1. Lec 23: Single View Modeling
CS4670/5670: Computer Vision
Kavita Bala
Lec 23: Single View Modeling

2. Projective geometry Readings
Ames Room
Readings
Mundy, J.L. and Zisserman, A., Geometric Invariance in Computer Vision, Appendix: Projective Geometry for Machine Vision, MIT Press, Cambridge, MA, 1992, (read  , 23.10)
available online:

3. Ames Room

4. Announcements Prelims at 2:30 in hand back room
Solutions on CMS tonight
Average: 77.5, std dev 12.8

6. Projective geometry—what’s it good for?
Uses of projective geometry
Drawing
Measurements
Mathematics for projection
Undistorting images
Camera pose estimation
Object recognition
Paolo Uccello

7. Homogeneous coordinates
(X,Y,W) same as (SX,SY,SW)
Line in projective plane
ax + by + c = 0
aX + bY + cW = 0
uTp = pTu =0
u = [a,b,c] T and p = [X,Y,W] Tp
(x,y) Euclidean = (X/W,Y/W)
W = 0? Ideal points, points at infinity (X,Y,0)

8. Ideal points Ideal point (“point at infinity”) p  (X, Y, 0) -y
(Xx,Y,0) -z x
image plane
Ideal point (“point at infinity”)
p  (X, Y, 0)

9. Points and lines Intersection of two lines Given two points p1 and p2
u1 = (a1, b1, c1), u2 = (a2, b2, c2)
p = (b1 c2 –b2c1, a2c1-a1c2, a1b2-a2b1)
p = u1 x u2
If u1 parallel to u2
p = (b1c2-b2c1,a2c1-a1c2,0)
Given two points p1 and p2
Line through them u = p1 x p2

10. Can consider the case of an origin. A plane \pi at a distance of 1
Can consider the case of an origin. A plane \pi at a distance of 1. Points on the plane are 3D points and represent rays from the origin.

11. Point and line duality A line l is a homogeneous 3-vector
What is the line l spanned by rays p1 and p2 ?
l is  to p1 and p2  l = p1  p2
l can be interpreted as a plane normal
A ray through origin: A X1 + B Y1 + C W1 = 0
A plane through origin with normal (A, B, C) has the same equation.
u.P = ut p = 0
What is the intersection of two lines l1 and l2 ?
p is  to l1 and l2  p = l1  l2
Points and lines are dual in projective space

13. Vanishing points (2D) image plane vanishing point camera center
line on ground plane

14. Vanishing points Properties
image plane
vanishing point V
line on ground plane
camera
center C
line on ground plane
Properties
Any two parallel lines (in 3D) have the same vanishing point v
The ray from C through v is parallel to the lines
An image may have more than one vanishing point
in fact, every image point is a potential vanishing point
How to determine which lines in the scene vanish at a given pixel?

15. Two point perspective vy vx

16. Vanishing lines Multiple Vanishing Points
Any set of parallel lines on the plane define a vanishing point
The union of all of these vanishing points is the horizon line
also called vanishing line
Note that different planes (can) define different vanishing lines

17. Vanishing lines Multiple Vanishing Points
Any set of parallel lines on the plane define a vanishing point
The union of all of these vanishing points is the horizon line
also called vanishing line
Note that different planes (can) define different vanishing lines

18. Three point perspective

19. Computing vanishing lines
ground plane
Properties
l is intersection of horizontal plane through C with image plane
Compute l from two sets of parallel lines on ground plane
All points at same height as C project to l
points higher than C project above l
Provides way of comparing height of objects in the scene

20. Is this parachuter higher or lower than the person taking this picture?
Lower—he is below the horizon
What about the parachute?

21. Fun with vanishing points

22. Perspective cues

23. Slide by Steve Seitz
Perspective cues

24. Perspective cues

25. Slide by Steve Seitz
Perspective cues

26. Perspective cues

27. Vanishing points are useful
Recover size
Camera calibration …

28. Comparing heights
Vanishing
Point

29. Measuring height 5.4 5 4 3.3 3 2.8 2 1 How high is the camera?
Camera height
2.8