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A Flexible New Technique for Camera Calibration Zhengyou Zhang. Sung Huh CSPS 643 Individual Presentation 1 February 25, 2009. Outline. Introduction Equations and Constraints Calibration and Procedure Experimental Results Conclusion. Outline. Introduction Equations and Constraints

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A flexible new technique for camera calibration zhengyou zhang

A Flexible New Technique for Camera CalibrationZhengyou Zhang

Sung Huh

CSPS 643 Individual Presentation 1

February 25, 2009


Outline
Outline

  • Introduction

  • Equations and Constraints

  • Calibration and Procedure

  • Experimental Results

  • Conclusion


Outline1
Outline

  • Introduction

  • Equations and Constraints

  • Calibration and Procedure

  • Experimental Results

  • Conclusion


Introduction
Introduction

  • Extract metric information from 2D images

  • Much work has been done by photogrammetry and computer vision community

    • Photogrammetric calibration

    • Self-calibration


Photogrammetric calibration three dimensional reference object based calibration
Photogrammetric Calibration(Three-dimensional reference object-based calibration)

  • Observing a calibration object with known geometry in 3D space

  • Can be done very efficiently

  • Calibration object usually consists of two or three planes orthogonal to each other

    • A plane undergoing a precisely known translation is also used

  • Expensive calibration apparatus and elaborate setup required


Self calibration
Self-Calibration

  • Do not use any calibration object

  • Moving camera in static scene

  • The rigidity of the scene provides constraints on camera’s internal parameters

  • Correspondences b/w images are sufficient to recover both internal and external parameters

    • Allow to reconstruct 3D structure up to a similarity

  • Very flexible, but not mature

    • Cannot always obtain reliable results due to many parameters to estimate


Other techniques
Other Techniques

  • Vanishing points for orthogonal directions

  • Calibration from pure rotation


New technique from author
New Technique from Author

  • Focused on a desktop vision system (DVS)

  • Considered flexibility, robustness, and low cost

  • Only require the camera to observe a planar pattern shown at a few (minimum 2) different orientations

    • Pattern can be printed and attached on planer surface

    • Either camera or planar pattern can be moved by hand

  • More flexible and robust than traditional techniques

    • Easy setup

    • Anyone can make calibration pattern


Outline2
Outline

  • Introduction

  • Equations and Constraints

  • Calibration and Procedure

  • Experimental Results

  • Conclusion


Notation
Notation

  • 2D point,

  • 3D point,

  • Augmented Vector,

  • Relationship b/w 3D point M and image projection m

(1)


Notation1
Notation

  • s: extrinsic parameters that relates the world coord. system to the camera coord. System

  • A: Camera intrinsic matrix

  • (u0,v0): coordinates of the principal point

  • α,β: scale factors in image u and v axes

  • γ: parameter describing the skew of the two image


Homography b w the model plane and its image
Homography b/w the Model Plane and Its Image

  • Assume the model plane is on Z = 0

  • Denote ith column of the rotation matrix R by ri

  • Relation b/w model point Mand image m

  • His homography and defined up to a scale factor

(2)


Constraints on intrinsic parameters
Constraints on Intrinsic Parameters

  • Let H be H = [h1 h2 h3]

  • Homography has 8 degrees of freedom & 6 extrinsic parameters

  • Two basic constraints on intrinsic parameter

(3)

(4)


Geometric interpretation
Geometric Interpretation

  • Model plane described in camera coordinate system

  • Model plane intersects the plane at infinity at a line


Geometric interpretation1
Geometric Interpretation

  • x∞is circular point and satisfy , or

    a2 + b2 = 0

  • Two intersection points

  • This point is invariant to Euclidean transformation


Geometric interpretation2
Geometric Interpretation

  • Projection of x∞ in the image plane

  • Point is on the image of the absolute conic, described by A-TA-1

  • Setting zero on both real and imaginary parts yield two intrinsic parameter constraints


Outline3
Outline

  • Introduction

  • Equations and Constraints

  • Calibration and Procedure

  • Experimental Results

  • Conclusion


Calibration
Calibration

  • Analytical solution

  • Nonlinear optimization technique based on the maximum-likelihood criterion


Closed form solution
Closed-Form Solution

  • Define B = A-TA-1≡

  • B is defined by 6D vector b

(5)

(6)


Closed form solution1
Closed-Form Solution

  • ith column of H = hi

  • Following relation hold

(7)


Closed form solution2
Closed-Form Solution

  • Two fundamental constraints, from homography, become

  • If observed n images of model plane

  • V is 2n x 6 matrix

  • Solution of Vb = 0 is the eigenvector of VTV associated w/ smallest eigenvalue

  • Therefore, we can estimate b

(8)

(9)


Closed form solution3
Closed-Form Solution

  • If n ≥ 3, unique solution b defined up to a scale factor

  • If n = 2, impose skewless constraint γ = 0

  • If n = 1, can only solve two camera intrinsic parameters, αandβ, assumingu0andv0are known and γ = 0


Closed form solution4
Closed-Form Solution

  • Estimate B up to scale factor, B = λATA-1

  • B is symmetric matrix defined by b

  • B in terms of intrinsic parameter is known

  • Intrinsic parameters are then


Closed form solution5
Closed-Form Solution

  • Calculating extrinsic parameter from Homography H = [h1 h2 h3] = λA[r1 r2 t]

  • R = [r1r2r3] does not, in general, satisfy properties of a rotation matrix because of noise in data

  • R can be obtained through singular value decomposition


Maximum likelihood estimation
Maximum-Likelihood Estimation

  • Given n images of model plane with m points on model plane

  • Assumption

    • Corrupted Image points by independent and identically distributed noise

  • Minimizing following function yield maximum likelihood estimate

(10)


Maximum likelihood estimation1
Maximum-Likelihood Estimation

  • is the projection of point Mj in image i

  • Ris parameterized by a vector of three parameters

    • Parallel to the rotation axis and magnitude is equal to the rotation angle

  • Rand r are related by the Rodrigues formula

  • Nonlinear minimization problem solved with Levenberg-Marquardt Algorithm

  • Require initial guess


Calibration procedure
Calibration Procedure

  • Print a pattern and attach to a planar surface

  • Take few images of the model plane under different orientations

  • Detect feature points in the images

  • Estimate five intrinsic parameters and all the extrinsic parameters using the closed-form solution

  • Refine all parameters by obtaining maximum-likelihood estimate


Outline4
Outline

  • Introduction

  • Equations and Constraints

  • Calibration and Procedure

  • Experimental Results

  • Conclusion


Experimental results
Experimental Results

  • Off-the-shelf PULNiX CCD camera w/ 6mm lense

  • 640 x 480 image resolution

  • 5 images at close range (set A)

  • 5 images at larger distance (set B)

  • Applied calibration algorithm on set A, set B and Set A+B


Experimental result
Experimental Result

Angle b/w image axes


Experimental result http research microsoft com en us um people zhang calib
Experimental Resulthttp://research.microsoft.com/en-us/um/people/zhang/calib/


Outline5
Outline

  • Introduction

  • Equations and Constraints

  • Calibration and Procedure

  • Experimental Results

  • Conclusion


Conclusion
Conclusion

  • Technique only requires the camera to observe a planar pattern from different orientation

  • Pattern could be anything, as long as the metric on the plane is known

  • Good test result obtained from both computer simulation and real data

  • Proposed technique gains considerable flexibility


Appendix estimating homography b w the model plane and its image
AppendixEstimating Homography b/w the Model Plane and its Image

  • Method based on a maximum-likelihood criterion (Other option available)

  • Let Mi and mi be the model and image point, respectively

  • Assume mi is corrupted by Gaussian noise with mean 0 and covariance matrix Λmi


Appendix
Appendix

  • Minimizing following function yield maximum-likelihood estimation of H

  • where with = ith row of H


Appendix1
Appendix

  • Assume for all i

  • Problem become nonlinear least-squares one, i.e.

  • Nonlinear minimization is conducted with Levenberg-Marquardt Algorithm that requires an initial guess with following procedure to obtain


Appendix2
Appendix

  • Let Then (2) become

  • n above equation with given n point and can be written in matrix equation as Lx = 0

  • L is 2n x 9 matrix

  • x is define dup to a scale factor

  • Solution of xLTL associated with the smallest eigenvalue


Appendix3
Appendix

  • Elements of L

    • Constant 1

    • Pixels

    • World coordinates

    • Multiplication of both


Possible future work
Possible Future Work

  • Improving distortion parameter caused by lens distortion



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