1. Recovery of affine and metric properties from images in 2D Projective space
Ko Dae-Won

2. Affine properties(line at infinity)
Recovery of affine and metric properties from images in 2D Projective space
Affine properties(line at infinity)
Parallelism
Parallel length ratios
Metric properties(circular points)
Angles
Length ratios
Recover the original shape

3. 1. Recovery of affine properties
Recovery of affine and metric properties from images in 2D Projective space
1. Recovery of affine properties
Homogeneous coordinates
equivalence class of vectors, any vector is representative
Set of all equivalence classes in R3(0,0,0)T forms P2
Homogeneous representation of points on
if and only if
The point x lies on the line l if and only if xTl=lTx=0
Homogeneous coordinates
Inhomogeneous coordinates
but only 2DOF

4. 1. Recovery of affine properties
Recovery of affine and metric properties from images in 2D Projective space
1. Recovery of affine properties
Points from lines and vice-versa
Intersections of lines
The intersection of two lines and is
Line joining two points
The line through two points and is
Intersections of parallel lines
Ideal points
Line at infinity

5. 1. Recovery of affine properties
Recovery of affine and metric properties from images in 2D Projective space
1. Recovery of affine properties
Duality
Duality principle:
To any theorem of 2-dimensional projective geometry there corresponds a dual theorem, which may be derived by interchanging the role of points and lines in the original theorem

6. 1. Recovery of affine properties
Recovery of affine and metric properties from images in 2D Projective space
1. Recovery of affine properties
Conics
Curve described by 2nd-degree equation in the plane
or homogenized
or in matrix form
with

7. 1. Recovery of affine properties
Recovery of affine and metric properties from images in 2D Projective space
1. Recovery of affine properties
Tangent lines to conics
The line l tangent to C at point x on C is given by l=Cx l x C

8. 1. Recovery of affine properties
Recovery of affine and metric properties from images in 2D Projective space
1. Recovery of affine properties
Dual conics
A line tangent to the conic C satisfies
In general :
Dual conics = line conics = conic envelopes

9. 1. Recovery of affine properties
Recovery of affine and metric properties from images in 2D Projective space
1. Recovery of affine properties
Projective transformations
Definition:
A projectivity is an invertible mapping h from P2 to itself such that three points x1,x2,x3 lie on the same line if and only if h(x1),h(x2),h(x3) do.
A mapping h:P2P2 is a projectivity if and only if there exist a non-singular 3x3 matrix H such that for any point in P2 represented by a vector x it is true that h(x)=Hx
Theorem:
Definition: Projective transformation or
projectivity=collineation=projective transformation=homography

10. 1. Recovery of affine properties
Recovery of affine and metric properties from images in 2D Projective space
1. Recovery of affine properties
The line at infinity

11. 1. Recovery of affine properties
Recovery of affine and metric properties from images in 2D Projective space
1. Recovery of affine properties
Affine properties from images

12. 1. Recovery of affine properties
Recovery of affine and metric properties from images in 2D Projective space
1. Recovery of affine properties

13. 1. Recovery of affine properties
Recovery of affine and metric properties from images in 2D Projective space
1. Recovery of affine properties

14. 1. Recovery of affine properties
Recovery of affine and metric properties from images in 2D Projective space
1. Recovery of affine properties
Distance ratio

15. 1. Recovery of affine properties
Recovery of affine and metric properties from images in 2D Projective space
1. Recovery of affine properties
Distance ratio

16. 2. Recovery of metric properties
Recovery of affine and metric properties from images in 2D Projective space
2. Recovery of metric properties
2. Recovery of metric properties

17. 2. Recovery of metric properties
Recovery of affine and metric properties from images in 2D Projective space
2. Recovery of metric properties
The circular points

18. 2. Recovery of metric properties
Recovery of affine and metric properties from images in 2D Projective space
2. Recovery of metric properties
The circular points

19. 2. Recovery of metric properties
Recovery of affine and metric properties from images in 2D Projective space
2. Recovery of metric properties
Conic dual to the circular points

20. 2. Recovery of metric properties
Recovery of affine and metric properties from images in 2D Projective space
2. Recovery of metric properties
Angles

21. 2. Recovery of metric properties
Recovery of affine and metric properties from images in 2D Projective space
2. Recovery of metric properties
Length ratios

22. 2. Recovery of metric properties
Recovery of affine and metric properties from images in 2D Projective space
2. Recovery of metric properties

23. 2. Recovery of metric properties
Recovery of affine and metric properties from images in 2D Projective space
2. Recovery of metric properties
Length ratios

24. 2. Recovery of metric properties
Recovery of affine and metric properties from images in 2D Projective space
2. Recovery of metric properties
Metric from affine

25. 2. Recovery of metric properties
Recovery of affine and metric properties from images in 2D Projective space
2. Recovery of metric properties
Metric from projective