1. Homogeneous Coordinates (Projective Space)
Let be a point in Euclidean space
Change to homogeneous coordinates:
Defined up to scale:
Can go back to non-homogeneous representation as follows:

2. 3-D Transformations: Translation
Ordinarily, a translation between points is expressed as a vector addition
Homogeneous coordinates allow it to be written as a matrix multiplication:

3. 3-D Rotations: Euler Angles
Can decompose rotation of about arbi-trary 3-D axis into rotations about the coordinate axes (“yaw-roll-pitch”)
, where:
the ordering is just a convention, and there are other representations
(Clockwise when looking
toward the origin)

4. 3-D Transformations: Rotation
A rotation of a point about an arbitrary axis normally expressed as a multiplication by the rotation matrix is written with homogeneous coordinates as follows:

5. 3-D Transformations: Change of Coordinates
Any rigid transformation can be written as a combined rotation and translation:

6. Pinhole Camera Model Image plane Optical axis Principal point
Focal length
Camera
center
Camera point
Image point

7. Pinhole Perspective Projection
Letting the camera coordinates of the projected point be leads by similar triangles to:
this is why more distant objects appear smaller, of course

8. Projection Matrix
Using homogeneous coordinates, we can describe perspective projection with a linear equation: (by the rule for converting between homogeneous and regular coordinates)

9. Example 1: 2D Translation
Q: How can we represent translation as a 3x3 matrix?
A: Using the rightmost column:

10. Translation Example of translation Homogeneous Coordinates
tx = 2 ty = 1

11. Basic 2D Transformations
Basic 2D transformations as 3x3 matrices
Translate
Scale
Rotate
Shear

12. Affine Transformations
Affine transformations are combinations of …
Linear transformations, and
Translations
Properties of affine transformations:
Origin does not necessarily map to origin
Lines map to lines
Parallel lines remain parallel
Ratios are preserved
Closed under composition
Models change of basis

13. Projective Transformations
Affine transformations, and
Projective warps
Properties of projective transformations:
Origin does not necessarily map to origin
Lines map to lines
Parallel lines do not necessarily remain parallel
Ratios are not preserved
Closed under composition
Models change of basis

14. Matrix Composition
Transformations can be combined by matrix multiplication
p’ = T(tx,ty) R(Q) S(sx,sy) p

15. Homography (Projective Transformation)
Definition: Projective transformation or
Recall (set f=1)
8DOF

16. Homography Estimation
Given N pairs
of corresponding
coordinates
(u,v) <> (u’,v’),
how to estimate
eight-parameter
transform H?

17. Computing the Homography
8 degrees of freedom in , so 4 pairs of 2-D points are sufficient to determine it
Other combinations of points and lines also work
3 collinear points in either image are a degenerate configuration preventing a unique solution
Direct Linear Transformation (DLT) algorithm: Least-squares method for estimating

18. Huang’s LS Method A

19. Direct Linear Transformation (DLT) Method
Since vectors are homogeneous, are parallel, so
Let be row j of , be stacked ‘s
Expanding and rearranging cross product above, we obtain , where
x’_i * first row + y’_i * second row = third row
the w’s are the homogeneous coordinate; i used lambda in my definition

20. DLT Homography Estimation: Solve System
Only 2 linearly independent equations in each , so leave out 3rd to make it 2 x 9
Stack every to get 2n x 9
Solve by computing singular value decomposition (SVD) ; is last column of
Solution is improved by normalizing image coordinates before applying DLT
x’_i * first row + y’_i * second row = third row

21. Applying Homographies to Remove Perspective Distortion
this could be helpful with trying to recognize a building from different points of view—convert to a common
one first. this sort of thing is also done to convert the ground plane in perspective to a bird’s eye view
from Hartley & Zisserman
4 point correspondences suffice for
the planar building facade

22. Homographies for Mosaicing
from Hartley & Zisserman