noisier2Inverse training scheme that tolerates correlated noise, and a novel validation-free early-stopping rule based on the Earth Mover's Distance (EMD). Experiments on simulated data show that the method consistently outperforms unsupervised baselines and approaches supervised performance.

Problem setting: finite-size detector blurring

The authors consider PAT in two spatial dimensions with integrating line detectors in circular scanning geometry, a configuration relevant to piezoelectric line arrays and optical detection via laser beams or elongated fiber sensors. With ideal point-like (3D) or line-like (2D) detectors, efficient inversion formulas exist—Fourier methods, filtered backprojection, time reversal—but these assume pointwise pressure data. When pressure is integrated over an extended aperture with angular sensitivity profile $w$, the measured data can be written as an angularly weighted average of ideal data.

The central structural result is a theorem stating that, under rotational invariance, extended-detector data $U_w f$ coincide exactly with ideal-operator data generated by an angularly blurred source:

$U_w f = U\,(C\,A_w\,P\,f),$

where $P$ maps to polar coordinates, $A_w$ is circular convolution along the angle with kernel $w$, and $C$ maps back to Cartesian coordinates. Consequently, applying any left inverse $V$ of the ideal operator followed by the polar transform reduces the full reconstruction problem to a pure deconvolution problem in the polar domain:

$g_P = A_w(f_P) + \eta_P.$

This reduction is exact for smooth compactly supported sources; it relies on the assumption of a fixed detector sensitivity profile independent of position and on rotational symmetry of the acquisition geometry.

Self-supervised deconvolution with Noisier2Inverse

A key technical obstacle is that the noise $\eta_P = P V \epsilon$ is correlated after application of the nonunitary inverse-and-polar-transform pipeline. This rules out self-supervised schemes such as Sparse2Inverse that require approximately i.i.d. noise. Noisier2Inverse, which the authors previously developed for general linear inverse problems with correlated, signal-independent additive noise, does not have this restriction.

The training procedure adds fresh noise drawn from the same model to already noisy polar-domain data and trains a U-Net–style encoder–decoder (four resolution levels, circular padding along the angular dimension) to map noisier inputs to targets of the form $U_w f$0. By the Noisier2Inverse identity, minimizing this loss is equivalent to minimizing the supervised risk against clean $U_w f$1, so no extrapolation step is needed at inference time. At test time the network is applied directly to $U_w f$2 rather than to noisier data, which the authors report works better empirically.

Automated early stopping via EMD

Because the loss lives in data space while the deconvolution is ill-posed—possessing a large null space and small singular values—full minimization overfits: reconstructions are sharp early in training but degrade into artifacts later. The paper proposes a statistically grounded stopping rule: at each iteration, compute the predicted residuals on a validation set and measure their distributional discrepancy from the known noise distribution using the 1-Wasserstein distance (EMD), solved as an optimal transport problem between empirical measures. Training stops when validation EMD attains its minimum.

Empirically, the validation EMD exhibits a consistent inverse relationship with test PSNR across all four blur kernels tested, so selecting the minimum-EMD iterate effectively selects high-PSNR models without access to ground truth. This is a notable practical contribution: baselines were granted oracle early stopping by validation/test PSNR, which is unavailable in practice, yet the proposed method still wins among unsupervised methods.

Numerical results

Experiments use the FIVES fundus dataset (800 images; 600/100/100 train/validation/test split), resized to $U_w f$3, with a $U_w f$4 Cartesian grid, 804 detector angles, and full temporal sampling per PAT sampling theory. Four periodic angular kernels are evaluated: box kernels corresponding to $U_w f$5 and $U_w f$6 apertures, and discrete Gaussians with $U_w f$7 samples. Noise levels are relative standard deviations $U_w f$8. Comparisons are against supervised learning (gold standard), SSLTV (self-supervised with total-variation penalty), and deep image prior (DIP), all using the same U-Net architecture.

Representative PSNR values (dB), computed against the sharp oracle inverse of the ground truth:

Method	Ind.-10 ($U_w f$9)	Ind.-20 ($U_w f = U\,(C\,A_w\,P\,f),$0)	Gauss-1 ($U_w f = U\,(C\,A_w\,P\,f),$1)	Gauss-2 ($U_w f = U\,(C\,A_w\,P\,f),$2)
DIP	26.62	24.81	26.70	22.78
SSLTV	26.25	25.06	26.19	22.89
Proposed	28.48	26.52	28.39	24.11

The proposed method achieves the best or near-best result in nearly all twelve kernel–noise combinations despite using automated stopping while baselines use oracle stopping. Gains are largest at higher noise levels—for example, at $U_w f = U\,(C\,A_w\,P\,f),$3 with Indicator-10, the method reaches 25.75 dB versus 23.65 dB (SSLTV) and 23.48 dB (DIP). Visual comparisons confirm reconstructions close to the supervised benchmark. These results imply that ground-truth-free recovery of detector-aperture-limited PAT images is feasible at quality approaching supervised learning, provided the angular point-spread function is known.

Limitations and open questions

Several limitations are stated plainly. All experiments are simulation-based; evaluation on real acquisitions has not been performed. The framework assumes a position-independent, rotationally symmetric detector profile and circular geometry, and extensions to point-like finite-area detectors in 3D are only discussed rather than demonstrated. The theoretical properties of the EMD-based stopping rule are not established—the paper observes its empirical correlation with PSNR but offers no convergence or consistency analysis. Additional resolution-limiting factors such as finite bandwidth and sparse sampling are not modeled, and benchmarking against Sparse2Inverse or other self-supervised CT-style methods remains open. Whether deconvolution should instead be performed directly in measurement space is also left unexamined.

Conclusion

The paper formulates finite-size-detector PAT reconstruction as angular deblurring in the polar domain and solves it with a ground-truth-free Noisier2Inverse scheme tailored to correlated post-inversion noise, complemented by an automated EMD-based early-stopping criterion. On simulated FIVES data across four blur kernels and three noise levels, the method outperforms DIP and SSLTV baselines—even those granted oracle stopping—and approaches supervised accuracy. Its practical relevance depends on validation on real data and on extension beyond idealized circular geometry, both of which remain open.