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A Flexible New Technique for Camera Calibration Zhengyou Zhang

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### A Flexible New Technique for Camera CalibrationZhengyou Zhang

OutlineOutlineOutline

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
- Calibration and Procedure
- Experimental Results
- Conclusion

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)

- 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

- 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

- Vanishing points for orthogonal directions
- Calibration from pure rotation

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

Outline

- Introduction
- Equations and Constraints
- Calibration and Procedure
- Experimental Results
- Conclusion

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

- 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

- 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

- Model plane described in camera coordinate system
- Model plane intersects the plane at infinity at a line

Geometric Interpretation

- x∞is circular point and satisfy , or

a2 + b2 = 0

- Two intersection points
- This point is invariant to Euclidean transformation

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

- Introduction
- Equations and Constraints
- Calibration and Procedure
- Experimental Results
- Conclusion

Calibration

- Analytical solution
- Nonlinear optimization technique based on the maximum-likelihood criterion

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 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 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 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

- 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 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

- 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

- Introduction
- Equations and Constraints
- Calibration and Procedure
- Experimental Results
- Conclusion

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

Angle b/w image axes

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

- Introduction
- Equations and Constraints
- Calibration and Procedure
- Experimental Results
- 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

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

- Minimizing following function yield maximum-likelihood estimation of H
- where with = ith row of H

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

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

Appendix

- Elements of L
- Constant 1
- Pixels
- World coordinates
- Multiplication of both

Possible Future Work

- Improving distortion parameter caused by lens distortion

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