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Flexible Camera Calibration by Viewing a Plane from Unknown Orientations. Zhengyou Zhang Vision Technology Group Microsoft Research. Problem Statement. Determine the characteristics of a camera (focal length, aspect ratio, principal point) from visual information (images). Motivations.

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Flexible Camera Calibration by Viewing a Plane from Unknown Orientations

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Flexible camera calibration by viewing a plane from unknown orientations l.jpg

Flexible Camera Calibration by Viewing a Plane from Unknown Orientations

Zhengyou Zhang

Vision Technology Group

Microsoft Research


Problem statement l.jpg

Problem Statement

  • Determine the characteristics of a camera (focal length, aspect ratio, principal point) from visual information (images)


Motivations l.jpg

Motivations

  • Recovery of 3D Euclidean structure from images is essential for many applications.

  • This requires camera calibration.

  • Look for a flexible and robust technique, suitable for desktop vision systems.

    (such that it can be used by the general public)


Classical approach photogrammetry l.jpg

Classical Approach(Photogrammetry)

  • Use precisely known 3D points

Known displacement

  • Shortcomings:Not flexible

    • very expensive to make such a calibration apparatus.


Futuristic approach self calibration l.jpg

Futuristic Approach(Self-calibration)

  • Move the camera in a static environment

    • match feature points across images

    • make use of rigidity constraint

  • Shortcoming:Not always reliable

    • too many parameters to estimate


Realistic approach my new method l.jpg

Realistic Approach(my new method)

  • Use only one plane

    • Print a pattern on a paper

    • Attach the paper on a planar surface

    • Show the plane freely a few times to the camera

  • Advantages:

    • Flexible!

    • Robust?

Yes. See RESULTS


Camera model l.jpg

m

C

Camera Model


Plane projection l.jpg

C

m

with

Plane projection

  • For convenience, assume the plane at z = 0.

  • The relation between image points and model points is then given by:


What do we get from one image l.jpg

Given H, which is defined up to a scale factor,

And let

, we have

What do we get from one image?

  • We can obtain two equations in 6 intermediate homogeneous parameters.

This yields


Geometric interpretation l.jpg

Absolute conic

Geometric interpretation

Plane at infinity

C


Linear equations l.jpg

Linear Equations

  • Let

  • Define

    up to a scale factor

  • Rewrite

    as linear equations:

symmetric


What do we get from 2 images l.jpg

What do we get from 2 images?

  • If we impose  = 0, which is usually the case with modern cameras, we can solve all the other camera intrinsic parameters.

How about more images?

Better! More constraints than unknowns.


Solution l.jpg

Solution

  • Show the plane under n different orientations (n > 1)

  • Estimate the n homography matrices

    (analytic solution followed by MLE)

  • Solve analytically the 6 intermediate parameters (defined up to a scale factor)

  • Extract the five intrinsic parameters

  • Compute the extrinsic parameters

  • Refine all parameters with MLE


Experimental results l.jpg

Experimental results


Extracted corner points l.jpg

Extracted corner points


Result 1 l.jpg

Result (1)


Result 2 l.jpg

Result (2)


Correction of radial distortion l.jpg

Original image

Correction of Radial Distortion

Corrected image


Errors vs noise levels in data l.jpg

Errors vs. Noise Levels in data


Errors vs number of planes l.jpg

Errors vs. Number of Planes


Errors vs angle of the plane l.jpg

Errors vs. Angle of the plane


Errors vs noise in model points l.jpg

Errors vs. Noise in model points


Errors vs spherical non planarity l.jpg

Errors vs. Spherical non-planarity


Errors vs cylindrical non planarity l.jpg

Errors vs. Cylindrical non-planarity


Application to object modeling l.jpg

Application to object modeling


Reconstructed vrml model l.jpg

Reconstructed VRML Model


Conclusion l.jpg

Conclusion

  • We have developed a flexible and robust technique for camera calibration.

  • Analytical solution exists.

  • MLE improves the analytical solution.

  • We need at least two images if c = 0.

  • We can use as many images of the plane as possible to improve the accuracy.


It really works l.jpg

It really works!

  • Currently used routinely in both Vision and Graphics Groups.

  • Binary executable will be distributed on the Web to the public soon.

  • Source code will also be made available.


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