1. Study the mathematical relations between corresponding image points.
Objective
3-D Scene u’ u
Study the mathematical relations between corresponding image points.
“Corresponding” means originated from the same 3D point.

2. Two-views geometry Outline
Background: Camera, Projection models
Necessary tools: A taste of projective geometry
Two view geometry:
Planar scene (homography ).
Non-planar scene (epipolar geometry).
3D reconstruction (stereo).

3. A few words about Cameras
Camera obscura dates from 15th century
First photograph on record shown in the book – 1826
The human eye functions very much like a camera

4. History Camera Obscura
solar eclipse
"Reinerus Gemma-Frisius, observed an eclipse of the sun at Louvain on January 24, 1544, and later he used this illustration of the event in his book De Radio Astronomica et Geometrica, It is thought to be the first published illustration of a camera obscura..."
Hammond, John H., The Camera Obscura, A Chronicle

5. The first “photograph” www.hrc.utexas.edu/exhibitions/permanent/wfp/
Joseph Nicéphore Niépce. View from the Window at Le Gras.

6. A few words about Cameras
Current cameras contain a lens and a recording device (film, CCD, CMOS)
Basic abstraction is the pinhole camera

7. A few words about Lenses Ideal Lenses
Lens acts as a pinhole (for 3D points at the focal depth).

8. Regular Lenses E.g., the cameras in our lab.
To learn more on lens-distortion see Hartley & Zisserman Sec. 7.4 p.189.
Not part of this class.

9. Modeling a Pinhole Camera (or projection)

10. Single View Geometry ∏ f

11. Modeling a Pinhole Camera (or projection)

12. Perspective Projection
Origin (0,0,0) is the Focal center
X,Y (x,y) axis are along the image axis (height / width).
Z is depth = distance along the Optical axis
f – Focal length

13. Projection
P=(X,Y,Z) y f Y X f Z

14. Projection
P=(X,Y,Z) y f Y X f Z

15. Orthographic Projection
Projection rays are parallel
Image plane is fronto-parallel
(orthogonal to rays)
Focal center at infinity

16. Scaled Orthographic Projection
Also called “weak perspective”

17. Pros and Cons of Projection Models
Weak perspective has simpler math.
Accurate when object is small and distant.
Useful for object recognition.
When accuracy really matters (SFM), we must model the real camera (Pinhole / perspective ):
Perspective projection, calibration parameters (later), and all other issues (radial distortion).

18. Two-views geometry Outline
Background: Camera, Projection
Necessary tools: A taste of projective geometry
Two view geometry:
Planar scene (homography ).
Non-planar scene (epipolar geometry).
3D reconstruction from two views (Stereo algorithms)
Hartley & Zisserman:
Sec. 2 Proj. Geom. of 2D.
Sec. 3 Proj. Geom. of 3D.

19. Reading Hartley & Zisserman: Sec. 2 Proj. Geo. of 2D:
point lines in 2D
transformations
2.7 line at infinity
Sec. 3 Proj. Geo. of 3D.
3.1 – point planes & lines.
transformations

20. Same shapes are related by rotation and translation
Why not Euclidian Geometry(Motivation)
Euclidean Geometry is good for
questions like:
what objects have the same shape (= congruent)
Same shapes are related by rotation and translation

21. Why Projective Geometry (Motivation)
Parallel lines meet at the horizon (“vanishing line”)
Where do parallel lines meet?

22. Coordinates in Euclidean Line R1
Not in space ∞

23. Coordinates in Projective Line P1
Take R2 –{0,0} and look at scale equivalence class (rays/lines trough the origin).
Realization: Points on a line P1
“Ideal point”
k(-1,1)
k(0,1)
k(1,1)
k(2,1) ∞
Rays vs. lines rigorous vs. clear We will use rays.
k(1,0)

24. Coordinates in Projective Plane P2
Take R3 –{0,0,0} and look at scale equivalence class
(rays/lines trough the origin).
k(0,1,1)
k(1,1,1)
“Ideal point”
k(0,0,1)
k(1,0,1)
k(x,y,0)

25. Projective Line vs. the Real Line
Symbol
R^2 – {0,0}
The real line
Space
Equivalence classes (2D “rays”)
points
Objects (points)
Intersection with line y=1
Realization
“Ideal point”
k(-1,1)
k(0,1)
k(1,1)
k(2,1) ∞
k(1,0)

26. Projective Plane vs Euclidian plane
Symbol
R3 – {0,0,0}
The real plane
Space
Equivalence classes (3D rays)
point
Objects (points)
Intersection with plane z=1
Realization
k(0,0,1)
k(x,y,0)
k(1,1,1)
k(1,0,1)
k(0,1,1)
“Ideal line”

27. 2D Projective Geometry: Basics
A point:
A line:
we denote a line with a 3-vector
Line coordinates are homogenous
Points and lines are dual: p is on l if
Intersection of two lines/points

28. Cross Product Every entry is a determinant of the two other entries
Area of parallelogram bounded by u and v
Hartley & Zisserman p. 581

29. Cross Product in matrix notation [ ]x
Hartley & Zisserman p. 581

30. Example: Intersection of parallel lines
Q: How many ideal points are there in P2?
A: 1 degree of freedom family – the line at infinity

31. Projective Transformationsu u’

32. Transformations of the projective line
A perspective mapping is a projective transformation T:P1  P1
Pencil of rays
Perspective mapping
One of two or more lines that have a point in common.
Perceptivity is a special projective mapping Hartley & Zisserman p. 632
Lines connecting corresponding points are “concurrent”

33. Perspectivities Projectivities Perspectivities are not a groupL
Slide Objective: Why perspective projection do not suffices l1 l2

34. Projective transformations of the projective line
Given a 2D linear transformation G:R2  R2
Study the induced transformation on the Equivalents classes.
On the realization y=1 we get:

35. Properties: Invertible (T-1 exists)
Composable (To G is a projective transformation)
Closed under composition
Has 4 parameters
3 degrees of freedom
Defined by 3 points
Every point defines 1 constraint

36. Ideal points and projective transformations
Projective transformation can map ∞ to a real point

37. Plane Perspective

38. Euclidean Transformations (Isometries)
Rotation:
Translation:

39. Hierarchy of 2D Transformations
Projective
Affine
Similarity
Rigid (Isometry)
Translation:
Rotation:
Scale
Hartley & Zisserman p. Sec. 2.4