1. EECS 274 Computer Vision Projective Structure from Motion

2. Projective structure from motion The Projective Structure from Motion Problem Elements of Projective Geometry Projective Structure and Motion from Two Images Projective Motion from Fundamental Matrices Projective Structure and Motion from Multiple Images From Projective to Euclidean Images Reading: FP Chapter 8

3. Perspective projection Preserve straight lines and cross-ratio of colinear points

4. Perspective projection Recall

5. The projective structure-from-motion problem Given m perspective images of n fixed points P we can write Problem: estimate the m 3×4 matrices M and the n positions P from the mn correspondences p. i j ij 2mn equations in 11m+3n unknowns Overconstrained problem, that can be solved using (non-linear) least squares! j

6. The projective ambiguity of projective SFM If M and P are solutions, i j So are M’ and P’ where i j and Q is an arbitrary non-singular 4×4 matrix. When the intrinsic and extrinsic parameters are unknown Q is a projective transformation.

7. Projective ambiguity Take ambiguity into account, the problem admits a finite number of solution as 2 mn ≥ 11m+3n-15 For two-view (m=2), 7 point correspondences are sufficient to solve projective structure from motion

8. Projective spaces: (semi-formal) definition

9. A model of P( R ) 3 The rays R A, R B, R C associated with vectors v A, v B, v C are mapped to points A, B, C The vectors v A, v B, v C, are linearly independent, and so are (by definition) the points A, B, C The projective plane P( R 3 ) can be constructed by adding to π a one- dimensional set of points at inifity The ray R D parallel to π maps to the point at infinity

10. Projective subspaces and projective coordinates

11. Projective coordinates defined in terms of points Projective coordinates P A* is linear dependent on A i m+1 linearly independent points

12. Projective coordinates Determined by the m+1 fundamental points A i and the unit point A* Coordinate vectors in the projective frame have a simple form

13. Projective subspaces Given a choice of coordinate frame Line:Plane: projective line defined by two points projective frame defined by two points and one unit point projective plane defined by three points projective frame defined by three points and one unit point

14. Affine and projective spaces

15. Affine and projective coordinates

16. Cross-ratios Collinear points Pencil of coplanar lines Pencil of planes {A,B;C,D}= sin(  +  )sin(  +  ) sin(  +  +  )sin 

17. Cross-ratios and projective coordinates Along a line equipped with the basis In a plane equipped with the basis In 3-space equipped with the basis *

18. Projective transformations Bijective linear map: Projective transformation: ( i.e., homography ) Projective transformations map projective subspaces onto projective subspaces and preserve projective coordinates. Projective transformations map lines onto lines and preserve cross-ratios.

19. Perspective projections induce projective transformations between planes. Any point A in scene plane is mapped onto the intersection of AO with the second plane

20. Reference

21. Motion estimation from fundamental matrices Q Facts: b’ can be found using LLS. Once M and M’ are known, P can be computed with LLS. skew-symmetric matrix

22. Projective structure from motion and factorization Factorization?? Algorithm (Sturm and Triggs, 1996) Guess the depths; (e.g., set initial depth values to 1 or estimate from epipolar geometry) Factorize D ; Iterate. Does it converge? (Mahamud and Hebert, 2000)

23. Bundle adjustment (photogrammetry) Minimize with respect to the matrices M i and the point positions P j. Nonlinear least squares minimization Expensive but offers the advantage of combining all measurements to minimize error measure Mean square error between the actual image point position and those predicted using the estimated scene structure and camera motion See “Bundle Adjustment —A Modern Synthesis” by Triggs et al.

24. Absolute scale cannot be recovered! The Euclidean shape (defined up to an arbitrary similitude) is the best that can be recovered. From projective to Euclidean images If z, P, R and t are solutions, so are z, P, R and t.

25. From uncalibrated to calibrated cameras Perspective camera: Calibrated camera: Problem: what is Q ?

26. From uncalibrated to calibrated cameras II Perspective camera: Calibrated camera: Problem: what is Q ? Example: known image center

27. (Pollefeys, Koch and Van Gool, 1999) Reprinted from “Self-Calibration and Metric 3D Reconstruction from Uncalibrated Image Sequences,” by M. Pollefeys, PhD Thesis, Katholieke Universiteit, Leuven (1999).