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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.

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EECS 274 Computer Vision

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## 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

Quantitative Measurements and Calibration

Euclidean Geometry

Euclidean Coordinate Systems

Planes

homogenous coordinate

Coordinate Changes: Pure Translations

OBP = OBOA + OAP ,BP = BOA+ AP

Coordinate Changes: Pure Rotations

1st column:

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

3rd row:

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

Coordinate Changes: Rotations about the z Axis

### Rotation matrix

Elementary rotation

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

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.

### 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

Coordinate Changes: Pure Rotations

Coordinate Changes: Rigid Transformations

Block Matrix Multiplication

What is AB ?

Homogeneous Representation of Rigid Transformations

Rigid Transformations as Mappings

Rigid Transformations as Mappings: Rotation about the k Axis

### Affine transformation

• Images are subject to geometric distortion introduced by perspective projection

• Alter the apparent dimensions of the scene geometry

### 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

Horizontal shear

Vertical shear

### 3D transformation

Idealized coordinate system

### 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

The Intrinsic Parameters of a Camera

Units:

k,l :pixel/m

f :m

a,b

: pixel

Physical Image Coordinates (f ≠1)

Normalized Image

Coordinates

The Intrinsic Parameters of a Camera

Calibration Matrix

The Perspective

Projection Equation

### 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

Extrinsic Parameters

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

Explicit Form of the Projection Matrix

Note:

: i-th row of R

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

Theorem (Faugeras, 1993)

• Projection equation

• 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