1. Announcements
list – let me know if you have NOT been getting mail
Assignment hand in procedure
web page
at least 3 panoramas:
(1) sample sequence, (2) w/Kaidan, (3) handheld
hand procedure will be posted online
Readings for Monday (via web site)
Seminar on Thursday, Jan 18
Ramin Zabih:
“Energy Minimization for Computer Vision via Graph Cuts “
Correction to radial distortion term

2. Projective geometry Readings
Ames Room
Readings
Mundy, J.L. and Zisserman, A., Geometric Invariance in Computer Vision, Chapter 23: Appendix: Projective Geometry for Machine Vision, MIT Press, Cambridge, MA, 1992, pp   (for this week, read  , 23.10)
available online:

3. Projective geometry—what’s it good for?
Uses of projective geometry
Drawing
Measurements
Mathematics for projection
Undistorting images
Focus of expansion
Camera pose estimation, match move
Object recognition via invariants
Today: single-view projective geometry
Projective representation
Point-line duality
Vanishing points/lines
Homographies
The Cross-Ratio
Later: multi-view geometry

4. Applications of projective geometry
Vermeer’s Music Lesson
Criminisi et al., “Single View Metrology”, ICCV 1999
Other methods
Horry et al., “Tour Into the Picture”, SIGGRAPH 96
Shum et al., CVPR 98
...

5. Measurements on planes1 2 3 4
Approach: unwarp then measure
What kind of warp is this?
A Homography

6. Image rectification To unwarp (rectify) an image p’ p
solve for homography H given p and p’
solve equations of the form: wp’’ = Hp
linear in unknowns: w and coefficients of H
H is defined up to an arbitrary scale factor
can assume H[2,2] = 1
how many points are necessary to solve for H?
work out on board

7. The projective plane Why do we need homogeneous coordinates?
represent points at infinity, homographies, perspective projection, multi-view relationships
What is the geometric intuition?
a point in the image is a ray in projective space
image plane
(x,y,1) y
(sx,sy,s)
(0,0,0) x z
Each point (x,y) on the plane is represented by a ray (sx,sy,s)
all points on the ray are equivalent: (x, y, 1)  (sx, sy, s)

8. Projective lines What is a line in projective space? l p
A line is a plane of rays through origin
all rays (x,y,z) satisfying: ax + by + cz = 0
A line is also represented as a homogeneous 3-vector l l p

9. Point and line duality What is the line l spanned by rays p1 and p2 ?
A line l is a homogeneous 3-vector (a ray)
It is  to every point (ray) p on the line: l p=0 l l1 l2 p p2 p1
What is the line l spanned by rays p1 and p2 ?
l is  to p1 and p2  l = p1  p2
l is the plane normal
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
every property of points also applies to lines

10. Ideal points and lines Ideal line Ideal point (“point at infinity”)
l  (a, b, 0) – parallel to image plane
Corresponds to a line in the image (finite coordinates)
(a,b,0) y x z
image plane
(sx,sy,0) y x z
image plane
Ideal point (“point at infinity”)
p  (x, y, 0) – parallel to image plane
It has infinite image coordinates

11. Homographies of points and lines
Computed by 3x3 matrix multiplication
To transform a point: p’ = Hp
To transform a line: lp=0  l’p’=0
0 = lp = lH-1Hp = lH-1p’  l’ = lH-1
lines are transformed by premultiplication of H-1

12. 3D projective geometry These concepts generalize naturally to 3D
Homogeneous coordinates
Projective 3D points have four coords: P = (X,Y,Z,W)
Duality
A plane N is also represented by a 4-vector
Points and planes are dual in 3D: N P=0
Projective transformations
Represented by 4x4 matrices T: P’ = TP, N’ = N T-1
However
Can’t use cross-products in 4D. We need new tools
Grassman-Cayley Algebra
generalization of cross product, allows interactions between points, lines, and planes via “meet” and “join” operators
Won’t get into this stuff today

13. 3D to 2D: “Perspective” Projection
Matrix Projection:
Preserves
Lines
Incidence
Does not preserve
Lengths
Angles
Parallelism

14. Vanishing Points Vanishing point projection of a point at infinity
image plane
vanishing point
camera
center
ground plane
Vanishing point
projection of a point at infinity

15. Vanishing Points (2D) image plane vanishing point camera center
line on ground plane

16. Vanishing Points Properties
image plane
vanishing point V
line on ground plane
camera
center C
line on ground plane
Properties
Any two parallel lines have the same vanishing point
The ray from C through v point is parallel to the lines
An image may have more than one vanishing point

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 define different vanishing lines

18. 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 define different vanishing lines

19. Computing Vanishing PointsD
Properties
P is a point at infinity, v is its projection
They depend only on line direction
Parallel lines P0 + tD, P1 + tD intersect at X

20. 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
Provides way of comparing height of objects in the scene

21. Fun With Vanishing Points

22. Perspective Cues

23. Perspective Cues

24. Perspective Cues

25. Comparing Heights
Vanishing
Point

26. Measuring Height
5.4 1 2 3 4 5
3.3
Camera height
2.8

27. Measuring Height Without a RulerY C
ground plane
Compute Y from image measurements
Need more than vanishing points to do this

28. The Cross Ratio The Cross-Ratio of 4 Collinear Points P4 P3 P2 P1
A Projective Invariant
Something that does not change under projective transformations (including perspective projection) P1 P2 P3 P4
The Cross-Ratio of 4 Collinear Points
Can permute the point ordering
4! = 24 different orders (but only 6 distinct values)
This is the fundamental invariant of projective geometry

29. Measuring Height  Scene Cross Ratio Image Cross Ratio
T (top of object) C vZ vL t b
Image Cross Ratio
L (height of camera on the this line) Y
B (bottom of object)
ground plane

30. Measuring Height Algebraic Derivation Eliminating  and  yields
reference plane
Algebraic Derivation
Eliminating  and  yields
Can calculate  given a known height in scene

31. So Far
Can measure
Positions within a plane
Height
More generally—distance between any two parallel planes
These capabilities provide sufficient tools for single-view modeling
To do:
Compute camera projection matrix

32. Vanishing Points and Projection Matrix
= vx (X vanishing point)
Not So Fast! We only know v’s up to a scale factor
Can fully specify by providing 3 reference lengths

33. 3D Modeling from a Photograph

34. 3D Modeling from a Photograph

35. Conics Conic (= quadratic curve) Defined by matrix equation
Can also be defined using the cross-ratio of lines
“Preserved” under projective transformations
Homography of a circle is a…
3D version: quadric surfaces are preserved
Other interesting properties (Sec in Mundy/Zisserman)
Tangency
Bipolar lines
Silhouettes