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EECS 274 Computer Vision

EECS 274 Computer Vision. Geometric Camera Calibration. GEOMETRIC CAMERA CALIBRATION. Camera calibration problem Least-squares techniques Linear calibration from points Analytical photogrammetry Reading: Chapter 3 of FP, Chapters 2,6 of S. Calibration.

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EECS 274 Computer Vision

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  1. EECS 274 Computer Vision Geometric Camera Calibration

  2. GEOMETRIC CAMERA CALIBRATION • Camera calibration problem • Least-squares techniques • Linear calibration from points • Analytical photogrammetry • Reading: Chapter 3 of FP, Chapters 2,6 of S

  3. Calibration • Determine the intrinsic and extrinsic parameters • Assume that the camera observe a set of features (points, or lines) with known positions • Calibration: modeled as an optimization to minimize the discrepancy between the observed image features and their theoretical projections (using the perspective projection equations)

  4. Calibration Problem Given n points, P1, …, Pn with known positions and their images points, p1, …, pn, find ξ

  5. Linear Systems Square system: A x b • unique solution • Gaussian elimination = Rectangular system ?? • underconstrained: • infinity of solutions A x b = • overconstrained: • no solution Minimize |Ax-b| 2

  6. How do you solve overconstrained linear equations ?

  7. In matrix form Can be derived from the perspective projection matrix

  8. Homogeneous Linear Systems Square system: A x 0 • unique solution: 0 • unless Det(A)=0 = Rectangular system ?? • 0 is always a solution A x 0 = 2 Minimize |Ax| under the constraint |x| =1 2

  9. How do you solve overconstrained homogeneous linear equations ? The solution is e . 1

  10. Example: Line Fitting Problem: minimize with respect to (a,b,d). • Minimize E with respect to d: • Minimize E with respect to a,b: where • Solution is the unit eigenvector with minimum eigenvalue

  11. Note: • Matrix of second moments of inertia • Axis of least inertia in mechanics

  12. Linear Camera Calibration min |Pm|2, |m|=1

  13. Once M is known, you still got to recover the intrinsic and extrinsic parameters ! This is a decomposition problem, not an estimation problem. r • Intrinsic parameters • Extrinsic parameters

  14. Decomposition of M As the recovered Orthonormal basis vector θ is close to π/2 and has positive sine

  15. Degenerate Point Configurations Are there other solutions besides M ? • One solution: (l,m,n)=(m1, m2, m3) • Consider the points Pi all lie in some plane, s.t., P∙Pi=0 for some P • Coplanar points: choose (l,m,n)=(P,0,0) or (0,P,0) or (0,0,P ), or • any linear combination of these vectors yields a solution Does not (usually) happen for 6 or more random points!

  16. Radial distortion • Depends on the distance separating the optical axis from the point of interest, d Barrel distortion Corners are detected by fitting lines in each square Using estimated distortion parameters

  17. Correct radial distortion • Tsai’s algorithm (1987) exploits radial alignment constraints for estimating extrinsic parameters

  18. Analytical Photogrammetry Given n points, P1, …, Pn with known positions and their images situations, p1, …, pn, find ξ Non-Linear Least-Squares Methods • Newton • Gauss-Newton • Levenberg-Marquardt Iterative, quadratically convergent in favorable situations

  19. Mobile Robot Localization (Devy et al., 1997)

  20. Calibration • Numerous ways that exploits properties of projective geometry • E.g. calibration using lines, calibration circular controlled points

  21. Camera calibration toolbox • Excellent MATLAB toolbox by Jean-Yves Bouguet http://www.vision.caltech.edu/bouguetj/calib_doc/ • Steps: • Generate calibration board • Collect images under different views • Select extreme points • Find corner points • Solve optimization problem

  22. Calibration images

  23. Extreme points

  24. Guessed grid corners

  25. Corner extraction

  26. Repeat for all other images

  27. Solving optimization problem

  28. Reprojected corners

  29. Camera centered view

  30. World centered view

  31. Applications • Augmented reality • Image registration • Image stitching • Panoramic image

  32. Panoramic image

  33. Notes • Camera pose estimation • Multi-camera calibration • Auto/self calibration • Multi-camera self calibration • Projective geometry • Multi-view geometry • RANSAC (RANdom Sample Consensus)

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