{"id":"dc8524ae-f0b4-4682-895e-a2dd0afc1c4e","arxiv_id":"2501.12497","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"MMGKS-OF jointly estimates motion via optical flow and reconstructs dynamic tomography images using an MMGKS solver with automatic regularization parameter selection, outperforming non-motion baselines.","lead":"This paper introduces MMGKS-OF, an algorithm that combines optical flow motion estimation with edge-preserving regularization for dynamic X-ray tomography. The method automatically selects regularization parameters and shows improved reconstructions on limited-angle and single-shot CT benchmarks.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Algorithm definition is internally inconsistent: Eq. (4.1) uses an ℓ2 motion-fidelity term, whereas the implemented subproblem (4.5)/(4.7) and MMGKS weighting minimize the ℓ1 norm of the same term; without code, the claimed results cannot be tied to the proposed method.","rationale":"The paper attacks a real problem and the experimental campaign is broad, with inverse-crime mitigation and noise added. The MMGKS-embedded optical-flow idea is a reasonable extension of prior work. However, the central claim rests entirely on numerical results, and the algorithm as written is ambiguous at a formative point. The reader's weakest-assumption (optical-flow validity) is a generalizability concern that the authors explicitly acknowledge in Section 6; it does not by itself invalidate the reported experiments, which include large-motion cases where the method still wins. By contrast, the ℓ1-versus-ℓ2 discrepancy between Eq. (4.1) and Eqs. (4.5)/(4.7) means the paper does not unambiguously state which functional was minimized. Since the MMGKS machinery and the P_epsilon weighting are for ℓ1, and the final combined form (4.7) is ℓ1, it is likely that the implemented method is ℓ1 and Eq. (4.1) has a typo; but a typo in the formal definition of the proposed method is still a correctness defect for a numerical-analysis paper. In addition, fixing λ1=λ2 without justification should be tested. The proposed test (code inspection and re-running Test 1 under both norms) would settle whether the reported RRE/SSIM values are stable across these choices. If they are stable, the concern becomes a matter of presentation; if not, the superiority claim depends on choices not described in the paper. Thus the verdict remains CONDITIONAL: the issue is concrete and fixable, and the underlying numerical evidence may survive correction.","tokens_in":21255,"tokens_out":9234,"duration_ms":88152,"concrete_test":"After code release, inspect the objective minimized in the image-reconstruction subproblem to determine whether the motion term is implemented as ℓ1 (matching Eq. 4.5/4.7) or ℓ2 (matching Eq. 4.1), then rerun Test 1 (nviews=3) using each norm. If the two runs yield materially different RRE/SSIM, the reported results are tied to an unspecified variant of the algorithm, and the central claim is not reproducible until the inconsistency is resolved.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 4 defines the joint objective (4.1) with λ2∥M_bar(s,s')u∥2 (an ℓ2 norm) as the motion-fidelity term, while the image-reconstruction subproblem (4.5) writes λ2∥M_bar(k)u∥1, and (4.7) combines both regularizers into ∥Hu−b∥2^2 + λ∥Θ(k)u∥1, with Θ(k) = [Ψ; M_bar(k)]. The MMGKS majorizer in Algorithm B.1 and Section 4.1.1 (weighting P_epsilon using (y^2+ε^2)^{1/2}) is designed for the ℓ1 setting. Thus the formal objective and the algorithm actually described/solved are different optimization problems. Since the paper's only evidence for 'superior reconstructions' is the numerical experiments, and the code is not yet public, a reader cannot determine which objective was minimized to produce Tables 5.2–5.5. This is a load-bearing ambiguity, not a stylistic quibble: ℓ1 and ℓ2 regularization of the motion residual behave differently (edge-preserving vs. smooth), and the results could depend on that choice. Relatedly, Eq. (4.5) still carries two parameters λ1 and λ2, but Section 4.1.1 fixes λ:=λ1=λ2; the ratio of spatial to temporal regularization is thus a hand-set modeling choice, which tempers the 'automatic parameter selection' claim even if the single remaining λ is chosen automatically.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes MMGKS-OF, an alternating algorithm for dynamic image reconstruction in which the optical flow between consecutive frames is estimated from the current reconstruction, encoded as a linear warping operator M(s), and then used as a temporal regularization term alongside edge-enhancing spatial regularization. Both the optical flow subproblem and the image reconstruction subproblem are solved with a majorization-minimization generalized Krylov subspace method, with regularization parameters selected automatically by the discrepancy principle or generalized cross-validation. The numerical section reports consistent improvements in relative reconstruction error and SSIM for MMGKS-OF over non-optical-flow baselines on limited-angle CT (synthetic and real emoji data) and single-shot CT (two synthetic datasets).","tokens_in":21664,"tokens_out":5029,"duration_ms":48839,"significance":"If the method is correctly specified and the reported experiments are representative, MMGKS-OF is a useful and computationally attractive contribution to joint motion estimation and dynamic tomographic reconstruction: it avoids expensive inner proximal loops, uses a Krylov subspace reduction, and replaces manual tuning of the main regularization parameter by automatic DP/GCV selection. The paper also contains several practical engineering choices, such as periodic optical flow updates, downsampling, reverse-flow approximation, and parallelization, which are relevant for large-scale problems, and it mitigates inverse crime in the synthetic tests by using mismatched projectors and added noise. The main strengths are therefore the algorithmic design and the breadth of numerical comparisons. However, as detailed below, a load-bearing ambiguity about which objective was actually minimized, a post-hoc choice of the optical flow norm exponents, and the partially hand-set regularization ratios mean that the numerical evidence does not yet fully establish the central claims.","major_comments":[{"comment":"The formal joint objective and the algorithm actually described minimize different functionals. Equation (4.1) defines the motion-fidelity term as λ2 ||M_bar(s,s')u||_2 (an ℓ2 norm), but the image reconstruction subproblem in Eq. (4.5) and its combined form in Eq. (4.7) use the ℓ1 norm ||M_bar^(k)u||_1, and the MMGKS weighting used in Algorithm 4.1 step 8 and Appendix B is designed for the ℓ1 setting. Because the code is not yet public, a reader cannot determine which objective produced the results in Tables 5.2–5.5. This ambiguity is load-bearing: ℓ1 and ℓ2 regularization of the motion residual have different edge-preservation properties, and the reported reconstruction gains could depend on that choice. Please state explicitly which objective is minimized in the implementation and make the code available for review.","section":"§4, Eq. (4.1) vs. §4.1.1, Eqs. (4.5), (4.7)"},{"comment":"The choice p=q=2 for the optical flow norm appears to be made after observing test performance. The text states that for a standalone optical flow problem p=q=1 gives optimal results, but that for the limited-angle and single-shot tomography problems \"choosing p=q=2 resulted in more accurate reconstructed image sequence. Hence, for all the experiments we use the later.\" Since p and q are free parameters of the proposed method and were selected on the same experiments used to demonstrate improvement, the reported gains may be partially due to selection bias. The paper provides no error bars, no repeated-run statistics, and no held-out tuning set. Please report results for both (p,q) choices, or otherwise demonstrate that the conclusions are robust to this choice.","section":"§4.1.2 and §5"},{"comment":"The claim of automatic regularization parameter selection is overstated. Equation (4.5) contains two regularization parameters, and Section 4.1.1 fixes λ := λ1 = λ2, so the ratio of spatial to temporal regularization is a hand-set modeling choice; the optical flow subproblem (4.3) carries additional parameters γ and q, with (p,q) selected manually as noted above. Thus only a single effective λ is selected automatically, via DP or GCV, once these ratios and exponents are fixed. Please temper the abstract and contribution statements, or demonstrate that the reconstruction quality is insensitive to the fixed λ1/λ2 ratio and to the other hand-set choices.","section":"§4.1.1 and §4.1.3"},{"comment":"Section 5 states that all test problems and algorithm implementations \"will be made publicly available... once the manuscript is accepted to the journal,\" which means the numerical experiments are not independently verifiable during review. Given that the reported improvements are entirely empirical and that the objective implemented is currently ambiguous (see the first major comment), a reviewer-accessible supplement or code archive is needed to confirm that the experiments correspond to the proposed algorithm as written.","section":"§5, reproducibility"}],"minor_comments":[{"comment":"The sentence preceding Eq. (2.4) says the right-hand side of (2.3) is linearly approximated, but the expansion is performed on the left-hand side, u(x_i + s_x ∆t, y_i + s_y ∆t, t + ∆t); please correct this to \"left-hand side.\"","section":"§2.2, Eq. (2.4)"},{"comment":"The phrase \"γ ∈ R is and a regularization parameter\" is garbled; it should read \"γ > 0 is a regularization parameter.\"","section":"§2.2, Eq. (2.7)"},{"comment":"In Eq. (4.4), the left-hand side reads \"s′(k) argmin\" and is missing an equals sign; it should be \"s′(k) = argmin.\"","section":"§4, Eq. (4.4)"},{"comment":"The definition of the weighting matrix P_ε is inconsistent across the paper: Eq. (4.8) uses P_ε = (diag(1/φ_ε(z)))^{1/2}, Algorithm 4.1 step 8 uses (diag((ν^(k))^2 + ε^2))^{-1/4}, and Algorithm B.1 step 6 uses (diag(w_ε))^{1/2} with w_ε = ((y^(k))^2 + ε^2)^{1/2}; please unify these definitions and relate them to the ℓ1 majorizer explicitly.","section":"§4.1.1, Algorithm 4.1, Appendix B"},{"comment":"In Table 5.2, the SSIM rows for nviews = 5 and nviews = 7 are identical to the corresponding RRE rows, which is almost certainly a copy-paste error; the correct SSIM values should be reported.","section":"§5.1.1, Table 5.2"},{"comment":"In Test 3, the evaluation is necessarily qualitative because no ground truth is available; the text says the OF methods \"from observation... outperform\" the others, but it would strengthen the comparison to quantify this, for example with a no-reference metric or by reporting the estimated optical flow quality.","section":"§5.1.3"}],"recommendation":"major_revision","confidential_remarks":"The central algorithmic idea is sound and the numerical results are consistently in favor of the proposed method, but the reviewer cannot yet verify which objective was actually minimized in the experiments. I recommend requesting the code as a supplement and asking the authors to clarify the ℓ1/ℓ2 issue before the claims can be accepted. The post-hoc selection of p,q and the hand-set λ1=λ2 ratio should also be addressed, as they affect the interpretation of the 'automatic parameter selection' selling point. The paper is within scope for the journal and the topic is timely."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a solid, well-written numerical paper that deserves a serious referee, but the stress-test concern is right. The joint objective in Eq. (4.1) uses an ℓ2 norm for the motion-fidelity term, while the image reconstruction subproblem in (4.5)/(4.7) uses ℓ1. Those are different optimization problems. Since the code is only promised upon acceptance, the favorable RRE/SSIM tables cannot be unambiguously tied to the stated method. That is load-bearing, not a stylistic quibble.\n\nWhat is genuinely new: the linear matrix encoding of optical flow (Eqs. (3.2)–(3.5)), the reverse-flow approximation, and the adaptation of MMGKS to a combined spatial-plus-temporal ℓ1 regularizer with automatic parameter selection via DP/GCV. The experiments are broad: limited-angle and single-shot CT, synthetic and real data, and they take inverse crime seriously (different projector, misaligned geometry). The consistent improvement of the OF variants over their non-OF counterparts is plausible, and the diminishing returns as projection angles increase make sense. The comparison with the competing method's parameter sensitivity in Appendix A is a nice touch.\n\nSoft spots, in order: (1) the ℓ2/ℓ1 inconsistency above; (2) the choice p=q=2 for the optical flow norm was made after seeing test performance, which is selection bias—report sensitivity to p,q instead; (3) setting λ=λ1=λ2 means only the overall weight is automatic, not the balance between spatial and temporal regularization; (4) no error bars or repeated noise realizations, so the reported margins (sometimes small) lack uncertainty quantification; (5) the small-motion assumption is acknowledged in the conclusion, so that is not a hidden flaw.\n\nThe central claim—motion regularization helps in severely limited-data dynamic tomography—is well supported. The issues are reproducibility and reporting quality, not a failed method. The citation pattern is normal; the self-citations are to the actual MMGKS and Trips-Py works being used.\n\nThis paper is for people in dynamic CT, motion-aware reconstruction, or Krylov methods for ℓ1 problems. I'd send it to peer review. The referee should require the code, a corrected statement of the objective (or a justification for ℓ1), and a sensitivity check for p,q and the λ1/λ2 balance. With those, it is a solid contribution.","headline":"Useful and honest numerical paper on optical-flow-regularized dynamic CT, but the mismatch between the stated ℓ2 objective and the implemented ℓ1 subproblem must be resolved before the reported gains are taken at face value.","tokens_in":22113,"tokens_out":8185,"would_cite":true,"duration_ms":71692,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["65F22","65F10","65K10"],"pacs":[],"model":"deepseek-v4-flash","headline":"A new algorithm, MMGKS-OF, reconstructs moving objects in dynamic tomography by estimating optical flow between frames and using it as an extra temporal regularizer; across limited-angle and single-shot CT tests, the motion-aware…","keywords":["dynamic inverse problems","generalized Krylov subspace","computerized tomography","motion estimation","optical flow","spatio-temporal regularization","ill-posed problem","regularization parameter selection"],"falsifier":"Run the published algorithm on a dynamic phantom whose true per-frame displacement exceeds about one pixel, or whose motion rotates or shears the object so that brightness constancy fails, under the same limited-angle geometry; if the motion-augmented reconstructions do not beat the non-motion baseline in relative error and SSIM, the central claim fails.","tokens_in":21109,"feed_emoji":"🩻","tokens_out":8810,"duration_ms":83761,"temperature":0.7,"pith_summary":"The paper proposes a way to handle dynamic inverse problems—where the object moves while projection data are collected—by estimating that motion with optical flow and using it as an additional temporal regularizer in the reconstruction. The central claim is that the motion-aware term, combined with an edge-preserving spatial regularizer and solved by a generalized Krylov subspace method, gives better reconstructions than the same solvers without motion information. The experiments cover limited-angle and single-shot computerized tomography, exactly the severely undersampled settings where temporal coherence should matter most. If the claim holds, the method offers a computationally practical route to reconstructing moving targets from sparse projection data, with regularization parameters selected automatically rather than tuned by hand.","feed_headline":"Optical flow helps dynamic CT beat motion-free methods","feed_subtitle":"It folds motion estimates into the reconstruction and picks its own tuning parameters for sharper sparse-scan images.","key_machinery":"The load-bearing object is the linearized optical-flow regularization matrix $\\bar{M}(s)$, a block matrix whose diagonal blocks are the identity and whose off-diagonal blocks are motion warping matrices $M(s(t))$; $M(s(t))$ is built either by rounding estimated velocities to integer pixel shifts (making it a permutation) or by bilinear warping weights. The identity that carries the argument is $\\bar{M}(s)u = \\Upsilon(u)s + u_t$, which says that applying the motion matrix to the image sequence is the same as evaluating the standard optical flow constraint. That identity turns a nonlinear motion model into a linear regularizer, which can be merged with the $\\ell^1$ edge-preserving spatial operator into a single operator $\\Theta^{(k)}$ and solved by majorization-minimization inside a growing Krylov subspace; the same MMGKS machinery solves the optical flow subproblem and selects its regularization parameter automatically.","core_discovery":"The paper's central claim is that the optical flow constraint can be encoded as a linear operator on the whole image sequence, so that motion information becomes a drop-in regularization term rather than a separate nonlinear model. Given estimated velocity fields, the matrix $\\bar{M}(s)$ is a block bidiagonal operator built from image warping blocks $M(s(t))$, and the identity $\\bar{M}(s)u \\approx \\Upsilon(u)s + u_t$ connects this matrix product to the classical optical flow equation. The paper solves the resulting functional by alternating between optical flow estimation and image reconstruction, using the MMGKS (majorization-minimization generalized Krylov subspace) method for both subproblems with automatic regularization parameter selection by discrepancy principle or generalized cross-validation. In the four test settings, the optical-flow variants consistently outperform their non-motion counterparts in relative error and structural similarity, with the largest gains in severely limited-angle and binned-angle scenarios and diminishing returns as more projection angles are added.","pith_inferences":["Because the optical flow term is encoded as a fixed linear operator once velocities are estimated, the same $\\bar{M}(s)$ matrix could be plugged into other variational solvers, not only Krylov-subspace ones; testing this would show whether the motion encoding itself or the MMGKS solver drives the reported gains.","The observed diminishing returns with more projection angles suggest the method's value is concentrated in extremely sparse sampling; a natural experiment is to apply it to gapped or non-uniform angular sampling patterns beyond the binning case studied here.","The reverse-flow approximation, which derives backward velocities from forward velocities instead of solving the optical flow problem in reverse, is a cost-saving shortcut whose accuracy should degrade as motion grows; measuring that degradation would define when the shortcut is safe.","A learned or parametric motion model could replace the brightness-constancy velocity field, potentially extending the approach to non-rigid deformation and larger time steps, which the paper names as future work."],"forward_implications":["In severely limited-angle tomography, adding the optical-flow term produces substantially lower relative reconstruction error and higher SSIM than the same spatial regularizers alone; the advantage shrinks as more projection angles are available.","The method applies to single-shot tomography, where only one projection angle is measured per time step, recovering rigidly moving objects from extremely sparse data.","Regularization parameters for both the image and motion subproblems are selected automatically, either by the discrepancy principle or by generalized cross-validation, removing a manual tuning step that the paper shows is a sensitivity bottleneck in an alternating-method baseline.","The algorithm keeps computational cost comparable to existing joint motion-estimation methods by projecting onto a low-dimensional Krylov subspace, recomputing optical flow only every 20 iterations, downsampling images before flow estimation, and deriving reverse motion from forward motion rather than solving the backward problem.","The motion-aware regularizer is compatible with several spatial priors, and in every tested combination the optical-flow version beats the non-motion version."],"supporting_citations":[{"why":"Introduces the optical flow constraint and the smoothness regularizer that the temporal model is built on.","marker":"[27]"},{"why":"Supplies the MMGKS solver that solves both the image and motion subproblems and selects regularization parameters automatically.","marker":"[31]"},{"why":"Provides a variational motion-model reconstruction baseline, the bilinear warping encoding the paper adapts, and the single-shot pinball comparison dataset.","marker":"[7]"},{"why":"Defines the dynamic inverse problem framework, tensor notation, spatial regularizers, and the emoji dataset used in the real-data test.","marker":"[42]"},{"why":"Provides the forward projector used to generate the fan-beam projection geometries in the numerical experiments.","marker":"[41]"},{"why":"Is the joint motion estimation and image reconstruction code whose regularization-parameter sensitivity is demonstrated in the appendix.","marker":"[17]"},{"why":"Underlies the majorization-minimization generalized Krylov subspace method that MMGKS-OF extends to the optical-flow setting.","marker":"[28]"}],"fun_headline_variants":["Optical flow regularization sharpens dynamic CT","Motion-aware reconstruction wins over static methods","Krylov solver embeds optical flow in CT reconstruction","Dynamic CT gets sharper with optical flow prior","Optical flow boosts limited-angle CT clarity"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The method assumes pixel intensities stay constant along motion paths and that frames are close enough in time for the first-order Taylor expansion of the optical flow equation to be accurate; if displacements are large, objects deform, or illumination changes, the temporal regularizer is mis-specified and can degrade the reconstruction.","fun_headline_variants_meta":{"raw":{"variants":["Optical flow regularization sharpens dynamic CT","Motion-aware reconstruction wins over static methods","Krylov solver embeds optical flow in CT reconstruction","Dynamic CT gets sharper with optical flow prior","Optical flow boosts limited-angle CT clarity"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000144,"raw_usage":{"total_tokens":1139,"prompt_tokens":871,"completion_tokens":268,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":487,"completion_tokens_details":{"reasoning_tokens":200}},"tokens_in":487,"tokens_out":268,"duration_ms":2954,"temperature":1.0,"reasoning_tokens":200,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T17:07:07.191299+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the published algorithm on a dynamic phantom whose true per-frame displacement exceeds about one pixel, or whose motion rotates or shears the object so that brightness constancy fails, under the same limited-angle geometry; if the motion-augmented reconstructions do not beat the non-motion baseline in relative error and SSIM, the central claim fails.","supporting_citations":[{"cited_title":"Lanza, S","cited_arxiv_id":null,"evidence_quote":"Supplies the MMGKS solver that solves both the image and motion subproblems and selects regularization parameters automatically."},{"cited_title":"Burger, H","cited_arxiv_id":null,"evidence_quote":"Provides a variational motion-model reconstruction baseline, the bilinear warping encoding the paper adapts, and the single-shot pinball comparison dataset."},{"cited_title":"Pasha, A","cited_arxiv_id":null,"evidence_quote":"Defines the dynamic inverse problem framework, tensor notation, spatial regularizers, and the emoji dataset used in the real-data test."},{"cited_title":"Pasha, S","cited_arxiv_id":null,"evidence_quote":"Provides the forward projector used to generate the fan-beam projection geometries in the numerical experiments."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Is the joint motion estimation and image reconstruction code whose regularization-parameter sensitivity is demonstrated in the appendix."},{"cited_title":"Huang, A","cited_arxiv_id":null,"evidence_quote":"Underlies the majorization-minimization generalized Krylov subspace method that MMGKS-OF extends to the optical-flow setting."}],"review_version":1}