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Last Lecture

y. x. Last Lecture. ( X,Y,Z ). P. origin. p. ( x,y ). principal point. (optical center). Camera calibration. Camera calibration. Estimate both intrinsic and extrinsic parameters Mainly, two categories: Using objects with known geometry as reference

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Last Lecture

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  1. y x Last Lecture (X,Y,Z) P origin p (x,y) principal point (optical center)

  2. Camera calibration

  3. Camera calibration • Estimate both intrinsic and extrinsic parameters • Mainly, two categories: • Using objects with known geometry as reference • Self calibration (structure from motion)

  4. One app of camera pose application • Virtual gaming • http://www.livestream.com/emtech/video?clipId=pla_74103098-95fb-4704-99f0-d07339dc16a1

  5. Camera calibration approaches • Directly estimate 11 unknowns in the M matrix using known 3D points (Xi,Yi,Zi) and measured feature positions (ui,vi)

  6. Linear regression

  7. Linear regression

  8. Linear regression Solve for Projection Matrix M using least-square techniques

  9. Normal equation (Geometric Interpretation) Given an overdetermined system the normal equation is that which minimizes the sum of the square differences between left and right sides

  10. Normal equation (Differential Interpretation) nxm, n equations, m variables

  11. Normal equation Carl Friedrich Gauss

  12. Any issues with the method?

  13. Nonlinear optimization • A probabilistic view of least square • Feature measurement equations • Likelihood of M given {(ui,vi)}

  14. Optimal estimation • Log likelihood of M given {(ui,vi)} • It is a least square problem (but not necessarily linear least square) • How do we minimize C?

  15. Nonlinear least square methods

  16. Least square fitting number of data points number of parameters

  17. Nonlinear least square fitting

  18. It is very hard to solve in general. Here, we only consider a simpler problem of finding local minimum. Function minimization Least square is related to function minimization.

  19. Function minimization

  20. Quadratic functions Approximate the function with a quadratic function within a small neighborhood

  21. Function minimization

  22. Computing gradient and Hessian Gradient Hessian

  23. Computing gradient and Hessian Gradient Hessian

  24. Computing gradient and Hessian Gradient Hessian

  25. Computing gradient and Hessian Gradient Hessian

  26. Computing gradient and Hessian Gradient Hessian

  27. Searching for update h Gradient Hessian Idea 1: Steepest Descent

  28. Steepest descent method isocontour gradient

  29. It has good performance in the initial stage of the iterative process. Converge very slow with a linear rate. Steepest descent method

  30. Searching for update h Gradient Hessian Idea 2: minimizing the quadric directly Converge faster but needs to solve the linear system

  31. Recap: Calibration • Directly estimate 11 unknowns in the M matrix using known 3D points (Xi,Yi,Zi) and measured feature positions (ui,vi) Camera Model:

  32. Recap: Calibration • Directly estimate 11 unknowns in the M matrix using known 3D points (Xi,Yi,Zi) and measured feature positions (ui,vi) Linear Approach:

  33. Recap: Calibration • Directly estimate 11 unknowns in the M matrix using known 3D points (Xi,Yi,Zi) and measured feature positions (ui,vi) NonLinear Approach:

  34. Practical Issue is hard to make and the 3D feature positions are difficult to measure!

  35. A popular calibration tool

  36. Multi-plane calibration Images courtesy Jean-Yves Bouguet, Intel Corp. • Advantage • Only requires a plane • Don’t have to know positions/orientations • Good code available online! • Intel’s OpenCV library:http://www.intel.com/research/mrl/research/opencv/ • Matlab version by Jean-Yves Bouget: http://www.vision.caltech.edu/bouguetj/calib_doc/index.html • Zhengyou Zhang’s web site: http://research.microsoft.com/~zhang/Calib/

  37. Step 1: data acquisition

  38. Step 2: specify corner order

  39. Step 3: corner extraction

  40. Step 3: corner extraction

  41. Step 4: minimize projection error

  42. Step 4: camera calibration

  43. Step 4: camera calibration

  44. Step 5: refinement

  45. Next Image Mosaics and Panorama • Today’s Readings • Szeliski and Shum paper http://www.acm.org/pubs/citations/proceedings/graph/258734/p251-szeliski/ Full screen panoramas (cubic): http://www.panoramas.dk/ Mars: http://www.panoramas.dk/fullscreen3/f2_mars97.html 2003 New Years Eve: http://www.panoramas.dk/fullscreen3/f1.html

  46. Why Mosaic? • Are you getting the whole picture? • Compact Camera FOV = 50 x 35° Slide from Brown & Lowe

  47. Why Mosaic? • Are you getting the whole picture? • Compact Camera FOV = 50 x 35° • Human FOV = 200 x 135° Slide from Brown & Lowe

  48. Why Mosaic? • Are you getting the whole picture? • Compact Camera FOV = 50 x 35° • Human FOV = 200 x 135° • Panoramic Mosaic = 360 x 180° Slide from Brown & Lowe

  49. Mosaics: stitching images together Creating virtual wide-angle camera

  50. Auto Stitch: the State of Art Method • Demo • Project 2 is a striped-down AutoStitch

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