{"id":"00f495ae-0348-4ff1-a6a3-91d9caa29f19","arxiv_id":"2501.07742","paper_version":3,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"New efficient minimal solvers estimate relative camera pose jointly with unknown depth scale and shift, improving speed and often accuracy over prior depth-aware solvers.","lead":"This paper proposes new minimal solvers that estimate the relative pose of two cameras from image matches plus monocular depth estimates, jointly recovering unknown depth scale and shift. They test across datasets and depth estimators, finding that depth-aware solvers beat classic point-only methods when depth is accurate, but that modeling shifts is not always necessary.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The affine depth model (Eq. 2) is load-bearing and mismatched for inverse-depth MDEs; the SM's inverse solver test is too narrow to settle it.","rationale":"The central claim has two parts: a solver-construction claim in Section 3.1 and an empirical claim of superior speed and accuracy. The algebraic derivation is internally coherent: subtracting pairs removes translation, norm constraints eliminate rotation, and the three equations in s, u, v reduce via c = s^2 to a quartic. I do not see a defect in that derivation. The weakest load-bearing point is the depth model itself. The main experiments run DA v2 and MiDaS, which are generally treated as inverse-depth networks, with the affine-depth solver 3PTsuv. The paper derived an inverse-depth solver but only reports a narrow GC-RANSAC comparison in SM Table 6, without local optimization, on one dataset, and without focal-length variants. That is not sufficient to establish that the main PoseLib/hybrid results are unaffected by the model mismatch. This is exactly the reader's weakest_assumption, and I agree with it. A concrete check is to run the inverse solver under the main protocols; the outcome would either justify the paper's choice or require recomputing several central comparisons. The reader's CONDITIONAL verdict remains appropriate, so no change is recommended. I additionally note that the abstract's 'outperform . . . in terms of speed and accuracy' is not uniformly supported by Tables 3-5, but the depth-model mismatch is the more fundamental concern because it bears on whether the experimental comparisons are valid for two of the five tested depth sources.","tokens_in":54962,"tokens_out":11475,"duration_ms":122682,"concrete_test":"Re-run the ETH3D, Phototourism, and ScanNet experiments for MiDaS and DA v2 using the 3PTsuv(inverse) solver inside the same PoseLib-with-Sampson and hybrid-RANSAC protocols used in Tables 3-5, and compare mAA and median pose error against the reported 3PTsuv results. Also derive and test inverse-depth analogues for the shared- and varying-focal-length solvers, since no such comparison is currently reported. If the inverse solver improves any configuration by more than the reported margins, the main depth-model choice is biased; if it does not, the mismatch is empirically settled.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing assumption is Eq. (2): true depths are a global affine function of the MDE output in each image, with one unknown scale and shift per image. This does not match the affine-invariant inverse-depth outputs of MiDaS and Depth Anything v2, which are used in the main experiments, unless the solver accounts for the reciprocal relation. The paper derives 3PTsuv(inverse) in SM Sec. 6.2 but does not use it in the main tables. SM Table 6 reports that this inverse solver does not beat 3PTsuv under GC-RANSAC on Phototourism with DA v2, which is real evidence against the concern. However, that test uses a different robustness protocol (no local optimization), a single dataset, and only the calibrated case; it does not cover the PoseLib-with-Sampson or hybrid-RANSAC protocols used in Tables 3-5, nor the focal-length settings. If the inverse solver performs better under those protocols, the main DA v2/MiDaS results, and the claim that shift modeling is unnecessary for them, would need to be recomputed. The algebra of 3PTsuv itself appears internally coherent; the open question is experimental validity for a subset of depth sources.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a family of minimal solvers for relative pose estimation from point correspondences augmented with monocular depth estimates. The key modeling assumption (Eq. 2) is that the true depth is a global affine function of the MDE output within each image, with one unknown scale and shift per image; the paper derives calibrated, shared-focal-length, and different-focal-length solvers that jointly estimate the relative pose with these depth parameters, using distance constraints to eliminate rotation and translation. The main calibrated solver (3PTsuv) reduces to a quartic, while the focal-length solvers (4PTsuvf, 4PTsuvf1,2) use Gröbner-basis or eigenvalue techniques. Experiments on ETH3D, Phototourism, and ScanNet with several MDEs, matchers, and RANSAC protocols compare the proposed solvers against prior depth-aware solvers (Madpose, Rel3PT, 3p3d, 4p4d) and classical point-based solvers, and the paper concludes that depth-aware solvers can outperform point-based methods when depth estimates are reliable, with scale-only solvers often matching or exceeding shift-modeling solvers.","tokens_in":55171,"tokens_out":4363,"duration_ms":42500,"significance":"If the results are sound, this is a useful contribution: it provides faster minimal solvers than the concurrent Madpose work, gives a systematic comparison across depth models and RANSAC variants, and offers evidence that modeling unknown shift is not always necessary. The algebraic derivations are plausible and the experimental scope is broad. The paper's main limitation is that the central affine-depth model is mismatched for inverse-depth MDEs used in the main experiments, and the experiments lack statistical validation of the small performance differences that support the headline claims.","major_comments":[{"comment":"","section":"Section 3, Eq. (2); SM Sec. 6.2, Table 6"},{"comment":"","section":"Tables 3-5; SM Tables 8-15"},{"comment":"","section":"Section 4.3, Figure 3"}],"minor_comments":[{"comment":"","section":"Section 1, Figure 1"},{"comment":"","section":"Table 1"},{"comment":"","section":"Section 3.1"},{"comment":"","section":"Section 3.2"},{"comment":"","section":"SM Sec. 6.2"}],"recommendation":"major_revision","confidential_remarks":"The paper competes closely with concurrent work by Yu et al. (Madpose), and the comparison is generally fair. The main technical novelty is the smaller/faster solvers and the finding that scale-only solvers can match shift-modeling ones in some settings. The load-bearing issue is the inverse-depth mismatch for DA v2/MiDaS; the current SM evidence is too narrow to settle it. The lack of error bars also weakens the 'outperform' claim. If the authors address these points, the paper could be a solid contribution."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a solid, useful minimal-solver paper. The new material is real — scale-invariant zero-shift solvers, an inverse-depth solver, and faster variants of the affine-invariant solvers that Madpose introduced in parallel. The experiments are extensive and mostly back the claim that depth-aware solvers can beat 5-point/6-point/7-point when depth is good, and that zero-shift solvers often match shift-modeling ones at lower cost.\n\nWhere I'd push back: the depth-model mismatch for inverse-depth MDEs is a genuine soft spot, not a fatal one. The paper classifies MiDaS and DA v2 as inverse-depth networks in the intro, then runs the main tables with 3PTsuv, which assumes depth is affine in the network output. The inverse solver is derived and tested only in the SM, under GC-RANSAC without local optimization, on one dataset. That is too narrow to fully settle whether the main DA v2/MiDaS numbers are fair. The authors should either add a reciprocal pre-processing step, use the inverse solver in the main comparisons, or provide a broader ablation. I don't think this sinks the paper — the affine-invariant solver still wins or ties on several depth sources, and the empirical guidance about when depth helps is useful — but it should be fixed in revision.\n\nOther soft spots are minor: code is promised but not available, there are no error bars on the medians, and some solver derivations are condensed in the SM. The algebraic core seems coherent; I didn't find a circular argument, and the comparison to Madpose is handled well.\n\nWho should read it: anyone working on minimal solvers, depth-based pose, or RANSAC pipelines. It deserves serious peer review. I'd send it out, with the request that the inverse-depth discrepancy be clarified experimentally.","headline":"Solid minimal-solver paper with real new variants; the inverse-depth mismatch is a fixable soft spot, not a fatal flaw.","tokens_in":55740,"tokens_out":3573,"would_cite":true,"duration_ms":35271,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper claims that monocular depth predictions, modeled as affine-unknown per image, reduce calibrated relative pose estimation to solving a quartic and yield faster, more accurate solvers.","keywords":["relative pose estimation","monocular depth","minimal solvers","affine-invariant depth","scale and shift estimation","focal length estimation","RANSAC","camera geometry"],"falsifier":"Run the calibrated 3-point affine-invariant solver on real image pairs whose depth maps come from an estimator known to produce per-pixel scale variation or inverse-depth outputs, and compare median rotation and translation error against the 5-point point-only solver on the same matches; if the depth-aware solver is not consistently more accurate when depth is reliable, the central claim that affine-invariant monocular depth improves relative pose fails.","tokens_in":54778,"feed_emoji":"📷","tokens_out":6891,"duration_ms":65282,"temperature":0.7,"pith_summary":"This paper asks whether monocular depth estimates from modern networks actually improve two-view relative pose estimation over classical point-only methods. It answers yes, provided the depth is modeled as defined up to an unknown scale and shift per image, and it derives minimal solvers that jointly estimate those depth parameters with the pose. For calibrated cameras the problem reduces to a single quartic equation; for cameras with unknown shared or different focal lengths it reduces to small polynomial systems solved by Groebner basis elimination. On real datasets, the new solvers are faster and often more accurate than prior depth-aware solvers, and the scale-only variants can match or beat shift-modeling variants when the depth source is reliable.","feed_headline":"Depth maps cut camera-pose math from 5 points to 3","feed_subtitle":"New solvers jointly estimate unknown depth scale and shift, beating point-only methods when depth is reliable.","key_machinery":"The load-bearing mechanism is distance-constraint elimination. Instead of solving directly for rotation and translation, the paper lifts each correspondence to a 3D point via its monocular depth and writes the unknown relative scale $s$ and shifts $u,v$ into the 3D coordinates. Pairwise differences cancel translation, and rotation is cancelled by observing that norms are rotation-invariant, leaving polynomial equations in $s,u,v$ (and focal lengths when they are unknown). For calibrated cameras, the substitution $c=s^2$ and Gauss-Jordan elimination express $\\{cv^2, cv, c\\}$ as quadratics in $u$; the consistency identity $g_2^2=g_1 g_3$ produces the quartic. This is the step that makes the solver minimal, closed-form, and fast.","core_discovery":"The core discovery is an algebraic reduction: substituting the affine depth model $\\eta_i=s_1(\\alpha_i+u)$, $\\lambda_i=s_2(\\beta_i+v)$ into the projection equation, dividing by $s_1$, and subtracting equations eliminates translation, while taking norms of both sides eliminates rotation. The resulting pairwise-distance constraints contain only the relative scale $s$, the two shifts $u,v$ (or a subset), and the focal lengths. After Gauss-Jordan elimination, the calibrated case collapses to the identity $(cv)^2 = c(cv^2)$, which yields a quartic in $u$ that can be solved in closed form; the focal-length cases become small Groebner-basis solvers. The paper claims these solvers outperform the concurrent depth-aware solvers in speed and accuracy, and that in many realistic settings a scale-only solver is enough.","pith_inferences":["Beyond the paper, if the affine-per-image depth model is violated—for example, by estimators that return inverse depth or that apply a different scale to different image regions—the quartic reduction no longer holds exactly; a testable extension would be a solver that estimates per-correspondence scale weights or fits the affine model robustly inside RANSAC.","Beyond the paper, the reported speed gain for the scale-only variant suggests a practical rule for pipeline designers: prefer the cheaper scale-only solver when the depth source is metric or reliably scale-calibrated, and reserve shift modeling for affine-invariant networks.","Beyond the paper, the same elimination-by-distance idea could be adapted to other two-view problems with depth priors, such as absolute pose with known scene depth or multi-camera rigs, though the paper does not test these settings."],"forward_implications":["When monocular depths are accurate, the calibrated 3-point affine-invariant solver is competitive with or better than the 5-point point-only solver while being faster per RANSAC iteration.","Modeling an unknown shift is not always necessary: the scale-only 3-point solvers often match or beat shift-modeling solvers when depth estimates are good, contradicting the concurrent claim that shift modeling is always beneficial.","The proposed focal-length solvers recover both relative pose and unknown shared or different focal lengths from four depth-tagged correspondences, with smaller elimination templates than prior work.","An inverse-depth variant exists but is slower and less accurate inside RANSAC, so the paper does not recommend it for current affine-invariant inverse-depth networks.","Across several datasets and matchers, depth-aware solvers improve over point-only baselines mainly when depth is reliable; with poor depth or very strong matches, point-only solvers can still win."],"supporting_citations":[{"why":"Classical five-point point-only baseline that the depth-aware solvers must beat in accuracy and speed.","marker":"[49]"},{"why":"Concurrent affine-invariant depth solvers that provide the main comparison and the hybrid RANSAC scheme used in experiments.","marker":"[67]"},{"why":"Prior depth-aware fundamental-matrix solvers using relative depths only; this paper extends them by modeling unknown shifts.","marker":"[17]"},{"why":"Closed-form calibrated solver using relative depth; a direct baseline for the calibrated case.","marker":"[1]"},{"why":"P3P solver used for scale-invariant depth with a single image's depth; a strong baseline and a component of the comparison.","marker":"[15]"},{"why":"Automatic Groebner-basis solver generator used to build the focal-length solvers.","marker":"[38]"},{"why":"Hidden-variable polynomial-eigenvalue technique used for the faster focal-length solver variants.","marker":"[35]"},{"why":"Baseline six-point solver for the shared unknown focal length case.","marker":"[26]"},{"why":"Baseline seven-point fundamental-matrix solver for two unknown focal lengths.","marker":"[27]"}],"fun_headline_variants":["Depth cues shrink pose solver to 3 points, closed-form quartic","Monocular depth + scale/shift joint solve: faster, fewer points","RePoseD: use depth to cut relative pose to a quartic","Depth-aware pose: 3 points, joint scale-shift, state-of-the-art speed","From 5 points to 3 with depth: faster pose, handles scale"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that, in each image, the true depth is a global affine function of the estimated depth with a single unknown scale and a single unknown shift; if a depth estimator violates this—for instance by returning inverse depth or varying its scale across the image—the solver's estimates can be biased.","fun_headline_variants_meta":{"raw":{"variants":["Depth cues shrink pose solver to 3 points, closed-form quartic","Monocular depth + scale/shift joint solve: faster, fewer points","RePoseD: use depth to cut relative pose to a quartic","Depth-aware pose: 3 points, joint scale-shift, state-of-the-art speed","From 5 points to 3 with depth: faster pose, handles scale"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000788,"raw_usage":{"total_tokens":3461,"prompt_tokens":919,"completion_tokens":2542,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":535,"completion_tokens_details":{"reasoning_tokens":2440}},"tokens_in":535,"tokens_out":2542,"duration_ms":19374,"temperature":1.0,"reasoning_tokens":2440,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T20:35:48.922959+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the calibrated 3-point affine-invariant solver on real image pairs whose depth maps come from an estimator known to produce per-pixel scale variation or inverse-depth outputs, and compare median rotation and translation error against the 5-point point-only solver on the same matches; if the depth-aware solver is not consistently more accurate when depth is reliable, the central claim that affine-invariant monocular depth improves relative pose fails.","supporting_citations":[{"cited_title":"An efficient hidden vari- able approach to minimal-case camera motion estimation","cited_arxiv_id":null,"evidence_quote":"Baseline six-point solver for the shared unknown focal length case."},{"cited_title":"An efficient solution to the five-point relative pose problem","cited_arxiv_id":null,"evidence_quote":"Classical five-point point-only baseline that the depth-aware solvers must beat in accuracy and speed."},{"cited_title":"Fundamental matrix es- timation using relative depths","cited_arxiv_id":null,"evidence_quote":"Prior depth-aware fundamental-matrix solvers using relative depths only; this paper extends them by modeling unknown shifts."},{"cited_title":"Fast relative pose estimation using relative depth","cited_arxiv_id":null,"evidence_quote":"Closed-form calibrated solver using relative depth; a direct baseline for the calibrated case."},{"cited_title":"Revisiting the p3p problem","cited_arxiv_id":null,"evidence_quote":"P3P solver used for scale-invariant depth with a single image's depth; a strong baseline and a component of the comparison."},{"cited_title":"Effi- cient solvers for minimal problems by syzygy-based reduc- tion","cited_arxiv_id":null,"evidence_quote":"Automatic Groebner-basis solver generator used to build the focal-length solvers."},{"cited_title":"Poly- nomial eigenvalue solutions to minimal problems in com- puter vision","cited_arxiv_id":null,"evidence_quote":"Hidden-variable polynomial-eigenvalue technique used for the faster focal-length solver variants."}],"review_version":1}