Learned backprojection for sparse and limited view photoacoustic tomography

Johannes Schwab; Markus Haltmeier; Stephan Antholzer

REVIEW 3 major objections 6 minor 28 references

Learned backprojection for sparse and limited view photoacoustic tomography

T0 review · 3 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Adding learned weight factors to the universal backprojection formula roughly halves the reconstruction error in limited-view and sparse photoacoustic tomography.

desk verdict A modest, clearly written methods paper: learning the weight field inside UBP roughly halves simulated reconstruction error on held-out phantoms, but the evaluation is narrow and the generalization claim is untested. read the letter →

arxiv 1908.00593 v1 pith:WKOLGMZW submitted 2019-08-01 eess.IV

Johannes Schwab , Stephan Antholzer , Markus Haltmeier This is my paper · ORCID

classification eess.IV MSC 92C5565R3268T07

keywords photoacoustictomographyuniversalbackprojectionfilteredlimitedviewsparsesamplingdetectordirectivitylearnedreconstructioninverseproblems

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper argues that the standard filtered-backprojection formula used in photoacoustic tomography can be materially improved for incomplete data by making its angular weight factors learnable rather than leaving them constant. The authors train those weights on 1,800 simulated source–data pairs that include a direction-dependent detector sensitivity, then use the learned weights in the same backprojection formula. Across three incomplete-data scenarios, the learned weighted UBP cuts the average relative squared reconstruction error roughly in half (from 0.20/0.35/0.35 to 0.09/0.18/0.16). If the approach transfers beyond simulation, it matters because a small, interpretable modification of an exact analytic inversion formula gives a fast reconstruction method that also handles detector directivity, which standard UBP ignores.

What carries the argument

The central object is the weighted universal backprojection operator $$\Psi(w,g)(x)=-\frac{2}{\pi}\int_\Gamma w(x,s)^2 \langle \nu_s, x-s\rangle \int_{\|x-s\|}^\infty \frac{\partial_t($t^{{-1}}$g(s,t))}{\sqrt{$t^{2}$-\|x-s\|^2}}\,dt\,ds,$$ which maps pressure data $g$ to a reconstruction by integrating over detector positions on the measurement surface. The standard UBP is $w\equiv 1$; replacing that constant with a learned weight tensor $W\in\mathbb{R}^{N_x\times N_x\times N_s}$ is the entire trainable modification. The paper generates training data with a directivity model $\phi(s,\omega)=\cos^2(\alpha)$ for $|\alpha|<\pi/2$, so the simulations already contain the angular sensitivity the learned weights must counteract. The weights are optimized by stochastic gradient descent on the mean squared error objective over 1,800 phantom–data pairs; in the solutions, the learned weights show angle-dependent structure tied to the detector arc and the source–detector angle, which is the mechanism that suppresses artifacts from missing views and coarse sampling.

What would settle it

Apply the same learned weights to experimentally recorded photoacoustic data from a tissue phantom under the same half-circle geometry and compare the average relative squared error over a held-out set. If the error is not close to half that of plain UBP—or if plain UBP matches it—the learned weights are tied to the simulation rather than to the imaging problem.

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Extended reading notes

Core claim

The paper's central claim is that the universal backprojection formula, normally used with constant weight $w\equiv 1$, can be turned into a learned reconstruction operator by optimizing a weight for every pair of reconstruction point and detector position. The training objective is the mean squared error between the weighted backprojection and the true source on a simulated dataset with 1,800 pairs, and the evaluation is on 200 held-out phantoms. Quantitatively, the learned UBP reduces the average relative squared $\ell^2$ error from 0.2002 to 0.0912 on half-circle limited-view data, from 0.3461 to 0.1806 on 20-detector sparse data, and from 0.3546 to 0.1649 when both limitations are combined. The paper also argues the recipe is not tied to a specific geometry: the weights can be optimized for any measurement surface, and the learned backprojection can serve as the first layer of a deeper convolutional network.

Load-bearing premise

The load-bearing premise is that weights trained on noiseless simulated data from one family of synthetic phantoms and one model of detector sensitivity will transfer to real photoacoustic measurements; if that transfer fails, the factor-of-two improvement may disappear outside the simulation.

Editorial extensions

If this is right

For a fixed detector geometry, reconstruction with the learned weights remains a single backprojection pass, so the speed of standard UBP is retained while accuracy improves.
In the reported experiments, held-out phantoms show relative squared error drops from 0.2002 to 0.0912 (limited view), 0.3461 to 0.1806 (sparse), and 0.3546 to 0.1649 (combined).
Because the weight factors are optimized rather than derived, the same training recipe applies to measurement surfaces where no exact inversion formula is known.
The learned backprojection can serve as the first layer of a deep convolutional network, composing model-based reconstruction with learned artifact removal.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Testable extension: train the same weighted UBP with additive measurement noise and compare the optimized weight fields; if the weights shift substantially, the correction is learning noise statistics rather than geometry alone.
A consequence the authors leave implicit is that if the learned weights are smooth in the detector angle, the improvement could be compressed into a small analytic correction factor, letting existing filtered-backprojection codes adopt it without a training stage.
The weight tensor holds one scalar per reconstruction pixel per detector position, so extending the recipe to three-dimensional point-detector data would need much larger tensors; weight sharing across detector positions or radial symmetry would be required to keep training feasible.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Worth a look if you work on photoacoustic or limited-view tomography. The new thing here is simple: instead of hand-designing angle weights for the universal backprojection formula, the authors make the full weight field w(x,s) learnable and fit it on simulated incomplete-data/ground-truth pairs. That direct optimization of UBP weights is not in the cited PAT literature, which uses hand-crafted weights or post-processing networks. The idea is a natural composition of known pieces, but it is a legitimate new application and the reported effect is large: roughly half the relative L2 error on a held-out set of 200 simulated phantoms across limited-view, sparse, and combined scenarios. The paper is clearly written, the setup is reproducible in principle, and the claims are stated without overreach. Credit where due: the held-out test set is independent of training, so the headline result is not circular.

The soft spots are real but mostly about scope, not about a flawed core. The model has about 2 million free parameters trained on 1800 phantoms; the test phantoms come from the same generator, the same cos^2 directivity model, the same geometry, and no noise. That means the factor-of-two gain may be partly a fit to the statistics of this phantom family rather than a general correction for incomplete-data artifacts. The paper does not test transfer to other source classes, other detector models, or real data, and it does not compare against iterative baselines or time-reversal, which are strong competitors for limited-view PAT. There are also no error bars on the Table 1 numbers, and no code or data released, so the practical value, while plausible, is not yet pinned down.

The central argument, however, holds up on its own terms: within the stated simulation setup, learned UBP beats standard UBP. The main limitation is that the paper generalizes beyond that setup in its conclusions without evidence. That is a standard weakness in early deep-learning-for-inverse-problems papers, and it is not a fatal flaw.

Who is this for? Someone building fast reconstruction methods for PAT or other tomographic modalities where you have a good analytic backprojection and want a cheap learned correction. It is a useful building block, not a breakthrough. I would send it to peer review, but I would ask the authors for code, error bars, and at least one experiment with noise or a different phantom class. If those are added, the result becomes solid.

Recommendation: engage with it; assign a referee familiar with PAT reconstruction and with the pitfalls of learning on simulated data.

Referee Report

3 major / 6 minor

Summary. The manuscript proposes a learned version of the universal backprojection (UBP) formula for two-dimensional photoacoustic tomography. Instead of the constant weight w≡1 in Eq. (7), the authors introduce a position- and detector-dependent weight tensor W and train it on simulated incomplete-data/ground-truth pairs for limited-view, sparse-sampling, and combined scenarios. The optimization is carried out with stochastic gradient descent in a Keras/TensorFlow layer that performs the time filtering and weighted backprojection. For a randomized Shepp-Logan phantom family, a cos^2 detector directivity model, and no noise, the authors report that the learned UBP roughly halves the relative l2 error compared with standard UBP on 200 held-out phantoms (Table 1), and they show learned weight images and example reconstructions.

Significance. The idea is simple, elegant, and computationally appealing: it keeps the fast analytic backprojection structure and merely replaces a constant by learned weights, so inference remains a single weighted backprojection pass. The evaluation is honest in that the 200 test phantoms were not used for training, and the reported gain is consistent across all three scenarios. I also credit the authors for explicitly addressing detector directivity, which standard UBP ignores. The main limitation is that the experiments stay entirely within one simulated distribution, so the general significance is conditional on transfer to other source classes, noise levels, and real detector characteristics. If such transfer is demonstrated, or if the claims are appropriately narrowed, the method is a useful addition to learned reconstruction for PAT.

major comments (3)

[Section 3 and Table 1] The claim that learned weights generally improve reconstruction quality in PAT is not established, because both training and test sets are generated by the same randomized Shepp-Logan procedure, with the same cos^2 directivity model (Eq. (6)), the same circular geometry, and no measurement noise. With roughly 2 million free parameters in W and only 1800 training examples, the held-out 200 phantoms test independence of the particular samples but not independence of the distribution; the learned weights could be compensating for the specific statistics of this phantom family and directivity kernel. Please add out-of-distribution tests (a different phantom class, noisy data, a different directivity model, or experimental measurements), or restrict the conclusions to the simulated setting.
[Section 3.4] The quantitative evaluation compares weighted UBP only to unweighted UBP. To support the practical significance claimed in the introduction, the method should be compared with existing iterative limited-view and sparse-data reconstructions (e.g., refs. [7,13]) and with the adaptive weighted FBP of refs. [18,21], as well as with a standard learned post-processing CNN. Without such baselines, the factor-of-two improvement over plain UBP does not tell the reader whether the method is competitive with current practice.
[Section 3 and 3.4] The reported averages in Table 1 are not accompanied by standard deviations, confidence intervals, or the number of independent training runs, and the optimization setup is underspecified (no learning rate, no regularization, no details of the discretization of Eq. (7), no policy for random seeds). Since the training is stochastic and the high-capacity weight tensor may be sensitive to initialization, the quantitative claim needs error bars and enough implementation detail to be reproducible.

minor comments (6)

[Abstract] Typo: 'in this papers' should be 'in this paper'.
[Section 2.2] Typo: 'Discetizing' should be 'Discretizing'.
[Section 2.2] The phrase 'given on a subset of the boundary Γ ⊂ Ω' should probably read 'Γ ⊂ ∂Ω', since Γ is a subset of the detection surface.
[Introduction] There is an empty citation in the sentence '...have been developed []'; the missing reference should be supplied.
[Figure 4 caption] The caption says 'The third row displays the learned weights...', but the figure appears to have two rows of images; the row reference should be corrected.
[General] The authors do not mention code or data availability; providing the trained weights or the data-generation code would substantially improve reproducibility.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the learned-UBP improvement is tested on held-out phantoms and is not equivalent to its training objective.

full rationale

The paper's central numerical claim is that weights W in the discretized weighted UBP formula (7), trained by minimizing the mean squared error (10) over M=1800 simulated data-source pairs, roughly halve the relative L2 error relative to standard UBP. The quantitative evaluation in Section 3.4 computes the averaged relative L2 error over 200 reconstructions of phantoms explicitly stated as 'not contained in the training set'. The reported improvement is therefore an independent evaluation of the fitted weights rather than a fitted parameter being reported as a prediction. The learned weights are fitted parameters, but the headline comparison in Table 1 compares standard UBP and weighted UBP on held-out data from the same simulator, so the result is not forced by construction. The paper's self-citations (e.g., references to prior work on inversion formulas, directivity models, and deep learning for PAT) are contextual and are not the load-bearing justification for the observed error reduction, which rests on the numerical experiment itself. The concern that weights trained on one simulated phantom family and directivity model may not transfer to real data or other source classes is a legitimate generalization risk, but it is not a circularity: the paper never defines the target result in terms of those inputs, nor does it disguise a training error as a test error. No equation or claim reduces to its own input by definition.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The method rests on standard wave propagation and UBP theory, plus two domain-specific modeling choices: the cosine-squared directivity model and the use of deformed Shepp-Logan phantoms as a stand-in for real sources. The learned weight vector is a large set of fitted parameters, not a derived constant. No invented entities are introduced.

free parameters (1)

Learned UBP weight vector W = Not reported (about 2 million parameters)
Optimized with SGD over 100 epochs on 1800 training pairs. This high-dimensional weight vector is the entire mechanism of the proposed method and is not provided in the paper.

assumptions (4)

standard math The photoacoustic pressure wave satisfies the wave equation with initial value f (Section 2.1, Eq. (1)).
Standard mathematical model for PAT; assumed throughout.
standard math Universal backprojection is an exact inversion formula for full circular data (Section 2.2, Eq. (7)).
The starting algorithm; correctness on full data is established in the cited literature.
domain assumption Detector directivity is modeled by the cosine-squared angular weighting in Eq. (6).
A specific sensitivity model from prior work; the learned weights are optimized under this model.
ad hoc to paper The synthetic Shepp-Logan phantom distribution is representative of real PAT sources.
Training and evaluation both use simulated phantoms; no real data validation is provided, making this a load-bearing untested premise.

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Cite this review

Pith. "Pith review of Learned backprojection for sparse and limited view photoacoustic tomography." pith.science (2026). https://pith.science/paper/WKOLGMZW

@misc{pith2026190800593,
  author       = {Pith},
  title        = {Pith review of: Learned backprojection for sparse and limited view photoacoustic tomography},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WKOLGMZW}},
  note         = {Machine review of arXiv:1908.00593}
}

read the original abstract

Filtered backprojection (FBP) is an efficient and popular class of tomographic image reconstruction methods. In photoacoustic tomography, these algorithms are based on theoretically exact analytic inversion formulas which results in accurate reconstructions. However, photoacoustic measurement data are often incomplete (limited detection view and sparse sampling), which results in artefacts in the images reconstructed with FBP. In addition to that, properties such as directivity of the acoustic detectors are not accounted for in standard FBP, which affects the reconstruction quality, too. To account for these issues, in this papers we propose to improve FBP algorithms based on machine learning techniques. In the proposed method, we include additional weight factors in the FBP, that are optimized on a set of incomplete data and the corresponding ground truth photoacoustic source. Numerical tests show that the learned FBP improves the reconstruction quality compared to the standard FBP.

Figures

Figures reproduced from arXiv: 1908.00593 by the authors.

**Figure 2.** Considered measurement scenarios. The detectors [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 3.** Reconstruction results for the limited view case in the reconstruction domain [−1, 1]2 . In the top row a simulated phantom not contained in the training set (left), the reconstruction using the UBP without trained weights (center) and the reconstruction using the weighted UBP (right) are shown. The bottom row displays the learned weights for measurements on a half circle (left) and the absolute differences from the… view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: Reconstruction results for sparse data on the reconstruction domain [−1, 1]2 . In the top row a simulated phantom not contained in the training set (left), the reconstruction using the UBP without trained weights (center) and the reconstruction using the weighted UBP …

**Figure 5.** Figure 5: Reconstruction results for limited view and sparse data on the reconstruction domain [−1, 1]2 . In the top row a simulated phantom not contained in the training set, the reconstruction using the UBP without trained weights and the reconstruction using the weighted UBP …

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

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Johannes Bauer-Marschallinger, Karoline Felbermayer, and Thomas Berer. All-optical photoacoustic projection imaging.Biomedical optics express, 8(9):3938–3951, 2017

work page 2017

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Hu Chen, Yi Zhang, Weihua Zhang, Peixi Liao, Ke Li, Jiliu Zhou, and Ge Wang. Low-dose ct via convolutional neural network.Biomedical optics express, 8(2):679–694, 2017

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