Efficient Circle-Based Camera Pose Tracking Free of PnP

Fulin Tang; Yihong Wu

arxiv: 1907.10219 · v1 · pith:KI4YJGNY · submitted 2019-07-24 · cs.CV

Efficient Circle-Based Camera Pose Tracking Free of PnP

Fulin Tang , Yihong Wu This is my paper

Reviewed by Pith T0 review T1 audit T2 compute T3 formal T4 kernel 2026-05-24 17:18 UTCgrok-4.3pith:KI4YJGNY record.json open to challenge →

classification cs.CV

keywords camera pose trackingcircular markersprojective invariancePnP-free estimationpose optimizationcircle edge detectionreal-time tracking

0 comments

The pith

Camera pose is computed directly from circle edges via projective invariance without point matching or PnP.

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

The paper develops a camera pose tracking method based on specially designed circular markers. It derives 6D pose in concise analytical form from the projective invariance properties of each imaged circle edge. This approach eliminates the need to identify and match individual points, so PnP solvers and RANSAC are not required. A subsequent nonlinear optimization refines the pose using a polar-n-direction geometric distance. The resulting tracker shows improved stability under noise, motion blur, and increased camera-to-marker distance while running near 100 FPS on CPU.

Core claim

The authors show that 6D camera pose can be represented analytically and unifiedly in concise forms directly from each circular marker via projective invariance on the imaged circle edges, without requiring point matching or PnP.

What carries the argument

Projective invariance formulas that map imaged circle edges to 6D camera pose analytically from each marker.

If this is right

Pose estimation remains stable when cameras move fast or are distant from markers.
Tracking accuracy improves without dependence on RANSAC to reject incorrect point matches.
Real-time performance near 100 FPS is achieved on standard CPU hardware.
Robustness to noise and blur exceeds that of conventional point-based PnP pipelines.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same invariance approach could be tested on other closed conic sections to broaden marker design options.
Integration with existing visual odometry pipelines might reduce reliance on feature tracking during rapid motion.
The polar-n-direction cost function could serve as a drop-in replacement for reprojection error in other geometric solvers.

Load-bearing premise

Imaged circle edges can be extracted reliably enough to apply the projective-invariance formulas accurately even under motion blur or large distances.

What would settle it

A sequence of frames with strong motion blur where the extracted circle edges produce an analytical pose that deviates by more than a few degrees or centimeters from ground truth even after the polar-n-direction optimization.

Figures

Figures reproduced from arXiv: 1907.10219 by Fulin Tang, Yihong Wu.

**Figure 2.** Figure 2: A point m and its polar line l = Cm related to a conic C are not needed. We use projective invariance to identify the origin and the orthogonal coordinate axes of world coordinate system from images. The camera pose is analytically and unifiedly represented as concise forms. Furthermore, we establish a nonlinear cost function based on a polar-ndirection geometric distance [28] to optimize the analytical … view at source ↗

**Figure 3.** Figure 3: The designed markers Given a point m in 2D homogeneous coordinates, let Cm represent a line (i.e. Cm is as line coordinate). Then m and Cm are of polarity relationship related to C. The relationship is invariant under a projective transformation. Cm is called the polar line of m related to C and inversely m is called the pole of Cm [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗

**Figure 4.** Figure 4: World coordinate system for marker (a) in Figure 3 [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗

**Figure 5.** Figure 5: Markers for experimental comparison. (a) is ARToolkit [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗

**Figure 6.** Figure 6: Evaluation VS. different Gaussian noise levels: (a) AR [PITH_FULL_IMAGE:figures/full_fig_p005_6.png] view at source ↗

**Figure 7.** Figure 7: Evaluation VS. different Gaussian blur levels: (a) AR [PITH_FULL_IMAGE:figures/full_fig_p006_7.png] view at source ↗

**Figure 9.** Figure 9: Augmented reality with different illuminations, where the airplane is virtual. From (a) to (d), illuminations are changed. The [PITH_FULL_IMAGE:figures/full_fig_p007_9.png] view at source ↗

**Figure 10.** Figure 10: Augmented reality with different blur levels, where the airplane is virtual. From (a) to (d), images are more and more blur in a [PITH_FULL_IMAGE:figures/full_fig_p007_10.png] view at source ↗

**Figure 11.** Figure 11: Augmented reality on two videos captured by a rolling shutter camera. (a) and (b) are from a video with weak texture. (c) and [PITH_FULL_IMAGE:figures/full_fig_p008_11.png] view at source ↗

**Figure 12.** Figure 12: Estimated trajectories of our method and ORBSLAM: [PITH_FULL_IMAGE:figures/full_fig_p008_12.png] view at source ↗

read the original abstract

Camera pose tracking attracts much interest both from academic and industrial communities, of which the methods based on planar markers are easy to be implemented. However, most of the existing methods need to identify multiple points in the marker images for matching to space points. Then, PnP and RANSAC are used to compute the camera pose. If cameras move fast or are far away from markers, matching is easy to generate errors and even RANSAC cannot remove incorrect matching. Then, the result by PnP cannot have good performance. To solve this problem, we design circular markers and represent 6D camera pose analytically and unifiedly as very concise forms from each of the marker by projective invariance. Afterwards, the pose is further optimized by a proposed nonlinear cost function based on a polar-n-direction geometric distance. The method is from imaged circle edges and without PnP/RANSAC, making pose tracking robust and accurate. Experimental results show that the proposed method outperforms the state of the arts in terms of noise, blur, and distance from camera to marker. Simultaneously, it can still run at about 100 FPS on a consumer computer with only CPU.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a PnP-free analytical pose from circular markers via projective invariance plus a polar-n-direction refinement, which targets real failure cases but rests on clean edge extraction.

read the letter

The core idea is to treat circular markers as conics and pull 6D pose directly from projective invariance in closed form per marker, then polish it with a nonlinear cost that measures geometric distance in polar-n directions. This skips point detection, matching, PnP, and RANSAC entirely. That combination is the actual novelty; earlier ellipse or circle work usually still fell back on correspondences or iterative fitting that reintroduces the same fragility. The approach is practical for the stated regimes of fast motion and larger distances, where standard pipelines lose tracks. The reported 100 FPS on CPU and the outperformance claims on noise, blur, and distance are the parts worth checking first in the experiments. The soft spot is exactly the one the stress-test note flags: everything downstream depends on the imaged circle edges being clean enough for the invariance formulas to land near the true pose. If Canny or whatever detector they use drops arcs or shifts under motion blur, the closed-form step has no outlier rejection to fall back on, and the refinement can only do so much. The paper frames the experiments as showing gains in those conditions, so the question is whether the test imagery actually stresses edge quality the way real deployments would. Minor implementation details like how they initialize the nonlinear step or handle partial occlusions would also need scrutiny, but those are fixable. This is aimed at engineers building marker-based tracking for AR or robotics who already know PnP pain points. It is worth sending to peer review because the geometric framing is coherent and the problem is concrete; a referee can check whether the edge-extraction assumption holds in the supplied results and whether the derivations are free of hidden fitting.

Referee Report

2 major / 2 minor

Summary. The paper proposes a camera pose tracking approach based on circular markers that computes 6D pose in closed form from each marker via projective invariance applied to imaged circle edges, avoiding point correspondences, PnP, and RANSAC entirely. An initial analytical pose is then refined by nonlinear optimization of a polar-n-direction geometric distance cost; the method is claimed to be robust to noise, motion blur, and large camera-marker distances while running at ~100 FPS on CPU.

Significance. If the projective-invariance derivations are correct and the edge-based initialization remains accurate without correspondence filtering, the approach would provide a genuinely PnP-free pipeline that could improve robustness in fast-motion or distant-marker scenarios common in AR and robotics; the reported speed and claimed outperformance would make it practically attractive.

major comments (2)

[Experimental results / §4] The central claim that imaged circle edges directly yield accurate 6D pose via projective invariance (without any RANSAC or point filtering) rests on the unverified assumption that standard edge detectors produce sufficiently clean conics under motion blur and large distance. No quantitative evaluation of edge-extraction error (e.g., geometric distance of fitted ellipses to ground-truth contours) is supplied in the experimental section to confirm that the invariance identities receive input within their required tolerance.
[Method / §3] The abstract and method description state that the nonlinear polar-n-direction refinement only “polishes” an already reasonable initialization, yet no ablation or table reports the pose error of the analytical projective-invariance stage alone versus the final refined result. Without this, it is impossible to determine how much of the claimed robustness is actually supplied by the closed-form step versus the subsequent optimization.

minor comments (2)

[Abstract] The abstract claims “outperforms the state of the arts” but does not name the competing methods or cite their papers; the comparison table (presumably in §4) should explicitly list the baselines.
[Method] Notation for the polar-n-direction distance is introduced without an accompanying equation or diagram; a short derivation or figure would clarify the geometric distance used in the cost function.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments. We respond to each major point below and indicate the revisions planned for the next version of the manuscript.

read point-by-point responses

Referee: [Experimental results / §4] The central claim that imaged circle edges directly yield accurate 6D pose via projective invariance (without any RANSAC or point filtering) rests on the unverified assumption that standard edge detectors produce sufficiently clean conics under motion blur and large distance. No quantitative evaluation of edge-extraction error (e.g., geometric distance of fitted ellipses to ground-truth contours) is supplied in the experimental section to confirm that the invariance identities receive input within their required tolerance.

Authors: We agree that an explicit quantitative evaluation of edge-extraction accuracy would strengthen the robustness claims. While the end-to-end pose accuracy results already demonstrate superior performance under blur and distance, isolating the geometric fitting error of the extracted conics would directly verify the tolerance of the projective-invariance identities. In the revised manuscript we will add this analysis, reporting mean geometric distances of fitted ellipses to ground-truth contours across controlled blur levels and camera-marker distances. revision: yes
Referee: [Method / §3] The abstract and method description state that the nonlinear polar-n-direction refinement only “polishes” an already reasonable initialization, yet no ablation or table reports the pose error of the analytical projective-invariance stage alone versus the final refined result. Without this, it is impossible to determine how much of the claimed robustness is actually supplied by the closed-form step versus the subsequent optimization.

Authors: We acknowledge that an ablation comparing the analytical initialization against the final optimized pose would clarify the relative contributions of each stage. Although the closed-form solution is intended to be sufficiently accurate on its own, quantifying the improvement provided by the polar-n-direction refinement under noise, blur, and distance would be informative. We will include such an ablation table or figure in the experimental section of the revised manuscript. revision: yes

Circularity Check

0 steps flagged

No circularity; analytical pose via projective invariance is independent of fitted inputs or self-citation chains

full rationale

The paper frames its core contribution as a direct application of projective geometry to imaged circles, yielding closed-form 6D pose per marker without PnP. No equations reduce by construction to data fits, and no load-bearing uniqueness theorem or ansatz is imported via self-citation. The derivation chain remains self-contained against external geometric facts; performance numbers are presented as empirical outcomes rather than forced predictions. This matches the default expectation of non-circularity for geometry-based methods.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard projective geometry for circles and the assumption that edge detection remains usable; no new physical entities are introduced and no parameters are shown to be fitted inside the abstract.

axioms (2)

domain assumption Projective invariance properties of circles under perspective projection yield a unique 6D pose per marker
Invoked when the paper states that pose is represented analytically from each marker by projective invariance.
domain assumption Circle edge detection remains sufficiently accurate under the tested noise, blur, and distance conditions
Required for the method to operate from imaged circle edges without PnP or RANSAC.

pith-pipeline@v0.9.0 · 5725 in / 1316 out tokens · 23097 ms · 2026-05-24T17:18:30.845711+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

30 extracted references · 30 canonical work pages

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H. Kato and M. Billinghurst. Marker tracking and hmd cal- ibration for a video-based augmented reality conferencing system. In Augmented Reality, 1999.(IWAR’99) Proceed- ings. 2nd IEEE and ACM International Workshop on, pages 85–94. IEEE, 1999

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R. Mu ˜noz-Salinas, M. J. Mar´ın-Jimenez, E. Yeguas-Bolivar, and R. Medina-Carnicer. Mapping and localization from pla- nar markers. Pattern Recognition, 73:158–171, 2018

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M. Toyoura, H. Aruga, M. Turk, and X. Mao. Mono- spectrum marker: an ar marker robust to image blur and de- focus. The Visual Computer, 30(9):1035–1044, 2014

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D. Wagner, G. Reitmayr, A. Mulloni, T. Drummond, and D. Schmalstieg. Real-time detection and tracking for aug- mented reality on mobile phones. IEEE transactions on vi- sualization and computer graphics, 16(3):355–368, 2010. 4329

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Wagner and D

D. Wagner and D. Schmalstieg. Artoolkitplus for pose track- ing on mobile devices. Proceedings of 12th Computer Vision Winter Workshop, pages 139–146, 2007

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Wang and E

J. Wang and E. Olson. Apriltag 2: Efﬁcient and robust ﬁdu- cial detection. In IROS, pages 4193–4198, 2016

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Y . Wu, H. Wang, F. Tang, and Z. Wang. Efﬁcient conic ﬁt- ting with an analytical polar-n-direction geometric distance. Pattern Recognition, 90:415–423, 2019

work page 2019
[30]

Y . Wu, H. Zhu, Z. Hu, and F. Wu. Camera calibration from the quasi-afﬁne invariance of two parallel circles. In European Conference on Computer Vision, pages 190–202. Springer, 2004. 4330

work page 2004

[1] [1]

Ababsa and M

F.-e. Ababsa and M. Mallem. Robust camera pose estimation using 2d ﬁducials tracking for real-time augmented reality systems. In Proceedings of the 2004 ACM SIGGRAPH in- ternational conference on Virtual Reality continuum and its applications in industry, pages 431–435. ACM, 2004

work page 2004

[2] [2]

Agarwal, K

S. Agarwal, K. Mierle, et al. Ceres solver. URL http://ceres- solver.org, 2012

work page 2012

[3] [3]

Bergamasco, A

F. Bergamasco, A. Albarelli, L. Cosmo, E. Rodola, and A. Torsello. An accurate and robust artiﬁcial marker based on cyclic codes. IEEE Trans. Pattern Anal. Mach. Intell. , 38(12):2359–2373, 2016

work page 2016

[4] [4]

Bergamasco, A

F. Bergamasco, A. Albarelli, and A. Torsello. Pi-tag: a fast image-space marker design based on projective invariants. Machine vision and applications, 24(6):1295–1310, 2013

work page 2013

[5] [5]

Calvet, P

L. Calvet, P. Gurdjos, and V . Charvillat. Camera track- ing based on circular point factorization. In Proceedings of the 21st International Conference on Pattern Recognition (ICPR2012), pages 2128–2131. IEEE, 2012

work page 2012

[6] [6]

Calvet, P

L. Calvet, P. Gurdjos, C. Griwodz, and S. Gasparini. De- tection and accurate localization of circular ﬁducials under highly challenging conditions. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition , pages 562–570, 2016

work page 2016

[7] [7]

Q. Chen, H. Wu, and T. Wada. Camera calibration with two arbitrary coplanar circles. In European Conference on Com- puter Vision, pages 521–532. Springer, 2004

work page 2004

[8] [8]

Claus and A

D. Claus and A. W. Fitzgibbon. Reliable automatic calibra- tion of a marker-based position tracking system. In Applica- tion of Computer Vision, 2005. WACV/MOTIONS’05 Volume

work page 2005

[9] [9]

IEEE, 2005

Seventh IEEE Workshops on , volume 1, pages 300–305. IEEE, 2005

work page 2005

[10] [10]

DeGol, T

J. DeGol, T. Bretl, and D. Hoiem. Chromatag: A colored marker and fast detection algorithm. In ICCV, 2017

work page 2017

[11] [11]

M. Fiala. Artag, a ﬁducial marker system using digital tech- niques. In Computer Vision and Pattern Recognition, 2005. CVPR 2005. IEEE Computer Society Conference on , vol- ume 2, pages 590–596. IEEE, 2005

work page 2005

[12] [12]

M. Fiala. Designing highly reliable ﬁducial markers. IEEE Transactions on Pattern analysis and machine intelligence , 32(7):1317–1324, 2010

work page 2010

[13] [13]

L. B. Gatrell, W. A. Hoff, and C. W. Sklair. Robust image features: Concentric contrasting circles and their image ex- traction. In Cooperative Intelligent Robotics in Space II, vol- ume 1612, pages 235–245. International Society for Optics and Photonics, 1992

work page 1992

[14] [14]

Kato and M

H. Kato and M. Billinghurst. Marker tracking and hmd cal- ibration for a video-based augmented reality conferencing system. In Augmented Reality, 1999.(IWAR’99) Proceed- ings. 2nd IEEE and ACM International Workshop on, pages 85–94. IEEE, 1999

work page 1999

[15] [15]

V . A. Knyaz. The development of new coded targets for au- tomated point identiﬁcation and non-contact 3d surface mea- surements. IAPRS, 5:80–85, 1998

work page 1998

[16] [16]

Maidi, J.-Y

M. Maidi, J.-Y . Didier, F. Ababsa, and M. Mallem. A performance study for camera pose estimation using visual marker based tracking. Machine Vision and Applications , 21(3):365–376, 2010

work page 2010

[17] [17]

Mu ˜noz-Salinas, M

R. Mu ˜noz-Salinas, M. J. Mar ´ın-Jimenez, and R. Medina- Carnicer. Spm-slam: Simultaneous localization and mapping with squared planar markers. Pattern Recognition, 86:156– 171, 2019

work page 2019

[18] [18]

Mu ˜noz-Salinas, M

R. Mu ˜noz-Salinas, M. J. Mar´ın-Jimenez, E. Yeguas-Bolivar, and R. Medina-Carnicer. Mapping and localization from pla- nar markers. Pattern Recognition, 73:158–171, 2018

work page 2018

[19] [19]

Mur-Artal, J

R. Mur-Artal, J. M. M. Montiel, and J. D. Tardos. Orb-slam: a versatile and accurate monocular slam system.IEEE Trans- actions on Robotics, 31(5):1147–1163, 2015

work page 2015

[20] [20]

Mur-Artal and J

R. Mur-Artal and J. D. Tard ´os. Orb-slam2: An open-source slam system for monocular, stereo, and rgb-d cameras.IEEE Transactions on Robotics, 33(5):1255–1262, 2017

work page 2017

[21] [21]

Naimark and E

L. Naimark and E. Foxlin. Circular data matrix ﬁducial system and robust image processing for a wearable vision- inertial self-tracker. In Proceedings of the 1st International Symposium on Mixed and Augmented Reality, page 27. IEEE Computer Society, 2002

work page 2002

[22] [22]

Y . C. J. L. U. Neumann. A multi-ring color ﬁducial sys- tem and a rule-based detection method for scalable ﬁducial- tracking augmented reality. Proceedings of International Workshop on Augmented Reality, 1998

work page 1998

[23] [23]

E. Olson. Apriltag: A robust and ﬂexible visual ﬁducial sys- tem. In Robotics and Automation (ICRA), 2011 IEEE Inter- national Conference on, pages 3400–3407. IEEE, 2011

work page 2011

[24] [24]

M. G. Prasad, S. Chandran, and M. S. Brown. A motion blur resilient ﬁducial for quadcopter imaging. In Applications of Computer Vision (WACV), 2015 IEEE Winter Conference on, pages 254–261. IEEE, 2015

work page 2015

[25] [25]

Toyoura, H

M. Toyoura, H. Aruga, M. Turk, and X. Mao. Mono- spectrum marker: an ar marker robust to image blur and de- focus. The Visual Computer, 30(9):1035–1044, 2014

work page 2014

[26] [26]

Wagner, G

D. Wagner, G. Reitmayr, A. Mulloni, T. Drummond, and D. Schmalstieg. Real-time detection and tracking for aug- mented reality on mobile phones. IEEE transactions on vi- sualization and computer graphics, 16(3):355–368, 2010. 4329

work page 2010

[27] [27]

Wagner and D

D. Wagner and D. Schmalstieg. Artoolkitplus for pose track- ing on mobile devices. Proceedings of 12th Computer Vision Winter Workshop, pages 139–146, 2007

work page 2007

[28] [28]

Wang and E

J. Wang and E. Olson. Apriltag 2: Efﬁcient and robust ﬁdu- cial detection. In IROS, pages 4193–4198, 2016

work page 2016

[29] [29]

Y . Wu, H. Wang, F. Tang, and Z. Wang. Efﬁcient conic ﬁt- ting with an analytical polar-n-direction geometric distance. Pattern Recognition, 90:415–423, 2019

work page 2019

[30] [30]

Y . Wu, H. Zhu, Z. Hu, and F. Wu. Camera calibration from the quasi-afﬁne invariance of two parallel circles. In European Conference on Computer Vision, pages 190–202. Springer, 2004. 4330

work page 2004