Revisiting $\Psi$DONet: microlocally inspired filters for incomplete-data tomographic reconstructions

Andrea Sebastiani; Luca Ratti; Tatiana A. Bubba

REVIEW 3 major objections 6 minor 37 references

Revisiting $\Psi$DONet: microlocally inspired filters for incomplete-data tomographic reconstructions

T0 review · 3 major / 6 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read Masked filters restricted to the visible cone match or slightly beat the original ΨDONet's reconstruction quality with far fewer learnable parameters.

desk verdict Honest, useful extension of ΨDONet with clever masked filters and a plausible microlocal story; the parity claim still needs error bars and code, and the "prevention" wording outruns the math. read the letter →

arxiv 2501.18219 v1 pith:UNTBEHSI submitted 2025-01-30 math.OC cs.CVcs.LG

Tatiana A. Bubba , Luca Ratti , Andrea Sebastiani This is my paper · ORCID

classification math.OCcs.CVcs.LG MSC 65R1068T07

keywords limited-angletomographysparse-anglemicrolocalanalysiswaveletframesunrolledoptimizationpseudodifferentialoperatorsstreakartifactscomputed

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

This paper revisits ΨDONet, a neural network built by unrolling iterative soft-thresholding for tomographic reconstruction, and gives a fuller microlocal account of what that network does: it can introduce edges that are invisible in the incomplete sinogram, and it can prevent streak artifacts rather than only dampening them, by smoothing the kernel of the incomplete-data backprojection operator. The paper's numerical claim is that the learnable filters can be masked down to the cone of visible directions (Ψbow) or to wedges along the boundary of the missing wedge (Ψx, Ψspa) with no loss, and sometimes a small gain, in PSNR and SSIM on limited-angle and sparse-angle test images, at considerably lower parameter counts. This matters because it shows that theoretical knowledge of where artifacts come from can be converted directly into cheaper learned reconstruction networks of equal quality.

What carries the argument

The carrying identity is the wavelet-domain representation $[W R_A^* R_A W^* w + \Lambda_k w]_\iota = \sum_{\iota'} (\kappa^{\iota,\iota'}_A + \tilde\kappa^{\iota,\iota'}_k) * w_{\iota'}$, where $\kappa^{\iota,\iota'}_A$ are the wavelet subband kernels of $R_A^* R_A$ and $\tilde\kappa^{\iota,\iota'}_k$ are the learned filter functions. The argument works by comparing the full normal operator, whose kernel $1/\|x-y\|$ identifies it as a pseudodifferential operator, with the incomplete-data kernel $\kappa_A(x-y) = \|x-y\|^{-1}\chi^c_A(x-y)$, whose jumps along the cone directions are the source of streak artifacts. The three masked filter geometries (bowtie in the visible cone, cross along the boundary directions, and sparse along the sampled directions) place learnable parameters exactly where kernel jumps occur, implementing the smoothing with minimal parameter count.

What would settle it

Compute the frame bounds (or their ratio) of the translation-invariant Haar frame with scales $J_0=4$ and $J=7$ used in Section 5; if the upper and lower bounds are far apart, the continuous kernel-smoothing mechanism analyzed in Section 4.3 does not describe the implemented network. A second check would be to retrain the masked variants on a wavelet frame certified to be Parseval and see whether the masked filters still match the full-square filter quality; if they do not, the proposed mechanism is not the source of the numerical result.

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Extended reading notes

Core claim

The central claim is that ΨDONet's correction term can be understood, in a continuous semi-discrete wavelet formulation, as adding learned convolutional filters to the wavelet-domain kernels of the incomplete-data normal operator $R_A^* R_A$; when those filters smooth the discontinuities introduced by the truncation cone $\chi^c_A$ inside the kernel $\kappa_A$, the corrected operator behaves like a pseudodifferential operator and streak singularities are removed at their source rather than only reduced in amplitude. A second claim is that the pointwise soft-thresholding at each wavelet subband creates new singularities, which explains how invisible edges may reappear, although the analysis cannot predict where they will appear. The numerical consequence is that filters only need to be learned where the kernel or its jumps live: bowtie-shaped supports inside the visible cone for limited-angle tomography, and thin wedges along the directions $\partial A$ for both limited- and sparse-angle tomography. Reported experiments on synthetic ellipse data show these masked variants preserve or slightly improve PSNR and SSIM relative to the original square filters while using far fewer learnable parameters.

Load-bearing premise

The continuous analysis assumes the wavelet dictionary is Parseval, meaning that reconstructing from wavelet coefficients is exact, but the Haar implementation with scales $J_0=4$ to $J=7$ used in the experiments is not shown to satisfy this, so the operator analyzed may differ from the operator actually trained.

Editorial extensions

If this is right

The masked variants cut the number of learnable parameters substantially while keeping PSNR and SSIM unchanged, so limited-angle and sparse-angle ΨDONet can be trained and deployed at lower cost without sacrificing reconstruction quality.
In the 120-degree missing-wedge experiment the masked variants slightly outperform the original square filters, and the reported images show better preservation of small, low-contrast features.
For 6-angle and 12-angle sparse-angle data, the Ψspa filters do not worsen the metrics and, in the reported images, reduce the magnitude of partial streak artifacts compared with the square-filter baseline.
The theoretical analysis implies that the soft-thresholding nonlinearity can introduce singularities not present in the data, so some invisible edges may be recovered, but the current architecture cannot predict or control where those edges appear.
If the learned filters really smooth the kernel discontinuities as argued, ΨDONet prevents streak artifacts from forming in the first place, rather than only attenuating them after they appear.

Reading between the lines

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

An untested extension, natural from the same reasoning, is to apply the mask-construction principle to other limited-data geometries such as exterior or region-of-interest tomography, where the artifact directions are likewise determined by the boundary of the available angular set.
The parameter savings could be reinvested in larger filter radii or more unrolled layers at a fixed total parameter count, which may improve fidelity in settings where the ten-layer architecture is the bottleneck.
Because the present analysis tracks only singular support, not direction, extending the architecture to directional dictionaries such as shearlets or curvelets would be the natural route to predicting which invisible edges are recovered; the paper itself notes that this would require redesigning the network.
The reported gains are established on synthetic ellipse images only; whether the same parameter reduction preserves quality on real experimental computed-tomography measurements remains an open empirical question.

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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

Short version: this is a solid, honest follow-up to the ΨDONet line, with two genuinely new things — the masked filter geometries (bowtie, cross, sparse-angle) and the wavelet-domain kernel-smoothing argument for why a learned correction can suppress streak artifacts. The sparse-angle proof-of-concept is also new. I believe the paper deserves a serious referee, but the numerical parity claim is not yet nailed down and the theory is a bit oversold.

What it does well: the masked filters are a clean, microlocally motivated way to cut parameters, and Tables 1–3 show the quality holds up (or slightly improves) across limited-angle and sparse-angle settings. The authors are transparent that the sparse-angle part is a proof of concept. Section 4.3's observation that the learned filters act directly on the discontinuous kernels κ_{A,ι,ι′} in the wavelet domain is a nice formalization, and restricting the analysis to singular support rather than wavefront is an honest choice given that wavelets do not resolve direction.

Where I'd push back. First, the numbers. All PSNR/SSIM values are averages over a test set with no error bars, no seeds, and no code. Differences of 0.1–0.3 dB between the original and masked variants are exactly the sort that can flip with initialization. Without at least a few seeds or per-image distributions, the central claim — same quality at lower parameter count — is plausible but unproven. Second, the theoretical "prevention" claim in the abstract and conclusion is stronger than Section 4.3: the text says learned filters "might" smooth the kernel, and Figure 6 still shows partial streaks for Ψspa. The math shows it is possible in principle, not that training finds such filters. Third, the continuous formulation in Section 4.1 assumes the finitely scaled, continuously translated wavelet frame is Parseval. That is not established for the Haar wavelets with J0=4 to J=7 actually used, and the discrete network uses a critically sampled orthogonal transform, not the continuous frame. So the microlocal analysis may be describing a slightly different operator. This is a real gap, though probably patchable.

Also minor: parameter counts for the masked variants are never reported explicitly, and there are no external baselines, so the practical significance is bounded. The circularity burden is real but not damning — building on one's own architecture and importing [3, Thm 4.11] is fine, and the mask designs are new.

Bottom line: this is a useful, readable paper for the learned-tomography audience. I'd send it to review, with requests for code, seeds/error bars, explicit parameter counts, and a toned-down conclusion. The central idea deserves to be published; it just needs the evidence to match the language.

Referee Report

3 major / 6 minor

Summary. The paper revisits the ΨDONet architecture for limited-angle and sparse-angle tomography. It introduces a continuous wavelet-domain formulation of the unrolled ISTA scheme, uses microlocal analysis (following [3]) to argue that soft-thresholding can introduce new edges and that learned convolutional corrections can prevent streak artifacts by smoothing the kernel of R_A^* R_A, and proposes three masked-filter variants (Ψbow, Ψx, Ψspa) whose supports are informed by the visible cone and the missing-wedge boundary. Numerical experiments on synthetic ellipse data report that these variants achieve nearly identical or slightly better PSNR/SSIM than the original full-square ΨDONet while using considerably fewer learnable parameters, and provide a proof-of-concept for sparse-angle data.

Significance. If the claims are supported, the paper makes a useful contribution: it gives a theoretically motivated way to reduce the parameter count of ΨDONet without sacrificing reconstruction quality, and it extends the architecture to sparse-angle tomography. The continuous formulation and the convolutional-kernel representation in Eq. (10) are valuable steps, and the paper correctly credits the tools imported from [3]. However, the central numerical claim is not statistically established: the reported improvements are small (0.03–0.32 dB) and no error bars, seeds, or multiple-run comparisons are given. The theoretical claim of 'prevention' of streak artifacts is only conditional in Section 4.3, and the Parseval-frame assumption used in Section 4.1 is not verified for the Haar wavelet system actually implemented. These gaps are load-bearing for the paper's main conclusions.

major comments (3)

[Section 5, Tables 1–3] The central numerical claim is that the masked-filter variants preserve or slightly improve reconstruction quality, but the reported differences are very small (e.g., +0.18 dB and +0.32 dB PSNR for Ψx in Tables 1 and 2, +0.03 and +0.16 dB for Ψspa in Table 3) and no error bars, number of training runs, random seeds, or statistical significance tests are provided. With a single run of a 10-block unrolled network trained by Adam, these gaps are plausibly within run-to-run variance. To support the parity/improvement claim, the authors should report multiple-seed experiments (at least 3–5 runs) with mean and standard deviation, or provide per-image test-set distributions and a paired test. Without this, the practical contribution of equal quality at lower parameter count is not established.
[Section 4.3, Eq. (10), and Section 6] The theoretical argument in Section 4.3 only proves the convolutional representation of W R_A^* R_A W^* + Λ_k and then states that the learned filters 'might' smooth the discontinuities of κ_A. It does not prove that training finds such a smoothing correction, nor does it characterize the trained filters. The conclusion in Section 6, however, states that ΨDONet 'can prevent' streak artifacts from appearing, which is stronger than what is shown. Figure 6 even reports residual partial streak artifacts for the Ψspa reconstruction in the 6-angle case. The authors should either soften the claim to 'reduce' or 'dampen' artifacts, or provide direct evidence (e.g., inspecting the learned filters or the effective kernel) that the trained correction indeed smooths κ_A.
[Section 4.1, Eq. (8), and Section 5] The continuous formulation in Section 4.1 assumes that the translation-invariant, finitely scaled wavelet frame is Parseval, so that W^* acts as a reconstruction operator and Eq. (8) faithfully represents the discrete network. The numerical experiments, however, use Haar wavelets with scales J0=4 and J=7 (Section 5), a setting for which the Parseval property is not established. If the frame is not Parseval, the microlocal analysis in Sections 4.2 and 4.3 describes a different operator from the one actually trained and evaluated. The authors should either prove the Parseval property for the implemented wavelet system (including the finite-scale truncation) or explicitly state this as an idealization that limits the scope of the theoretical conclusions.

minor comments (6)

[Section 2.2, Figure 1 caption] The caption contains a typo: 'when only limited data are are available' should be 'when only limited data are available'.
[Section 4, opening paragraph] The phrase 'a new continuos formulations' should be 'a new continuous formulation'.
[Section 5.1, text after Table 2] The sentence 'We report the average PSNR and SSIM in Table 1' refers to the second limited-angle experiment but should refer to Table 2, not Table 1.
[Section 5, first paragraph] The metric is referred to as 'structured similarity index'; the standard name is 'structural similarity index' (SSIM).
[Section 6, Conclusions] There is a typo in 'different visisble wedges'; it should be 'different visible wedges'.
[Section 5] The repository is stated to be made available only upon acceptance. For a paper whose main contribution is numerical, it would strengthen reproducibility to provide the code or at least detailed training and evaluation scripts during the review process.

Circularity Check

0 steps flagged · score 0.0 of 10

No definitional circularity: the numerical comparisons are empirical, the theoretical claims are explicitly conditional, and the prior-work citations provide independent published support rather than a self-referential derivation chain.

full rationale

The paper's central numerical claim is that the masked-filter variants Ψbow, Ψx and Ψspa preserve or slightly improve reconstruction quality at a considerably lower parameter count (Section 5, Tables 1–3). This is an empirical comparison on a held-out test set: the masks are constructed from independently stated microlocal principles (Principle 2 and the support of χcA in κA), and the reported PSNR/SSIM values are evaluation metrics, not quantities fitted and then repackaged as predictions. The theoretical claim in Section 4.3 derives the convolutional representation (10) and then states only that the learned filters 'might' dampen the discontinuities introduced by χcA; the conclusion's stronger wording 'can prevent' is an unsupported overstatement, and the Parseval-frame assumption is explicitly invoked but not verified for the Haar implementation used in the experiments. These are gaps in support and correctness, not circularity. The paper imports the ΨDONet architecture and its convergence properties from the authors' prior work [9], but that prior work is a published, code-released architecture that serves here as the baseline Ψo to which the new variants are compared, so the self-citation is not used to forbid alternatives or to smuggle in the conclusion. The microlocal ReLU analysis is imported from the external reference [3], not from the present authors. No step in the derivation reduces by construction to its own inputs.

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

The paper introduces no new physical or mathematical entities. The theoretical results rest on prior microlocal theorems from [3], [8], and [33], and on two unproven structural assumptions: the Parseval frame property of the semi-discrete wavelet dictionary, and the equivalence between the discrete trained network and the continuous model. The numerical claims depend on several hyperparameters chosen by hand or inherited from [9].

free parameters (4)

Stripe half-width q in boolean masks = 1, 2, or 3
Chosen per experiment to compensate for line discretization; variants with q=1,2,3 are tested and the best-performing one is emphasized (e.g., Ψ1x in Table 1, Ψ3x in Table 2). This is a post-hoc selection affecting the numerical comparison.
Filter size = 33x33 limited-angle, 65x65 sparse-angle
Limited-angle size fixed; sparse-angle size increased because 33x33 masks covered the entire support, citing approximation analysis from [9, Section 5.2].
Network hyperparameters = 10 unrolled blocks, Adam lr=1e-3, batch 25, 15 epochs
Inherited from [9] or chosen without sensitivity analysis; the theoretical claims do not depend on them, but the empirical results do.
Learnable parameters αk, βk, γk and filter weights ζk = trained on 10,000 synthetic ellipse images
Standard supervised fitting. The claim that ΨDONet 'can prevent' artifacts implicitly refers to this trained instance, whose filters are never analyzed directly.

assumptions (4)

domain assumption The translation-invariant wavelet frame with finite scales J0..J and continuous offsets is Parseval, so W* is a reconstruction operator.
Invoked in Section 4.1 to equate the continuous iteration (8) with the discrete network (7). For Haar wavelets with two scales this is not automatic and is not verified.
standard math The Nemytskii operators for ReLU and soft-thresholding can be dampened to act on distributions, as in [3].
Used in Section 4.2 to apply [3, Theorem 4.11] and obtain Corollary 4.
domain assumption Streak-generating singularities of the convolution kernel κA lie close to the boundary directions ∂A and can be captured by local masks.
Motivates the Ψbow, Ψx, Ψspa filter supports in Section 4.4; the paper relies on numerical validation rather than proof.
ad hoc to paper The learned filters at convergence actually smooth the discontinuities of κA rather than introducing other behavior.
Section 4.3 states the filters 'might be such that' they dampen discontinuities; no analysis of trained filters is provided, and residual streaks appear in Figure 6.

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Cite this review

Pith. "Pith review of Revisiting $\Psi$DONet: microlocally inspired filters for incomplete-data tomographic reconstructions." pith.science (2026). https://pith.science/paper/UNTBEHSI

@misc{pith2026250118219,
  author       = {Pith},
  title        = {Pith review of: Revisiting $\Psi$DONet: microlocally inspired filters for incomplete-data tomographic reconstructions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UNTBEHSI}},
  note         = {Machine review of arXiv:2501.18219}
}

abstract

In this paper, we revisit a supervised learning approach based on unrolling, known as $\Psi$DONet, by providing a deeper microlocal interpretation for its theoretical analysis, and extending its study to the case of sparse-angle tomography. Furthermore, we refine the implementation of the original $\Psi$DONet considering special filters whose structure is specifically inspired by the streak artifact singularities characterizing tomographic reconstructions from incomplete data. This allows to considerably lower the number of (learnable) parameters while preserving (or even slightly improving) the same quality for the reconstructions from limited-angle data and providing a proof-of-concept for the case of sparse-angle tomographic data.

Figures

Figures reproduced from arXiv: 2501.18219 by the authors.

**Figure 2.** Different filter geometries for ΨDONet. (a) ΨDONet filters initially proposed in [9]. (b) ΨDONet-bow [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗

**Figure 3.** Visualization of the reconstructions of two images from the test set with [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: Visualization of the reconstructions of two images from the test set with [PITH_FULL_IMAGE:figures/full_fig_p015_4.png]

**Figure 5.** Figure 5: Visualization of the reconstructions of two images with 12 angles. [PITH_FULL_IMAGE:figures/full_fig_p016_5.png]

**Figure 6.** Figure 6: Visualization of the reconstructions of two images with 6 angles. [PITH_FULL_IMAGE:figures/full_fig_p016_6.png]

Discussion (0). Continue with ORCID to comment.

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