Title: Cycle-Sync: Robust Global Camera Pose Estimation through Enhanced Cycle-Consistent Synchronization
Venue: NeurIPS 2025
Accepted: yes
Keywords: Robust Synchronization

Camera Location Estimation

Structure from Motion

Computer Vision
Abstract: We introduce Cycle-Sync, a robust and global framework for estimating camera poses (both rotations and locations). Our core innovation is a location solver that adapts message-passing least squares (MPLS) - originally developed for group synchronization - to the camera localization setting. We modify MPLS to emphasize cycle-consistent information, redefine cycle consistencies using estimated distances from previous iterations, and incorporate a Welsch-type robust loss. We establish the strongest known deterministic exact-recovery guarantee for camera location estimation, demonstrating that cycle consistency alone enables the lowest sample complexity to date. To further boost robustness, we introduce a plug-and-play outlier rejection module inspired by robust subspace recovery, and we fully integrate cycle consistency into MPLS for rotation averaging. Our global approach avoids the need for bundle adjustment. Experiments on synthetic and real datasets show that Cycle-Sync consistently outperforms leading pose estimators, including full structure-from-motion pipelines with bundle adjustment.

Cycle-Sync: Robust Global Camera Pose Estimation through Enhanced Cycle-Consistent Synchronization

Abstract

We introduce Cycle-Sync, a robust and global framework for estimating camera poses (both rotations and locations). Our core innovation is a location solver that adapts message-passing least squares (MPLS)-originally developed for group synchronization-to camera location estimation. We modify MPLS to emphasize cycle-consistent information, redefine cycle consistencies using estimated distances from previous iterations, and incorporate a Welsch-type robust loss. We establish the strongest known deterministic exact-recovery guarantee for camera location estimation, showing that cycle consistency alone-without access to inter-camera distances-suffices to achieve the lowest sample complexity currently known. To further enhance robustness, we introduce a plug-and-play outlier rejection module inspired by robust subspace recovery, and we fully integrate cycle consistency into MPLSfor rotation synchronization. Our global approach avoids the need for bundle adjustment. Experiments on synthetic and real datasets show that Cycle-Sync consistently outperforms leading pose estimators, including full structure-frommotion pipelines with bundle adjustment.

1 Introduction

Structure-from-Motion (SfM) is a central task in 3D computer vision [21], aimed at reconstructing the 3D structure of a scene from 2D images captured by cameras with unknown poses. It plays a critical role in applications such as virtual and augmented reality, robotics, and autonomous driving. Acore challenge in SfM is accurate camera pose estimation, which is also foundational to modern 3D techniques such as neural radiance fields [17] and Gaussian splatting [8], where camera parameters serve as priors or inputs for rendering and synthesis.

A typical SfM pipeline begins by estimating essential matrices between pairs of cameras, from which local pose information is extracted. It then infers absolute camera orientations from relative ones-a step commonly referred to as rotation synchronization (or averaging). Next, it estimates camera locations from relative direction vectors. Once the camera poses are determined, a standard final step is to recover the 3D structure of the scene. The aim of this work is to revisit both pose estimation tasks and propose global solutions that eliminate reliance on the highly incremental and computationally intensive bundle adjustment step [31], which is ubiquitous in current SfM pipelines. We refer to our robust location solver as Cycle-Sync 1 , and use the same name for the full pipeline that incorporates this solver along with robust direction and rotation estimation.

Mathematically, these problems are formulated on a graph G ([ n ] , E ) , where nodes correspond to cameras, and an edge ( i, j ) ∈ E indicates the availability of relative pose information between cameras i and j . The goal of rotation synchronization is to estimate the ground-truth camera rotations

{ R ∗ i } i ∈ [ n ] from noisy relative rotations { R ij } ij ∈ E , whose clean counterparts satisfy R ∗ ij = R ∗ i R ∗⊤ j . This recovery is up to an arbitrary global rotation. The subsequent localization step seeks to estimate the absolute positions { t ∗ i } n i =1 from noisy unit direction vectors { γ ij } ij ∈ E , which approximate the ground-truth directions:

These locations are only recoverable up to a global translation and scale.

Camera location estimation is significantly more challenging than rotation estimation, and is therefore the primary focus of our work. First, the direction vectors derived from essential matrices lack scale information. If scale were available, the problem would reduce to a special case of group synchronization over the translation group, making it considerably easier. Second, in modern SfM pipelines, the estimated direction vectors are often highly corrupted due to failures in feature matching or RANSAC, as well as error propagation from earlier stages-particularly from rotation estimates. Lastly, unlike rotations, the space of camera locations lacks a group structure and is non-compact, which makes location estimation more sensitive and numerically unstable in the presence of corruption and noise.

1.1 Relevant previous works

Since our main contributions focus on camera location estimation, we review the most relevant work on this task, while also briefly noting related advances in rotation synchronization. Early location solvers based on ℓ 2 minimization [1, 2, 6, 20, 32, 33] or ℓ ∞ minimization [18] are highly sensitive to outliers and unsuitable for real-world Structure-from-Motion (SfM) data. More robust approaches, including Least Unsquared Deviations (LUD) [19] and ShapeFit [7, 5], use convex ℓ 1 objectives, solved via Iteratively Reweighted Least Squares (IRLS) and the Alternating Direction Method of Multipliers (ADMM), respectively. These methods minimize the distance between t i -t j and the line defined by γ ij , which can overweight long edges and become unstable when edge lengths vary significantly or are corrupted. BATA [36] instead minimizes the sine of the angle between t i -t j and γ ij , offering robustness to edge-length variation. However, it treats all edges equally and thus underutilizes information from clean long edges. Fused Translation Averaging (fused-TA) [15] alternates between LUD- and BATA-type objectives and merges their outputs using uncertainty estimates, but may underperform both in practice, with no clear winner among the three. Both 1-Dimensional SfM (1DSfM) [34] and All-About-that-Base (AAB) [25] exploit 3-cycle consistency to detect outliers. 1DSfM projects directions to 1D and applies a heuristic combinatorial algorithm, without theoretical guarantees. AAB enforces coplanarity of γ ∗ ij , γ ∗ jk , and γ ∗ ki and combines this constraint with message passing. AAB outperforms 1DSfM empirically but is unstable in nearcolinear configurations and only achieves approximate guarantees under a specific probabilistic model.

The only deterministic recovery guarantees for location estimation under adversarial corruption are those for ShapeFit [7] and LUD [12], under Gaussian location priors and Erd˝ os-Rényi measurement graphs with number of nodes approaching infinity and certain bounds on the connection probability and the number of corrupted incident edges per node.

In the related problem of group synchronization, stronger guarantees exist. Cycle-Edge Message Passing (CEMP) [11] handles adversarial and probabilistic corruption in rotation synchronization. It was used to create Message-Passing Least Squares (MPLS) [26], which empirically improves performance on camera orientation estimation. However, MPLS progressively downweights cycle information and cannot be extended to location estimation due to the lack of inter-camera distance measurements. Similarly inspired by CEMP, DESC [28] estimates edge corruption levels via quadratic programming, but it is computationally slow and its theoretical guarantees do not cover adversarial corruption. DDS [16] is able to address adversarial corruption by exploiting Tukey depth in the tangent space of SO ( d ) . Other extensions attempt to generalize CEMP or MPLS to permutation synchronization [27] and partial permutation synchronization [14], yet these methods and their theory also do not naturally extend to the location estimation problem.

Finally, several global SfM pipelines incorporate location solvers. These include LUD [19] (used as a full pipeline), Theia [30], and GLOMAP [22]. All of them benefit from non-global bundle adjustment [31], which is explicitly integrated into Theia and GLOMAP.

1.2 Contributions

Nearly all existing methods for location estimation fall within the IRLS framework and do not fully exploit cycle consistency information. As a result, they struggle to handle cycle-consistent corruption (i.e., situations when the corrupted edges exhibit cycle-consistent behavior), which frequently arises in real-world SfM datasets.

The goal of this work is to propose a new location estimation method that is robust to severe corruption and accommodates missing and highly variable edge lengths. We also revisit the global pipeline for pose estimation and improve several of its core components.

Our contributions to camera location estimation are summarized as follows:

We introduce a novel formulation with a Welsch-type objective function that directly addresses key limitations of LUD-type and BATA-type objectives, particularly in the presence of large variations in edge distances.

We propose a new MPLS framework to optimize the Welsch objective for location estimation. This framework is designed to fully exploit cycle-consistent information. To handle missing distance data, we redefine cycle consistencies using distances estimated in previous iterations.

We establish the strongest known deterministic exact-recovery guarantee for location estimation under adversarial corruption. Under standard probabilistic models (e.g., i.i.d. Gaussian locations with Erd˝ os-Rényi connectivity), our theoretical sample complexity improves over all prior work (see Table 1).

Our additional contributions to global pose estimation include:

We extend the full-cycle MPLS framework to the rotation synchronization problem, yielding significantly reduced orientation error compared to the strongest existing baselines.

We introduce a plug-and-play outlier rejection module inspired by robust subspace recovery. This module significantly improves the performance of existing location estimators.

Our global pose estimation pipeline eliminates the need for bundle adjustment. Experiments on both synthetic and real datasets show that Cycle-Sync consistently outperforms leading pose estimators, including full structure-from-motion pipelines that rely on bundle adjustment.

2 The Cycle-Sync Framework

We first present the Cycle-Sync location solver: §2.1 introduces its optimization formulation, §2.2 details its algorithmic implementation with cycle-consistent weighting, and §2.3 establishes theoretical recovery guarantees. Finally, §2.4 integrates the solver into a full camera pose estimation pipeline.

2.1 New optimization formulation for location estimation

We note that all major solutions to the camera location estimation problem are either special cases or variants of the following formulation:

where ρ ( x ) is a certain function to be specified later. Here, α ij is interpreted as an auxiliary variable representing the distance between t i and t j . Its constraint, i.e., α ij ≥ 1 , aims to avoid the trivial solution t i = 0 for all i 's. This general formulation captures several existing methods as special cases. For example, constrained least squares [32, 33] uses ρ ( x ) = x 2 and LUD [19] uses ρ ( x ) = | x | . Furthermore, ShapeFit [7] replaces ∥ t i -t j -α ij γ ij ∥ by the projection distance from t i -t j to the line of γ ij . It also switches the α ij ≥ 1 to a weaker linear constraint ∑ ij ∈ E ⟨ t i -t j , γ ij ⟩ = 1 . Let θ ij denote the angle between the true relative location t i -t j and direction γ ij . By choosing the optimal α ij , the LUD objective function (with ρ ( x ) = | x | ) reduces to the ShapeFit one:

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which scales linearly with the length of edges. Therefore, such an objective function may suffer from long and corrupted edges. BATA [36] changes the distance to ρ (sin θ ij ) in order to make it independent of ∥ t i -t j ∥ . However, it may not sufficiently benefit from long and clean edges that have a stronger effect on the global distribution of the locations.

As opposed to previous robust formulation which used ρ ( x ) = | x | , we propose minimizing (1) with

where a is a fixed parameter. Throughout this paper we fix a = 4 . We remark that this ρ ( x ) is similar to the Welsch objective function ρ ( x ) = 1 -e -ax 2 . The latter objective functions put less emphasis on longer edges, but still grow (with a very small rate) as ∥ t i -t j ∥ grows. Therefore, they are much less sensitive to large variations of distances. This avoids the limitations of both LUD and BATA. A comparison of these ρ ( x ) choices appears in Figure 1.

On the other hand, unlike the traditional Welsch function, our new energy function inherits from LUD the non-smoothness at x = 0 . This non-differentiability of ρ ( x ) at 0 makes the exact recovery of ground truth locations possible under corrupted directions [12].

2.2 Solution of the proposed optimization with emphasis on cycles

The formulation proposed in (1) and (2) can be addressed by an IRLS-type approach. Indeed, let r ij := ∥ t i -t j -α ij γ ij ∥ , then our objective function is F ( { t i } n i =1 , { α ij } ij ∈ E ) := ∑ ij ∈ E ρ ( r ij ) . As demonstrated in [3], the problem can be tackled with an IRLS scheme:

Typically, w ij,t +1 is updated by f ( r ij,t ) , where residual r ij,t = ∥ t i,t -t j,t -α ij,t γ ij ∥ , and the reweighting function

Here δ is a small number to avoid the zero denominator. However, our choice of ρ ( x ) in (2) is nonconvex, making IRLS highly sensitive to weight initialization. Even with good initialization and convex ρ , IRLS may fail to recover the ground truth under high corruption. Indeed, in the highly corrupted scenario, the residuals r ij,t may fail to reflect true corruption levels (defined up to an unknown fixed scale):

making IRLS easily get stuck in local minima. This limitation motivates a much more robust iterative estimator for s ∗ ij , using the cycle-consistency information. Following the MPLS strategy [26] we approximate s ∗ ij using a weighted average of cycle inconsistencies:

where β > 0 is a parameter, Z ij,t is the normalization factor ensuring convex combinations, and

measuring the cycle inconsistency. We note that with clean edges and true locations, d ijk,t = 0 . Furthermore, as β →∞ and t i,t → t ∗ i , s ij,t → s ∗ ij .

We note that in (6), the corruption level is no longer approximated by a single residual. Instead, it aggregates information from 3-cycles (indexed by ijk ) and incorporates residuals from neighboring edges, making s ij,t significantly more stable and robust to outliers.

In early iterations, when distance estimates ∥ t i,t -t j,t ∥ are unreliable, we blend the residuals r ij,t with the corruption scores s ij,t to define the edge weights. As estimates improve, the weighting gradually shifts toward s ij,t . This message-passing mechanism enables cycle consistency to refine edge weights, while the weighted least squares updates, in turn, refine cycle inconsistencies. This bi-directional communication between cycles and edges facilitates global information propagation and significantly reduces the risk of getting trapped in local minima. We refer to our method as Cycle-Sync, and its information propagation is illustrated in Figure 2.

Weight initialization. In the first iteration, when locations are unknown, weights can be initialized as w ij, 0 = exp( -20 ˜ s ij ) , where ˜ s ij estimates the angular corruption level. Its ground-truth counterpart is defined as

Since this initialization precedes weighted least squares, distances, residuals, and the cycle inconsistency d ijk,t are unavailable. Instead, ˜ s ∗ ij can be estimated using a variant of the update rule in (6), replacing residuals with previous ˜ s ij,t values and substituting d ijk,t with an AAB-style cycle inconsistency:

where γ is constrained to satisfy

We refer the reader to [25] for a closed-form expression of ˜ d ij,k . When computing ˜ s ij , we use only well-shaped triangles-those where the angle between γ ik and γ jk lies in [arcsin(0 . 6) , π -arcsin(0 . 6)] -since extreme angles can make ˜ d ij,k unstable, as discussed in our theory. We refer to this modified version of AAB as truncated AAB (T-AAB) .

Although this initialization cannot exactly estimate ˜ s ∗ ij for corrupted edges without location information, ˜ s ij,t is expected to approach zero on clean edges under the assumption of no additive noise. This allows for reliable separation of clean and corrupted edges, enabling exact location recovery.

Finally, we remark that Cycle-Sync is not sensitive to initialization. Nonetheless, the AAB-based initialization is lightweight, independent of least squares, and essentially a free improvement. A complete description of our location solver is given below.

In all of our experiments, we choose t max = 20 , β = 20 and λ t = t/ ( t + 10) . We remark that these parameters are not fine-tuned, but the performance of our method is already superior with the suboptimal parameters.

Our MPLS procedure differs from the original version for rotation estimation in several key aspects. First, unlike the rotation setting, we must address missing distance information, and our cycleinconsistencies depend on iteratively updated distance estimates. Second, we adopt a Welsch-type objective, which better handles the large variation in residual scales typical in location estimation, whereas the original MPLS uses ℓ 1 / 2 minimization. Lastly, our weighting parameter satisfies λ t → 1 , gradually emphasizing cycle information over residuals, in contrast to the original one where λ t → 0 . Further discussion and experiments on the annealing parameter λ t are included in Section I.3 in the supplementary material.

Algorithm 1 Cycle-Sync

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2.3 Why we need cycle consistency: theory for exact location recovery

To motivate our use of cycle information, we present a theoretical result showing that AAB-style cycle inconsistency, when combined with iterative reweighting, can exactly separate clean and corrupted directions-even without access to inter-camera distances. Intuitively, this holds because corrupted directions tend to violate 3-cycle consistency, while clean directions remain geometrically consistent. Formally, under mild conditions, we show that the estimator ˜ s ij,t converges to zero on clean edges and remains bounded away from zero on corrupted ones, enabling exact location recovery . Under a probabilistic corruption model, our result also yields the lowest known sample complexity for exact recovery.

We assume that the edge set E is partitioned into good (clean) edges E g and bad (corrupted) edges E b . For ij ∈ E g , γ ij = γ ∗ ij , and for ij ∈ E b , γ ij is an arbitrary unit vector distinct from γ ∗ ij . Define N ij = { k ∈ [ n ] : ik, jk ∈ E } , G ij = { k ∈ N ij : ik, jk ∈ E g } , and B ij = N ij \ G ij . Let λ = max ij ∈ E | B ij | / | N ij | , µ = min ij ∈ E b ∑ k ∈ G ij ˜ d ij,k / ( | G ij | s ∗ ij ) and θ ij,k denote the angle between γ ik and γ jk .

Theorem 2.1. Assume there exists α > 0 such that for all ij ∈ E g and k ∈ N ij , α < θ ij,k < π -α , and λ < 1 + eC α /µ -√ eC α (2 µ + eC α ) /µ , where C α = 2(cos α + √ 5 -4 cos 2 α ) / sin 2 α . Then, for ˜ s ij,t computed by the iteratively reweighted AAB algorithm [25] using β 0 ≤ 1 2 λ and β t +1 = rβ t with 1 < r < µ (1 -λ ) 2 / (2 eC α λ ) , it holds for all t > 0 that

In Theorem 2.1, the separation between clean and corrupted edges arises because the upper bound for clean edges, 1 / (2 β 0 r t ) , vanishes as iteration t →∞ (note that r > 1 ), while the lower bound for bad edges remains strictly positive, proportional to their corruption levels. This result makes no assumptions on the distribution of corrupted directions, allowing fully adversarial and cycle-consistent

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corruption. The angle condition on θ ij,k addresses instability in ˜ d ij,k caused by ill-shaped triangles. To ensure this condition, we exclude triangles with θ ij,k < α or θ ij,k > π -α when computing ˜ d ij,k . In the probabilistic setting with i.i.d. Gaussian locations, Erd˝ os-Rényi edge probability p , and independent edge corruption probability q , our result achieves the strongest known recovery guarantee (see Table 1), where nϵ b is the maximum degree of the corrupted subgraph. The proof appears in the supplementary material.

2.4 Our full pipeline for camera pose estimation

We describe the full pipeline for camera pose estimation. Having covered location estimation, we focus here on the preceding steps: rotation and direction estimation. Given image keypoint matches (e.g., from SIFT or deep features), we estimate essential matrices for camera pairs using RANSAC, assuming calibrated cameras. The relative rotations R ij are inferred from the essential matrices.

Rotation averaging via MPLS-cycle. Absolute rotations are recovered from relative ones using the MPLS algorithm. Unlike the original version, we fix λ t = 1 to emphasize cycle consistency and disable IRLS reweighting. This variant, which we call MPLS-cycle , yields significantly lower orientation error on real SfM datasets. We report these results in the supplementary material.

Direction estimation via robust subspace recovery. Following [19], each ground-truth direction γ ∗ ij is orthogonal to the vectors v k ij = R i η k i × R j η k j , where η k i and η k j are the normalized homogeneous coordinates of corresponding keypoints. Thus, γ ∗ ij lies in the orthogonal complement of the subspace spanned by { v k ij } k . However, many v k ij vectors may be corrupted due to outlier matches. To robustly recover this subspace and estimate direction vectors (a problem reviewed in [9]), we use STE [35, 13], which significantly outperforms the REAPER method [10] used in the LUD pipeline [19].

Outlier detection for directions. As an optional filtering step, we use STE to reject direction estimates γ ij with low inlier numbers. This plug-and-play module improves the accuracy of downstream location solvers. We observe notable performance gains with this preprocessing. In our real data experiment we use 20 as our minimum number of inliers.

3 Synthetic data experiments

We generate synthetic data under the uniform corruption model (UCM), where n is the number of cameras, q is the corruption probability per edge, σ is the noise level, and p is the probability of edge connection in the Erd˝ os-Rényi viewing graph. Ground-truth camera locations t ∗ i are sampled independently from N ( 0 , I 3 × 3 ) . For each edge ij in the generated graph, the observed direction is given by:

where ϵ ij ∼ N ( 0 , I 3 × 3 ) independently.

We also consider an adversarial corruption model where corrupted directions remain cycle-consistent:

where { t c i } is a set of alternative locations used to generate coherent corrupted directions. To avoid ambiguity with the true structure, we require q < 0 . 5 in this model.

We also test the robustness to corruption for different methods. We fix n = 100 , p = 0 . 5 . We obtain the estimated absolute locations ˆ t i by our Cycle-Sync solver and compare the estimation error with

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existing works ShapeFit [5], BATA [36], LUD [19], FusedTA [15]. Since absolute locations can only be estimated up to a global translation and scaling, we remove these ambiguities by computing the minimizer ( c ∗ , t ∗ ) of min c ∈ R , t ∈ R 3 ∑ i ∈ [ n ] ∥ t ∗ i -( c ˆ t i + t ) ∥ 2 .

We compute the absolute translation error for camera i as ∥ t ∗ i -( c ∗ ˆ t i + t ∗ ) ∥ 2 and report the median error over all cameras in Figure 3. We generate 10 instances of the synthetic data and report the standard deviation of these statistics with error bars (no error bars with log scales).

We say a method achieves 'exact recovery" if it estimates the absolute camera locations with less than 10 -4 median error. On UCM with σ = 0 , exact recovery is achieved only when q ≤ 0 . 3 except our method and ShapeFit, while our method exactly recovers when q ≤ 0 . 8 . Therefore, our method improves the phase transition threshold for exact recovery by a large margin. For the adversarial setting, Cycle-Sync remains robust up to corruption rates near the theoretical limit ( q < 0 . 5 ), while all baselines quickly deteriorate. These results validate the effectiveness of our cycle-based reweighting and its ability to leverage global consistency for accurate location recovery in both stochastic and adversarial scenarios.

4 Real data experiments

We conduct real-world experiments on 13 ETH3D stereo datasets [23, 24], using a personal laptop equipped with an 11th Gen Intel(R) Core(TM) i9-11900H processor (2.50 GHz, 8 cores, 16 threads) and 16 GB physical memory. This dataset contains 13 sets of undistorted images taken from different indoor and outdoor scenes. It is challenging because of its occlusions, viewpoint changes and lighting differences which results in highly corrupted pairwise directions. From raw images, we estimate the initial keypoint matches using SIFT feature matching and perform geometric verification with RANSAC. With these initial keypoint matches as the input, we go through our full camera pose estimation pipeline and compute the output absolute location estimates { ˆ t i } i ∈ [ n ] and the absolute rotation estimates { ˆ R i } i ∈ [ n ] . Weremark that if the viewing graph contains multiple weakly connected components that cannot be merged into a single scene, we retain only the largest component for rotation and location estimation, and exclude the others from computation and evaluation.

We use the millimeter-accurate, laser-scanned camera poses from the dataset as the ground truth. To eliminate the translation and scale ambiguity in camera locations, we preprocess the camera locations by translating them so that the geometric mean of all camera positions is centered at the origin, followed by scaling to ensure that the median distance of the cameras from the origin is 1. We name the preprocessed ground truth camera locations as { t ∗ i } i ∈ [ n ] and camera orientations as { R ∗ i } i ∈ [ n ] . To evaluate the output, we first align the rotation estimates with R align, the minimizer of the L2 rotation alignment error min R ∈ R 3 × 3 ∑ i ∈ [ n ] ∥ R ∗ i -RR i ∥ 2 F . Then, to remove the global scale and translation ambiguities for camera locations, we compute the minimizer of the L1 alignment

error as ( c ∗ , t ∗ ) : min c ∈ R ,t ∈ R 3 ∑ i ∈ [ n ] ∥ t ∗ i -( c R align ˆ t i + t ) ∥ 2 . For each camera i , we compute the translation error as ∥ t ∗ i -( c ∗ R align ˆ t i + t ∗ ) ∥ 2 . For each dataset, we report the median error over cameras. We test our method Cycle-Sync, BATA [36], FusedTA [15], ShapeFit [5] and LUD [19] for comparison of different location estimation methods, while the camera orientation method is fixed to MPLS-Cycle. Also, to compare with existing global SfM pipelines, we report the translation error of camera poses from LUD, Theia [30] and GLOMAP [22]. In addition, we conduct ablation studies. Starting from LUD+IRLS, the original LUD pipeline, we gradually add each building block of our full pipeline to demonstrate the effectiveness of each block. We summarize the results in Figure 4. We highlight some key comparisons derived from our experiments.

■ Cycle-Sync

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Comparison among SfM pipelines (top panel). We observe that our full pipeline significantly outperforms existing SfM pipelines when averaging across datasets. Here, LUD refers to the original unmodified version proposed in the literature. Our method reduces the median location error to below 0.05, whereas other pipelines yield considerably less accurate results, with median errors exceeding 0.2 on average. Despite bypassing bundle adjustment, our method yields more accurate camera locations on average, whereas Theia and GLOMAP, even with bundle adjustment, suffer from many outliers and large alignment errors on several datasets. Although Theia performs better on the majority of datasets, our method consistently avoids failure cases and achieves the lowest average error across all datasets. Both metrics-the number of datasets with superior performance and the average performance across all datasets-are informative: the former reflects robustness across the majority of scenarios, while the latter highlights stability and resilience against more challenging or adversarial datasets.

Comparison of location estimation algorithms (middle panel). In this comparison, we fix all preceding steps across all location solvers for fairness. Specifically, for all baselines, we employ MPLS-Cycle for rotation estimation and STE for direction estimation and filtering. Under this standardized setup, our method consistently outperforms others on 10 out of the 14 datasets. In terms of median location error averaged over all datasets, our approach (with STE) achieves a reduction of 60 . 9% compared to LUD (with STE), 66 . 1% compared to BATA, 89 . 8% compared to ShapeFit, and 90 . 0% compared to FusedTA.

Ablation studies (bottom panel). We observe that each component of our method consistently reduces the median location error. Starting from the LUD pipeline (LUD for location + REAPER for pairwise direction + IRLS [3] for rotation averaging), upgrading IRLS to MPLS reduces median location error by 17 . 8% ; upgrading MPLS to MPLS-cycle reduces location error further by 9 . 0% . This is because better camera orientations improve the initial pairwise direction estimates. We remark that STE is an optional module for Cycle-Sync. Cycle-Sync outperforms all baselines (under the same preprocessing) both with and without STE. Specifically, upgrading LUD (without STE) to Cycle-Sync (without STE) reduces the location error by 66 . 0% , and the above panel indicates error reduction by 60 . 9% when using STE for both LUD and Cycle-Sync. Finally, replacing REAPER with STE within Cycle-Sync yields a 70 . 5% improvement. This significant gain stems from STE's effectiveness not just as an algorithm, but also from our specific use of STE in filtering outlying directions.

For other details, we refer the readers to the table in the supplementary material.

5 Conclusion and Limitations

We proposed a global framework for camera pose estimation that fully exploits cycle-consistency information. Our method simultaneously addresses the challenges posed by large variations in pairwise distances and highly corrupted directions. We establish the strongest known exact recovery guarantee for location estimation under adversarial corruption, which also yields improved sample complexity under standard probabilistic models. Empirically, our method is the only algorithm that consistently outperforms all state-of-the-art baselines on real datasets.

Several limitations remain. Our theoretical guarantees apply only to the initialization phase and do not extend to the convergence of the full nonconvex optimization procedure. Establishing guarantees for the entire location synchronization algorithm-and eventually the full pose estimation pipeline-is an important direction for future work. In addition, the method relies on the presence of well-shaped 3-cycles, and performance may degrade on sparse or structured graphs lacking such cycles. All hyperparameters (e.g., reweighting schedules and robust loss parameters) are manually selected, and while not overly sensitive, it would be valuable to develop more adaptive or learnable reweighting strategies.

Broader Impact Our work advances the robustness and theoretical understanding of structurefrom-motion (SfM) pipelines, with potential applications in robotics, autonomous navigation, digital reconstruction of cultural heritage, and low-cost 3D mapping. By improving pose estimation under high corruption and without bundle adjustment, our method may make SfM more accessible and reliable in challenging or resource-constrained environments.

Acknowledgement G. Lerman and S. Li were supported by NSF DMS-2152766. Y. Shi was supported by NSF DMS-2514152.

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Solving for a and b gives

Therefore γ p = x -yz 1 -z 2 γ ki + y -xz 1 -z 2 γ jk .

Case 1: γ p ̸∈ Ω( γ jk , γ ki ) . In this case, since ˜ d ij,k ≤ 1 , we only need to prove ˜ s ∗ ik + ˜ s ∗ jk > 1 /C α .

Note that by the definition of Ω( γ jk , γ ki ) we have either a < 0 or b < 0 , which implies x -yz > 0 or y -xz > 0 . On the other hand, since the ground truth directions γ ∗ ij , γ ∗ jk , γ ∗ ki are cycle consistent, we know that the projection of γ ∗ ij onto Span ( γ ∗ ki , γ ∗ jk ) is in the set Ω( γ ∗ ki , γ ∗ jk ) . Therefore we also have x ∗ -y ∗ z ∗ < 0 and y ∗ -x ∗ z ∗ < 0 , where x ∗ = γ T ij γ ∗ ki , y ∗ = γ T ij γ ∗ jk and z ∗ = γ ∗ T jk γ ∗ ki . Without loss of generality, we assume the case x -yz > 0 . If max(˜ s ∗ ik , ˜ s ∗ jk ) ≥ 1 C α , then the lemma is trivial. If max(˜ s ∗ ik , ˜ s ∗ jk ) < 1 C α , we first verify two claims.

Claim 1: x ∗ -y ∗ z ∗ < -sin α ∗ , where α ∗ is the angle such that cos α ∗ = cos α + 2 C α .

By the definition of ˜ s ∗ ik and ˜ s ∗ jk , we have the following inequality:

Supplementary Material

We provide additional theoretical and experimental details that complement the main paper. In Section A, we present full proofs of Theorem 2.1. Section B explains the sample complexity analysis that underlies the bounds for T-AAB reported in Table 1. Sections C-G provide additional experimental results: 3D reconstructions (Section C and D), supplementary figures to the ETH3D dataset (Section E), runtime comparisons (Section F), and extended results on rotation synchronization (Section G), and generalization to the IMC-PT dataset (Section H).

All equation, figure, table, and theorem numbers continue from the main paper.

A Proofs of theory

We first establish Theorem A.1 and then conclude the main theorem.

Theorem A.1. Assume there exists an absolute α > 0 such that for any ij ∈ E g and k ∈ N ij , α < θ ij,k < π -α . Then for ij ∈ E g , we have ˜ d ij,k ≤ C α (˜ s ∗ ik + ˜ s ∗ jk ) , where C α = 2(cos α + √ 5 -4 cos 2 α ) sin 2 α .

We note that the triangle is ill-shaped whenever θ ij,k ≈ 0 or θ ij,k ≈ π . In practice, we want d ij,k to be small for a clean edge ij ∈ E g whenever the other two edges are relatively clean with small ˜ s ∗ ik and ˜ s ∗ jk . However, in these two ill-shaped cases, C α in the theorem goes to infinity, and there is no effective upper bound to control ˜ d ij,k .

Proof. Let γ p be the projected vector of γ ij onto Span ( γ ik , γ kj ) . Denote x = γ T ij γ ki , y = γ T ij γ jk and z = γ T jk γ ki . Since γ p is in Span ( γ ki , γ kj ) , there exists constants a, b such that γ p = aγ ki + bγ jk . By the definition of γ p , we have

By linearity of vector inner products, we have

Also, α < θ ij,k < π -α is equivalent to | γ T jk γ ki | = | cos θ ij,k | < cos α . Combining this with equation (14) gives

On the other hand, we know that a ∗ = x ∗ -y ∗ z ∗ 1 -z ∗ 2 , and a ∗ = -| a ∗ | ≤ -sin α ∗ . This implies that x ∗ -y ∗ z ∗ < -sin α ∗ 1 = -sin α ∗ .

Claim 2: Let δ ij = γ ij -γ ∗ ij , δ jk = γ jk -γ ∗ jk and δ ki = γ ki -γ ∗ ki . Suppose max( | δ ij | , | δ jk | , | δ ki | ) = δ . Then | ( x ∗ -y ∗ z ∗ ) -( x -yz ) | ≤ 6 δ ; if ij ∈ E g (i.e. δ ij = 0 ), then | ( x ∗ -y ∗ z ∗ ) -( x -yz ) | ≤ 4 δ

In fact, by the definition of x, x , y, y , z, z

. ∗ ∗ ∗ , we have the following estimate:

By the fact that all γ 's are unit vectors, the right hand side of the equation above is at most 6 δ in general; if ij ∈ E g (i.e. δ ij = 0 ) then it is at most 4 δ .

Combining claim 1 and claim 2, we know that 0 < x -yz ≤ ( x ∗ -y ∗ z ∗ )+ | ( x -yz ) -( x ∗ -y ∗ z ∗ ) | ≤ 4 δ -sin α ∗ . This yields δ > sin α ∗ 4 . Note that by ij ∈ E g , we know that δ ij = 0 . Therefore δ = max( | δ ij | , | δ jk | , | δ ki | ) = max( | δ jk | , | δ ki | ) ≤ | δ jk | + | δ ki | = 2sin ˜ s ∗ jk 2 +2sin s ∗ ki 2 ≤ ˜ s ∗ jk + s ∗ ki . By C α = 2(cos α + √ 5 -4 cos 2 α ) sin 2 α , we know that sin α ∗ 4 = 1 C α , therefore the theorem is proved.

Case 2: γ p ∈ Ω( γ ik , γ kj ) . In this case, let δ ik = γ ik -γ ∗ ik and δ jk = γ jk -γ ∗ jk . Then ˜ d ij,k = | γ ik × γ kj · γ ij | sin θ ikj . By the fact that γ ∗ ij , γ ∗ jk , γ ∗ ki are coplanar and γ ij = γ ∗ ij , we know that γ ∗ ik × γ ∗ kj · γ ij = 0 . We have the following inequalities:

Note that | δ ik | = 2sin ˜ s ∗ ik 2 ≤ ˜ s ∗ ik , and similarly | δ jk | = 2sin ˜ s ∗ jk 2 ≤ ˜ s ∗ jk . Therefore ˜ d ij,k ≤ 1 sin α (˜ s ∗ ik + ˜ s ∗ jk ) ≤ C α (˜ s ∗ ik + ˜ s ∗ jk ) , where the latter inequality comes from the fact that C α ≥ 1 sin α .

Proof of the main theorem.

Recall the statement of Theorem 2.1:

Main Theorem (Theorem 2.1). Assume there exists α > 0 such that for all ij ∈ E g and k ∈ N ij , α < θ ij,k < π -α , and λ < 1 + eC α /µ -√ eC α (2 µ + eC α ) /µ , where C α = 2(cos α + √ 5 -4 cos 2 α ) / sin 2 α . Then, for ˜ s ij,t computed by the iteratively reweighted AAB algorithm [25] using β 0 ≤ 1 2 λ and β t +1 = rβ t with 1 < r < µ (1 -λ ) 2 / (2 eC α λ ) , it holds for all t > 0 that

Proof . We prove the main theorem by induction. For t = 0 , the definition of λ imply that for all ij ∈ E ,

Furthermore, by the fact that 0 ≤ ˜ d ij,k ≤ 1 we have for all ij ∈ E g ,

Therefore the theorem is proved when t = 0 .

Next, we assume the theorem holds true for 0 , 1 , · · · , t , and show that it also holds true for t +1 . By the definition of ˜ s ( t +1) ij and the induction assumption 1 2 β t ≥ max ij ∈ E g ˜ s ij,t , we have the following inequalities for any ij ∈ E b :

Next we bound ˜ s ( t +1) ij for ij ∈ E g . By the definition of ˜ s ( t +1) ij , the fact that ˜ d ij,k = 0 for k ∈ G ij , and Theorem A.1 we know that

By the induction assumption that ˜ s ( t ) ij ≥ µ (1 -λ ) e ˜ s ∗ ij for all ij ∈ E , we know that

Note that xe -cx < 1 ce for any c > 0 and x > 0 . Let c = β t µ (1 -λ ) e and x = ˜ s ∗ ik + ˜ s ∗ jk , we have

Also, by the induction assumption that 1 2 β t ≥ max ij ∈ E g ˜ s ( t ) ij and the nonnegativity of ˜ s ( t ) ij 's, we have

Combining (20), (22), (23), (24) and the definition of λ , we have

By the assumption that λ < 1 + eC α µ -√ eC α µ (2 + eC α µ ) , we know that 2 eλ µ (1 -λ ) 2 < 1 . Therefore by taking 1 < r < µ (1 -λ ) 2 2 eλ , we guarantee that for any ij ∈ E g , ˜ s ( t +1) ij ≤ 1 2 β t +1 = 1 2 β 0 r t +1 . This and (19) concludes our theorem.

Comment on the order of µ : Weremark that in the theorem µ = min ij ∈ E b ∑ k ∈ G ij ˜ d ij,k / ( | G ij | ˜ s ∗ ij ) , which implies that for all ij ∈ E ,

We would like to investigate the dependence of µ on n . That is, we estimate the magnitude of µ such that (26) holds for all edges. First of all, it is safe to claim that (26) holds for all ij whose ˜ s ∗ ij > 0 . 5 when µ is a positive constant (i.e., µ = Θ(1) ). That is, the left-hand side of (26) is lower bounded by a positive constant. Let n ij,k be the normal vector of the plane Span { t ∗ k -t ∗ i , t ∗ k -t ∗ j } , where t ∗ i follows the standard Gaussian distribution. For ˜ s ∗ ij ≤ 0 . 5 , one can show that

for some absolute constant c ′ , c , which suggests

In (27), the first inequality follows from the definition of ˜ d ij,k , the first equality follows from the definition of ˜ s ∗ ij and the last equality is due to the assumption ˜ s ∗ ij ≤ 0 . 5 . The second inequality is commonly assumed for all ij ∈ E in [7, 12, 25], which they call the c/ √ log n -well-distributed condition. It is proved in [7] that the if t ∗ i is i.i.d. with standard Gaussian, then c/ √ log n -welldistributed condition holds with high probability.

B Explanation of the Order of Complexity for T-AAB in Table 1

We assume the Erd˝ os-Rényi graph G ( n, p ) , where p is the probability of connecting two nodes, with edge corruption probability q . We show that the recovery guarantee in Theorem 2.1 holds under this probabilistic model, provided

and

We note that given (28), (29) is equivalent to

We first verify (30), where we note that it is sufficient to focus on the worst case

That is, we show that this choice is sufficient for exact recovery with high probability.

We prove exact recovery by establishing with high probability the sufficient condition of Theorem 2.1:

We control the ratio of bad cycles as follows. For any fixed edge ij ∈ E , λ ij = | B ij | / | N ij | is the average of the Bernoulli random variables X k = 1 { k ∈ B ij } where k ∈ B ij with probability 1 -(1 -q ) 2 . Consequently,

Next, we investigate the concentration bound for λ ij and then for λ = max ij λ ij . We recall the following one-sided Chernoff bound [4] for independent Bernoulli random variables { X l } n l =1 with means { p l } n l =1 , ¯ p = ∑ n l =1 p l /n , and any η ≥ 1 :

Applying (32) with the random variables X k = 1 { k ∈ B ij } and η = 1 ,

To control the size of | N ij | in above probability bound, we use the following Chernoff bound [4] for i.i.d. Bernoulli random variables { X l } l m =1 with means µ and any 0 < η < 1 :

We note that by applying (34) with the random variables 1 { k ∈ N ij } and η = 1 / 2 , we obtain that

By combining the bounds in (33) and (35), we have for sufficiently large n

By applying a union bound over ij ∈ E , we have

where λ = max ij λ ij . Therefore, with q = c 1 / √ log n and sufficiently large n , we have

Finally, we show for a proper constant c 1 , (31) holds with high probability, and the exact recovery is concluded. We note that the RHS of (31) is lower bounded by

Combining this estimate of µ with (39), we obtain

Therefore, to guarantee (31) it suffices to let RHS of (38) be bounded from above by the RHS of (40). Namely, we require that

which can be easily satisfied by setting c 1 < c 12 eC α . Therefore, with the order of q = ( c/ (12 eC α √ log n )) and equivalently ϵ b = ( cp/ (12 eC α √ log n )) , we can guarantee (31) and hence exact recovery with the probability specified in (38). Consequently, we verify that (30) is sufficient for exact recovery with the latter probability.

We finally note that assuming (28), that is, p ≥ c 0 n -1 / 2 log 1 / 2 n for sufficiently large constant c 0 , the probability specified in (38) is high. Indeed,

Thus, with high probability, the recovery conditions in Theorem 2.1 are satisfied when (28) and (29) hold. We thus verify the bounds reported for T-AAB in Table 1.

C Visualization of 3D sparse models on ETH3D

We compare 3D sparse point cloud reconstructions of Cycle-Sync, GLOMAP and Theia. For GLOMAP and Theia, we use their default reconstruction parameters. For the Cycle-Sync pipeline, we feed the resulting camera poses to the 3D point triangulator in COLMAP (it uses bundle adjustment for the triangulations, while fixing camera pose estimators) and return the sparse 3D model. The latter was done quickly without careful tuning of parameters. Table 2 compares the number of triangulated 3D points for different SfM pipelines and some 3D sparse models on ETH3D. Tables 3 and 4 demonstrate the actual 3D sparse models by these methods.

We observe from Table 2 that for 9 out of the 13 datasets, Cycle-Sync improves the number of triangulated 3D points. This leads to an improvement of the overall quality of reconstruction for these datasets as noticed in Tables 3 and 4. This is due to the improvement on initial camera poses thanks to Cycle-Sync. For the other 4 datasets, Cycle-Sync fails to recover a meaningful 3D sparse model. For two of these datasets (meadow and office) all three methods are not performing well. For one dataset (relief) Theia is the only one which performs well and for the last dataset (relief_2) GLOMAP performs better than the other two methods.

Dataset

Cycle-Sync

GLOMAP

Theia

courtyard

delivery_area

electro

facade

kicker

meadow

office

pipes

playground

D Visualization of Camera Pose Estimation

We demonstrate pose estimation results for four scenes: courtyard , meadow , office , and pipes . For each scene, we compare the ground-truth camera poses with those estimated by GLOMAP, Theia, and Cycle-Sync. Figures 5-8 illustrate these comparisons. For the first three scenes, Cycle-Sync produces camera poses that align more closely with the ground truth, while GLOMAP and Theia exhibit larger misalignments. In the pipes scene, both Cycle-Sync and GLOMAP achieve good alignment, whereas Theia fails to produce meaningful results.

Dataset

Cycle-Sync

GLOMAP

Theia

relief

relief_2

terrace

terrains

E Supplementary Tables for ETH3D

We provide additional tables and figures to demonstrate the pose estimation quality of Cycle-Sync. Table 5 demonstrates the location error for each SfM pipeline. Table 6 demonstrates the location error for different location estimation algorithms. Table 7 demonstrates the effect of each building block of Cycle-Sync by beginning with the LUD pipeline, and gradually adding MPLS, MPLS-cycle, Cycle-Sync and STE.

F Runtime

Table 8 compares the runtime of location estimation methods on ETH3D. We observe that STE-based methods are significantly faster than non-STE methods, including LUD+IRLS (the old LUD pipeline). In particular, Cycle-Sync runtime is 48% lower than that of the LUD pipeline. Although Cycle-Sync is slower than common location estimation algorithms such as BATA, ShapeFit, and FusedTA, its

A

40

35

IRLS

30

25

MPLS

20

15

MPLS cycle only

10

5

0

courtyard

electro

facade

kicker

meadow

office

pipes

relief_2

terrace

terrains

Average

delivery_area

playground

runtime remains within the same order of magnitude, while achieving superior accuracy and stability in camera pose estimation.

G Table and Figures for Rotation Synchronization

In this section we show the tables and figures for rotation errors. Figure 9 demonstrates the rotation errors on ETH3D across different rotation synchronization methods. Table 9 demonstrates the rotation errors on ETH3D across different pipelines.

We observe that MPLS-cycle (used in our Cycle-Sync) greatly improves rotation accuracy over existing pipelines. On average, MPLS-cycle reduces the median rotation error by 62 . 8% and mean rotation error by 74 . 1% , compared to the best existing pipeline GLOMAP. Also, our proposed MPLScycle reduces the mean rotation error of MPLS by 56 . 6% and median rotation error by 64 . 8% . It is worth noting that Cycle-Sync outperforms GLOMAP and Theia even without bundle adjustment. This demonstrates that even without bundle adjustment, our approach outperforms baselines that rely on it.

H Additional Experiment for IMC-PT

In this section we compare the camera location estimation results for different location estimation algorithms on IMC-PT. This dataset consists of 9 city-scale image sets, as well as ground truth camera poses estimated by aligning COLMAP SfM model with a LiDAR scan. We generate image matches using LoFTR [29], a deep learning feature matching method instead of SIFT. We use LoFTR since

0.7

Medain Location Error

0.6

0.5

0.4

0.3

0.2

0.1

0

brandenburg

taj_mahal

Average

buckingham_palace

LUD

colosseum_exterior

grand_place_brussels

BATA

palace_of_ westminster

ShapeFit

pantheon_exterior

FusedTA

temple_nara_japan

■Cycle-Sync

westminster_abbey

it is proved to be effective on popular homography estimation, relative pose estimation and visual localization benchmarks. Table 10 and Figure 10 demonstrate the location error of different location estimation algorithms, where rotation synchronization method is MPLS-cycle and all methods use STE.

We observe that Cycle-Sync achieves the smallest mean and median location error averaging on all datasets. For 3 out of 9 datasets, Cycle-Sync achieves both the smallest mean and median error. For other datasets, Cycle-Sync achieves no significantly larger mean and median error. The largest difference from Cycle-Sync to the best method in mean and median location error is 0.05, which is small compared to the average scale of location. The improvement of Cycle-Sync is smaller than that in ETH3D. We believe the reason is that LoFTR is not a good feature for this dataset, since it tends to overfit repetitive structures such as windows and facades. To sum up, while the gains over baselines are smaller than on ETH3D, Cycle-Sync still achieves the best mean and median across most scenes.

I Additional Supplementary Tables for ETH3D

I.1 Sensitivity to Initialization

In tables 12 and 11 we report the mean and median location error on ETH3D data (averaged over different scenes) after several iterations for T-AAB and trivial initialization schemes. While the T-AAB initialization accelerates convergence by providing better starting weights, it does not significantly influence the final accuracy. Even trivial initialization using uniform weights performs similarly after sufficient iterations. Therefore, our method is robust even to trivial initialization, let alone variations in the T-AAB parameter.

I.2 Replacing LUD with BATA in Our Pipeline

In table 13 we show results of replacing LUD with BATA within our reweighting framework. Indeed, integrating LUD under Cycle-Sync's reweighting leads to better performance compared to integrating BATA. This is likely due to the stronger constraint of LUD for preventing collapsed trivial solution (the constraint is enforced on every edge). Moreover, our Welsh-type objective function already accounts

for large variations in distances, making angle-based methods such as BATA less advantageous in this case.

I.3 Annealing Schedule λ t

In tables 14 and 15 we include synthetic experiments for different annealing schedules λ t in settings with both additive noise and high corruption. Table 14 uses uniform corruption with q = 0 . 7 and higher noise level σ = 0 . 2 .

Table 15 the adversarial corruption with q = 0 . 45 (close to the theoretical limit q = 0 . 5 ) and higher noise level σ = 0 . 2 .

Weobserve that our choice of λ t has the lowest median error for both settings. Therefore, our proposed schedule strikes a good balance between residual-driven updates early on and cycle-consistency emphasis later. This schedule has consistently outperformed alternatives, particularly in settings with both additive noise and high corruption.

We remark that using only the residual for reweighting (i.e., setting λ t = 0 , which corresponds to IRLS) often leads to significantly higher errors compared to cycle-based reweighting methods. The underlying issue is that, in noisy settings, some bad edges may coincidentally exhibit low residuals. When this occurs, the aggressive reweighting imposed by the Welsch objective can assign them disproportionately large weights, thereby amplifying their adverse impact. In contrast, our cyclebased reweighting effectively overcomes this limitation: it is extremely unlikely for a bad edge to exhibit a low average cycle inconsistency, unless all cycles it participates in are consistent-an event that is highly improbable for corrupted measurements. Overall, we find that our annealing strategy consistently achieves the best performance across most scenarios.

We also observe this trend in the context of rotation synchronization. Figure G illustrates that emphasizing cycle-consistency can significantly reduce orientation error. However, for rotation there is no need for annealing as it does not rely on distance estimation.

J Additional Experiment on 1DSfM Datasets

We report the mean and median location errors for 1DSfM dataset (averaged over different scenes), with all methods consistently preprocessed using STE and MPLS-cycle in table 16. Note that the ground truth camera poses in 1DSfM are generated by Bundler, which is considered outdated compared to modern tools like COLMAP (and even COLMAP may lack the accuracy of laser scans). We remark that this could be less reliable when benchmarking high-precision location solvers.

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Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper proposes a method for camera location synchronization which is resistant to corrupted data which is present in real-world Structure-from-Motion (SfM) applications. By using redesigned Message Passing Least Squares (MLPS) to minimize a Welsch function the paper shows high robustness, great performance and solves the limitations of Least Unsquared Deviations (LUD) and BATA-type objectives. The paper also establishes a deterministic exact-recovery guarantee under adversarial corruption which outperforms all prior work under standard probabilistic model constraints. The framework is then extended to the rotation synchronization problem to reduce the orientation error. Lastly the proposed model in the paper removes any need for bundle adjustment which are computationally expensive but present in all referenced SfM pipelines.\n\nSTRENGTHS AND WEAKNESSES:\n1. Quality - The paper is well written with all claims being well supported and methods used being increments of previous works. The paper provides a clear overview of the method and how it is used and what its drawbacks are. The two main downsides to the quality is that in some places there are missing references for statements that are clear to the authors but might not be clear to a third party. The second downside is that some graphs (i.e. Fig 3, Fig 4) are a bit illegible due to the absolute differences of values being shown so a better representation of those would make a difference.\n2. Clarity - The paper is clearly written with all theoretical setups providing assumptions and proofs. The introduction, related work and methodology provide clear overviews of what the paper proposes. The experimental setup is properly noted and reproducible.\n3. Significance - The paper provides a method that consistently outperforms referenced methods in the task it was tested for. It advances the topic and provides insights into sensor synchronization, not only for location but for rotation and direction.\n4. Originality - The paper builds upon previous methods' limitations and solves them in a novel way, providing valuable insights into the topic.\n\nQUESTIONS:\nNone.\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 4, "ORIGINALITY": 3, "DATE": "2025-06-18"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes Cycle-Sync, a robust translation averaging method.\n\nThe key contributions are twofold:\n\n1. A Welsch-type loss function, which is less sensitive to large variations of distances and preserves non-smoothness at the origin.\n2. A message-passing least squares (MPLS) framework that integrates cycle-consistency information across 3-cycles.\n\nExperiments on both synthetic and the ETH3D real datasets show that Cycle-Sync outperforms other translation averaging and SfM baselines.\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths:**\n1. Translation averaging is indeed an important component in SfM, and the paper tackles the challenging case of cycle-consistent corruption, where other methods underperform.\n2. The proposed translation averaging fully exploits cycle consistency using message-passing least squares.\n3. The Welsch-type loss provides both nonlinearity and robustness, offering advantages over L1/L2 losses.\n\n**Weaknesses:**\n1. The geometric intuition of cycle consistency is not clearly described. It would be helpful to Including intuitive geometric visualizations.\n2. The choice of hyperparameters, such as setting the Welsch loss parameter $a=4$ in line 118, some parameters in line 169, and weight initialization as $w_{ij,0} = exp(-20\\tilde{s}_{ij})$ in line 152, lacks sufficient justification or empirical support. Although the method is claimed to be insensitive to these choices, it would be beneficial to include an ablation study or comparative analysis to support this claim.\n3. Fig. 2 is not self-explanatory and the propagation mechanism in the bottom-left part is unclear. It would be helpful to provide a more detailed textual description.\n4. In Algorithm 1, it is not explained how the 3-cycle inconsistencies $d_{ij, k}_{k\\inC_{ij}}$ are obtained or whether they are known a priori or estimated.\n5. The plug-and-play outlier rejection module is claimed as a contribution, but it simply uses STE in line 215. While effective, the novelty may be overstated unless further integration or modification is described.\n6. The paper only verifies the performance of real data on ETH3D dataset, while ShapeFit, BATA, Fused-TA, LUD and other methods include results on the widely used 1DSfM dataset [26]. It would be beneficial to provide the comparisons on that dataset.\n7. In Fig. 3, the robustness is only demonstrated under mild noise $\\sigma = 0.05$. It would be valuable to explore performance under higher noise levels to validate the claim of robustness.\n8. The All-About-that-Base (AAB) [20] is discussed as a related work that also leverages 3-cycle consistency, but not included in the experiments. Since AAB is most similar in spirit to Cycle-Sync, a direct experimental comparison would be highly relevant.\n\nQUESTIONS:\n1. Add geometric illustrations or figures to intuitively explain what cycle consistency captures, and how it improves robustness in practice.\n2. Clarify $d_{ij, k}$ in Algorithm 1. Are these precomputed from input directions or estimated during optimization?\n3. Justify hyperparameter choices, especially the value of $a=4$ in the loss, other parameters, and the weight initialization. Including a ablation study would be helpful.\n4. Compare with AAB [20] in the experiments. Since both methods rely on 3-cycle consistency, a direct comparison is necessary.\n5. Consider evaluating on the 1DSfM dataset [26] to improve comparability with prior work.\n6. Evaluate performance under higher noise (e.g., σ = 0.1 or higher) to further support the claim of robustness.\n7. Clarify what is novel in the \"plug-and-play outlier rejection module\" beyond simply using STE.\n\nConsidering the above concerns, I give a borderline reject at this stage. However, I would like to increase my rating if the concerns are addressed.\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 2, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-06-24"}

comments: SUMMARY:
Cycle-Sync is a global camera pose estimation method based on message-passing least squares for cycle consistency and Welsch-like robust loss, instead of L1 or L2 loss, for camera location estimation. With unknown inter-camera distance, Cycle-Sync adapts MPLS on translation estimation. Such a method avoids bundle adjustment in the classical Structure from Motion pipelines while showing superior performance.

STRENGTHS AND WEAKNESSES:
Strengths: 
1. A novel global location estimation pipeline based on cycle consistency, showing decent improvement over the SOTA structure-from-motion methods with bundle adjustment. 
2. A new optimization formulation, solved by cycle consistency, leading to reduced corruption level.
3. The paper is well-written and easy to read. 

Weakness: 
1. The contribution claimed to deal with the highly variant edges in distance. In principle, yes. However, there is only one real-world dataset,  ETH3D, used in the evaluation with its precise ground truth from the laser, which is not sufficient to conclude eliminating bundle adjustment for all.  
2. Missing discussion. It is unclear how fast or how expensive the process of iteratively optimizing the cycle inconsistencies over the estimated locations is.

QUESTIONS:
1. How do you define well-shaped triangles? How do you choose the lower and upper bounds used in truncated AAB (T-AAB)?.
2. Why does the proposed loss function use $\alpha$=4? How sensitive is the optimization w.r.t. the choice of $\alpha$? Or it can be any constant.
3. It seems the performance decreases a lot when removing the STE outlier filter in some scenes. Have you tried using STE for the baselines?
4. Others: see the weakness

LIMITATIONS:
Yes
TITLE: Official Review
CLARITY: 4
IMPACT: 3
ORIGINALITY: 4
DATE: 2025-07-01

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes a framework for robust global location estimation of cameras, given pairwise direction estimates in the context of the Structure-from-Motion (SfM) problem. The approach formulates a LUD-type cost function using a robust Welsch loss and leverages distance-based cycle-consistency over cycles, where the distances are iteratively estimated. In the initial iteration, the weights are derived using angle-based cycle-consistency computed via the AAB method [20], which relies solely on the input relative directions.\n\nTo obtain more accurate global rotations during the rotation averaging step, the authors introduce a variant of the MPLS method that progressively prioritizes cycle-consistency over residual minimization. These improved global rotations are crucial for enhancing the quality of relative translation directions in the subsequent translation averaging stage.\n\nAdditionally, the authors propose a technique called STE, which robustly estimates the relative translation directions using a subspace recovery method [27]. A theoretical guarantee is provided, demonstrating that the proposed method can accurately recover the corruption levels of both inlier and outlier edges.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n\n1. The authors propose a comprehensive framework for robust camera translation estimation by:\n\n     (a) Introducing a novel technique based on distance-based cycle-consistency, enabling accurate estimation of camera locations from input translation directions.\n\n     (b) Applying robust modifications to each step that influences the accuracy of translation directions, including (i) a modified cycle-consistency based rotation averaging step, and (ii) the use of matched point coordinates to estimate robust relative translation directions via subspace recovery (STE).\n\n2. The paper is generally clear, well-organized, and accessible to readers familiar with Structure-from-Motion pipelines.\n\n3. Strong theoretical guarantees are provided, demonstrating that the method can accurately recover corruption levels of relative translations even under arbitrary adversarial corruption.\n\n4. The experimental evaluation is extensive, with results reported on both synthetic and real datasets, across a variety of corruption models (uniform and adversarial). The method is compared both as a standalone location solver and within end-to-end SfM pipelines that include bundle adjustment, providing a comprehensive assessment of its practical value.\n\nWeaknesses:\n1. Theorem 2.1 lacks intuitive interpretation, particularly in terms of the values taken by the right-hand side (RHS) of the inequalities involving $s_{ij}^{(t)}$ for inlier and outlier edges. For the recovery guarantee to be meaningful, the RHS for inliers should ideally tend toward zero, while that for outliers should be significantly larger—indicating a clear separation between the two. However, the paper does not provide any insight into the typical magnitudes or range of these RHS terms, making it difficult to assess how strong or practical the theoretical guarantee really is. A small illustrative example or further discussion quantifying the behavior of the RHS terms would greatly improve the clarity and usefulness of this result.\n2. The method relies on a carefully chosen initialization of weights $w_{ij}$ using $\\tilde{s}_{ij}$ from T-AAB, but it’s unclear whether such meticulous design is truly necessary. If so, this should be justified through a sensitivity analysis showing how performance varies under alternative initializations.\n3. The ablation studies are somewhat incomplete, as they do not evaluate the effect of removing or replacing individual components in the pipeline. For example, it remains unclear how a variant such as STE + LUD + MPLS-cycle would perform. Based on the results, it appears that a significant portion of the improvement may be attributable to the STE step rather than the proposed cycle-consistency refinement, which is positioned as the primary contribution. Additionally, it would be informative to evaluate the performance of BATA in place of LUD within the pipeline, especially since BATA is known to yield better solutions in scenarios with high variability in edge lengths.\n4. In the comparison against SfM pipelines such as GloMap and Theia, the results shown in Figure 9 of the supplementary material (a corrected version of upper subfigure of Figure 4 in the main paper) indicate that Theia actually outperforms the proposed method on 9 out of 13 datasets. The average error metric, which favors the proposed method, appears to be disproportionately influenced by a few datasets where Theia performs poorly. Given the variability across datasets, the average may not be a reliable indicator of overall performance. In this context, the frequency of outperformance across datasets would be a more meaningful and fair comparison metric.\n5. The performance of the proposed Cycle-sync method appears to depend on the specific annealing schedule for $\\lambda_t$, which is currently fixed as $\\frac{t}{t+10}$. It would be insightful to evaluate how the method behaves under different trajectories of $\\lambda_t$. In particular, it would be useful to experiment with a schedule where $\\lambda_t \\rightarrow 0$ as $t$ increases—similar to MPLS—thereby placing more emphasis on cycle-consistency in early iterations and gradually shifting focus to the residuals. Additionally, analyzing the performance under a fixed $\\lambda_t$ could help isolate the benefit of annealing itself. Such experiments would clarify the impact of the annealing strategy on the overall robustness and accuracy of the method.\n\nQUESTIONS:\n1. When $\\lambda_t$ increases from 0 to 1, the weighting shifts from residuals to cycle-inconsistency. While it’s understandable that cycle-inconsistency may be unreliable in early iterations due to dependence on the initial (and inaccurate) distance estimates, the residuals are also computed from the same initialization and are therefore equally unreliable. This raises a key question: what is the rationale for prioritizing residuals early on if they too are sensitive to initialization? I think that addressing weakness-5 may address this concern also.\n2. It is unclear why the authors specifically advocate for the Welsch loss function. Other parametric robust loss functions, such as Geman-McClure or Huber, can exhibit similar behavior in handling variability in edge lengths when tuned appropriately. Can you correct me if this is wrong?\n3. In MPLS vs MPLS-cycle for rotations averaging, what is meant by \"disabling reweighting\" in Lines 206-207?\n4. Typo: In Algorithm-1, in the first two steps, should it be $\\tilde{s_{ij}}$ instead of $s_{ij}$?\n5. Although the 1DSfM datasets are older, they typically exhibit higher corruption levels than ETH3D. Given that the translation errors in Table 6 (supplementary) for ETH3D are already low, it’s unclear whether improvements in translation averaging significantly impact the final SfM output. As an optional evaluation, it would be valuable to test the method on more challenging datasets like 1DSfM to better highlight its robustness.\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-03"}