1 / 33

# EECS 274 Computer Vision - PowerPoint PPT Presentation

EECS 274 Computer Vision. Geometric Camera Models. 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.

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

## PowerPoint Slideshow about ' EECS 274 Computer Vision' - fionan

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

### EECS 274 Computer Vision

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

Euclidean Geometry

homogenous coordinate

OBP = OBOA + OAP ,BP = BOA+ AP

1st column:

iA in the basis of (iB, jB, kB)

3rd row:

kB in the basis of (iA, jA, kA)

Elementary rotation

R=R x R y R z , described by three angles

• 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.

Rotation group and SO(3) properties:

• 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

Block Matrix Multiplication properties:

What is AB ?

Homogeneous Representation of Rigid Transformations

Affine transformation properties:

• Images are subject to geometric distortion introduced by perspective projection

• Alter the apparent dimensions of the scene geometry

Affine transformation properties:

• 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

Shear matrix properties:

Horizontal shear

Vertical shear

3D transformation properties:

Camera parameters properties:

• 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

The Intrinsic Parameters of a Camera properties:

Units:

k,l :pixel/m

f :m

a,b

: pixel

Physical Image Coordinates (f ≠1)

Normalized Image

Coordinates

The Intrinsic Parameters of a Camera properties:

Calibration Matrix

The Perspective

Projection Equation

In reality properties:

• 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

Explicit Form of the Projection Matrix properties:

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

Explicit Form of the Projection Matrix properties:

Note:

: i-th row of R

M is only defined up to scale in this setting!!

• Projection equation properties:

• The projection matrix models the cumulative effect of all parameters

• Useful to decompose into a series of operations

identity matrix

intrinsics

projection

rotation

translation

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”

• Definitions are not completely standardized

• especially intrinsics—varies from one book to another