Camera calibration from multiple view of a 2D object, using a global non linear

minimization method

Computer EngineeringYOO GWI HYEON

Camera calibration• Abstract

– We propose this paper a which is based on a camera model that incorporates lens distortion

– Which involves a non-linear minimization technique

– which can be performed using multiple view of a single 2D object, and subpixel features extraction

Camera calibration• Two factors demanded from camera

calibration

1. Intrinsic parameters– The process of determining the internal camera

geometric and optical characteristics

2. Extrinsic parameters– The 3D position and orientation of the camera

with respect to some predefined world coordinate system

Camera calibration• The achieved calibration accuracy depends

– The accuracy of the 3D calibration object or accuracy of the xyz translation stage for moving a planar calibration tile

– The accuracy provided on the attributes of 2D features extracted from the image

– The numerical method used for estimating the camera parameters

Camera models• Pinhole model

– The most widely used camera model

– The coordinate of a 3D point P = [x, y, z]T in a world coordinate system and its retinal coordinates P = [u, v]T are related by a perfect perspective transformation

(su, sv, s)T = M (X, Y, Z, 1)T

Numerical methods • Three categories of techniques

1. Closed-Form Solution techniques– The Parameters values are computed directly form

analytical formulas 2. Global Nonlinear Minimization techniques

– The parameters are estimated by an iterative algorithm used to minimize linear (or non-linear) criterion provided by set of measurement equations

3. Two- step Method a closed- form solution– Derived for most of the calibration parameters and some

iterative solution is used for the other parameters

Numerical methods • Minimization methods

1. Initialization step– An approximative solution is computed for all the

parameters by a simple technique using a simple camera model

2. Estimation step – The nonlinear optimization is started with this

approximative solution as an initial guess

Camera models• The Pinhole camera model

Camera models• The Pinhole camera model

Camera models• The Pinhole camera model

Numerical methods • Levenberg-Marquardt method

– This method can be used with a very approximative initial guess

– Main drawback is that it does not provide an accuracy estimate on the computed parameter ɵ

Numerical methods • Extended Kalman Filter(EKF)

– It is a recursive method

– A well-known drawback of the EKF method is that it is very sensitive about the initial guess and many measurements are often required to ensure a good convergence

Numerical methods • Extended Kalman Filter(EKF)

Numerical methods • Application of Extended KalmanFilter(EKF)

– Integrated Navigation System– GPS– Target tracking– Attitude determination– Orbit determination

Numerical methods • Bard-Deming algorithm

– Bard-Deming algorithm Can be prefered to the EKF method, for off line processing, like the calibration step

– It is iterative method which considers globally all measurements

Experimental set-up• These Global nonlinear Minimization

techniques could be performed in different contexts

– Classical calibration• The parameter vector ɵ contains the I intrinsic parameters

and only one E transform– Mutiview calibration

• The parameter vector contains the I intrinsic parameters and as many Ei transforms as the object position

– Object calibration • The parameter vector contains only one E transform

Experimental set-up

Experimental set-up• The calibration function

Experimental set-up• Grey level model

– The grey level model of a cross line centered in is given by a mathematical function which depends on 11 parameter

– The precise location of the cross center is given by the estimation of these parameters with a non linear minimization of the difference between grey levels of the theoretical shape and the observed one

Experimental results• The evaluation criteria of a calibration method

– The stability of the intrinsic parameters when several runs are performed with different positions of the calibration object

– several residues provided by the minimization method, i.e. the value of the measurement equation computed with the final parameter estimates

– The best criteria: the application.

Experimental results

Experimental results

Experimental results

Conclusion• Several Nonlinear Minimization methods

– The best one of the parameter accuracy seems to be the Bard-Deming algorithm, but it is too expensive in memory requirements and in computation time

– Levenberg-Marquardt minimization algorithm converges faster

– The Extended Kalman Filter requires a good initial guess, but allow to detect the outliers with a probabilistic test, provided that realistic uncertainty model can be exhibited for the extracted point and for the object model