pith. sign in

arxiv: 1907.01759 · v1 · pith:JOZGXGU5new · submitted 2019-07-03 · 📡 eess.IV · cs.CV

Calibration of fisheye camera using entrance pupil

Peter Fasogbon , Emre Aksu This is my paper

Pith reviewed 2026-05-25 10:07 UTC · model grok-4.3

classification 📡 eess.IV cs.CV
keywords fisheye camera calibrationentrance pupilsingle viewpointthin lens modelbundle adjustmentnon-single viewpointintrinsic parametersimage formation model
0
0 comments X

The pith

Modeling entrance pupil shifts with thin-lens equations lets fisheye systems be calibrated as single-viewpoint cameras with improved parameter accuracy.

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

The paper establishes a formation model that treats varying entrance pupil position in fisheye lenses as a thin-lens effect. This adjustment converts a non-single-viewpoint system into one that behaves as if it had a fixed viewpoint. The method adds entrance pupil parameters to the standard projection equations and solves for them together with the usual intrinsics via bundle adjustment. Experiments report modestly lower reprojection errors than conventional single-viewpoint calibrations. The procedure integrates directly into existing thin-lens image formation pipelines.

Core claim

By expressing entrance pupil location as a function of field angle through thin-lens ray tracing, the image formation equation is rewritten so that the effective projection center remains fixed; nonlinear optimization then recovers both the conventional intrinsics and the pupil trajectory coefficients, producing calibration results whose reprojection error is smaller than that obtained under a strict single-viewpoint assumption.

What carries the argument

Entrance pupil trajectory expressed via thin-lens equations, inserted into the projection model to enforce single-viewpoint geometry across the field of view.

If this is right

  • Intrinsic parameters for fisheye lenses become more stable across different calibration poses.
  • Reprojection error decreases compared with standard single-viewpoint methods.
  • The same pupil-correction terms can be added to any other thin-lens formation model without changing the optimizer.
  • Calibration remains simple to implement because only a few extra scalar parameters are introduced.

Where Pith is reading between the lines

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

  • The corrected model may reduce systematic errors in downstream tasks such as depth estimation or panoramic stitching from fisheye sequences.
  • The approach could be tested on other non-central cameras, such as catadioptric systems, to check whether thin-lens pupil modeling generalizes.
  • If the thin-lens assumption holds only approximately, residual pupil variation might be absorbed by allowing higher-order terms in the pupil trajectory function.

Load-bearing premise

The actual path of the entrance pupil in a real fisheye lens follows the trajectory predicted by the thin-lens approximation for every field angle.

What would settle it

Direct optical measurement of entrance pupil position at multiple field angles that deviates from the thin-lens curve by more than the reported calibration uncertainty would falsify the model.

read the original abstract

Most conventional camera calibration algorithms assume that the imaging device has a Single Viewpoint (SVP). This is not necessarily true for special imaging device such as fisheye lenses. As a consequence, the intrinsic camera calibration result is not always reliable. In this paper, we propose a new formation model that tends to relax this assumption so that a Non-Single Viewpoint (NSVP) system is corrected to always maintain a SVP, by taking into account the variation of the Entrance Pupil (EP) using thin lens modeling. In addition, we present a calibration procedure for the image formation to estimate these EP parameters using non linear optimization procedure with bundle adjustment. From experiments, we are able to obtain slightly better re-projection error than traditional methods, and the camera parameters are better estimated. The proposed calibration procedure is simple and can easily be integrated to any other thin lens image formation model.

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.

Referee Report

3 major / 0 minor

Summary. The paper proposes a camera formation model that uses thin-lens equations to account for entrance-pupil (EP) variation in fisheye lenses, thereby algebraically correcting a non-single-viewpoint (NSVP) system to single-viewpoint (SVP) behavior. It further presents a bundle-adjustment procedure to estimate the additional EP-shift parameters and reports that the resulting calibration yields slightly lower reprojection error and better parameter estimates than conventional methods; the procedure is claimed to be simple and integrable with other thin-lens models.

Significance. If the thin-lens EP trajectory were shown to match real fisheye optics and the reported improvement were demonstrated with proper controls, the work would supply a practical route to reliable intrinsic calibration of NSVP fisheye systems, directly benefiting downstream tasks such as panoramic stitching and metric 3-D reconstruction. The absence of an external validation step and of quantitative comparison data, however, prevents a firm assessment of practical impact.

major comments (3)
  1. [Abstract] Abstract: the claim of “slightly better re-projection error” and “better estimated” parameters is presented without error bars, dataset size or composition, or a side-by-side numerical comparison against the exact baseline equations that omit the EP correction; this leaves the central empirical claim unsupported.
  2. [Model and calibration procedure] Model and calibration sections: the new EP-shift parameters are introduced and then fitted inside the same bundle-adjustment objective used to report the improvement; without an independent measurement of pupil position versus field angle (or a parameter-free derivation), the reported gain risks being circular.
  3. [Experiments] Experiments: no separate empirical check is supplied that the thin-lens EP trajectory actually reproduces the measured entrance-pupil locus of the fisheye lens under test; any systematic mismatch would invalidate both the SVP correction and the claimed parameter improvement.

Simulated Author's Rebuttal

3 responses · 1 unresolved

We thank the referee for the thorough review and constructive criticism. We agree that the empirical claims require more rigorous presentation and that additional validation would be beneficial. We respond to each major comment below and indicate the revisions we will make.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the claim of “slightly better re-projection error” and “better estimated” parameters is presented without error bars, dataset size or composition, or a side-by-side numerical comparison against the exact baseline equations that omit the EP correction; this leaves the central empirical claim unsupported.

    Authors: We agree with this observation. The abstract will be revised to specify the dataset details (number of images and target type), include standard deviations or error bars on the reported reprojection errors, and provide a side-by-side numerical comparison of the reprojection errors with and without the EP correction terms. This will be done in the revised manuscript. revision: yes

  2. Referee: [Model and calibration procedure] Model and calibration sections: the new EP-shift parameters are introduced and then fitted inside the same bundle-adjustment objective used to report the improvement; without an independent measurement of pupil position versus field angle (or a parameter-free derivation), the reported gain risks being circular.

    Authors: The bundle adjustment is the standard method for estimating all intrinsic parameters simultaneously to minimize reprojection error. The EP-shift parameters are derived from the thin-lens model and their estimation leads to improved fit, which is the evidence provided. We concede that this is not an independent validation of the EP trajectory. In the revision, we will clarify this point and add text explaining that the improvement demonstrates consistency with the model but does not replace direct optical measurement. revision: partial

  3. Referee: [Experiments] Experiments: no separate empirical check is supplied that the thin-lens EP trajectory actually reproduces the measured entrance-pupil locus of the fisheye lens under test; any systematic mismatch would invalidate both the SVP correction and the claimed parameter improvement.

    Authors: The current experiments evaluate the calibration through reprojection error on the calibration images rather than direct measurement of pupil positions. We do not have independent measurements of the entrance pupil locus for the lens used. This is a genuine limitation of the presented work. revision: no

standing simulated objections not resolved
  • Absence of an independent empirical validation of the thin-lens entrance pupil trajectory against direct measurements of the pupil locus.

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper introduces a thin-lens-based entrance-pupil model as an explicit modeling assumption to convert NSVP fisheye geometry to SVP behavior, then estimates the added parameters via standard bundle adjustment. No equations reduce the claimed correction or parameter estimates to the inputs by construction, no self-citation chain is load-bearing, and no fitted quantity is relabeled as an independent prediction. The reported improvement in reprojection error follows directly from the additional degrees of freedom in the optimizer and is presented as an empirical outcome rather than a derived necessity. The derivation therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the thin-lens entrance-pupil trajectory being an adequate description of real fisheye optics and on the optimization recovering the true parameters; both are introduced without external validation in the abstract.

free parameters (1)
  • entrance-pupil shift parameters
    Fitted by nonlinear optimization; their number and functional form are not stated in the abstract.
axioms (1)
  • domain assumption Thin-lens model accurately represents entrance-pupil movement in fisheye lenses
    Invoked to convert NSVP to SVP; location is the proposed formation model paragraph of the abstract.

pith-pipeline@v0.9.0 · 5676 in / 1212 out tokens · 21381 ms · 2026-05-25T10:07:06.276778+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

20 extracted references · 20 canonical work pages · 1 internal anchor

  1. [1]

    Calibration of fisheye camera using entrance pupil

    INTRODUCTION Camera calibration is the estimation of a camera’s mapping function between a set of known world points and their mea- sured image coordinates. The parameters that define this mapping are usually divided into two categories: intrinsic and extrinsic parameters. The intrinsic parameters represent the internal characteristics of the image lens an...

    work page internal anchor Pith review Pith/arXiv arXiv 1907

  2. [2]

    In the last section, we summarize the paper and ex- plain the future direction of this work

    In section 4, we perform experiments on calibration im- age data, and comparison with two representative state of art methods. In the last section, we summarize the paper and ex- plain the future direction of this work

    work page

  3. [3]

    IMAGE FORMA TION MODEL In the figure 2, i = 1,...,n represents the number of world points projected onto the lens. O(X,Y,Z ),O c(Xc,Y c,Z c) represent the world and camera origins.o(x,y ),o p(u,v ) rep- resent the image and pixel coordinates origin, f is the cam- era’s focal length and r is the distance between a projected image point p and origin o. In ad...

    work page

  4. [4]

    This tend to affect the accuracy of the calibration and proper estimation of the distortion parameters

    PROPOSED MODEL USING ENTRANCE PUPIL Some fisheye calibration methods [3, 11, 4] put a lot of un- necessary pressure on the optical lens model to simplify the distortions introduced as a result of the varying EP. This tend to affect the accuracy of the calibration and proper estimation of the distortion parameters. Most especially, closer object to the came...

    work page

  5. [5]

    The experiments have been done with a calibration target in a working distance of between 100to 150mm from the camera to be calibrated

    EXPERIMENTS The proposed calibration method has been evaluated on both real and synthetic dataset. The experiments have been done with a calibration target in a working distance of between 100to 150mm from the camera to be calibrated. This work- ing distance is required to be able to accurately measure the capability of the system in an extreme setup. The...

    work page arXiv 2048

  6. [6]

    We model the entrance pupil variation as part of extrinsic and thus separate it from the intrinsic that includes the conventional lens distor- tion models

    CONCLUSION We propose a new camera model that integrates entrance pupil variation for fisheye camera calibration. We model the entrance pupil variation as part of extrinsic and thus separate it from the intrinsic that includes the conventional lens distor- tion models. The proposed method gives accurate entrance pupil and other camera parameters estimate w...

    work page

  7. [7]

    A flexible new technique for camera calibra- tion,

    Z. Zhang, “A flexible new technique for camera calibra- tion,” IEEE Trans. Pattern Anal. Mach. Intell. , vol. 22, no. 11, pp. 1330–1334, Nov. 2000

    work page 2000

  8. [8]

    Measuring curved surfaces for robot vision,

    E. L. Hall, J. B. K. Tio, C. A. McPherson, and F. A. Sadjadi, “Measuring curved surfaces for robot vision,” in Computer Journal, Dec. 1982, vol. 15 (12), pp. 42– 54

    work page 1982

  9. [9]

    A generic camera model and calibration method for conventional, wide-angle, and fish-eye lenses,

    J. Kannala and S. S. Brandt, “A generic camera model and calibration method for conventional, wide-angle, and fish-eye lenses,” IEEE Transactions on Pattern Analysis and Machine Intelligence , vol. 28, pp. 1335– 1340, 2006

    work page 2006

  10. [10]

    Generic calibration of cam- eras with non-parallel optical elements,

    P. Fasogbon and L. Fan, “Generic calibration of cam- eras with non-parallel optical elements,” in Proceedings of the International Conference on Pattern Recognition , August 2018, pp. 1875–1881

    work page 2018

  11. [11]

    Cahvor camera model and its photogram- metric conversion for planetary applications,

    R. Li K. Di, “Cahvor camera model and its photogram- metric conversion for planetary applications,” inJournal of Geophysical Research, 2004, vol. 109, p. 108115

    work page 2004

  12. [12]

    A general imaging model and a method for finding its parameters,

    S.K. Nayar M.D. Grossberg, “A general imaging model and a method for finding its parameters,” in IEEE Int. Conf. on Computer Vision, 2014, vol. 2, p. 108115

    work page 2014

  13. [13]

    Caustics of catadioptric cameras,

    R. Swaminathan, M. D. Grossberg, and S. K. Nayar, “Caustics of catadioptric cameras,” in IEEE Int. Conf. on Computer Vision, 2001, vol. 2, pp. 2–9

    work page 2001

  14. [14]

    Generalized pupil-centric imaging and analytical calibration for a non-frontal camera,

    A. Kumar and N. Ahuja, “Generalized pupil-centric imaging and analytical calibration for a non-frontal camera,” in IEEE Conference on Computer Vision and Pattern Recognition, 2014, pp. 3970–3977

    work page 2014

  15. [15]

    On the equivalence of moving entrance pupil and radial distortion for camera calibra- tion,

    A. Kumar and N. Ahuja, “On the equivalence of moving entrance pupil and radial distortion for camera calibra- tion,” in IEEE International Conference on Computer Vision (ICCV), Dec 2015, pp. 2345–2353

    work page 2015

  16. [16]

    Generalized camera calibration includ- ing fish-eye lenses,

    D. B. Gennery, “Generalized camera calibration includ- ing fish-eye lenses,” International Journal of Computer Vision, vol. 68, no. 3, pp. 239–266, July 2006

    work page 2006

  17. [17]

    A toolbox for easily calibrating omnidirectional cameras,

    D. Scaramuzza, A. Martinelli, and R. Siegwart, “A toolbox for easily calibrating omnidirectional cameras,” in IEEE/RSJ International Conference on Intelligent Robots and Systems, Oct 2006, pp. 5695–5701

    work page 2006

  18. [18]

    Close-range camera calibration,

    D. C. Brown, “Close-range camera calibration,” in Pho- togrammetric Engineering, 1971, vol. 37 (8), pp. 855– 866

    work page 1971

  19. [19]

    Automatic feature extrac- tion for wide-angle and fish-eye camera calibration,

    P. Fasogbon and L. Fan, “Automatic feature extrac- tion for wide-angle and fish-eye camera calibration,” in Proceedings of the International Conference on Pattern Recognition, August 2018, pp. 2947–2952

    work page 2018

  20. [20]

    Bundle adjustment - a modern synthesis,

    B. Triggs, P. F. McLauchlan, R. I. Hartley, and A. W. Fitzgibbon, “Bundle adjustment - a modern synthesis,” in Proceedings of the International Workshop on Vision Algorithms: Theory and Practice , London, UK, UK, 2000, ICCV ’99, pp. 298–372, Springer-Verlag

    work page 2000