1. Projective geometry ECE 847: Digital Image Processing Stan Birchfield
Clemson University

2. Lines A line in 2D is described by two parameters: But vertical lines?
almost
A line in 2D is described by two parameters:
But vertical lines?
Only two parameters are sufficient, but requires nonlinear formulation: ^
slope
y-intercept

3. Lines A better parameterization can represent all lines:
Here the line is represented by 3 parameters:
But nonzero scalar multiple does not change the equation:
So we have only 2 degrees of freedom
To make this work, we have to introduce a non-intuitive definition:
I.e., the vector u and its scalar multiple are the same

4. Lines While we are at it, let us put the point into a vector, too:
Which leads to the beautiful expression:
Nonzero scalar multiple also does not change the point:
So we introduce an analogous non-intuitive definition:

5. Example Ques: What does the vector [4, 6, 2]T represent?
Ans: It depends.
If the vector is a 2D point, then the point is (4/2, 6/2) = (2, 3) -- divide by 3rd coordinate
If the vector is a 2D line, then the line is 4x + 6y + 2 = 0, or 2x + 3y + 1 = 0
Points and lines are represented in the same way. Context determines which.

6. Lines Ques.: Is the point p on the line u? Ans: Check whether pTu = 0
Ques.: Which line passes through two points p1 and p2?
Ans.: Compute u = p1 x p2
Ques.: Which points lies at the intersection of two lines?
Ans.: Compute p = u1 x u2

7. Euclidean transformation
2D Euclidean transformation:
is more conveniently represented as
Again, we use 3 numbers to represent 2D point (These are homogeneous coordinates)

8. Perspective projection
Nonlinear perspective projection
can be replaced by linear equation
where (x,y,w)T are homogeneous coordinates of (u,v,1)T:

9. Recap
Homogeneous coordinates of 2D point (x,y)T are p=(wx,wy,w)T where w ≠ 0
We have seen three reasons for homogeneous coordinates:
simple representation of points and lines, no special cases
simple representation of Euclidean transformation
simple representation of perspective projection

10. Q & A Questions: Answers:
Is there a unifying theory to explain homogeneous coordinates?
How can they be extended to 3D?
Are they useful for anything else?
Answers:
Projective geometry
Useful for planar warping, 3D reconstruction, image mosaicking, camera calibration, etc.

11. Euclidean  Projective
Start with 2D Euclidean point (x,y)
To convert to Projective,
Append 1 to the coordinates: p=(x,y,1)
Declare equivalence class: p=ap, a≠0
To convert back to Euclidean,
Divide by last coordinate: (u, v, w)  (u/w, v/w) x=u/w, y=v/w

12. Ideal points What if last coordinate is zero? (u,v,0)
Cannot divide by zero
Projective plane contains more points than the Euclidean plane:
All Euclidean planes, plus
Points at infinity (a.k.a. ideal points)
All ideal points lie on ideal line: (0, 0, 1)

13. Are ideal points special?
In pure projective geometry, there is no distinction between real points and ideal points
Transformations will often convert one to another
We will freely make use of this, and often ignore the distinction
However, distinction is necessary to convert back to Euclidean
Distinction will be made when we need to interpret results

14. Geometries
Every geometry has
transformations
invariants

15. Stratification of geometries
Euclidean
similarity
affine
projective

16. Stratification Euclidean Similarity Affine Projective allow parallel
projection
allow perspective
projection
allow scale
Euclidean
Similarity
Affine
Projective
one length
absolute points
ideal line

17. Cross ratio

18. Ray space

19. Unit hemisphere

20. Augmented affine plane∞ ∞ ∞ ℓ∞ ∞
The line at infinity ℓ∞ is “beyond infinity”

21. Intersection of parallel lines
y=mx+b
(where m is the same)
intersect at
(1, m, 0)
(1,m,0) ∞ ∞ ∞ ℓ∞ ∞
Note: Antipodal points are identified

22. Representing line at infinity
(1,m1,0) ∞ ∞ ∞ ℓ∞
(1,m2,0) ∞
Cross product of two points at infinity yields ℓ∞ = (0,0,1)

23. The strange world beyond infinity
The line at infinity ℓ∞ = (0,0,1) ax+by+c=0
This means 1 = 0 !

24. Line transformations If point transforms according to p’ = Ap
How does line transform? u’ = A-Tu

25. Conics Take picture of circle  ellipse
No distinction between types of conic sections in projective geometry

26. 3D Projective
Points and planes
Plucker coordinates for lines

27. Image formation 3D world point is (X,Y,Z,W)T
2D image point is (x,y,w)T
Therefore, perspective projection is a 3x4 matrix P

28. Perspective projection
Camera calibration matrix K

29. Homography Simple case is projection from plane to plane
Can be either world plane to image plane, or
image plane to another image plane, or
world plane to another world plane,
etc.
3x3 matrix is a projective transformation
Called a homography

30. Euclidean homography
Needs K

31. Essential and fundamental matrices

32. Relationship b/w FM and H
Fundamental matrix and homography

33. How to compute homography
Direct Linear Transform

34. Normalization
Important