1. Geometric Camera Models
EECS 274 Computer Vision
Geometric Camera Models

2. Geometric Camera Models
Elements of Euclidean geometry
Intrinsic camera parameters
Extrinsic camera parameters
General Form of the Perspective projection equation
Reading: Chapter 2 of FP, Chapter 2 of S

3. Quantitative Measurements and Calibration
Euclidean Geometry

4. Euclidean Coordinate Systems

5. Planes
homogenous coordinate

6. OBP = OBOA + OAP , BP = BOA+ AP
Coordinate Changes: Pure Translations
OBP = OBOA + OAP , BP = BOA+ AP

7. Coordinate Changes: Pure Rotations
1st column:
iA in the basis of (iB, jB, kB)
3rd row:
kB in the basis of (iA, jA, kA)

8. Coordinate Changes: Rotations about the z Axis

9. Rotation matrix Elementary rotation
R=R x R y R z , described by three angles

10. A rotation matrix is characterized by the following properties:
Its inverse is equal to its transpose, R-1=RT , and
its determinant is equal to 1.
Or equivalently:
Its rows (or columns) form a right-handed
orthonormal coordinate system.

11. Rotation group and SO(3)
Rotation group: the set of rotation matrices, with matrix product
Closure, associativity, identity, invertibility
SO(3): the rotation group in Euclidean space R3 whose determinant is 1
Preserve length of vectors
Preserve angles between two vectors
Preserve orientation of space

12. Coordinate Changes: Pure Rotations

13. Coordinate Changes: Rigid Transformations

14. Block Matrix Multiplication
What is AB ?
Homogeneous Representation of Rigid Transformations

15. Rigid Transformations as Mappings

16. Rigid Transformations as Mappings: Rotation about the k Axis

17. Affine transformation
Images are subject to geometric distortion introduced by perspective projection
Alter the apparent dimensions of the scene geometry

18. Affine transformation
In Euclidean space, preserve
Collinearity relation between points
3 points lie on a line continue to be collinear
Ratios of distance along a line
|p2-p1|/|p3-p2| is preserved

19. Shear matrix
Horizontal shear
Vertical shear

20. 2D planar transformations

21. 2D planar transformations

22. 2D planar transformations

23. 3D transformation

24. Idealized coordinate system

25. Camera parameters
Intrinsic: relate camera’s coordinate system to the idealized coordinated system
Extrinsic: relate the camera’s coordinate system to a fix world coordinate system
Ignore the lens and nonlinear aberrations for the moment

26. The Intrinsic Parameters of a Camera
Units:
k,l : pixel/m
f : m
a,b
: pixel
Physical Image Coordinates (f ≠1)
Normalized Image
Coordinates

27. The Intrinsic Parameters of a Camera
Calibration Matrix
The Perspective
Projection Equation

28. In reality
Physical size of pixel and skew are always fixed for a given camera, and in principal known during manufacturing
Focal length may vary for zoom lenses
Optical axis may not be perpendicular to image plane
Change focus affects the magnification factor
From now on, assume camera is focused at infinity

29. Extrinsic Parameters

30. Explicit Form of the Projection Matrix
denotes the i-th row of R, tx, ty, tz, are the coordinates of t
can be written in terms of the corresponding angles
R can be written as a product of three elementary rotations,
and described by three angles
M is 3 x 4 matrix with 11 parameters
5 intrinsic parameters: α, β, u0, v0, θ
6 extrinsic parameters: 3 angles defining R and 3 for t

31. Explicit Form of the Projection Matrix
Note:
: i-th row of R
M is only defined up to scale in this setting!!

32. Theorem (Faugeras, 1993)

33. Camera parameters A camera is described by several parameters
Translation T of the optical center from the origin of world coords
Rotation R of the image plane
focal length f, principle point (x’c, y’c), pixel size (sx, sy)
blue parameters are called “extrinsics,” red are “intrinsics”
Projection equation
The projection matrix models the cumulative effect of all parameters
Useful to decompose into a series of operations
projection
intrinsics
rotation
translation
identity matrix
Definitions are not completely standardized
especially intrinsics—varies from one book to another