Presentation on theme: "Flexible Camera Calibration by Viewing a Plane from Unknown Orientations Zhengyou Zhang Vision Technology Group Microsoft Research."— Presentation transcript:
Flexible Camera Calibration by Viewing a Plane from Unknown Orientations Zhengyou Zhang Vision Technology Group Microsoft Research
Problem Statement u Determine the characteristics of a camera (focal length, aspect ratio, principal point) from visual information (images)
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)
Classical Approach (Photogrammetry) u Use precisely known 3D points u Shortcomings: Not flexible –very expensive to make such a calibration apparatus. Known displacement
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
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
Camera Model C m
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.
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
Geometric interpretation Plane at infinity Absolute conic C
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
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.
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
Extracted corner points
Correction of Radial Distortion Corrected image Original image
Errors vs. Noise Levels in data
Errors vs. Number of Planes
Errors vs. Angle of the plane
Errors vs. Noise in model points
Errors vs. Spherical non-planarity
Errors vs. Cylindrical non-planarity
Application to object modeling
Reconstructed VRML Model
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.
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.