Chapter 14 Analytic Photogrammetry

1. Chapter 14Analytic Photogrammetry Presented by 王夏果 and Dr. Fuh R94922103@ntu.edu.tw 0937384214 Digital Camera and Computer Vision Laboratory Department of Computer Science and Information Engineering National Taiwan University, Taipei, Taiwan, R.O.C.

2. Analytic Photogrammetry • Make inferences about : • 3D position • Orientation • Length of the observed 3D object parts in a world reference frame from measurements of one or more 2D- perspective projections of a 3D object DC & CV Lab. NTU CSIE

3. Analytic Photogrammetry (cont.) • These inference problems can be construed as nonlinear least-square problems • Iteratively linearize the nonlinear functions from an initially given approximate solution DC & CV Lab. NTU CSIE

4. Photogrammetry • Provide a collection of methods for determining the position and orientation of cameras and range sensors in the scene and relating camera positions and range measurements to scene coordinates • GIS: Geographic Information System • GPS: Global Positioning System DC & CV Lab. NTU CSIE

5. DC & CV Lab. NTU CSIE

6. Exterior Orientation • Determine position and orientation of camera in absolute coordinate system from projections of calibration points in scene • The exterior orientation of the camera is specified by all parameters of camera pose, such as perspectivity center position, optical axis direction. DC & CV Lab. NTU CSIE

7. Exterior Orientation (cont.) • Exterior orientation specification: requires 3 rotation angles, 3 translations DC & CV Lab. NTU CSIE

8. DC & CV Lab. NTU CSIE

9. Interior Orientation • Determine internal geometry of camera • The interior orientation of camera is specified by all the parameters that determines the geometry of 3D rays from measured image coordinates DC & CV Lab. NTU CSIE

10. Interior Orientation (cont.) • The parameters of interior orientation relate the geometry of ideal perspective projection to the physics of a camera. • Parameters: camera constant, principal point, lens distortion, … DC & CV Lab. NTU CSIE

11. Interior Orientation (cont.) • With interior and external orientation, we can complete specify the camera orientation. DC & CV Lab. NTU CSIE

12. Relative Orientation • Determine relative position and orientation between 2 cameras from projections of calibration points in scene • Calibrate relation between two cameras for stereo • Relates coordinate systems of two cameras to each other, not knowing 3D points themselves, only their projections in image DC & CV Lab. NTU CSIE

13. Relative Orientation (cont.) • Assume interior orientation of each camera known • Specified by 5 parameters: 3 rotation angles, 2 translations DC & CV Lab. NTU CSIE

14. DC & CV Lab. NTU CSIE

15. Absolute Orientation • Determine transformation between 2 coordinate systems or position and orientation of range sensor in absolute coordinate system from coordinates of calibration points • Convert depth measurements in viewer-centered coordinates to absolute coordinate system for the scene DC & CV Lab. NTU CSIE

16. Absolute Orientation (cont.) • Orientation of stereo model in world reference frame • Determine scale, 3 translations, 3 rotations • Recovery of relation between two coordinate system DC & CV Lab. NTU CSIE

17. DC & CV Lab. NTU CSIE

18. Symbol Definition DC & CV Lab. NTU CSIE

19. Rotation Matrix DC & CV Lab. NTU CSIE

20. Rotation Matrix (cont.) DC & CV Lab. NTU CSIE

21. Rotation Matrix (cont.) DC & CV Lab. NTU CSIE

22. World Frame to Camera Frame • (x, y, z)’ in world frame represented by (p, q, s)’ in camera frame: DC & CV Lab. NTU CSIE

23. Pinhole Camera Projection • Pinhole camera with image at distance f from camera lens, projection: where f is a camera constant, related to focal length of lens DC & CV Lab. NTU CSIE

24. Principal Point • Origin of measurement image plane coordinate • Represented by (u0, v0) DC & CV Lab. NTU CSIE

25. Perspective Projection Equations • Collinearity equation: DC & CV Lab. NTU CSIE

26. Perspective Projection Equations (cont.) • Show that the relationship between the measured 2D-perspective projection coordinates and the 3D coordinates is a nonlinear function of u0, v0, x0, y0, z0, ω, ψ, and κ DC & CV Lab. NTU CSIE

27. Take a Break DC & CV Lab. NTU CSIE

28. Nonlinear Least-Square Solutions • Noise model: DC & CV Lab. NTU CSIE

29. Nonlinear Least-Square Solutions (cont.) • Maximum likelihood solution: β1, …, βM maximize Prob(α1, …, αk | β1, …, βM ) • In other words, this solution minimizes least-squares criterion: where DC & CV Lab. NTU CSIE

30. First-Order Taylor Series Expansion • First-order Taylor series expansion of gk taken around βt: DC & CV Lab. NTU CSIE

31. First-Order Taylor Series Expansion (cont.) DC & CV Lab. NTU CSIE

32. Exterior Orientation Problem • Determine the unknown rotation and translation that put the camera reference frame in the world reference frame. DC & CV Lab. NTU CSIE

33. Exterior Orientation Problem (cont.) DC & CV Lab. NTU CSIE

34. One Camera Exterior Orientation Problem • Known: (xn, yn, zn)’ and (un, vn)’ (un, vn)’ is the corresponding set of 2D-perspective projections, n = 1, …, N • Unknown: (ω,ψ,κ) and (x0, y0, z0)’ DC & CV Lab. NTU CSIE

35. Other Exterior Orientation Problem • Camera calibration problem: unknown position of camera in object frame • Object pose estimation problem: unknown object position in camera frame • Spatial resection problem in photogrammetries: 3D positions from 2D orientation DC & CV Lab. NTU CSIE

36. Nonlinear Transformation For Exterior Orientation DC & CV Lab. NTU CSIE

37. Standard Solution • By chain rule, DC & CV Lab. NTU CSIE

38. In matrix form, DC & CV Lab. NTU CSIE

39. DC & CV Lab. NTU CSIE

40. Standard Solution (cont.) DC & CV Lab. NTU CSIE

41. Auxiliary Solution • Not iteratively adjust the angles directly • Reorganize the calculation such that we iteratively adjust the three auxiliary parameters of a skew symmetric matrix associated with the rotation matrix • Then, we determine the adjustment of the angles DC & CV Lab. NTU CSIE

42. DC & CV Lab. NTU CSIE

43. Quaternion Representation • From any skew symmetric matrix, we can construct a rotation matrix R by choosing scalar d: R = (dI + S)(dI - S)-1 which guarantees that R’R = I DC & CV Lab. NTU CSIE

44. Quaternion Representation (cont.) • Expanding the equation for R: parameters a, b, c, d can be constrained to satisfy a2 + b2 + c2 + d2 = 1 DC & CV Lab. NTU CSIE

45. Quaternion Representation (cont.) DC & CV Lab. NTU CSIE

46. Take a Break DC & CV Lab. NTU CSIE

47. Relative Orientation • The transformation from one camera station to another can be represented by a rotation and a translation • The relation between the coordinates, rl and rr of a point P can be given by means of a rotation matrix and an offset vector DC & CV Lab. NTU CSIE

48. DC & CV Lab. NTU CSIE

49. Relative Orientation (cont.) • Relative orientation is typically with the determination of the position and orientation of one photograph with respect to another, given a set of corresponding image points DC & CV Lab. NTU CSIE

50. Relative Orientation (cont.) • Relative orientation specified by five parameters: (yR - yL), (zR - zL), (ωR - ωL), (ψR - ψL), (κR - κL) • Assumption: • Camera interior orientation known • Image positions expressed to identical scale and with respect to principal point DC & CV Lab. NTU CSIE