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Stanford CS223B Computer Vision, Winter 2006 Lecture 4 Camera Calibration

Stanford CS223B Computer Vision, Winter 2006 Lecture 4 Camera Calibration. Professor Sebastian Thrun CAs: Dan Maynes-Aminzade and Mitul Saha [with slides by D Forsyth, D. Lowe, M. Polleyfeys, C. Rasmussen, G. Loy, D. Jacobs, J. Rehg, A, Hanson, G. Bradski,…] . Today’s Goals.

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Stanford CS223B Computer Vision, Winter 2006 Lecture 4 Camera Calibration

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  1. Stanford CS223B Computer Vision, Winter 2006Lecture 4 Camera Calibration Professor Sebastian Thrun CAs: Dan Maynes-Aminzade and Mitul Saha [with slides by D Forsyth, D. Lowe, M. Polleyfeys, C. Rasmussen, G. Loy, D. Jacobs, J. Rehg, A, Hanson, G. Bradski,…]

  2. Today’s Goals • Calibration: Problem definition • Solution via Singular Value Decomposition • Solution by nonlinear Least Squares • Distortion

  3. Intrinsic Camera Parameters • Determine the intrinsic parameters of a camera (with lens) • What are Intrinsic Parameters?

  4. Perspective Projection, Remember? O X -x Z f

  5. Intrinsic Camera Parameters • Determine the intrinsic parameters of a camera (with lens) • Intrinsic Parameters: • Focal Length f • Pixel size sx ,sy • Distortion coefficients k1 ,k2… • Image center ox ,oy

  6. A Quiz • Can we determine all intrinsic parameters by … exposing the camera to many known objects?

  7. Example Calibration Pattern

  8. Our Calibration target

  9. Harris Corner Detector

  10. Another Quiz (the last today) • How Many Flat Calibration Targets are Needed for Calibration? 1: 2: 3: 4: 5: 10 • How Many Corner Points do we need in Total? 1: 2: 3: 4: 10: 20

  11. Experiment 1: Parallel Board

  12. Projective Perspective of Parallel Board 10cm 20cm 30cm

  13. Experiment 2: Tilted Board

  14. Projective Perspective of Tilted Board 10cm 20cm 30cm 50cm 100cm 500cm

  15. Perspective Camera Model Object Space

  16. Calibration: 2 steps • Step 1: Transform into camera coordinates • Step 2: Transform into image coordinates

  17. Calibration Model (extrinsic) Homogeneous Coordinates

  18. Homogeneous Coordinates • Idea: Most Operations Become Linear! • Extract Image Coordinates by Z-normalization

  19. Advantage of Homogeneous C’s i-th data point

  20. Calibration Model (intrinsic) Focal length Pixel size Image center

  21. Intrinsic Transformation

  22. Plugging the Model Together!

  23. Summary Parameters • Extrinsic • Rotation • Translation • Intrinsic • Focal length • Pixel size • Image center coordinates • (Distortion coefficients)

  24. Q: Can We recover all Intrinsic Params? • No

  25. Summary Parameters, Revisited • Focal length, in pixel units • Aspect ratio • Extrinsic • Rotation • Translation • Intrinsic • Focal length • Pixel size • Image center coordinates • (Distortion coefficients)

  26. Today’s Goals • Calibration: Problem definition • Solution via Singular Value Decomposition • Solution by nonlinear Least Squares • Distortion

  27. Calibration via SVD

  28. Calibration via SVD N>=7 points, not coplanar

  29. Calibration via SVD

  30. Calibration via SVD A has rank 7 (without proof)

  31. Calibration via SVD • Remaining Problem: • See book

  32. Summary, SVD Solution • Replace rotation matrix by arbitrary matrix • Transform into linear set of equations • Solve via SVD • Enforce rotation matrix (see book) • Solve for remaining parameters (see book) SVD solution: algebraic minimization, assume Gaussian noise in parameter space

  33. Today’s Goals • Calibration: Problem definition • Solution via Singular Value Decomposition • Solution by nonlinear Least Squares • Distortion

  34. Calibration by nonlinear Least Squares • Calibration Examples: …

  35. Calibration by nonlinear Least Squares • Least Squares

  36. Calibration by nonlinear Least Squares • Least Mean Square • Gradient descent:

  37. Summary Non-Linear Least Squares • Solve nonlinear equations via gradient descent • Assume Gaussian noise in image space, not parameter space

  38. SVD Versus LQ SVD Minimization of squared distance in parameter space Globally optimal Nonlin Least Squares Minimization of squared distance in Image space Locally optimal

  39. Q: How Many Images Do We Need? • Assumption: K images with M corners each • 4+6K parameters • 2KM constraints • 2KM  4+6K  M>3 and K 2/(M-3) • 2 images with 4 points, but will 1 images with 5 points work? • No, since points cannot be co-planar!

  40. Today’s Goals • Calibration: Problem definition • Solution via Singular Value Decomposition • Solution by nonlinear Least Squares • Distortion

  41. Advanced Calibration:Nonlinear Distortions • Barrel and Pincushion • Tangential

  42. Barrel and Pincushion Distortion wideangle tele

  43. Models of Radial Distortion distance from center

  44. Tangential Distortion cheap CMOS chip cheap lense image cheap glue cheap camera

  45. Image Rectification (to be continued)

  46. Summary • Calibration: Problem definition • Solution via Singular Value Decomposition • Solution by nonlinear Least Squares • Distortion

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