{"id":"f3af978b-ab83-4ddd-90dd-00d4a7373374","arxiv_id":"2506.10813","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":7,"one_line_summary":"SmoothProper unrolls a smoothed regularization layer with learned basis vectors into deformable image registration networks, achieving a 1.88 pixel target registration error on the FIRE retinal image dataset.","lead":"This paper introduces SmoothProper, a plug-and-play module for unsupervised deformable image registration that enforces smoothness and message passing during the neural network forward pass. On the FIRE retinal dataset, it reports a registration error of 1.88 pixels, claiming the first unsupervised DIR to handle both aperture and large-displacement challenges.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Eq. (7) is not the stationary condition of Eq. (5): differentiating C yields terms in ||v−b_i||² and (qb−v)·b_i, not α(p−q). The unrolled optimization claim is unsupported unless a corrected q update is supplied.","rationale":"The reader's weakest_assumption correctly identifies Eq. (7) as unsupported. My independent check goes further: the expression is not merely unproven but is contradicted by direct differentiation of the stated objective. The most load-bearing consequence is that the claimed smooth-reinforce/message-passing loop depends on the q update, and no valid update rule is provided. The empirical TRE of 1.88 could still be genuine if the implemented layer performs a different operation, but as written the paper does not establish the mechanism it claims. I agree with the reader's CONDITIONAL verdict: the concern is addressable by supplying the correct derivation, implementation details, and preferably code, but the current manuscript does not resolve it. I do not move the verdict because the concern was already the reader's stated weakest assumption; my analysis confirms rather than redirects it.","tokens_in":15235,"tokens_out":7713,"duration_ms":92436,"concrete_test":"Differentiate Eq. (5) with respect to q_i(x) and solve; verify that the stationarity condition is (1/(2α))[||v−b_i||² + 2(qb−v)·b_i] + 2(q_i−p_i)=0, not Eq. (7). Then inspect the released implementation at the stated repository to see which update is actually coded. If the code uses Eq. (7), the q update collapses to q≈p and no v-dependent message passing occurs; if it uses the true minimizer, Eq. (7) is a typo and the paper must be revised with the correct update and a numerical comparison of the two variants on FIRE.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central mechanism rests on the claim that the q subproblem in Eq. (6) has a closed-form update obtained by differentiating C and setting the result to zero, Eq. (7). Direct differentiation of Eq. (5) refutes this. With C = Σ_i q_i ||v−b_i||² + ||qb−v||², the derivative is ∂C/∂q_i = ||v−b_i||² + 2(qb−v)·b_i, with no p dependence. The full stationarity condition for the q subproblem is (1/(2α))[||v−b_i||² + 2(qb−v)·b_i] + 2(q_i−p_i)=0. Eq. (7) instead asserts ∂C/∂q_i = α(p_i−q_i), which would imply q=p at stationarity and remove the coupling to v entirely. The text never gives the actual closed-form q update, and no code is provided. If the implemented q update follows Eq. (7), the claimed message-passing and smooth-reinforce mechanism cannot operate as described; if it follows the true stationarity condition, then Eq. (7) is a serious typographical error that invalidates the derivation as written. Either way, the paper's theoretical foundation for SmoothProper is not currently supported. The Gaussian-blur approximation of the v subproblem is secondary and explicitly approximate, but the q update is presented as exact, making this the most load-bearing point.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes SmoothProper, a plug-and-play neural layer for unsupervised deformable image registration. The method is built on a quadratic-relaxation of the Horn-Schunck energy, with a learned non-negative coefficient field p, a learned basis matrix b, and alternating updates of a coefficient field q and an auxiliary displacement field v. The authors claim that unrolling this optimization in the network forward pass enforces smoothness and message passing, addressing the aperture and large-displacement problems in retinal vessel registration. On the FIRE dataset they report a TRE of 1.88 pixels at 2912x2912 resolution, outperforming a broad set of detector-based, detector-free, and learning-based baselines. The paper also introduces conditional basis vectors generated by two MLPs conditioned on a randomly sampled regularization strength, eliminating manual lambda tuning.","tokens_in":15541,"tokens_out":4079,"duration_ms":48427,"significance":"If the technical derivation were sound, the contribution would be valuable: SmoothProper is model-agnostic, adds modest parameters, and the landmark-based TRE evaluation is independent of the unsupervised training loss, which is a genuine strength. The reported accuracy on FIRE is substantially better than the baselines, and the idea of using the pairwise regularizer as a message-passing mechanism during the forward pass is interesting and well motivated. However, the central mathematical claim in Section 3.1 is incorrect as written, and no code or precise algorithmic specification is provided to resolve the discrepancy. The paper also leaves key implementation details of the approximate v update unspecified and does not clarify whether the hyperparameters used for the headline result were selected on the test set. These issues make the current manuscript unsuitable for acceptance without a major revision.","major_comments":[{"comment":"Equation (7) is not the stationary condition of the q subproblem in Eq. (6). Differentiating C = sum_i q_i ||v - b_i||^2 + ||q b - v||^2 with respect to q_j gives ||v - b_j||^2 + 2(q b - v) . b_j, which has no dependence on p. The full stationarity condition for the subproblem in Eq. (6) is (1/(2 alpha)) [||v - b_j||^2 + 2(q b - v) . b_j] + 2(q_j - p_j) = 0. Equation (7) instead asserts that the derivative of C equals alpha (p - q), which would imply q = p at stationarity and would remove the coupling between q and v entirely. This is load-bearing because the smooth-reinforce mechanism described in Section 3.1 depends on q being updated from v and b. Please provide the correct closed-form update, or state explicitly if the implemented update differs from the derivation.","section":"Sec. 3.1, Eq. (7)"},{"comment":"The paper states that the v subproblem is solved by fixed-point iteration and then approximated by applying Gaussian blurring to q b, with the blurring strength determined by beta. Neither the kernel size nor the standard deviation is specified, and no derivation shows that Gaussian blurring solves the regularized least-squares subproblem in Eq. (6). The citation to Diffeomorphic Demons motivates a smoothing step but does not justify this specific approximation in the proposed energy. Because the unrolled network is trained end-to-end, the approximation could be absorbed by learned parameters, but the claim that SmoothProper solves the stated optimization by coordinate descent is weakened. Please specify the blur operation and, ideally, verify that the approximation is faithful to the fixed-point solution.","section":"Sec. 3.1, v subproblem"},{"comment":"The headline TRE of 1.88 pixels is reported for m = 4x3^2, K = 6, and input size 1024x1024, with these parameters selected via the ablations shown in Fig. 6. The text does not state whether the ablations were performed on the validation split or on the test split. If the test set was used to choose m, K, and the input size, the reported TRE is an optimistically biased estimate of generalization. Please clarify the protocol and, if necessary, report results on a held-out test split after fixing hyperparameters.","section":"Sec. 4.3, Fig. 6"},{"comment":"The comparison protocol needs clarification. The text says all learning-based methods were initialized using the best-performing descriptor matching method, but Table 1 does not appear to show the initial alignment row that is described as highlighted in gray. Without this row, it is unclear how much of the reported improvement comes from SmoothProper rather than from the initial alignment. Please report the TRE of the initial alignment for each category and state whether all methods, including the detector-free and detector-based baselines, were evaluated under the same initialization protocol.","section":"Sec. 4.2, Table 1"}],"minor_comments":[{"comment":"There are several typos in this section: 'regularirizer' should be 'regularizer', 'convoloution' should be 'convolution', and 'wapred' should be 'warped'.","section":"Sec. 4.2"},{"comment":"The phrase 'from-matching strength lambda' is unclear. Please define what 'from-matching' refers to, or rephrase the sentence.","section":"Sec. 3.2"},{"comment":"The conclusion states that extending to 3D is as simple as replacing 2D convolution with 3D convolution, but the basis vectors b_i are d-dimensional and the MLP output dimensions would also need to change. Please qualify this claim.","section":"Sec. 5"},{"comment":"The abstract calls the results 'Preliminary results' while the conclusions state that SmoothProper 'addresses aperture and large displacement challenges' and achieves the first unsupervised DIR result of this kind. Please align the strength of these claims with the evidence and the limitations of the evaluation.","section":"Abstract and Sec. 5"},{"comment":"The boxplots in Fig. 6 would be more informative if the text stated whether the TRE values are computed on the validation set or the test set, and how many pairs are represented in each box.","section":"Fig. 6"}],"recommendation":"major_revision","confidential_remarks":"The core issue is the incorrect derivative in Eq. (7), which undermines the claimed optimization foundation. The paper would be substantially strengthened by releasing the source code, since the implemented q update is not derivable from the text. The empirical comparison is broad and the reported gains are large, but the hyperparameter selection protocol needs to be transparent before the 1.88 pixel claim can be assessed. Given the scope of the correction, I recommend major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First thing to know: SmoothProper is a real architectural idea and the 1.88 px TRE on FIRE is a big jump over the 3.08 px of GAMorph, so the paper deserves attention. The second thing: the reader's strongest complaint is correct. Eq. (7) is not the derivative of C. For C = Σ_i q_i||v-b_i||² + ||qb-v||², differentiating with respect to q_i gives ||v-b_i||² + 2(qb-v)·b_i, with no α(p-q). The stationarity condition for the q subproblem couples q and v; Eq. (7) does not. If the code follows Eq. (7), the claimed message passing cannot work; if it follows the true condition, the paper contains a serious typo in the central derivation. No code is provided, so I can't tell which. That's a load-bearing flaw, not a nitpick.\n\nNow the credit. The design is a new combination: learned conditional basis vectors, a directional-bias term, and a coupling term that acts like a smooth-reinforce loop during the forward pass. That is genuinely different from PDD-Net, GraDIRN, and VR-Net. The evaluation is broad, with many detector-based and detector-free baselines, and the TRE metric is independent of the unsupervised loss, so the central result is not circular. The ablations on m, K, and input size are useful. The reported improvement is large enough that even a piece of it would be interesting.\n\nSoft spots beyond Eq. (7): the v subproblem is \"approximated\" by a Gaussian blur, but the blur size is never specified; the method was initialized with a descriptor matcher, which weakens the \"first unsupervised\" phrasing; the ablations that select m, K, and image size appear to be run on FIRE without saying whether they used the val split; and the 1.88 figure has no error bars or test-size specification. All fixable, but they matter for trusting the number.\n\nWho is this for? Anyone working on unsupervised DIR, especially retinal or other low-texture registration. It deserves a serious referee. My recommendation: send to review, but with a request for a corrected derivation (or an explicit statement that Eq. (7) is a typo), code, and a clear validation protocol.","headline":"Genuinely new forward-pass smoothing layer with a big FIRE result, but Eq. (7)'s q-update derivation is wrong as written and the optimization story needs a fix before the accuracy claim can be trusted.","tokens_in":16098,"tokens_out":3633,"would_cite":false,"duration_ms":40185,"reading_group":"maybe","serious_thinker":"unclear","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"SmoothProper unrolls smoothness regularization into the forward pass so that unsupervised deformable image registration can handle sparse-feature images with aperture and large-displacement challenges.","keywords":["Nonparametric smoothing","Deformable image registration","Neural networks","Plug-and-play","Aperture problem","Large displacement","Unsupervised learning","Retinal vessel registration"],"falsifier":"Compare the closed-form q update against a numerical gradient of the q subproblem on random inputs; if they disagree, the unrolled layer is not solving the stated optimization. Independently, swap the Gaussian-blur v update for the exact fixed-point solve of the v subproblem and check whether the 1.88-pixel target registration error on FIRE persists; if it does not, the blur approximation is load-bearing rather than incidental.","tokens_in":15016,"feed_emoji":"👁️","tokens_out":6705,"duration_ms":72254,"temperature":0.7,"pith_summary":"The paper's claim is that the pairwise smoothness regularizer in deformable registration is more than a penalty: if unrolled into the network's forward pass as an optimization layer, it can do the message-passing work that lets flow estimates propagate from vessel edges into large featureless regions. The result is SmoothProper, a plug-and-play module that replaces the final convolution of a registration backbone, and on the FIRE retinal dataset it lowers target registration error to 1.88 pixels at 2912x2912 resolution. The paper reports this as the first unsupervised deformable registration method to handle both aperture and large-displacement challenges together. If the claim holds, unsupervised deformable registration can approach the accuracy of supervised and descriptor-based methods on sparse-feature medical images without labels or manual regularizer tuning.","feed_headline":"Unsupervised registration cuts retinal error to 1.88 px","feed_subtitle":"A forward-pass smoothing layer spreads flow signal across blank regions, beating all baselines on 2912×2912 retinal scans.","key_machinery":"The central object is the basis-constrained quadratic relaxation energy, a discretized objective involving a coefficient field p from the backbone network, an auxiliary field v, learned basis vectors b generated by two MLPs conditioned on a randomly sampled lambda, and the coupling term C(q,v,b). The q subproblem is solved pointwise in closed form, and the v subproblem is approximated by Gaussian blurring of qb. Alternating these updates for K iterations constitutes the unrolled SmoothProper layer, which carries the argument by converting smoothness regularization from a loss-term penalty into a forward-pass operation that propagates flow signals and preserves structural consistency.","core_discovery":"SmoothProper casts registration as a bi-level optimization in which the displacement field is produced by minimizing a quadratic-relaxation energy rather than by a single network forward pass. The energy couples a coefficient field q(x) to learned basis vectors, pulls an auxiliary field v(x) toward the linear combination q(x)b, and applies a diffusive regularizer to v. Unrolling alternating updates of q and v, with v obtained by Gaussian blurring of qb, makes the regularizer act as a message-passing mechanism: strong flow signals in vessel-rich regions are reinforced while their influence spreads into smooth regions. On FIRE, the method achieves a target registration error of 1.88 pixels, improving over the best unrolled baseline by 62.20% and over the best learning-based non-unrolled baseline by 39.01%. The paper attributes the gain to this smooth-reinforce feedback loop rather than to a larger receptive field.","pith_inferences":["An implication the paper leaves implicit is that the same unrolled smoothing layer could replace the final convolution in 3D registration backbones with minimal change, because the formulation is dimension-agnostic and only the convolution needs to become 3D.","The learned basis vectors are described as dictionary atoms; if that analogy holds, SmoothProper should be robust to the choice of backbone, but the paper only demonstrates it with a single three-level Laplacian pyramid backbone.","The smooth-reinforce explanation predicts a testable separation: removing the directional bias term should still leave smoothness but lose structural anchoring, so the error should degrade most on images with thin vessels and large blank areas, which is not isolated in the ablations."],"forward_implications":["Unsupervised deformable registration can reach sub-2-pixel accuracy on sparse-feature retinal images without label supervision, closing a gap previously filled only by supervised or descriptor-based methods.","Adding SmoothProper to an existing registration network requires only replacing the final convolution with the unrolled layer plus the two basis-vector MLPs, adding about 0.09 G multiply-add operations when the conditional basis vectors are not used.","Regularizer hyperparameter tuning becomes unnecessary, since the basis vectors are conditioned on a randomly sampled lambda during training and the smoothing strength adapts after amortized optimization.","The number of unrolled iterations has a sweet spot, with K=6 optimal on FIRE, matching earlier mean-field inference practice where further iterations give limited improvement."],"supporting_citations":[{"why":"Supplies the Horn-Schunck variational optical flow formulation whose pairwise regularizer is the message-passing mechanism SmoothProper unrolls.","marker":"[34]"},{"why":"Provides the duality-based optimization and quadratic-relaxation technique used to decouple data fidelity from the regularizer.","marker":"[8]"},{"why":"Motivates the large-displacement handling by showing convex subproblems can absorb large deformations without linearizing the data term around small steps.","marker":"[60]"},{"why":"Supports the approximation of the v subproblem by Gaussian blurring analogously to Diffeomorphic Demons.","marker":"[68]"},{"why":"Is the prior unrolling work that most directly enforces smoothness, which SmoothProper extends with learned basis vectors and interaction terms.","marker":"[28]"},{"why":"Supplies the unrolled model-based architecture that SmoothProper's structure resembles.","marker":"[1]"},{"why":"Provides the alpha schedule used in the alternating iterations.","marker":"[30]"}],"fun_headline_variants":["SmoothProper cuts unsupervised retinal registration error to 1.88 px","First unsupervised DIR tackles aperture and large displacement, hits 1.88 px","Unsupervised registration with built-in smoothing reaches 1.88 px error","New plug-and-play module lowers retinal registration error to 1.88 px"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"Everything rests on the unrolled update being a faithful solver of the stated subproblem: the q-update derivative in Eq. (7) is asserted rather than derived, and the v-update is replaced by a Gaussian blur without proof, so if either step does not minimize the energy the reported gains come from an unexplained modification rather than from the claimed optimization.","fun_headline_variants_meta":{"raw":{"variants":["SmoothProper cuts unsupervised retinal registration error to 1.88 px","First unsupervised DIR tackles aperture and large displacement, hits 1.88 px","Unsupervised registration with built-in smoothing reaches 1.88 px error","New plug-and-play module lowers retinal registration error to 1.88 px"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001351,"raw_usage":{"total_tokens":5493,"prompt_tokens":961,"completion_tokens":4532,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":577,"completion_tokens_details":{"reasoning_tokens":4450}},"tokens_in":577,"tokens_out":4532,"duration_ms":35110,"temperature":1.0,"reasoning_tokens":4450,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T04:17:26.946668+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compare the closed-form q update against a numerical gradient of the q subproblem on random inputs; if they disagree, the unrolled layer is not solving the stated optimization. Independently, swap the Gaussian-blur v update for the exact fixed-point solve of the v subproblem and check whether the 1.88-pixel target registration error on FIRE persists; if it does not, the blur approximation is load-bearing rather than incidental.","supporting_citations":[{"cited_title":"NeuroImage45(1), S61–S72 (2009)","cited_arxiv_id":null,"evidence_quote":"Supports the approximation of the v subproblem by Gaussian blurring analogously to Diffeomorphic Demons."},{"cited_title":"In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2019: 22nd International Confer- ence, Shenzhen, China, October 13–17, 2019, Proceedings, Part VI 22","cited_arxiv_id":null,"evidence_quote":"Is the prior unrolling work that most directly enforces smoothness, which SmoothProper extends with learned basis vectors and interaction terms."}],"review_version":1}