Linearly Solving Robust Rotation Estimation

Tianyu Huang; Yinlong Liu; Zhi-Xin Yang

REVIEW 3 major objections 7 minor 118 references

Linearly Solving Robust Rotation Estimation

T0 review · 3 major / 7 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper claims that robust rotation estimation reduces to a linear voting problem: each rotation constraint becomes a great circle on the quaternion sphere, and the rotation is the most-intersected projected curve.

desk verdict The quaternion-circle linearization is elegant but largely a repackaging of known linear attitude estimators; the real question is whether the GPU Hough voting's 99%-outlier claim survives structured outliers. read the letter →

arxiv 2506.11547 v1 pith:YROXF4TH submitted 2025-06-13 cs.CV cs.ROcs.SYeess.SY

Yinlong Liu , Tianyu Huang , Zhi-Xin Yang This is my paper · ORCID

classification cs.CVcs.ROcs.SYeess.SY

keywords rotationestimationrobustmodelfittingquaternioncircleHoughvotingstereographicprojectionoutlierrobustnessWahba'sproblemrigidpose

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Rotation estimation is normally treated as a nonlinear, non-convex optimization problem, but this paper seeks to show that it can be reformulated as a linear model-fitting problem with no constraints dropped and no singularities introduced. The key geometric fact is that the set of rotations carrying one unit vector to another forms a great circle on the unit quaternion sphere; equivalently, each such motion contributes two linear equations in the quaternion. Using a stereographic projection, these quaternion circles become ordinary circles in a 3D ball, so finding the rotation becomes finding the point where the most projected curves intersect. The authors build a GPU-friendly Hough-style voting algorithm around this, reporting linear-time behavior in the number of correspondences and empirical success at one million inputs with 99% outliers in under half a second. If the reformulation is right, the notoriously difficult robust rotation problem inherits the maturity of robust linear model fitting.

What carries the argument

The central object is the quaternion circle: for a given motion $Ra=b$, all quaternions representing compatible rotations form a great circle on the unit sphere $S^3$, spanned by two orthonormal quaternion basis vectors $q_{b1}$ and $q_{b2}$. The orthogonal complement space gives two linear equations per observation, so a rotation estimate becomes the least-squares null vector of a stacked linear system. For robust estimation, a stereographic projection from the north pole maps the half quaternion sphere into the unit 3D ball, and the circle-preserving property turns each quaternion circle into a circle or line in $\mathbb{R}^3$; a Hough-style accumulator then votes for the intersecting point, with resolution $\varepsilon$ controlling the grid. This three-step pipeline—quaternion circle, linear equations, projected voting—is what makes the rotation problem linear and embarrassingly parallel.

What would settle it

Run the voting algorithm at $\varepsilon=1/180$ on a synthetic problem with $N=10^5$ and 99% outliers engineered so that a large block of outlier quaternion circles all pass close to one fake rotation point (for instance, many same-axis rotations with angles chosen to cluster in the accumulator) and check whether the global vote maximum corresponds to the fake rather than the true rotation.

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Extended reading notes

Core claim

The paper's central claim is that the constraint $Ra=b$, instead of living on the nonconvex manifold $\mathrm{SO}(3)$, can be described exactly by a great circle on $S^3$, the quaternion circle. For each observation the circle is $C_{\mathrm{quat}} = \{q_{b1}\cos(\alpha/2) + q_{b2}\sin(\alpha/2)\}$; the two basis quaternions $q_{b1},q_{b2}$ are orthonormal, and the complementary vectors $q_{b3},q_{b4}$ give two homogeneous linear equations $[q_{b3}^T; q_{b4}^T]q=0$. Thus $N$ correspondences yield a $2N\times 4$ matrix $Q$ whose null vector is the sought quaternion; in the outlier-free case the rotation is the eigenvector belonging to the smallest eigenvalue of $Q^TQ$. In the outlier case, the same linear equations are projected by stereographic projection into 3D circles, and the rotation is recovered by voting for the point that intersects the greatest number of projected curves. The paper also claims this voting is the first linear-time algorithm for robust rotation estimation, with complexity $O(N/\varepsilon^3)$, and that it extends directly to multiple rotations and to 6D rigid pose via pairwise constraints.

Load-bearing premise

The extreme robustness results rest on the unproved assumption that at the chosen accumulator resolution $\varepsilon=1/180$, genuine inlier votes always form the global maximum while outlier votes stay dispersed, even for structured outlier mixes up to 99%; the paper supports this only empirically, with no worst-case or probabilistic guarantee.

Editorial extensions

If this is right

If the reformulation is correct, robust rotation estimation can be solved by the same tools as robust linear model fitting, instead of specialized nonconvex optimization.
The claimed $O(N/\varepsilon^3)$ complexity means the method scales linearly in input count at fixed accuracy, and the voting loop is embarrassingly parallel, so GPU speedups are direct.
At the tested resolution, the method reports 100% success at 5% inlier ratio with structured same-axis outliers, whereas decomposition-based axis-only methods can fail because they drop the angle constraint.
The method returns multiple high peaks in one pass, so multiple simultaneous rotations in one scene can be estimated without repeated RANSAC-style sequential fitting.
Embedded in a pairwise-decomposition pose pipeline, the method reaches comparable accuracy on 3DMatch, 3DLoMatch, and KITTI while running faster than compared baselines in the reported experiments.

Reading between the lines

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

The same quaternion-circle linearization could be applied to rotation averaging or translation voting, possibly converting iterative averaging into one-shot voting; the paper does not pursue this direction.
The 99% robustness is demonstrated empirically, not certified; an adversarial outlier distribution engineered to concentrate votes at a fake rotation would provide a sharper test of whether the accumulator's global maximum is guaranteed.
The resolution $\varepsilon$ and sampling number $J$ create a memory-accuracy tradeoff, and one could derive bounds on how fine $\varepsilon$ must be as a function of noise level and outlier fraction; the paper does not provide such bounds.
The paper's closing conjecture that a similar linear formulation may exist for $\mathrm{SE}(3)$ suggests a concrete next target: finding the analog of the quaternion circle for rotations plus translations.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Hi,

Quick take: this paper's algebraic core is sound but not new, and it says so itself in Appendices C and D. The genuinely interesting part is the Hough-style voting on stereographically projected quaternion circles, which gives very fast GPU rotation estimation in synthetic and real tests. If the speed and robustness are real, it's a useful engineering tool. But the headline 99%-outlier result is not yet established.

What it does well: the derivation of two linear constraints per correspondence is correct and standard (quaternion product, orthonormal basis, null space), and the closed-form solution matches SVD in the noise-free and noisy cases. The experiments are unusually broad: synthetic noise/outlier sweeps, panorama matching, 3DMatch/KITTI registration, multiple motions. The paper also deserves credit for acknowledging the overlap with OLAE [23] and dual-quaternion pairwise formulations [112,113]; that honesty makes the novelty claims easier to evaluate.

Soft spots, in order of importance:

1. The extreme-robustness claim is an extrapolation. At 99% outlier ratio, experiments only use random mismatches. The structured same-axis outliers are tested at η ≤ 40% with inlier ratios 5–20%, not 1% inliers. The voting argument needs inlier votes to dominate the true accumulator bin; no worst-case or probabilistic bound is given, and nothing in Algorithm 1 certifies the returned peak. So the abstract's "99% outlier ratio" sentence overstates what is demonstrated.

2. The complexity statement O(N/ε^3) does not follow from Algorithm 1. Per input the cost is O(J) plus O(1) accumulator update; the number of bins is O(1/ε^3) but that's memory, not per-input time. Maybe they meant O(NJ) with J = O(1/ε), but as written it is wrong.

3. Lemma 1 is stated without proof. It's trivial to prove from the orthonormal basis argument, so this is a minor presentation issue, not a substantive gap.

4. No code. The 10^6-point/0.5s runtime and 100% success rates cannot be independently verified.

Would I send this to peer review? Yes. The math is mostly sound, the empirical sweep is impressive, and the problem matters. But I'd make the authors either provide a formal robustness guarantee (or a certificate) or visibly narrow the claims, fix the complexity analysis, and release code. Roughly: a promising systems/speed paper wearing the clothes of a theory paper.

For a reading group, worth a maybe—good discussion of what "linear" means when you discretize a sphere and vote.

Referee Report

3 major / 7 minor

Summary. The paper proposes a reformulation of the rotation estimation problem Ra=b using quaternion circles: for each correspondence, the set of rotation quaternions satisfying the constraint lies on a great circle in S^3, which can be described by two linear equations. The authors use this to derive a closed-form linear least-squares solution for outlier-free cases and a Hough-style voting algorithm in a stereographically projected 3D space for outlier-robust cases. They claim linear time complexity O(N/ε^3), extreme robustness to 99% outliers at scale 10^6, and support these claims with synthetic and real-world experiments, including panoramic image stitching, 6D pose estimation, and multi-motion estimation.

Significance. If the central claims are correct, the quaternion-circle linearization is an elegant and potentially useful geometric insight that connects rotation estimation to linear model fitting and voting, with clear GPU parallelizability. The empirical results suggest the voting method is competitive with or faster than state-of-the-art robust estimators in several settings. The paper also demonstrates an interesting failure mode of decomposition-based axis-first methods. However, the significance is dampened by an incorrect complexity statement, an unproven central lemma in the main text, and a robustness claim that is experimentally supported only for random, unstructured outliers at the extreme 99% level, not for structured outlier geometries at that rate.

major comments (3)

[Section 4.3.2 and Contribution bullet (Section 3.1)] The stated time complexity O(N/ε^3) does not follow from Algorithm 1. In Algorithm 1, for each input the inner loop (lines 5–9) iterates over J samples and performs O(1) accumulator updates per sample; the algorithm never iterates over the O(1/ε^3) accumulator cells on a per-input basis. The total runtime is therefore O(NJ), independent of ε except through the memory footprint of the accumulator. The claimed complexity in the contribution bullet should be corrected to O(NJ) runtime with O(1/ε^3) memory, or the algorithm should be modified so that ε actually appears in the per-input cost.
[Section 5.3.1, Abstract, and Table 1] The headline robustness claim of handling 99% outlier ratios is only demonstrated for random mismatches (Fig. 6). The structured same-axis outlier experiments in Table 1 are run only up to η=40% and with inlier ratios of 5–20%, not at the 1% inlier level implied by 99% outliers. No worst-case or probabilistic argument is provided to guarantee that the accumulator's global maximum coincides with the true rotation at ε=1/180 under such extreme structured corruption. The abstract's statement that the method can solve 'severely corrupted (99% outlier ratio) rotation estimation problems' is therefore an extrapolation beyond the tested regime; the authors should either add experiments with structured or adversarial outlier geometries at ρ=99% or explicitly restrict the claim to the settings actually tested.
[Section 4.2, Lemma 1] Lemma 1 is the foundation of the two-linear-equations-per-correspondence formulation, but it is stated with only 'can be easily proved, and we omit trivial explanations.' While Appendix A derives related linear constraints via the eigen-decomposition of the matrix M, the main text should provide a self-contained proof of Lemma 1 or explicitly reference the appendix derivation and state the conditions under which the vectors q_b3 and q_b4 are well-defined, particularly in the degenerate cases a=b or a=-b. As written, the central linearization rests on an unproved lemma in the main text.

minor comments (7)

[Section 5.2] The statement that the linear method and SVD are 'identical' is supported only by empirical agreement; the text should be reworded to say the results are numerically indistinguishable in the tested settings, since no formal equivalence is proven.
[Throughout] There are several typos and wording issues, e.g., 'RASNAC' for RANSAC, 'homomorphically' for 'homeomorphically', 'quanternion' for 'quaternion', and 'explore the dual structure' where a more precise term may be intended. A careful proofread is recommended.
[Figure 6] The axis labels 'N=105' should read 'N=10^5' or 'N=10^5' with a superscript; similarly, the noise and error axes should include units (degrees and milliseconds) consistently.
[Table 1] The header layout is confusing: 'inlier ratio 20%' appears as a row label rather than as a column group label, and the same-axis ratio η is not clearly defined in the table itself. Please restructure the table for readability.
[Section 4.3.1] The paper claims the formulation introduces no singularities, but stereographic projection has a singularity at the north pole. The authors should explicitly state that the choice of the lower hemisphere avoids this point and discuss the boundary case q3=0.
[Section 4.3.2] The voting procedure returns only the center of the block with the highest count; no sub-grid refinement or peak verification is described. At 99% outlier ratios, spurious peaks could be a concern; a brief discussion or a refinement step would strengthen the method.
[Reproducibility] No code or data are released. Given that the 10^6-point/0.5-second runtime and 100% success rates are central to the paper's claims, providing an implementation would greatly aid verification.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the quaternion-circle linearization is a direct algebraic reformulation, and the robust-voting claims are supported by external baselines rather than by fitted inputs.

full rationale

The paper's central derivation is not circular. Section 4.1 obtains the quaternion circle q(alpha)=q_b1 cos(alpha/2)+q_b2 sin(alpha/2) directly from the Rodrigues composition formula, and Eqs. (26)-(28) verify that q_b1 and q_b2 are orthonormal; the one-dimensional circle structure is a consequence of quaternion algebra, not an assumed input. Lemma 1 states that a quaternion in such a circle is orthogonal to the complementary basis vectors, giving the two linear equations (33); the proof is omitted, but Appendix A independently re-derives the same two linear constraints from the eigenvalue structure of the quaternion matrix M, so the linear system is not imported from the target claim. The closed-form solution Qq=0 and the subsequent voting algorithm contain no fitted parameters that are then relabeled as predictions; the accumulator resolution and sampling number are algorithmic choices, not calibrated to the reported success criterion. Comparisons with SVD, ARCS, RANSAC, TEASER++, and GORE use ground-truth rotation error, so the empirical claims are externally checkable. The paper cites prior work by the same authors ([24], [66], [67], [71]) for standard rotation decompositions and for a compact summary of the quaternion-circle derivation, but the necessary algebra is reproduced in the paper and is parameter-free, so these citations are not load-bearing. The 99%-outlier robustness claim does rely on an unproved inlier-peak-dominance assumption and on empirical success rather than a worst-case guarantee; that is an evidence gap about correctness, not a circularity of the derivation. Overall, the derivation chain is self-contained and no prediction reduces by construction to its inputs.

Assumptions & free parameters 4 free parameters · 6 assumptions · 2 invented entities

The central derivation relies on standard quaternion algebra and a few hand-chosen discretization parameters; no hidden fitted constants were found in the theory, but the robust claims depend on unproved dispersion of outlier votes.

free parameters (4)

accumulator resolution epsilon = 1/180 (default)
Grid step in the 3D voting ball; determines angular accuracy and memory. Chosen by hand, not derived; affects all reported success rates.
sampling number J = 180 (default, 540 for multi-motion)
Number of points sampled per half quaternion circle for voting; trades runtime versus coverage; chosen by hand.
length-check threshold mu_t = not specified numerically
Threshold in Section 4.5.1 to prune outlier pairwise constraints in 6D pose experiments; hand-tuned.
translation accumulator step = 0.025
Discretization step for translation voting in Section 5.4.1; hand-chosen.

assumptions (6)

standard math Quaternion composition formula for rotations (Olinde Rodrigues 1840)
Used in Eq. (23)-(25) to build q_b1 and q_b2; standard result.
standard math Stereographic projection maps circles/lines to circles/lines and is homeomorphic
Invoked in Remark 1 and Section 4.3.1 to map quaternion circles into 3D curves.
standard math Eigenvalues of M are (1,1,-1,-1) for a single pair as derived in Appendix A
Core fact that yields two linear equations per correspondence; verified algebraically.
domain assumption Each input correspondence (x_i,y_i) is a pair of unit vectors with the true rotation satisfying R x_i = y_i for inliers
The formulation starts from Ra=b with unit vectors; the model assumes inliers plus random outliers.
ad hoc to paper The true rotation can be represented by a quaternion in the lower hemisphere q3 <= 0
Used to define stereographic projection into the unit ball; valid by antipodal symmetry but an arbitrary choice.
ad hoc to paper Discretization with epsilon=1/180 and J=180 yields sufficient accuracy for the reported tasks
No derivation links epsilon to a rotation error bound; these are default values used in all experiments.

invented entities (2)

quaternion circle independent evidence
purpose: Geometric representation of all rotations mapping a fixed vector pair (a,b); the basis for the linear equations and the voting curves.
Though named here, it is the intersection of a 2D eigenspace with S^3, fully determined by standard quaternion algebra and verifiable from Eq. (33)-(35); not an unobservable postulate.
half quaternion circle and its stereographic image independent evidence
purpose: Reduces the S^3 voting domain to a 3D ball for practical Hough voting.
Mathematically derived via circle-preserving stereographic projection; computationally verifiable, no extra physical assumption.

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Cite this review

Pith. "Pith review of Linearly Solving Robust Rotation Estimation." pith.science (2026). https://pith.science/paper/YROXF4TH

@misc{pith2026250611547,
  author       = {Pith},
  title        = {Pith review of: Linearly Solving Robust Rotation Estimation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YROXF4TH}},
  note         = {Machine review of arXiv:2506.11547}
}

abstract

Rotation estimation plays a fundamental role in computer vision and robot tasks, and extremely robust rotation estimation is significantly useful for safety-critical applications. Typically, estimating a rotation is considered a non-linear and non-convex optimization problem that requires careful design. However, in this paper, we provide some new perspectives that solving a rotation estimation problem can be reformulated as solving a linear model fitting problem without dropping any constraints and without introducing any singularities. In addition, we explore the dual structure of a rotation motion, revealing that it can be represented as a great circle on a quaternion sphere surface. Accordingly, we propose an easily understandable voting-based method to solve rotation estimation. The proposed method exhibits exceptional robustness to noise and outliers and can be computed in parallel with graphics processing units (GPUs) effortlessly. Particularly, leveraging the power of GPUs, the proposed method can obtain a satisfactory rotation solution for large-scale($10^6$) and severely corrupted (99$\%$ outlier ratio) rotation estimation problems under 0.5 seconds. Furthermore, to validate our theoretical framework and demonstrate the superiority of our proposed method, we conduct controlled experiments and real-world dataset experiments. These experiments provide compelling evidence supporting the effectiveness and robustness of our approach in solving rotation estimation problems.

Figures

Figures reproduced from arXiv: 2506.11547 by the authors.

**Figure 3.** Intersections between projected circles. (a) The intersections of [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

**Figure 4.** Illustration of multiple rotation motions after stereographic pro [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗

Figures from the paper (14 more)

**Figure 5.** Figure 5: Comparison with the de facto standard outlier-free rotation esti [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]

**Figure 6.** Figure 6: Controlled experiments for our proposed rotation voting method. The first row shows the rotation error (degree) in different settings, and the [PITH_FULL_IMAGE:figures/full_fig_p011_6.png]

**Figure 7.** Figure 7: The working flow of rotation-only panorama stitching. (a) The input image pair. (b) Constructing the putative correspondences between image [PITH_FULL_IMAGE:figures/full_fig_p011_7.png]

**Figure 8.** Figure 8: Experimental results on passta datasets using the rotation voting [PITH_FULL_IMAGE:figures/full_fig_p012_8.png]

**Figure 9.** Figure 9: Comparative experiment for full rigid pose estimation using synthetic data. The top row shows the rotation error (degree) in different settings; [PITH_FULL_IMAGE:figures/full_fig_p013_9.png]

**Figure 10.** Figure 10: Illustrating the typical scenes for rigid point cloud registration. [PITH_FULL_IMAGE:figures/full_fig_p013_10.png]

**Figure 11.** Figure 11: The 6D rigid pose estimation results on different datasets. The left figure shows the rotation accuracy. The middle figure shows the [PITH_FULL_IMAGE:figures/full_fig_p014_11.png]

**Figure 13.** Figure 13: Performance of the rotation voting method for multiple rotation [PITH_FULL_IMAGE:figures/full_fig_p014_13.png]

**Figure 12.** Figure 12: Recall curves on different real-world datasets (higher is better). [PITH_FULL_IMAGE:figures/full_fig_p014_12.png]

**Figure 14.** Figure 14: Multiple rigid motion estimation experiments using synthetic data. The figure on the left shows the median running time under 100 trials. [PITH_FULL_IMAGE:figures/full_fig_p016_14.png]

**Figure 15.** Figure 15: Typical point clouds from chair category of ModelNet10 [PITH_FULL_IMAGE:figures/full_fig_p016_15.png]

**Figure 16.** Figure 16: Multiple rigid motion estimation experiments using ModelNet10 [PITH_FULL_IMAGE:figures/full_fig_p016_16.png]

**Figure 17.** Figure 17: Illustration of incompatible case in R3 . The red curves are the stereographic projection of half quaternion circle. (a) and (b) The quaternion circles intersect in S 3 . Therefore, after stereographic projection, they still intersect in R3 . (c) and (d) The quaternio…

**Figure 18.** Figure 18: One incompatible example. Two input rotation constraints share the same rotation axis [PITH_FULL_IMAGE:figures/full_fig_p023_18.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

118 extracted references · 78 canonical work pages

[23]

Optimal linear attitude estimator,

D. Mortari, F. L. Markley, and P . Singla, “Optimal linear attitude estimator,”Journal of Guidance, Control, and Dynamics, vol. 30, no. 6, pp. 1619–1627, 2007

2007
[1]

Fast rotation search with stereographic projections for 3d registra- tion,

A. P . Bustos, T.-J. Chin, A. Eriksson, H. Li, and D. Suter, “Fast rotation search with stereographic projections for 3d registra- tion,”IEEE transactions on pattern analysis and machine intelligence, vol. 38, no. 11, pp. 2227–2240, 2016. JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2025 16 2 3 4 5 6 7 8 9 # model number 0 1000 2000 3000 4000 Medi...

2016
[2]

Hybrid camera pose estimation,

F. Camposeco, A. Cohen, M. Pollefeys, and T. Sattler, “Hybrid camera pose estimation,” inProceedings of the IEEE Conference on Computer Vision and Pattern Recognition, 2018, pp. 136–144

2018
[3]

Szeliski,Computer vision: algorithms and applications

R. Szeliski,Computer vision: algorithms and applications. Springer Nature, 2022

2022
[4]

I love my attitude problem,

F. L. Markley, “I love my attitude problem,” inAIAA Guidance, Navigation, and Control Conference, no. AIAA-2005-5927, 2006

2005
[5]

The generalized wahba problem,

M. D. Shuster, “The generalized wahba problem,”The Journal of the Astronautical Sciences, vol. 54, no. 2, pp. 245–259, 2006

2006
[6]

Equivalence of two solutions of wahba’s prob- lem,

F. L. Markley, “Equivalence of two solutions of wahba’s prob- lem,”The Journal of the Astronautical Sciences, vol. 60, pp. 303–312, 2013

2013
[7]

Numerical solution of a generalized wahba problem for a spinning spacecraft,

M. L. Psiaki and J. C. Hinks, “Numerical solution of a generalized wahba problem for a spinning spacecraft,”Journal of Guidance, Control, and Dynamics, vol. 35, no. 3, pp. 764–773, 2012

2012

Show all 118 references

[8]

30 years of wahba’s problem,

F. L. Markley, “30 years of wahba’s problem,” inFlight Mechanics, 1999

1999
[9]

Zhang,Star Identification: Methods, Techniques and Algorithms

G. Zhang,Star Identification: Methods, Techniques and Algorithms. Springer, 2016

2016
[10]

Rosia: Rotation-search-based star identification algorithm,

C.-K. Chng, A. P . Bustos, B. McCarthy, and T.-J. Chin, “Rosia: Rotation-search-based star identification algorithm,”IEEE Trans- actions on Aerospace and Electronic Systems, 2023

2023
[11]

Certifiably optimal outlier-robust ge- ometric perception: Semidefinite relaxations and scalable global optimization,

H. Yang and L. Carlone, “Certifiably optimal outlier-robust ge- ometric perception: Semidefinite relaxations and scalable global optimization,”IEEE Transactions on Pattern Analysis & Machine Intelligence, vol. 45, no. 03, pp. 2816–2834, 2023

2023
[12]

Chin and D

T.-J. Chin and D. Suter,The maximum consensus problem: recent algorithmic advances. Springer Nature, 2022

2022
[13]

Outlier- robust estimation: Hardness, minimally tuned algorithms, and applications,

P . Antonante, V . Tzoumas, H. Yang, and L. Carlone, “Outlier- robust estimation: Hardness, minimally tuned algorithms, and applications,”IEEE Transactions on Robotics, vol. 38, no. 1, pp. 281–301, 2021

2021
[14]

A review of sensor technologies for perception in automated driving,

E. Marti, M. A. De Miguel, F. Garcia, and J. Perez, “A review of sensor technologies for perception in automated driving,”IEEE Intelligent Transportation Systems Magazine, vol. 11, no. 4, pp. 94– 108, 2019

2019
[15]

Localization strategies for robotic endoscopic capsules: a re- view,

F. Bianchi, A. Masaracchia, E. Shojaei Barjuei, A. Menciassi, A. Arezzo, A. Koulaouzidis, D. Stoyanov, P . Dario, and G. Ciuti, “Localization strategies for robotic endoscopic capsules: a re- view,”Expert review of medical devices, vol. 16, no. 5, pp. 381–403, 2019

2019
[16]

2d-3d point set registra- tion based on global rotation search,

Y. Liu, Y. Dong, Z. Song, and M. Wang, “2d-3d point set registra- tion based on global rotation search,”IEEE Transactions on Image Processing, vol. 28, no. 5, pp. 2599–2613, 2018

2018
[17]

A quaternion-based certifiably optimal solution to the wahba problem with outliers,

H. Yang and L. Carlone, “A quaternion-based certifiably optimal solution to the wahba problem with outliers,” inProceedings of the IEEE/CVF International Conference on Computer Vision, 2019, pp. 1665–1674

2019
[18]

Branch-and-bound meth- ods for euclidean registration problems

C. Olsson, F. Kahl, and M. Oskarsson, “Branch-and-bound meth- ods for euclidean registration problems.”IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 31, no. 5, pp. 783– 794, 2009

2009
[19]

Global optimality for point set registration using semidefinite programming,

J. P . Iglesias, C. Olsson, and F. Kahl, “Global optimality for point set registration using semidefinite programming,” inProceedings of the IEEE/CVF conference on computer vision and pattern recogni- tion, 2020, pp. 8287–8295

2020
[20]

Robust rotation search in computer vision,

A. J. Parra Bustos, “Robust rotation search in computer vision,” Ph.D. dissertation, 2016

2016
[21]

Globally optimal consensus set maximization through rotation search,

J.-C. Bazin, Y. Seo, and M. Pollefeys, “Globally optimal consensus set maximization through rotation search,” inComputer Vision– ACCV 2012: 11th Asian Conference on Computer Vision, Daejeon, Korea, November 5-9, 2012, Revised Selected Papers, Part II 11. Springer, 2013, pp. 539–551

2012
[22]

Convex relaxations for pose JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2025 17 graph optimization with outliers,

L. Carlone and G. C. Calafiore, “Convex relaxations for pose JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2025 17 graph optimization with outliers,”IEEE Robotics and Automation Letters, vol. 3, no. 2, pp. 1160–1167, 2018

2025
[24]

Glob- ally optimal linear model fitting with unit-norm constraint,

Y. Liu, Y. Wang, M. Wang, G. Chen, A. Knoll, and Z. Song, “Glob- ally optimal linear model fitting with unit-norm constraint,” International Journal of Computer Vision, vol. 130, no. 4, pp. 933– 946, 2022

2022
[25]

How the hough transform was invented [dsp his- tory],

P . E. Hart, “How the hough transform was invented [dsp his- tory],”IEEE Signal Processing Magazine, vol. 26, no. 6, pp. 18–22, 2009

2009
[26]

A survey of hough transform,

P . Mukhopadhyay and B. B. Chaudhuri, “A survey of hough transform,”Pattern Recognition, vol. 48, no. 3, pp. 993–1010, 2015

2015
[27]

Efficient absolute orientation re- visited,

M. Lourakis and G. Terzakis, “Efficient absolute orientation re- visited,” in2018 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS). IEEE, 2018, pp. 5813–5818

2018
[28]

Estimating 3-d rigid body transformations: a comparison of four major algorithms,

D. W. Eggert, A. Lorusso, and R. B. Fisher, “Estimating 3-d rigid body transformations: a comparison of four major algorithms,” Machine vision and applications, vol. 9, no. 5-6, pp. 272–290, 1997

1997
[29]

Least-squares fitting of two 3-d point sets,

K. S. Arun, T. S. Huang, and S. D. Blostein, “Least-squares fitting of two 3-d point sets,”IEEE Transactions on pattern analysis and machine intelligence, no. 5, pp. 698–700, 1987

1987
[30]

Least-squares estimation of transformation pa- rameters between two point patterns,

S. Umeyama, “Least-squares estimation of transformation pa- rameters between two point patterns,”IEEE Transactions on Pat- tern Analysis & Machine Intelligence, vol. 13, no. 04, pp. 376–380, 1991

1991
[31]

Closed-form solution of absolute orientation using unit quaternions,

B. P . HORN, “Closed-form solution of absolute orientation using unit quaternions,”Journal of the Optical Society of America. A, Optics and image science, vol. 4, no. 4, pp. 629–642, 1987

1987
[32]

A survey on quaternion algebra and geo- metric algebra applications in engineering and computer science 1995–2020,

E. Bayro-Corrochano, “A survey on quaternion algebra and geo- metric algebra applications in engineering and computer science 1995–2020,”IEEE Access, vol. 9, pp. 104 326–104 355, 2021

1995
[33]

The iteration method for the wahba problem solu- tion,

I. Kruzhilov, “The iteration method for the wahba problem solu- tion,” inAIP Conference Proceedings, vol. 1637, no. 1. American Institute of Physics, 2014, pp. 513–522

2014
[34]

Arcs: Accurate rotation and correspondence search,

L. Peng, M. C. Tsakiris, and R. Vidal, “Arcs: Accurate rotation and correspondence search,” inProceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, 2022, pp. 11 153– 11 163

2022
[35]

Random sample consensus: a paradigm for model fitting with applications to image analysis and automated cartography,

M. A. Fischler and R. C. Bolles, “Random sample consensus: a paradigm for model fitting with applications to image analysis and automated cartography,”Communications of the ACM, vol. 24, no. 6, pp. 381–395, 1981

1981
[36]

ANSAC: adaptive non-minimal sample and consensus,

V . Fragoso, C. Sweeney, P . Sen, and M. A. Turk, “ANSAC: adaptive non-minimal sample and consensus,” inBritish Machine Vision Conference 2017, BMVC 2017, London, UK, September 4-7, 2017. BMVA Press, 2017. [Online]. Available: https://www.dropbox.com/s/sy3zrke54b4fq1k/0105.pdf

2017
[37]

Andersen,Modern methods for robust regression

R. Andersen,Modern methods for robust regression. Sage, 2008, no. 152

2008
[38]

Estimation contracts for outlier-robust geomet- ric perception,

L. Carloneet al., “Estimation contracts for outlier-robust geomet- ric perception,”Foundations and Trends® in Robotics, vol. 11, no. 2-3, pp. 90–224, 2023

2023
[39]

Certifiably optimal rotation and pose estimation based on the cayley map,

T. D. Barfoot, C. Holmes, and F. D ¨umbgen, “Certifiably optimal rotation and pose estimation based on the cayley map,” 2023

2023
[40]

Guaranteed outlier removal for point cloud registration with correspondences,

´A. Parra Bustos and T.-J. Chin, “Guaranteed outlier removal for point cloud registration with correspondences,”IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 40, no. 12, pp. 2868–2882, 2018

2018
[41]

Outlier removal using duality,

C. Olsson, A. Eriksson, and R. Hartley, “Outlier removal using duality,” in2010 IEEE Computer Society Conference on Computer Vision and Pattern Recognition. IEEE, 2010, pp. 1450–1457

2010
[42]

Efficient and glob- ally optimal camera orientation estimation with line correspon- dences,

T. Huang, Y. Liu, B. Yang, and Y.-H. Liu, “Efficient and glob- ally optimal camera orientation estimation with line correspon- dences,”IEEE Robotics and Automation Letters, vol. 9, no. 11, pp. 10 232–10 239, 2024

2024
[43]

Hartley and A

R. Hartley and A. Zisserman,Multiple view geometry in computer vision. Cambridge university press, 2003

2003
[44]

Global optimization through rotation space search,

R. I. Hartley and F. Kahl, “Global optimization through rotation space search,”International Journal of Computer Vision, vol. 82, no. 1, pp. 64–79, 2009

2009
[45]

The 3d-3d registration problem revisited,

H. Li and R. Hartley, “The 3d-3d registration problem revisited,” in2007 IEEE 11th international conference on computer vision. IEEE, 2007, pp. 1–8

2007
[46]

Globally-optimal inlier set maximisation for camera pose and correspondence estimation,

D. Campbell, L. Petersson, L. Kneip, and H. Li, “Globally-optimal inlier set maximisation for camera pose and correspondence estimation,”IEEE transactions on pattern analysis and machine intelligence, vol. 42, no. 2, pp. 328–342, 2020

2020
[47]

Efficient and robust point cloud registration via heuristics-guided parameter search,

T. Huang, H. Li, L. Peng, Y. Liu, and Y.-H. Liu, “Efficient and robust point cloud registration via heuristics-guided parameter search,”IEEE Transactions on Pattern Analysis and Machine Intelli- gence, vol. 46, no. 10, pp. 6966–6984, 2024

2024
[48]

Scalable 3d registra- tion via truncated entry-wise absolute residuals,

T. Huang, L. Peng, R. Vidal, and Y.-H. Liu, “Scalable 3d registra- tion via truncated entry-wise absolute residuals,” inIEEE/CVF Conference on Computer Vision and Pattern Recognition, 2024, pp. 27 477–27 487

2024
[49]

Teaser: Fast and certifiable point cloud registration,

H. Yang, J. Shi, and L. Carlone, “Teaser: Fast and certifiable point cloud registration,”IEEE Transactions on Robotics, vol. 37, no. 2, pp. 314–333, 2020

2020
[50]

Efficient global point cloud registration by matching rotation invariant features through translation search,

Y. Liu, C. Wang, Z. Song, and M. Wang, “Efficient global point cloud registration by matching rotation invariant features through translation search,” inProceedings of the European Confer- ence on Computer Vision (ECCV), 2018, pp. 448–463

2018
[51]

Qgore: Quadratic-time guaranteed outlier removal for point cloud registration,

J. Li, P . Shi, Q. Hu, and Y. Zhang, “Qgore: Quadratic-time guaranteed outlier removal for point cloud registration,”IEEE Transactions on Pattern Analysis and Machine Intelligence, 2023

2023
[52]

A practicalo(n 2)outlier removal method for correspondence-based point cloud registration,

J. Li, “A practicalo(n 2)outlier removal method for correspondence-based point cloud registration,”IEEE Transac- tions on Pattern Analysis and Machine Intelligence, vol. 44, no. 8, pp. 3926–3939, 2021

2021
[53]

Mac: Maximal cliques for 3d registration,

J. Yang, X. Zhang, P . Wang, Y. Guo, K. Sun, Q. Wu, S. Zhang, and Y. Zhang, “Mac: Maximal cliques for 3d registration,”IEEE Transactions on Pattern Analysis and Machine Intelligence, 2024

2024
[55]

Mutual voting for ranking 3d correspondences,

J. Yang, X. Zhang, S. Fan, C. Ren, and Y. Zhang, “Mutual voting for ranking 3d correspondences,”IEEE Transactions on Pattern Analysis & Machine Intelligence, no. 01, pp. 1–18, 2023

2023
[56]

Deterministic point cloud registration via novel transformation decomposition,

W. Chen, H. Li, Q. Nie, and Y.-H. Liu, “Deterministic point cloud registration via novel transformation decomposition,” in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, 2022, pp. 6348–6356

2022
[57]

Linear model selection and regular- ization,

G. James, D. Witten, T. Hastie, R. Tibshirani, G. James, D. Witten, T. Hastie, and R. Tibshirani, “Linear model selection and regular- ization,”An introduction to statistical learning: with applications in R, pp. 225–288, 2021

2021
[58]

Globally optimal vertical direction estimation in atlanta world,

Y. Liu, G. Chen, and A. Knoll, “Globally optimal vertical direction estimation in atlanta world,”IEEE Transactions on Pattern Analysis & Machine Intelligence, vol. 44, no. 04, pp. 1949–1962, 2022

1949
[59]

Brooks and H

F. Brooks and H. Kugler,No silver bullet. April, 1987

1987
[61]

Practical optimal registration of terrestrial lidar scan pairs,

Z. Cai, T.-J. Chin, A. P . Bustos, and K. Schindler, “Practical optimal registration of terrestrial lidar scan pairs,”ISPRS journal of photogrammetry and remote sensing, vol. 147, pp. 118–131, 2019

2019
[62]

Multi-model 3d registration: Finding multiple moving objects in cluttered point clouds,

D. Jin, S. Karmalkar, H. Zhang, and L. Carlone, “Multi-model 3d registration: Finding multiple moving objects in cluttered point clouds,” in2024 IEEE International Conference on Robotics and Automation (ICRA). IEEE, 2024, pp. 4990–4997

2024
[63]

Event-based motion segmentation by motion compensation,

T. Stoffregen, G. Gallego, T. Drummond, L. Kleeman, and D. Scaramuzza, “Event-based motion segmentation by motion compensation,” inProceedings of the IEEE/CVF International Con- ference on Computer Vision, 2019, pp. 7244–7253

2019
[64]

Great circular surfaces in the three-sphere,

S. Izumiya, T. Nagai, and K. Saji, “Great circular surfaces in the three-sphere,”Differential Geometry and its Applications, vol. 29, no. 3, pp. 409–425, 2011

2011
[65]

Loxodrome, orthodrome, stereodrome,

W. Immler, “Loxodrome, orthodrome, stereodrome,”The Interna- tional Hydrographic Review, 1936

1936
[66]

Globally optimal solutions for unit-norm constrained computer vision problems,

Y. Liu, “Globally optimal solutions for unit-norm constrained computer vision problems,” Ph.D. dissertation, Technische Uni- versit¨at M ¨unchen, 2022

2022
[67]

Absolute pose estimation with a known direction by motion decoupling,

Y. Liu, G. Chen, and A. Knoll, “Absolute pose estimation with a known direction by motion decoupling,”IEEE Transactions on Circuits and Systems for Video Technology, 2023

2023
[68]

Strang,Linear Algebra and Its Applications 4th ed., 2012

G. Strang,Linear Algebra and Its Applications 4th ed., 2012

2012
[69]

[Online]

Quaternions and spatial rotation. [Online]. Available: https: //en.wikipedia.org/wiki/Quaternions and spatial rotation
[70]

Rodrigues, “Des lois g ´eom´etriques qui r ´egissent les d´eplacements d’un syst `eme solide dans l’espace, et de la varia- JOURNAL OF LATEX CLASS FILES, VOL

O. Rodrigues, “Des lois g ´eom´etriques qui r ´egissent les d´eplacements d’un syst `eme solide dans l’espace, et de la varia- JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2025 18 tion des coordonn ´ees provenant de ces d ´eplacements consid´er´es ind´ependamment des c...

2025
[71]

Robustly solving pnl problem using clifford tori,

Y. Liu, X. Liu, S. Ding, and Z.-x. Yang, “Robustly solving pnl problem using clifford tori,”Pattern Recognition, p. 111659, 2025

2025
[72]

Efficient algorithms for outlier-robust regression,

A. Klivans, P . K. Kothari, and R. Meka, “Efficient algorithms for outlier-robust regression,” inConference On Learning Theory. PMLR, 2018, pp. 1420–1430

2018
[73]

Efficient large scale inlier voting for geometric vision problems,

D. Aiger, S. Lynen, J. Hosang, and B. Zeisl, “Efficient large scale inlier voting for geometric vision problems,” inProceedings of the IEEE/CVF International Conference on Computer Vision, 2021, pp. 3243–3251

2021
[74]

General techniques for approximate incidences and their ap- plication to the camera posing problem,

D. Aiger, H. Kaplan, E. Kokiopoulou, M. Sharir, and B. Zeisl, “General techniques for approximate incidences and their ap- plication to the camera posing problem,” in35th International Symposium on Computational Geometry (SoCG 2019). Schloss Dagstuhl-Leibniz-Zentrum fuer Infor...

2019
[75]

The geometry of m ¨obius transformations,

J. Olsen, “The geometry of m ¨obius transformations,”Rochester: University of Rochester, 2010

2010
[76]

On quaternion based parameterization of orientation in computer vision and robotics,

G. Terzakis, P . Culverhouse, G. Bugmann, S. Sharma, and R. Sut- ton, “On quaternion based parameterization of orientation in computer vision and robotics,”Journal of Engineering Science and Technology Review, vol. 7, no. 1, pp. 82–93, 2014

2014
[77]

Real-time detection of planar regions in unorganized point clouds,

F. A. Limberger and M. M. Oliveira, “Real-time detection of planar regions in unorganized point clouds,”Pattern Recognition, vol. 48, no. 6, pp. 2043–2053, 2015

2015
[78]

Robust detection of lines using the progressive probabilistic hough transform,

J. Matas, C. Galambos, and J. Kittler, “Robust detection of lines using the progressive probabilistic hough transform,”Computer vision and image understanding, vol. 78, no. 1, pp. 119–137, 2000

2000
[79]

Deep hough transform for semantic line detection,

K. Zhao, Q. Han, C.-B. Zhang, J. Xu, and M.-M. Cheng, “Deep hough transform for semantic line detection,”IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 44, no. 9, pp. 4793–4806, 2021

2021
[80]

Consensus set maximization with guaranteed global optimality for robust geometry estimation,

H. Li, “Consensus set maximization with guaranteed global optimality for robust geometry estimation,” in2009 IEEE 12th International Conference on Computer Vision. IEEE, 2009, pp. 1074– 1080

2009
[81]

Applying the expectation- maximization algorithm to multivariate signal extraction,

J. Livsey and T. McElroy, “Applying the expectation- maximization algorithm to multivariate signal extraction,”Jour- nal of Official Statistics, vol. 40, no. 4, pp. 660–684, 2024

2024
[82]

Em converges for a mixture of many linear regressions,

J. Kwon and C. Caramanis, “Em converges for a mixture of many linear regressions,”ArXiv, vol. abs/1905.12106, 2019. [Online]. Available: https://api.semanticscholar.org/CorpusID:168169665

1905 arXiv
[83]

Progressive-x: Efficient, anytime, multi- model fitting algorithm,

D. Barath and J. Matas, “Progressive-x: Efficient, anytime, multi- model fitting algorithm,” inProceedings of the IEEE/CVF interna- tional conference on computer vision, 2019, pp. 3780–3788

2019
[84]

Transformation decoupling strategy based on screw theory for deterministic point cloud registration with gravity prior,

X. Li, Z. Ma, Y. Liu, W. Zimmer, H. Cao, F. Zhang, and A. Knoll, “Transformation decoupling strategy based on screw theory for deterministic point cloud registration with gravity prior,”IEEE Transactions on Pattern Analysis & Machine Intelligence, no. 01, pp. 1–18, 2024

2024
[85]

A new outlier removal strategy based on reliability of correspondence graph for fast point cloud registration,

L. Yan, P . Wei, H. Xie, J. Dai, H. Wu, and M. Huang, “A new outlier removal strategy based on reliability of correspondence graph for fast point cloud registration,”IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 45, no. 07, pp. 7986– 8002, jul 2023

2023
[86]

Sc 2-pcr++: Rethinking the generation and selection for efficient and robust point cloud registration,

Z. Chen, K. Sun, F. Yang, L. Guo, and W. Tao, “Sc 2-pcr++: Rethinking the generation and selection for efficient and robust point cloud registration,”IEEE Transactions on Pattern Analysis and Machine Intelligence, 2023

2023
[87]

Deepftsg: Multi-stream asymmetric use-net trellis encoders with shared decoder feature fusion architecture for video motion segmentation,

G. Rahmon, K. Palaniappan, I. E. Toubal, F. Bunyak, R. Rao, and G. Seetharaman, “Deepftsg: Multi-stream asymmetric use-net trellis encoders with shared decoder feature fusion architecture for video motion segmentation,”International Journal of Computer Vision, vol. 132, no. 3,...

2024
[88]

Image alignment for panorama stitching in sparsely structured environments,

G. Meneghetti, M. Danelljan, M. Felsberg, and K. Nordberg, “Image alignment for panorama stitching in sparsely structured environments,” inImage Analysis: 19th Scandinavian Conference, SCIA 2015, Copenhagen, Denmark, June 15-17, 2015. Proceedings 19. Springer, 2015, pp. 428–439

2015
[89]

When to use what feature? sift, surf, orb, or a-kaze features for monocular visual odometry,

H.-J. Chien, C.-C. Chuang, C.-Y. Chen, and R. Klette, “When to use what feature? sift, surf, orb, or a-kaze features for monocular visual odometry,” in2016 International Conference on Image and Vision Computing New Zealand (IVCNZ). IEEE, 2016, pp. 1–6

2016
[90]

3dmatch: Learning local geometric descriptors from rgb-d reconstructions,

A. Zeng, S. Song, M. Nießner, M. Fisher, J. Xiao, and T. Funkhouser, “3dmatch: Learning local geometric descriptors from rgb-d reconstructions,” inProceedings of the IEEE conference on computer vision and pattern recognition, 2017, pp. 1802–1811

2017
[91]

Predator: Registration of 3d point clouds with low overlap,

S. Huang, Z. Gojcic, M. Usvyatsov, A. Wieser, and K. Schindler, “Predator: Registration of 3d point clouds with low overlap,” inProceedings of the IEEE/CVF Conference on computer vision and pattern recognition, 2021, pp. 4267–4276

2021
[92]

Are we ready for au- tonomous driving? the kitti vision benchmark suite,

A. Geiger, P . Lenz, and R. Urtasun, “Are we ready for au- tonomous driving? the kitti vision benchmark suite,” in2012 IEEE conference on computer vision and pattern recognition. IEEE, 2012, pp. 3354–3361

2012
[93]

The perfect match: 3d point cloud matching with smoothed densities,

Z. Gojcic, C. Zhou, J. D. Wegner, and A. Wieser, “The perfect match: 3d point cloud matching with smoothed densities,” in Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, 2019, pp. 5545–5554

2019
[94]

Geometric transformer for fast and robust point cloud registration,

Z. Qin, H. Yu, C. Wang, Y. Guo, Y. Peng, and K. Xu, “Geometric transformer for fast and robust point cloud registration,” inPro- ceedings of the IEEE/CVF conference on computer vision and pattern recognition, 2022, pp. 11 143–11 152

2022
[95]

Fast point feature his- tograms (fpfh) for 3d registration,

R. B. Rusu, N. Blodow, and M. Beetz, “Fast point feature his- tograms (fpfh) for 3d registration,” in2009 IEEE international conference on robotics and automation. IEEE, 2009, pp. 3212–3217

2009
[96]

Ransac for robotic applications: A survey,

J. M. Mart ´ınez-Otzeta, I. Rodr´ıguez-Moreno, I. Mendialdua, and B. Sierra, “Ransac for robotic applications: A survey,”Sensors, vol. 23, no. 1, p. 327, 2022

2022
[97]

3d shapenets: A deep representation for volumetric shapes,

Z. Wu, S. Song, A. Khosla, F. Yu, L. Zhang, X. Tang, and J. Xiao, “3d shapenets: A deep representation for volumetric shapes,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, 2015, pp. 1912–1920

2015
[98]

Accurate peak detection in multimodal optimization via approximated landscape learn- ing,

Z. Ma, H. Lian, W. Qiu, and Y.-J. Gong, “Accurate peak detection in multimodal optimization via approximated landscape learn- ing,”arXiv preprint arXiv:2503.18066, 2025

2025 arXiv
[99]

Three-axis attitude determination from vector observations,

M. D. Shuster and S. D. Oh, “Three-axis attitude determination from vector observations,”Journal of guidance and Control, vol. 4, no. 1, pp. 70–77, 1981

1981
[100]

Quaternions and rotations,

Y.-B. Jia, “Quaternions and rotations,”Com S, vol. 477, no. 577, p. 15, 2008

2008
[101]

Understanding quaternions,

R. Goldman, “Understanding quaternions,”Graphical models, vol. 73, no. 2, pp. 21–49, 2011

2011
[102]

J. B. Kuipers,Quaternions and rotation sequences: a primer with applications to orbits, aerospace, and virtual reality. Princeton university press, 1999

1999
[103]

Linear algebra: Theory and applica- tions,

W. Cheney and D. Kincaid, “Linear algebra: Theory and applica- tions,”The Australian Mathematical Society, vol. 110, pp. 544–550, 2009

2009
[104]

Optimal linear attitude estimators via geometric analysis,

D.-r. Gong, X.-w. Shao, W. Li, and D.-p. Duan, “Optimal linear attitude estimators via geometric analysis,”Journal of Zhejiang University-SCIENCE A, vol. 12, pp. 873–882, 2011

2011
[105]

Second estimator of the optimal quaternion,

D. Mortari, “Second estimator of the optimal quaternion,”Journal of Guidance, Control, and Dynamics, vol. 23, no. 5, pp. 885–888, 2000

2000
[106]

Generalized n-dimensional rigid registra- tion: Theory and applications,

J. Wu, M. Wang, H. Fourati, H. Li, Y. Zhu, C. Zhang, Y. Jiang, X. Hu, and M. Liu, “Generalized n-dimensional rigid registra- tion: Theory and applications,”IEEE Transactions on Cybernetics, vol. 53, no. 2, pp. 927–940, 2022

2022
[107]

On cayley’s fac- torization with an application to the orthonormalization of noisy rotation matrices,

S. Sarabandi, A. Perez-Gracia, and F. Thomas, “On cayley’s fac- torization with an application to the orthonormalization of noisy rotation matrices,”Advances in Applied Clifford Algebras, vol. 29, no. 3, p. 49, 2019

2019
[108]

Rotation averaging,

R. Hartley, J. Trumpf, Y. Dai, and H. Li, “Rotation averaging,” International journal of computer vision, vol. 103, pp. 267–305, 2013

2013
[109]

Frank,Modern Robotics-Mechanics, Planning, and Control

C. Frank,Modern Robotics-Mechanics, Planning, and Control. Cam- bridge University Press, 2017

2017
[110]

babaiasl

M. babaiasl. Cayley-rodrigues parameters to express orientations in robotics. [Online]. Available: https://mecharithm.com/learning/lesson/ cayley-rodrigues-parameters-to-express-orientations-in-robotics-13
[111]

Cayley transform: Quaternion homography

Wikipedia. Cayley transform: Quaternion homography. [Online]. Available: https://en.wikipedia.org/wiki/Cayley transform
[112]

Estimating se (3) elements using a dual quaternion based linear kalman filter

R. A. Srivatsan, G. T. Rosen, D. F. N. Mohamed, and H. Choset, “Estimating se (3) elements using a dual quaternion based linear kalman filter.” inRobotics: Science and systems, 2016

2016
[113]

Registration with a small number of sparse measurements,

R. Arun Srivatsan, N. Zevallos, P . Vagdargi, and H. Choset, “Registration with a small number of sparse measurements,”The International Journal of Robotics Research, vol. 38, no. 12-13, pp. 1403–1419, 2019. JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2025 19 APPENDIX...

2019
[116]

A.1 Special Example Furthermore, we here directly provide an eigenvector matrix ofM, which can be verified by mathematical software, e.g., MATLAB11 and SymPy 12

Moreover,qis orthogonal to the other eigenvectors corresponding to eigenvalue−1. A.1 Special Example Furthermore, we here directly provide an eigenvector matrix ofM, which can be verified by mathematical software, e.g., MATLAB11 and SymPy 12. The matrix is however not orthogon...
[117]

https://www.mathworks.com/
[118]

0 a−b # =−1·

https://www.sympy.org/en/index.html JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2025 21 In addition [105], M " 0 a−b # =−1· " 0 a−b # (85) Therefore, 0a−b T is an eigenvector ofM, who is corresponding to eigenvalue−1. Notably, whena−b=0, then 0a−b T is a trivial solut...

2025
[119]

In this case, the intersection of two hyper planes cannot intersect the sphere surface

Intersect only at the origin. In this case, the intersection of two hyper planes cannot intersect the sphere surface. Therefore, the two quaternion circles cannot intersect
[120]

In this case, the intersecting line will meet the unit sphere surface at two symmetric points±q ∗, which are also the intersection points of two quaternion circles

Intersect in a straight line (cross the origin point). In this case, the intersecting line will meet the unit sphere surface at two symmetric points±q ∗, which are also the intersection points of two quaternion circles. JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2025...

2025
[2018]

Available: http://arxiv.org/abs/1812.11307

[Online]. Available: http://arxiv.org/abs/1812.11307

arXiv
[2019]

Available: http://arxiv.org/abs/1902.01534

[Online]. Available: http://arxiv.org/abs/1902.01534

1902 arXiv

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [23]

Optimal linear attitude estimator,

D. Mortari, F. L. Markley, and P . Singla, “Optimal linear attitude estimator,”Journal of Guidance, Control, and Dynamics, vol. 30, no. 6, pp. 1619–1627, 2007

2007

[2] [1]

Fast rotation search with stereographic projections for 3d registra- tion,

A. P . Bustos, T.-J. Chin, A. Eriksson, H. Li, and D. Suter, “Fast rotation search with stereographic projections for 3d registra- tion,”IEEE transactions on pattern analysis and machine intelligence, vol. 38, no. 11, pp. 2227–2240, 2016. JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2025 16 2 3 4 5 6 7 8 9 # model number 0 1000 2000 3000 4000 Medi...

2016

[3] [2]

Hybrid camera pose estimation,

F. Camposeco, A. Cohen, M. Pollefeys, and T. Sattler, “Hybrid camera pose estimation,” inProceedings of the IEEE Conference on Computer Vision and Pattern Recognition, 2018, pp. 136–144

2018

[4] [3]

Szeliski,Computer vision: algorithms and applications

R. Szeliski,Computer vision: algorithms and applications. Springer Nature, 2022

2022

[5] [4]

I love my attitude problem,

F. L. Markley, “I love my attitude problem,” inAIAA Guidance, Navigation, and Control Conference, no. AIAA-2005-5927, 2006

2005

[6] [5]

The generalized wahba problem,

M. D. Shuster, “The generalized wahba problem,”The Journal of the Astronautical Sciences, vol. 54, no. 2, pp. 245–259, 2006

2006

[7] [6]

Equivalence of two solutions of wahba’s prob- lem,

F. L. Markley, “Equivalence of two solutions of wahba’s prob- lem,”The Journal of the Astronautical Sciences, vol. 60, pp. 303–312, 2013

2013

[8] [7]

Numerical solution of a generalized wahba problem for a spinning spacecraft,

M. L. Psiaki and J. C. Hinks, “Numerical solution of a generalized wahba problem for a spinning spacecraft,”Journal of Guidance, Control, and Dynamics, vol. 35, no. 3, pp. 764–773, 2012

2012