A minimal solution to the autocalibration oA minimal solution to the autocalibration of radial distortion f radial distortion

Young Ki Baik (CV Lab.)Young Ki Baik (CV Lab.)2007. 8. 29 (Wed)2007. 8. 29 (Wed)

A minimal solution to the autocalibration of rA minimal solution to the autocalibration of radial distortionadial distortion

ReferencesReferencesA minimal solution to the autocalibration of radial distortion

• Zuzana Kukelova and Tomas Pajdla (CVPR2007)

Recent Developments on Direct Relative Orientation• H. Stewenius, C. Engels and D. Nister, Kurt cornelis, Luc Van Gool (I

SPRS Journal of Photogrammetry and Remote Sensing 2006)

A minimal solution to the autocalibration of rA minimal solution to the autocalibration of radial distortionadial distortion

Why?Why?… did I select this paper?

• Is fundamental matrix is really best material of real 3D reconstruction?

• Is there any other good solution to replace

fundamental matrix?

A minimal solution to the autocalibration of rA minimal solution to the autocalibration of radial distortionadial distortion

Why?Why?… did I select this paper?

• If fundamental matrix is best solution for 3D reconstruction…

• How can we compute…

accurate fundamental matrix?

A minimal solution to the autocalibration of rA minimal solution to the autocalibration of radial distortionadial distortion

Why?Why?… did I select this paper?

• A recent trend of the MVG is to add …

some constraints!!!

• Autocalibration via Rank-Constrained Estimation of the Absolute Quadric

• M. Chandraker, D. Nister. et. al. (CVPR 2007)

• Minimal Solutions for Panoramic Stitching• Matthew Brown, Richard Hartley, and D. Nister (CVPR 2007)

• An Efficient Minimal Solution for Infinitesimal Camera Motion• Henrik Stewenius, Chris Engels, and D. Nister (CVPR 2007)

A minimal solution to the autocalibration of rA minimal solution to the autocalibration of radial distortionadial distortion

What?What?… is the purpose of this paper?

• Correcting radial distortion

from a pair of distorted real images!!

A minimal solution to the autocalibration of rA minimal solution to the autocalibration of radial distortionadial distortion

Previous work…Previous work…Simultaneous linear estimation of multiple view

geometry and lens distortion • A. Fitzgibbon (CVPR 2001)

A non-iterative method for correcting lens distortion from nine-point correspondences

• H. Li and R. Hartley (OMNIVIS 2005)

A minimal solution to the autocalibration of rA minimal solution to the autocalibration of radial distortionadial distortion

Fitzgibbon’s work (CVPR2001)Fitzgibbon’s work (CVPR2001)Assumption

• Radial distortion model (Division model)

21~

d

du r

xx

positionpoint dundistorte:1,, uuu yxx

positionpoint distorted : 1,, ddd yxx

distortion ofcenter thefrom distance : 222ddd yxr

parameter distortion :

• Square pixelSquare pixel• Known Known center of distortioncenter of distortion

A minimal solution to the autocalibration of rA minimal solution to the autocalibration of radial distortionadial distortion

Fitzgibbon’s work (CVPR2001)Fitzgibbon’s work (CVPR2001)Assumption

• Fundamental matrix

8,,1 ,0 iTu

Tu iixFx

333231

232221

131211

fff

fff

fff

F

Scale factorScale factor

Final factor can not be zero !!!Final factor can not be zero !!!

A minimal solution to the autocalibration of rA minimal solution to the autocalibration of radial distortionadial distortion

Fitzgibbon’s work (CVPR2001)Fitzgibbon’s work (CVPR2001)Proposed linear model

• 9 parameters

2311 ,,, ff

9 points algorithm9 points algorithm

•Simultaneous linear estimation of multiple view geometry and lens distortion

- A. Fitzgibbon (CVPR 2001)

A minimal solution to the autocalibration of rA minimal solution to the autocalibration of radial distortionadial distortion

Fitzgibbon’s work (CVPR2001)Fitzgibbon’s work (CVPR2001)Using two real distorted imagesFinding initial correspondences

• Cross-correlation• Window size (100x100)

Proposed linear model• Radial distortion param.• MVG param. (F)

RANSAC• Find correct correspondances• Find radial distortion param.• Find geometrical property

A minimal solution to the autocalibration of rA minimal solution to the autocalibration of radial distortionadial distortion

What?What?… is the difference …

between Fitzgibbon’s work and this paper?

0det F

A minimal solution to the autocalibration of rA minimal solution to the autocalibration of radial distortionadial distortion

What?What?… is the difference …

between Fitzgibbon’s work and this paper?

• If they succeed their proposed algorithm,If they succeed their proposed algorithm,

8 Points Algorithm8 Points Algorithm

A minimal solution to the autocalibration of rA minimal solution to the autocalibration of radial distortionadial distortion

What?What?… is the Problem …

to unify constraints?

0det F

0TuTu iixFx 21

~d

du r

xx

Linear Linear equationequation

Too complicatedToo complicated

polynomial polynomial

equationequation

A minimal solution to the autocalibration of rA minimal solution to the autocalibration of radial distortionadial distortion

How?How?… can solve the complicated polynomial

equation?

•Recent Developments on Direct Relative Orientation

H. Stewenius, C. Engels and D. Nister, Kurt cornelis, Luc Van Gool ( ISPRS Journal of Photogrammetry and Remote Sensing 2006)

A minimal solution to the autocalibration of rA minimal solution to the autocalibration of radial distortionadial distortion

Stewenius’ workRelative position

E 02 EEEEEE TT trace

Also complicated polynomial equationAlso complicated polynomial equation

A minimal solution to the autocalibration of rA minimal solution to the autocalibration of radial distortionadial distortion

Stewenius’ workRelative position

• Algebraic geometry tools

• Gröbner basis methodGröbner basis method

• Using Algebraic Geometry•D. Cox, J. Little, and D. O’Shea D. Cox, J. Little, and D. O’Shea (Springer-Verlag, 2005)(Springer-Verlag, 2005)

• Ideals, Varieties, and Algorithms•D. Cox, J. Little, and D. O’Shea D. Cox, J. Little, and D. O’Shea (Springer-Verlag, 2005)(Springer-Verlag, 2005)

A minimal solution to the autocalibration of rA minimal solution to the autocalibration of radial distortionadial distortion

Features of Proposed methodFeatures of Proposed methodUsing an additional constraint

Solving polynomial equations

0det F

Gröbner basis method

A minimal solution to the autocalibration of rA minimal solution to the autocalibration of radial distortionadial distortion

Quantitative results of estimating Quantitative results of estimating Synthetic dataSynthetic data

A minimal solution to the autocalibration of rA minimal solution to the autocalibration of radial distortionadial distortion

Results of real data Results of real data

Distorted imageDistorted image Corrected imageCorrected image

A minimal solution to the autocalibration of rA minimal solution to the autocalibration of radial distortionadial distortion

Contribution of this paperContribution of this paper

Realize the minimal solution• previous 9-point algorithm → 8-point algorithm

Obtain more accurate and stable results

•Additional constraint give more …

A minimal solution to the autocalibration of rA minimal solution to the autocalibration of radial distortionadial distortion

Why?Why?… should this paper have been accepted?

• Idea and contributions of this paper are

not excellent.(-)

•Numerical formulation and results are

good for practical point of view. (+)

•Previous work is well described. (+)

•Paper is well written. (+)