1. Single View Geometry Course web page: vision.cis.udel.edu/cv April 9, 2003  Lecture 20

2. Announcements Class next Monday is cancelled due to scheduling commitments for departmental review Read Forsyth & Ponce, Chapter 10- 10.1.1 and Hartley & Zisserman, Chapter 8-8.3 (skip 8.2.2), 8.6-8.6.1 on stereo for Friday

3. Outline Single view metrology –Cross ratio Distances between planes –Homology (homography) Lengths & areas on planes

4. The Cross Ratio Definition: Ratio of ratio of distances between four collinear points In affine coordinates, handle points at infinity with convention 1/1 = 1, a/1 = 0, 1/a = 1 Key property: Invariant under projective transformations x1x1 x2x2 x3x3 x4x4

5. Single View Metrology: Background Vanishing line l of reference plane ¼ Vanishing point v of reference line (not parallel to ¼ —typically but not necessarily orthogonal) from Criminisi et al. ¦

6. Vanishing Line Partitions Scene Space Image points above vanishing line back-project to 3-D points which are farther from reference plane than camera Points below vanishing line are closer to reference plane Points on line are same distance from reference plane as camera from Criminisi et al.

7. Plane Correspondences 3-D points on separate planes parallel to reference plane correspond if line joining them is parallel to reference direction from Criminisi et al. C X X’X’ ¦ ¦’¦’

8. Plane Correspondences The images of corresponding 3-D points are collinear with the vanishing point from Criminisi et al. C X X’X’ x x’x’ c v Point same distance from reference plane as camera

9. Measuring Distances between Planes: Four Collinear Points World coordinatesImage coordinates from Criminisi et al. ZcZc

10. A Real Image from Criminisi et al. v out of image

11. Measuring Distances between Planes: Cross Ratio from Criminisi et al. measurable from image

12. Computing the Distance Z between ¦ and ¦’ From the cross ratio’s invariance, we know that: But V is a point at infinity, so we have:

13. Computing the Distance Z between ¦ and ¦’ In terms of quantities we are interested in, this is: which after some algebra yields: So with a known far-plane-to-camera distance Z c or near- plane-to-camera distance Z c - Z, we can use the cross ratio measured from the image to compute the plane-to- plane distance Z, and vice versa ZcZc Z c - Z from Criminisi et al.

14. Using Reference Distances The plane-to-camera distance may be difficult or inconvenient to measure Instead, we can use a known plane-to- plane distance (in the reference direction) to derive the unknown plane- to-plane distance

15. Transferring Distances through Linked Parallel Planes If Z r known, apply cross ratio to get plane ¦ r -to-camera distance Alternately apply cross ratio to get plane-to-plane distances, plane-to-camera distances (near or far as necessary) along links until Z is obtained World coordinatesImage coordinates from Criminisi et al. unknown known

16. Example: Distance between Planes Original image Image with vanishing line, cross ratio points, computed plane-to-plane distance from Criminisi et al.

17. Example: Distance between Planes from Criminisi et al.

18. Measurements on a Plane For one plane, we can compute the following from image measurements after affine rectification (vanishing line mapped to l 1 ): –Ratios of lengths of parallel line segments –Ratios of areas from Criminisi et al.

19. Affine Rectification Before (foreshortened)After (no foreshortening) from Hartley & Zisserman vanishing line (before rectification)

20. Measurements within Different Planes Cross ratio approach tells us distance between planes, but how to compare shapes on each plane? from Criminisi et al.

21. 2-D Homography (aka Projectivity) 2-D to 2-D projective transformation mapping points from plane to plane (e.g., image of a plane) 3 x 3 homogeneous matrix defines homo- graphy such that for any pair of corresponding points and, from Hartley & Zisserman

22. Some Image Transformations Described by Homographies Planar surfaces under general camera motion General scene when camera motion is rotation about camera center

23. Computing a Homography 8 degrees of freedom in H, so at least n = 4 pairs of 2-D points are sufficient to determine it –Other combinations of points and lines also work Use Direct Linear Transformation (DLT) algorithm to compute H –Stacked matrix A is 2 n x 9 vs. 2 n x 12 for camera matrix P estimation because points are 2-D 3 collinear points in either image is a degenerate configuration preventing a unique solution

24. Rectification: Removing Planar Perspective Distortions from Hartley & Zisserman Take rectangle corners as corresponding points

25. Example: Rectification for Extracting Building Façade Textures from Hartley & Zisserman

26. Homographies for Mosaicing from Hartley & Zisserman

27. Planar Homology Definition: Homography between planes related by a perspectivity (lines joining points are concurrent, meeting at an apex) 5 DOF (general homographies have 8) True of parallel planes: Apex is point at infinity from Criminisi et al.

28. Computing the Homology for Parallel Planes What is needed: –Apex: Vanishing point of reference direction v –Axis: Vanishing line l of planes –One point correspondence (r, r’) between the planes Compute and solve for ¹ from correspondence condition given by from Criminisi et al.

29. Comparing Quantities on Parallel Planes 1.Compute homology between planes ¦, ¦’ 2.Map points from one plane to the other 3.Compare quantities on one plane vs. mapped plane as though they are from same plane (after affine rectification) from Criminisi et al.

30. Algebraic Derivation All of these relationships can be derived algebraically (the cross ratio is a geometric construction), with the following key advantage: –Can overconstrain computations: Multiple reference distances can be used to decrease uncertainty by a known amount Details in paper and Criminisi et al., 2001 IJCV paper from Criminisi et al. Multiple measurements reduce error