Uncertainty Quantification for Visual Object Pose Estimation: S-Lemma Ellipsoidal Bounds

Charis Georgiou; Lorenzo Shaikewitz; Luca Carlone

REVIEW 2 major objections 4 minor 1 cited by

Uncertainty Quantification for Visual Object Pose Estimation: S-Lemma Ellipsoidal Bounds

T0 review · 2 major / 4 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read A convex program turns keypoint error bounds into ellipsoidal pose uncertainty sets guaranteed to contain the true pose with high probability.

desk verdict SLUE is a genuinely useful convex relaxation for shape-adaptive ellipsoidal pose bounds, but the paper's 'high probability' pose-coverage claim outruns the math: the guarantee is conditional on all keypoint bounds holding. read the letter →

arxiv 2511.21666 v2 pith:OYUCD7UQ submitted 2025-11-26 cs.RO cs.CV

Lorenzo Shaikewitz , Charis Georgiou , Luca Carlone This is my paper

classification cs.ROcs.CV

keywords poseuncertaintyellipsoidalboundsS-lemmasum-of-squaresrelaxationconformalpredictionkeypointdetectionmonocularestimationdistribution-free

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

This paper aims to give a single RGB image a statistically honest statement about where an object is: not just a pose estimate, but an ellipsoidal region that is guaranteed to contain the true pose with a chosen confidence, under almost no distributional assumptions. The only input is a set of pixel keypoint detections each equipped with a high-probability error bound, as obtained from conformal prediction. The paper shows that the non-convex set of poses consistent with those keypoint bounds can be outer-approximated by the minimum-volume ellipsoid, computed by a convex program based on the generalized S-lemma; a sum-of-squares hierarchy provably converges to the true minimum-volume ellipsoid. The resulting bound projects onto independent translation and axis-angle ellipsoids, and experiments on three real datasets show the translation bounds are orders of magnitude tighter than prior fixed-shape methods at a fraction of the runtime.

What carries the argument

The central object is the generalized S-lemma: it states that if a quadratic form W is nonnegative on a set defined by quadratic inequalities and equalities, then W can be written as a nonnegative combination of those constraint quadratics plus a sum-of-squares polynomial. The paper uses the one-way implication to relax the minimum-volume bounding ellipsoid problem into a convex SDP, with the log-determinant of the ellipsoid matrix as the objective. An SOS hierarchy, where the dual multiplier polynomials are allowed to grow in degree, refines this relaxation and is guaranteed to converge to the true minimum-volume ellipsoid as the order goes to infinity. The infinity-norm keypoint bound and

What would settle it

Run SLUE on an exchangeable test set of several hundred frames from a calibrated camera with a known textured object; if the ground-truth pose is inside the SLUE ellipsoid less often than the chosen confidence even in frames where every keypoint lies inside its conformal bound, the method's central guarantee fails.

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Extended reading notes

Core claim

SLUE computes an ellipsoid H such that any pose consistent with the keypoint noise bounds (and hence, with high probability, the true pose) lies inside the ellipsoid centered at the estimate. The construction reformulates the reprojection and chirality constraints as quadratic inequalities, then invokes the generalized S-lemma to dualize them into a single matrix inequality; maximizing log det(H) under that inequality is a convex semidefinite program whose solution is an outer ellipsoidal bound. Raising the relaxation order via sum-of-squares multipliers yields a hierarchy of bounds that provably converges, as the order grows, to the minimum-volume ellipsoid enclosing the constraint set. The

Load-bearing premise

The guarantee collapses if the per-keypoint error bounds do not all hold simultaneously for the same frame, or if the calibration data used to set those bounds is not representative of the test images.

Editorial extensions

If this is right

Any keypoint-based pose estimator can be wrapped with a distribution-free, confidence-labeled ellipsoidal uncertainty set without retraining or changing the front-end.
The joint ellipsoid provides correlated translation-orientation uncertainty, which can be projected to separate translation and axis-angle bounds for planners that treat them independently.
The SOS hierarchy gives an explicit compute-accuracy trade-off: first-order for real-time, higher-order for tighter sets, with guaranteed convergence to the minimum-volume bound.
Because the ellipsoid is free to stretch along the optical axis, it automatically captures monocular scale ambiguity that fixed-shape methods miss.
The bound is a conservative outer approximation, so downstream safety checks can treat it as a certificate rather than a heuristic scatter.

Reading between the lines

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

The practical value of SLUE hinges on the conformal calibration step that produces the keypoint bounds; if those bounds are replaced by heuristic estimates, the same SDP machinery will still output an ellipsoid, but its coverage guarantee will be only as good as the input.
The paper's worst-case multi-keypoint bound (beta >= 1 - N alpha) is nearly vacuous for typical N, which suggests a natural extension: an outlier-aware or subset-of-keypoints formulation that does not require every keypoint to be in-bounds simultaneously.
The ellipsoid volume could serve as an active-perception reward, since it directly quantifies remaining pose ambiguity in a form that is cheap to evaluate.
For near-symmetric objects, the ellipsoid will stretch along the symmetry directions but cannot represent disconnected pose components (such as a bowl rotated 180 degrees); a union-of-ellipsoids or multimodal representation would be needed to be faithful.

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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

Two things to know before reading. The algorithmic core is solid and genuinely useful: SLUE turns infinity-norm keypoint constraints into a log-det SDP whose solution is a true outer ellipsoid of the implied pose constraint set, and it runs in tens of milliseconds. The second thing is that the advertised statistical guarantee does not survive contact with the paper's own numbers. The S-lemma/SOS chain is correct as geometry — the ellipsoid does outer-bound the set P∞. But P∞ contains the true pose only when every keypoint bound holds simultaneously, and Proposition 1 gives worst-case β = 1 − Nα, which is zero for N=10 at α=0.1 and about 35% under independence. Table I shows the gap: at α=0.4, P∞ pose coverage is 5–15%, SLUE 19–46%. The abstract's 'guaranteed to contain the true pose with high probability' is conditional, not distribution-free at the pose level. The Limitations section admits they don't use the probabilistic nature of keypoint bounds; that's the right place to fix it.

Credit where due. The novelty is the combination: shape-adaptive ellipsoidal bounds via generalized S-lemma, the SOS hierarchy with convergence to the minimum-volume ellipsoid, the quaternion encoding, and clean projection to translation and axis-angle bounds. I checked the quaternion identities and constraint matrices; they are consistent. The runtime comparison is favorable, and releasing code helps reproducibility. Self-citation is not an issue; the foundations are properly attributed.

Soft spots beyond the guarantee. The headline volume comparison is confounded: SLUE consumes p=∞ bounds, baselines use p=2, and 2–6% of frames are dropped where some method fails. The YCB-V synthetic calibration is an acknowledged exchangeability violation and under-covers at α=0.4. These are addressable but should be reported directly.

Who this is for: robotics readers who want practical, interpretable pose uncertainty without distributional assumptions, and optimization people interested in S-lemma/SOS applications. It deserves a serious referee — the method is new, checkable, and faster than prior work — but the statistical claims need reworking and the comparisons need cleaning up before publication. Send it out.

Referee Report

2 major / 4 minor

Summary. The paper develops SLUE, a convex-optimization method for computing ellipsoidal uncertainty bounds on 6-DOF object pose from per-keypoint pixel uncertainty sets. The authors assume each 2D keypoint detection has a high-probability infinity-norm error bound, derive a non-convex pose constraint set, and relax the minimum-volume bounding ellipsoid problem using a generalized S-lemma/SOS hierarchy. They also provide a projection to translation-only and axis-angle ellipsoids. Experiments on LM-O, YCB-V and a drone tracking dataset compare bound volumes and runtime against RANSAG and GRCC. The paper claims the resulting ellipsoid is guaranteed to contain the true pose with high probability and that the SOS hierarchy converges to the minimum-volume ellipsoid.

Significance. The geometric core of the paper is solid and useful: given a set of quadratic pose constraints, the SDP (8) returns a genuine outer ellipsoid, and the SOS hierarchy (11) plausibly converges under compactness. The quaternion implementation, explicit constraint matrices, released code, and careful runtime experiments are strengths. If the statistical claim were correct, this would be a significant step in distribution-free pose UQ. However, the advertised high-probability pose guarantee is not established; the theory only guarantees coverage conditional on all keypoint bounds holding, and the experiments show substantially lower pose-level coverage. The contribution is better viewed as a fast, shape-optimized geometric outer-bounding method for a given keypoint uncertainty set, rather than a distribution-free pose confidence set.

major comments (2)

[Section IV-A, Prop. 1, Table I] The headline guarantee in the Abstract and Problem 1 — a single ellipsoid that "is guaranteed to contain the true object pose with high probability" — is not delivered. Prop. 1 gives beta ≥ 1 − sum_i alpha_i for the constraint set (P_inf); for N≈10 and alpha_i=0.1 this is only beta>0, and under independence beta=(0.9)^10≈35%. Since Prop. 2's S-lemma dual only outer-bounds (P_inf), the SLUE ellipsoid inherits this weak coverage. Table I confirms the gap: at α=0.4, (P_inf) coverage is 5.4–15.1% and SLUE 19.5–45.8%; at α=0.1, SLUE is 86.5–91.1%, below the nominal 90% on CAST. The Limitations section explicitly acknowledges the multiple-testing issue. The guarantee can be repaired by using a joint calibration (e.g., α_i = α/N) or a joint conformal score, or by restating the claim as conditional on all keypoint bounds holding; as written the abstract's claim is unsupported.
[Appendix C, proof of Theorem 6] The convergence proof relies on Theorem A8, which requires the set (6) to be compact, but compactness is never established. For fixed R, the backprojection constraints are affine in t; if all detected keypoints lie within r_i of the principal point in infinity norm, the feasible set is unbounded along the optical axis. In that case no finite ellipsoid can contain (P_inf), the minimum-volume problem is ill-posed, and the SDP (11) may be infeasible. The authors should state sufficient conditions for compactness (e.g., at least one keypoint with max(|u_i−c_x|,|v_i−c_y|) > r_i) or describe how unbounded/infeasible cases are handled.

minor comments (4)

[Section V-B and Section VII-C] The text states "volume is inversely proportional to log det(H)"; the correct relation is volume ∝ det(H)^{-1/2}. Likewise, "translation volume is (4π/3) log det(H_t)" is dimensionally inconsistent. If a log-volume metric is intended, the sign should be −0.5 log det, and this should be stated explicitly.
[Appendix C] Typo: "Therorem" appears twice (Theorem A8 and Theorem 6 proof).
[Section III] The notation S^n is used both for the set of symmetric n×n matrices and for the unit sphere. Please disambiguate to avoid confusion.
[Section VII-C] The acronym "RANSAG" is used but never expanded; consider defining it at first use or using the original method name.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: SLUE's ellipsoid is a forward SDP outer approximation of the pose constraint set; the only fitted inputs are the declared conformal keypoint radii, and the statistical gap is an acknowledged validity caveat, not a circular step.

full rationale

The derivation chain is self-contained as a geometric propagation: eq. (3) bounds each keypoint error with probability 1-alpha_i; Proposition 1 builds the pose constraint set P_infinity with coverage beta; eq. (6) rewrites P_infinity as quadratic constraints; Propositions 3 and 5 use the generalized S-lemma/SOS dual to produce an outer ellipsoid via the SDPs (8) and (11); Theorem 6's convergence is supported by an external Positivstellensatz ([25]) and an appendix proof. No step fits a parameter to the quantity it later claims to predict, and no quantity is defined in terms of its own output. The conformal radii r_i(alpha) are calibrated inputs and the paper explicitly says 'In this paper we treat r_i(alpha) as a given constant'; the ellipsoid is computed forward from them. The self-citations ([15] for the CAST dataset, [43] for quaternion product matrices) are data/algebra sources and are not load-bearing for the central derivation. The known statistical limitation—pose coverage of P_infinity can be far below 1-alpha (Proposition 1 and Table I), and Section VIII states 'we do not make use of the high-probability nature of the keypoint bounds'—is a correctness/validity caveat about the claimed high-probability guarantee, not a circularity. The paper also openly notes in Section VII-A that synthetic calibration 'may violate exchangeability.' These are honest limitations rather than hidden reductions, so no circular step is present.

Assumptions & free parameters 5 free parameters · 7 assumptions · 0 invented entities

The paper's actual contribution is a forward relaxation from given constraint matrices; the only fitted inputs are the conformal radii. The ledger is therefore dominated by domain assumptions on the input layer (correct correspondences, exchangeability, simultaneous keypoint coverage), not by hidden parameters in the derivation. No new entities are postulated: the ellipsoid is an output representation, and the S-lemma/SOS machinery is imported standard mathematics.

free parameters (5)

Keypoint bound radii r_i(alpha) = adjusted (1-alpha)(1+1/n) quantile of c_i*||y_i - z_i||_p, divided by c_i
Data-fitted from calibration images; the whole pose-bound guarantee and all reported volumes are conditional on these radii being valid (Section VII-A, eq. 19).
Confidence-weighted score metric s(y_i,c_i; z_i) = c_i*||y_i - z_i||_p with c_i the detector confidence (fixed to 1 for CAST)
Design choice defining what the conformal radii bound; changing the weight changes both radii and reported volumes (Section VII-A).
Confidence level alpha = 0.1 and 0.4
User-chosen; every coverage number and volume result is conditional on it; at alpha=0.4 the advertised 'high probability' becomes 19.5-45.8% pose coverage in Table I.
Calibration source and size = 200 BOP images (LM-O), 200 random frames (CAST), 2000 BlenderProc synthetic (YCB-V)
The choice determines exchangeability; the synthetic choice violates it and produces 47.4% keypoint coverage at alpha=0.4 versus the 60% target (Section VII-A).
Relaxation order kappa = kappa=1 (matrix form, 39 ms), kappa=2 (quaternion form, 548 ms), kappa=3 ablation
Compute/tightness knob; no data-driven selection; kappa=2 is the recommended operating point (Section VII-D, Appendix D).

assumptions (7)

domain assumption Measurement model (2): each detected keypoint y_i equals the projected 3D model keypoint plus noise epsilon_i.
Invoked in Section IV-A to derive the reprojection constraint (4); requires correct keypoint correspondences (no outliers) and a pinhole camera; broken detections violate (P_inf).
domain assumption Keypoint noise bound (3): P(||epsilon_i||_p <= r_i(alpha)) >= 1 - alpha_i, obtained by split conformal prediction.
The entire coverage chain (Prop. 1) starts here; validity requires exchangeability of calibration and test data, which is deliberately violated for YCB-V (Section VII-A).
standard math Generalized (lossy) S-lemma: dual implies primal (Prop. 2, from [12], [30]).
The ellipsoid outer-bound guarantee in (8) is conditional on this implication; the relaxation gap means volume optimality is not guaranteed at finite order.
standard math Positivstellensatz (Theorem A8 of [25]) and compactness of the constraint set (6).
Used in the proof of Thm. 6 (App. C) for hierarchy convergence; compactness of (6) is assumed without proof.
standard math The 15 quadratic equalities Q_1..Q_15 exactly encode R in SO(3).
Appendix B: unit column norms, orthogonality, and right-hand-rule cross products; standard and correct encoding.
standard math Conformal quantile theorem [28, Thm. 3.2]: the adjusted (1-alpha)(1+1/n) quantile yields 1-alpha marginal coverage.
Imported theorem that justifies (3) under exchangeability (Section VII-A); the guarantee inherits its finite-sample character.
domain assumption theta <= 90 deg restriction for the matrix-form axis-angle marginalization (Prop. 7).
Needed to recover omega and theta from omega sin(theta); the quaternion variant (App. D) removes it, so this constrains the interpretable output form, not the containment guarantee.

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Cite this review

Pith. "Pith review of Uncertainty Quantification for Visual Object Pose Estimation: S-Lemma Ellipsoidal Bounds." pith.science (2026). https://pith.science/paper/OYUCD7UQ

@misc{pith2026251121666,
  author       = {Pith},
  title        = {Pith review of: Uncertainty Quantification for Visual Object Pose Estimation: S-Lemma Ellipsoidal Bounds},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OYUCD7UQ}},
  note         = {Machine review of arXiv:2511.21666}
}

read the original abstract

Quantifying the uncertainty of an object's pose estimate is essential for robust control and planning. Although pose estimation is a well-studied robotics problem, attaching statistically rigorous uncertainty is not well understood without strict distributional assumptions. We develop distribution-free pose uncertainty bounds about a given pose estimate in the monocular setting. Our pose uncertainty only requires high probability noise bounds on pixel detections of 2D semantic keypoints on a known object. This noise model induces an implicit, non-convex set of pose uncertainty constraints. Our key contribution is SLUE (S-Lemma Uncertainty Estimation), a convex program to reduce this set to a single ellipsoidal uncertainty bound that is guaranteed to contain the true object pose with high probability. SLUE solves a relaxation of the minimum volume bounding ellipsoid problem inspired by the celebrated S-lemma. It requires no initial guess of the bound's shape or size and is guaranteed to contain the true object pose with high probability. For tighter uncertainty bounds at the same confidence, we extend SLUE to a sum-of-squares relaxation hierarchy which is guaranteed to converge to the minimum volume ellipsoidal uncertainty bound for a given set of keypoint constraints. We show this pose uncertainty bound can easily be projected to independent translation and axis-angle orientation bounds. We evaluate SLUE on two pose estimation datasets and a real-world drone tracking scenario. Compared to prior work, SLUE generates substantially smaller translation bounds and competitive orientation bounds. We release code at https://github.com/MIT-SPARK/PoseUncertaintySets.

Figures

Figures reproduced from arXiv: 2511.21666 by the authors.

**Figure 1.** Conformal Pose and Uncertainty Estimation. Given an RGB image of an object (a), we extract 2D semantic keypoints and conformal uncertainty sets (b) which contain the ground truth keypoint with high probability. These sets imply a non-convex set of quadratic constraints on the object pose. We use a generalization of the S-Lemma and a projection scheme to reduce this set to an explicit bound (c) containing the true o… view at source ↗

**Figure 2.** Keypoint measurements. Given a 3D model with annotated 3D keypoints (left), we assume pixel detections of the location of each keypoint in the image frame (right). Pixel keypoint measurements also carry an uncertainty bound, shaded in blue. certifiably optimal scale estimation. In this paper, we give a new perspective on the sum-of-squares approach based on the generalized S-Lemma. Under our approach, the log determ… view at source ↗

**Figure 3.** Pose Uncertainty Constraint Set. The infinity-norm bounds on keypoint error (left) each imply a cone of backprojected 3D feasible keypoint positions (right). Combining the bounds for multiple keypoints and imposing object shape constraints yields an implicit pose uncertainty constraint set which contains many feasible poses. eqs. (2, 3), the following reprojection constraint holds with probability at least 1 − αi [… view at source ↗

Figures from the paper (6 more)

**Figure 4.** Figure 4: Hierarchy of Bounding Ellipsoids. We solve for an ellipsoidal representation (5), in blue, of a set defined by several quadratic constraints (P∞), in purple, which may be non-convex. Our approach (SLUE) admits a hierarchy of ellipsoidal bounds guaranteed to converge to…

**Figure 5.** Figure 5: Image-plane projections of ellipsoidal pose uncertainty. Plots show the set of possible poses in the second-order joint ellipsoidal bound for α = 0.1. Uncertainty is mostly concentrated along the optical axis. For CAST, we only show translation uncertainty about the po…

**Figure 6.** Figure 6: Volume of Translational and Angular Uncertainty Ellipsoids. We plot the cumulative distribution function (CDF) of each approach at first and second-order. Our translation and angular bounds are significantly tighter at first-order, while only the translation bounds are…

**Figure 7.** Figure 7: Qualitative bounding sets in rotation and translation space. Projections of the first and second-order uncertainty ellipsoids generated by SLUE and RANSAG [10] for the duck object on LM-O (α = 0.1). SLUE optimizes directly for the shape of the ellipsoid, leading to exp…

**Figure 8.** Figure 8: Image-plane projections of ellipsoidal pose uncertainty. Plots show the set of possible poses in the second-order joint ellipsoidal bound for α = 0.4, which are significantly smaller than α = 0.1. We omit CAST. (a) Translation uncertainty ellipsoid (α = 0.4). (b) Orien…

**Figure 9.** Figure 9: Qualitative bounding sets in rotation and translation space. Projections of the first and second-order SLUE uncertainty ellipses for the duck object on LM-O. The α = 0.4 case gives a smaller uncertainty set at the cost of lower confidence. We omit baselines for visual …

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Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Detecting Pose Estimation Failures via Keypoint Self-Consistency
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Keypoint self-consistency features fed to a logistic regression detect 6D pose estimation failures better than conformal keypoint uncertainty on LINEMOD Occluded.

Reference graph

Works this paper leans on

43 extracted references · 6 linked inside Pith · cited by 1 Pith paper

[1]

B. Wen, C. Mitash, B. Ren, and K. E. Bekris,se(3)-tracknet: Data- driven 6d pose tracking by calibrating image residuals in synthetic domains, IEEE/RSJ Intl. Conf. on Intelligent Robots and Systems (IROS), (2020), 10367–10373

2020
[2]

A. H. Li, P. Culbertson, V . Kurtz, and A. D. Ames,DROP: Dexterous Reorientation via Online Planning, preprint, (2025). arXiv:2409.14562

arXiv 2025
[3]

Peng,Deep learning for 3D Object Detection and Tracking in Autonomous Driving: A Brief Survey, preprint, (2023)

Y . Peng,Deep learning for 3D Object Detection and Tracking in Autonomous Driving: A Brief Survey, preprint, (2023)

2023
[4]

G. Shi, Y . Zhu, J. Tremblay, S. Birchfield, F. Ramos, A. Anandku- mar, and Y . Zhu,Fast Uncertainty Quantification for Deep Object Pose Estimation, preprint, (2021). arXiv:2011.07748

arXiv 2021
[5]

X. Deng, A. Mousavian, Y . Xiang, F. Xia, T. Bretl, and D. Fox, PoseRBPF: A Rao-Blackwellized Particle Filter for 6D Object Pose Tracking, Robotics: Science and Systems (RSS), (2019)

2019
[6]

Brachmann, F

E. Brachmann, F. Michel, A. Krull, M. Y . Yang, S. Gumhold, and c. Rother,Uncertainty-Driven 6D Pose Estimation of Objects and Scenes From a Single RGB Image, Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), (2016)

2016
[7]

Y . Lin, J. Tremblay, S. Tyree, P. A. Vela, and S. Birchfield, Keypoint-Based Category-Level Object Pose Tracking from an RGB Sequence with Uncertainty Estimation, 2022 International Confer- ence on Robotics and Automation (ICRA), (2022), 1258–1264

2022
[8]

Hoda ˇn, D

T. Hoda ˇn, D. Baráth, and J. Matas,EPOS: Estimating 6D Pose of Objects with Symmetries, IEEE Conference on Computer Vision and Pattern Recognition (CVPR) (2020)

2020

Show all 43 references

[9]

Okorn, M

B. Okorn, M. Xu, M. Hebert, and D. Held,Learning Orientation Distributions for Object Pose Estimation, 2020 IEEE/RSJ Inter- national Conference on Intelligent Robots and Systems (IROS), (2020), 10580–10587

2020
[10]

Yang and M

H. Yang and M. Pavone,Object Pose Estimation with Statistical Guarantees: Conformal Keypoint Detection and Geometric Uncer- tainty Propagation, preprint, (2023)

2023
[11]

Tang, J-B

Y . Tang, J-B. Lasserre, and H. Yang,Uncertainty quantification of set-membership estimation in control and perception: Revisiting the minimum enclosing ellipsoid, Proceedings of the 6th Annual Learning for Dynamics and Control Conference,242, Proceedings of Machine Learning R...

2024
[12]

Pólik and T

I. Pólik and T. Terlaky,A Survey of the S-Lemma, SIAM Review, 49(3) (2007) 371–418

2007
[13]

Brachmann, A

E. Brachmann, A. Krull, F. Michel, S. Gumhold, J. Shotton, and C. Rother,Learning 6d object pose estimation using 3d object coordinates, IEEE Conf. on Computer Vision and Pattern Recognition (CVPR), (2014), 536–551

2014
[14]

Xiang, T

Y . Xiang, T. Schmidt, V . Narayanan, and D. Fox,PoseCNN: A convolutional neural network for 6D object pose estimation in cluttered scenes, Robotics: Science and Systems (RSS), (2018)

2018
[15]

Shaikewitz, S

L. Shaikewitz, S. Ubellacker, and L. Carlone,A Certifiable Algorithm for Simultaneous Shape Estimation and Object Tracking, IEEE Robotics and Automation Letters (RA-L) (2024)

2024
[16]

Wursthorn, M

K. Wursthorn, M. Hillemann, and M. Ulrich,Uncertainty quan- tification with deep ensembles for 6d object pose estimation, arXiv preprint arXiv:2403.07741 (2024)

2024 arXiv
[17]

R. L. Haugaard and A. G. Buch,Surfemb: Dense and continuous correspondence distributions for object pose estimation with learnt surface embeddings, Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, (2022), 6749–6758

2022
[18]

J. Liu, W. Sun, H. Yang, Z. Zeng, C. Liu, J. Zheng, X. Liu, H. Rahmani, N. Sebe, and A. Mian,Deep learning-based object pose estimation: A comprehensive survey, arXiv preprint arXiv:2405.07801 (2024)

2024
[19]

M. Tian, L. Pan, M. H. Ang, and G. H. Lee,Robust 6d object pose estimation by learning rgb-d features, IEEE Intl. Conf. on Robotics and Automation (ICRA), (2020), 6218–6224

2020
[20]

B. Wen, W. Yang, J. Kautz, and S. Birchfield,Foundationpose: Unified 6d pose estimation and tracking of novel objects, Proceed- ings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, (2024), 17868–17879

2024
[21]

Shafer and V

G. Shafer and V . V ovk,A Tutorial on Conformal Prediction, J. of Machine Learning Research (2008) 51

2008
[22]

Y . Gao, Y . Tang, H. Qi, and H. Yang,CLOSURE: Fast Quan- tification of Pose Uncertainty Sets, Robotics: Science and Systems (RSS), (2024)

2024
[23]

J. B. Lasserre,A generalization of Löwner-John’s ellipsoid the- orem, Mathematical Programming,152(1) (2015) 559–591

2015
[24]

Ben-Tal and A

A. Ben-Tal and A. Nemirovski,Lectures on Modern Convex Optimization: Analysis, Algorithms, and Engineering Applications, (2001) Society for Industrial

2001
[25]

Nie and J

J. Nie and J. W. Demmel,Minimum Ellipsoid Bounds for Solutions of Polynomial Systems via Sum of Squares, Journal of Global Optimization,33(4) (2005) 511–525

2005
[26]

Casini, A

M. Casini, A. Garulli, and A. Vicino,Feasible Parameter Set Ap- proximation for Linear Models with Bounded Uncertain Regressors, IEEE Transactions on Automatic Control,59(11) (2014) 2910– 2920

2014
[27]

K. He, G. Gkioxari, P. Dollár, and R. Girshick,Mask R-CNN, Intl. Conf. on Computer Vision (ICCV), (2017), 2980–2988

2017
[28]

A. N. Angelopoulos, R. F. Barber, and S. Bates,Theo- retical Foundations of Conformal Prediction, preprint, (2024). arXiv:2411.11824

2024 arXiv
[29]

Terzakis and M

G. Terzakis and M. Lourakis,A consistently fast and globally optimal solution to the perspective-n-point problem, European Conf. on Computer Vision (ECCV), (2020), 478–494

2020
[30]

Lessard,The S-Lemma

L. Lessard,The S-Lemma. https://laurentlessard.com/teaching/ me7247/supplementary/S-lemma.pdf
[31]

Boyd and L

S. Boyd and L. Vandenberghe,Convex optimization, (2004) Cambridge University Press

2004
[32]

J. B. Lasserre,Global optimization with polynomials and the problem of moments, SIAM J. Optim.,11(3) (2001) 796–817

2001
[33]

Version 11.0.30(2025)

MOSEK ApS,The MOSEK Optimizer API for Julia. Version 11.0.30(2025). https://docs.mosek.com/latest/juliaapi/index.html

2025
[34]

Nie,Optimality conditions and finite convergence of Lasserre’s hierarchy, Mathematical programming,146(1-2) (2014) 97–121

J. Nie,Optimality conditions and finite convergence of Lasserre’s hierarchy, Mathematical programming,146(1-2) (2014) 97–121

2014
[35]

J. Wang, V . Magron, J. B. Lasserre, and N. H. A. Mai,CS- TSSOS: Correlative and term sparsity for large-scale polynomial optimization, arXiv preprint arXiv:2005.02828 (2020)

2005 arXiv
[36]

W. Karl, G. Verghese, and A. S. Willsky,Reconstructing ellip- soids from projections
[37]

Barfoot,State Estimation for Robotics, (2017) Cambridge University Press

T. Barfoot,State Estimation for Robotics, (2017) Cambridge University Press

2017
[38]

Hoda ˇn, M

T. Hoda ˇn, M. Sundermeyer, B. Drost, Y . Labbé, E. Brachmann, F. Michel, C. Rother, and J. Matas,BOP Challenge 2020 on 6D Object Localization, European Conference on Computer Vision Workshops (ECCVW) (2020)

2020
[39]

Schmeckpeper, P

K. Schmeckpeper, P. Osteen, Y . Wang, G. Pavlakos, K. Chaney, W. Jordan, X. Zhou, K. Derpanis, and K. Daniilidis,Semantic keypoint-based pose estimation from single RGB frames, arXiv preprint arXiv:2204.05864 (2022)

2022 arXiv
[40]

Hoda ˇn, F

T. Hoda ˇn, F. Michel, E. Brachmann, W. Kehl, A. G. Buch, D. Kraft, B. Drost, J. Vidal, S. Ihrke, X. Zabulis, C. Sahin, F. Manhardt, F. Tombari, T-K. Kim, J. Matas, and C. Rother,BOP: Benchmark for 6D Object Pose Estimation, European Conf. on Computer Vision (ECCV), (2018), 19–35

2018
[41]

K. He, X. Zhang, S. Ren, and J. Sun,Deep residual learning for image recognition(2016) 770–778

2016
[42]

Denninger, D

M. Denninger, D. Winkelbauer, M. Sundermeyer, W. Boerdijk, M. Knauer, K. H. Strobl, M. Humt, and R. Triebel,BlenderProc2: A Procedural Pipeline for Photorealistic Rendering, Journal of Open Source Software,8(82) (2023) 4901

2023
[43]

Yang and L

H. Yang and L. Carlone,A Quaternion-based Certifiably Optimal Solution to the Wahba Problem with Outliers, Intl. Conf. on Computer Vision (ICCV), (2019). APPENDIXA 2-NORMPOSEUNCERTAINTYCONSTRAINTSET For completeness and alignment with [10], we give the pose uncertainty constra...

2019

Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.

[1] [1]

B. Wen, C. Mitash, B. Ren, and K. E. Bekris,se(3)-tracknet: Data- driven 6d pose tracking by calibrating image residuals in synthetic domains, IEEE/RSJ Intl. Conf. on Intelligent Robots and Systems (IROS), (2020), 10367–10373

2020

[2] [2]

A. H. Li, P. Culbertson, V . Kurtz, and A. D. Ames,DROP: Dexterous Reorientation via Online Planning, preprint, (2025). arXiv:2409.14562

arXiv 2025

[3] [3]

Peng,Deep learning for 3D Object Detection and Tracking in Autonomous Driving: A Brief Survey, preprint, (2023)

Y . Peng,Deep learning for 3D Object Detection and Tracking in Autonomous Driving: A Brief Survey, preprint, (2023)

2023

[4] [4]

G. Shi, Y . Zhu, J. Tremblay, S. Birchfield, F. Ramos, A. Anandku- mar, and Y . Zhu,Fast Uncertainty Quantification for Deep Object Pose Estimation, preprint, (2021). arXiv:2011.07748

arXiv 2021

[5] [5]

X. Deng, A. Mousavian, Y . Xiang, F. Xia, T. Bretl, and D. Fox, PoseRBPF: A Rao-Blackwellized Particle Filter for 6D Object Pose Tracking, Robotics: Science and Systems (RSS), (2019)

2019

[6] [6]

Brachmann, F

E. Brachmann, F. Michel, A. Krull, M. Y . Yang, S. Gumhold, and c. Rother,Uncertainty-Driven 6D Pose Estimation of Objects and Scenes From a Single RGB Image, Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), (2016)

2016

[7] [7]

Y . Lin, J. Tremblay, S. Tyree, P. A. Vela, and S. Birchfield, Keypoint-Based Category-Level Object Pose Tracking from an RGB Sequence with Uncertainty Estimation, 2022 International Confer- ence on Robotics and Automation (ICRA), (2022), 1258–1264

2022

[8] [8]

Hoda ˇn, D

T. Hoda ˇn, D. Baráth, and J. Matas,EPOS: Estimating 6D Pose of Objects with Symmetries, IEEE Conference on Computer Vision and Pattern Recognition (CVPR) (2020)

2020