1. Geometric Transformation & Projective Geometry
by: Diako Mardanbegi, PhD student

2. Transformation and Projective Geometry
Computer Vision:
Relative movement of a camera
Object locations wrt. Camera
Graphics
Describe objects and cameras in a scene and how they should move in 3D scene.
User interfaces (like WPF) relies heavy placing components relative to each other (2D) and for its 3D graphics engine.

3. Outline 2-D Transformations Multiple frames 3-D Transformations
Representing Transformations
Homogeneous form
Multiple frames
Cartesian Coordinate System
Describe a local frame
Orientation
Translation
Mapping between multiple frames
3-D Transformations
Euler angles
Quaternion
Projective coordinate system
Homography

4. 2-D Translation y x

5. 2-D Translation

6. 2-D Rotation about the origin

7. 2-D Rotation about the origin
Rotation about the origin

8. 2-D Rotation about the origin
Rotation about the origin

9. 2-D Rotation about a specified point

10. 2-D Rotation about a specified point a b d c
Rotation about a specified point

11. 2-D Scaling relative to the origin
Vertical shift proportional to vertical position
Horizontal shift proportional to Horizontal position

12. 2-D Scaling relative to the originb
Scaling relative to the origin

13. 2-D Scaling relative to a specified point

14. Representing transformations,
Trick of additional coordinate makes this possible to express all transformations in a common linear form
Old way:
Cartesian coordinates
New way:
With additional coordinate
(Homogeneous coordinates)

15. Representing transformations
homogeneous transformation:
linear transformation of homogeneous coordinates
In homogeneous form
x‘ = H x
In homogeneous form

16. from Homogeneous coordinates to Cartesian coordinates
from Cartesian coordinates to Homogeneous coordinates
from Homogeneous coordinates to Cartesian coordinates

17. Advantages Easy to compose multiple transformations after each other
Combination of all transformations into a single matrix

18. Advantages
for example :General 2-D Pivot-Point Rotation  o’ a b d c

19. Inverse of transformations
Easy to invert

20. Transformations Don’t Commuteb 

21. Transformations Don’t Commuteb 

22. Other 2-D transformations
Reflection
Shear 2

23. Homography matrix
Full-generality 3 x 3 homogeneous transformation

24. Translation components
Homography matrix
Full-generality 3 x 3 homogeneous transformation
Translation components

25. Scale/rotation components
Homography matrix
Full-generality 3 x 3 homogeneous transformation
Scale/rotation components

26. Shear/rotation components
Homography matrix
Full-generality 3 x 3 homogeneous transformation
Shear/rotation components

27. Homogeneous scaling factor
Homography matrix
Full-generality 3 x 3 homogeneous transformation
Homogeneous scaling factor

28. Homography matrix Full-generality 3 x 3 homogeneous transformation
When these are zero (as they have been so far), H is an affine transformation

29. Why we are using only one fixed Cartesian system?
Why we don’t define a local coordinate system? o’  o’  a b d c

30. We can do it, but first know about:
What is exactly a cartesian coordinate system?
How to describe a local frame in a fixed frame?
How to express a point in each frame?

31. Cartesian Coordinate System
y-axis
A coordinate system in Rn is defined by an origin o and n orthogonal basis vectors
In R3, positive direction of each axis X, Y, Z is indicated by unit vector i, j, k
Let P = (x, y, z)T be a point in R3
Coordinate is length of projection of vector from origin to point onto axis basis vector
. P i j k
x-axis O
What do these values mean?
z-axis ?

32. Vector Projection
The projection of vector a onto u is that component of a in the direction of u

33. X’ Y’ B Y
. P X A

34. Describe a local frame in a fixed frameY X’ Y’
A,B X Y X’ Y’ 
A,B
homogeneous form A X Y X’ Y’
(x, y) B
: orientation of frame B relative to A
: translation of frame B relative to A

35. Mapping from frame to frameX Y X’ Y’
. P A B
Mapping
Transformation

36. Transformation in 3-Dimension

37. c = a x b = (ya zb -za yb, za xb - xa zb, xa yb- ya xb)T
Going from 2D to 3D
Vector cross product
Definition: If a = (xa, ya, za)T and b = (xb, yb, zb)T, then:
c = a x b = (ya zb -za yb, za xb - xa zb, xa yb- ya xb)T
c is orthogonal to both a and b (direction given by right-hand r with magnitude
k = i xj

38. 3-D Translation
P’ = T . P

39. 3-D Scaling

40. 3-D Rotation
In 2-D, we are always rotating in the plane of the image, but in 3-D the axis of rotation itself is a variable
Three canonical rotation axes are the coordinate axes X, Y, Z

41. 3-D Rotation
Similar to 2-D rotation matrices, but with coordinate corresponding to rotation axis held constant

42. 3-D Rotation
When object is to be rotated about an axis that is parallel to one of the coordinate axes
mesle mesale 2d

43. 3-D Rotation Easiest way ?!
When an object is to be rotated about an axis that is not parallel to one of the coordinate axes

44. Orientation in 3-D A B

45. Hierarchy of frames
+Yworld
+Xworld
+Zworld

46. 3-D Camera Coordinates +X right, -X left +Y down, -Y up
Right-handed system
From point of view of camera looking out into scene:
+X right, -X left
+Y down, -Y up
+Z in front of camera, -Z behind

47. +Yworld
+Xworld
+Zworld
Creating a simple WPF 3D space with camera:

48. Orientation in 3-D
in 2D we have 4 elements in orientation matrix but we can locate the local frame by having only one angle . (we know the axis)
in 3D we have 9 elements but how many angles do we need for locating the local frame in space?
only 3

49. Euler’s Theorem Euler’s Theorem :
“Any two independent orthonormal coordinate frames can be related by a sequence of rotations (not more than three) about coordinate axes, where no two successive rotations may be about the same axis”

50. Z-Y-X Euler angles
Start with the frame coincident with a known frame {A}. Rotate {B} first about ZB by an angle , then about YB by an angle  and, finally, about XB by an angle 

51. Euler angles representation
Advantages
Matrix representation
The mathematics is well-known
Disadvantages:
Describing a general rotation as rotations about the three basis axes is not natural for an animator.
Order is important
Gimbal lock

52. Quaternions
Euler:
“Any rotation of a rigid body can be described by defining an axis of rotation, often called the Euler rotation axis, and a rotation angle.”
William Rowan Hamilton formulated quaternions, utilizing this Euler’s theorem, as a method of representing rotations.

53. Quaternions Complex numbers are represented in the form
where , and a,b are real numbers.
Quaternion is an extended complex number

54. Rotation by Quaternions
Any unit quaternion has the form
for some angle and unit vector
We will represent a point p in space by the quaternion
We compute the rotation of point P about the unit vector I by an angle  by using the unit quaternion
and by Rodrigues formula :

55. Rotation by Quaternions
We can also combine two rotation by multiplication of the involved quaternions.
Example: 2

56. Quaternions representation
Advantages:
Simple composition Rotations
Obvious geometrical interpretation
Disanvantages:
Quaternions only represent rotation
Quaternion mathematics appears complicated

57. How does the model look in the view of the camera
2D Projective Geometry
How does the model look in the view of the camera
+Yworld
+Xworld
+Zworld
+Ycamera
+Xcamera
+Zcamera

58. Texture mapping on 3D surfaces

59. Projective Geometry Outline
Description of projective geometry in a plane
Descriptions of Points and lines
Projective transformations
Example of application

60. Projective Geometry Euclidean geometry describes shapes “as they are”
Properties of objects that are unchanged by rigid motions
Lengths
Angles
Parallelism
Projective geometry describes objects “as they appear”
Lengths, angles, parallelism become “distorted” when we look at objects

61. 2D Projective Geometry
Coordinates are called homogeneous or projective coordinates
R is the coordinate vector of point M and (x,y,w) are its homogeneous coordinates
the rays (x,y,w) and (λx, λy, λw) are the same and are mapped to the same point M
L is the coordinate vector of line l and (a,b,c) are its homogeneous coordinates

62. From Projective Plane to Euclidean Plane
How do we “land” back from the projective world to the 2D world of the plane?
for Point:
for line:

63. Projective Geometry in 2D
Point M belongs to line l if and only if:
Two lines L=(a,b,c) and L’=(a’,b’,c’) intersect in the point R=L×L’
The line through 2 points R and R’ is L=R×R’

64. Projective Geometry in 2D
The points R=(x,y,0) do not correspond to finite points in the plane. they are points at infinity, also called ideal points
The line L=(0,0,1) passes through all points at infinity, since L.R=0
Two parallel lines L=(a,b,c) and L’=(a,b,c’) intersect at the point R=L×L’=(c’-c)(b,-a,0), i.e. (b,-a,0)
Any line (a,b,c) intersects the line at infinity at (b,-a,0). so the line at infinity is the set of all points at infinity

65. Duality
Duality principle: To any theorem of 2-D projective geometry there corresponds a dual theorem, which may be derived by interchanging the role of points and lines in the original theorem

66. Projective Transformations in a Plane
Projectivity (Collineation or Homography)
Mapping from points in plane to points in plane
A homography is an invertible transformation from the real projective plane to the projective plane that maps straight lines to straight lines.
3 aligned points are mapped to 3 aligned points

67. Transformation of lines
A mapping is a projectivity if and only if the mapping consists of a linear transformation of homogeneous coordinates
Definition: Projective transformation or
8DOF
For a point transformation
Transformation for lines

68. Special Projectivities

69. A special case A plane to plane projective transformation
2D World plane
Observer
2D Image plane
(retina, film, canvas)

70. Example of Projective Transformations
Central projection maps planar scene points to image plane by a projectivity
The image of the same plannar scene from a second camera can be obtained from the image from the first camera by a projectivity

71. Applications of homographies

72. Mosaics

73. projectivity is from one plane to another plane
Remember that
projectivity is from one plane to another plane

74. Camera Model Homogeneous form Concise form
K: Camera Calibration Matrix

75. General mapping of a pinhole camera
Xc = R(Xw − Cw)
Xc = RXw − RCw
Translation
Homogeneous form of
Point on image
Rotation
Camera
Calibration

76. Corrected image (front-to-parallel)

77. Homography
The floor (enlarged)
Automatically
rectified floor

78. Analysing patterns and shapes
Automatic rectification
From Martin Kemp The Science of Art
(manual reconstruction)
2 patterns have been discovered !

79. Analysing patterns and shapes
What is the (complicated)
shape of the floor pattern?
Automatically rectified floor
St. Lucy Altarpiece, D. Veneziano

80. Analysing patterns and shapes
Automatic
rectification
From Martin Kemp, The Science of Art
(manual reconstruction)

81. The Ambassadors
H. Holbein

82. Applications of Homographies summary
Mosaics (Image processing)
Involves computing homographies between pairs of input images (image-image mappings)
Rendering textures (Computer graphics)
Require planar scene surface and image plane. (scene-image mapping)
Computing planar shadows (Computer graphics)
Require apply H between two surfaces inside a 3D scene having the light source as the center of projections (Scene-Scene mapping)
Remove perspective distortion (Computer vision)
Involves computing H between image and 3D scene surfaces (image-scene mapping)