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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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