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Published byAmelia Duncan Modified over 2 years ago

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Flexible Camera Calibration by Viewing a Plane from Unknown Orientations Zhengyou Zhang Vision Technology Group Microsoft Research

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

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Motivations u Recovery of 3D Euclidean structure from images is essential for many applications. u This requires camera calibration. u Look for a flexible and robust technique, suitable for desktop vision systems. (such that it can be used by the general public)

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Classical Approach (Photogrammetry) u Use precisely known 3D points u Shortcomings: Not flexible –very expensive to make such a calibration apparatus. Known displacement

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Futuristic Approach (Self-calibration) u Shortcoming: Not always reliable –too many parameters to estimate u Move the camera in a static environment –match feature points across images –make use of rigidity constraint

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Realistic Approach (my new method) u 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 u Advantages: –Flexible! –Robust? Yes. See RESULTS

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Camera Model C m

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C Plane projection u The relation between image points and model points is then given by: with m u For convenience, assume the plane at z = 0.

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What do we get from one image? u We can obtain two equations in 6 intermediate homogeneous parameters. Given H, which is defined up to a scale factor, And let, we have This yields

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Geometric interpretation Plane at infinity Absolute conic C

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Linear Equations u Let u Define up to a scale factor up to a scale factor u Rewrite as linear equations: as linear equations: symmetric

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What do we get from 2 images? u 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.

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Solution u Show the plane under n different orientations (n > 1) u Estimate the n homography matrices (analytic solution followed by MLE) u Solve analytically the 6 intermediate parameters (defined up to a scale factor) u Extract the five intrinsic parameters u Compute the extrinsic parameters u Refine all parameters with MLE

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Experimental results

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Extracted corner points

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Result (1)

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Result (2)

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Correction of Radial Distortion Corrected image Original image

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Errors vs. Noise Levels in data

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Errors vs. Number of Planes

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Errors vs. Angle of the plane

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Errors vs. Noise in model points

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Errors vs. Spherical non-planarity

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Errors vs. Cylindrical non-planarity

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Application to object modeling

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Reconstructed VRML Model

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Conclusion u We have developed a flexible and robust technique for camera calibration. u Analytical solution exists. u MLE improves the analytical solution. u We need at least two images if c = 0. u We can use as many images of the plane as possible to improve the accuracy.

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It really works! u Currently used routinely in both Vision and Graphics Groups. u Binary executable will be distributed on the Web to the public soon. u Source code will also be made available.

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