# Zhengyou Zhang Vision Technology Group Microsoft Research

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Zhengyou Zhang Vision Technology Group Microsoft Research
Flexible Camera Calibration by Viewing a Plane from Unknown Orientations Zhengyou Zhang Vision Technology Group Microsoft Research Please remind the audience that this is all completely confidential. We are in the midst of the patent process and certainly do not want to jeopardize that effort in any way. 1

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

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) Examples include: virtual world navigation. We are familiar with Euclidean world. If we don’t know the camera internal parameters, we can at best obtain projective structure. It would need a lot of training for human being to navigate successfully in a projective world. Vision-based user interface. How far is the person from the screen? Where is s/he looking at? Product advertisement. An on-line customer would prefer to check the exact Euclidean shape, rather than projective distorted one.

Classical Approach (Photogrammetry)
Use precisely known 3D points Decision theory group focuses on solving problems with probability and decision theory. Focus is on making application of these methods practical. Related methods, approximations, heuristics from AI, OR, Statistics, Economics are applied as needed. Groups’s work typically motivated by real-world problems, e.g., those posed by MS product and systems groups. Known displacement Shortcomings: Not flexible very expensive to make such a calibration apparatus. 3

Futuristic Approach (Self-calibration)
Move the camera in a static environment match feature points across images make use of rigidity constraint Decision theory group focuses on solving problems with probability and decision theory. Focus is on making application of these methods practical. Related methods, approximations, heuristics from AI, OR, Statistics, Economics are applied as needed. Groups’s work typically motivated by real-world problems, e.g., those posed by MS product and systems groups. Shortcoming: Not always reliable too many parameters to estimate 3

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 C m

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

What do we get from one image?
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 Let symmetric Define up to a scale factor Rewrite
as linear equations: symmetric

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 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

Extracted corner points

Result (1)

Result (2)

Original image Corrected 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 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! 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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