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

zyx

zyxOPOPzOPyOPx

Pkjikji

...

Planes

1

and where

00,],,[,],,[

00

zyx

dcbadczbyax

dOAcbazyxP

OAOPAPTT

PΠ

PΠnn

nnn

homogenous coordinate

Coordinate Changes: Pure Translations

OBP = OBOA + OAP , BP = BOA+ AP

Coordinate Changes: Pure Rotations

BABABA

BABABA

BABABABA R

kkkjkijkjjjiikijii

.........

AB

AB

AB kji

TB

A

TB

A

TB

A

kji

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

1000cossin0sincos

RBA

Rotation matrix

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

Elementary rotation

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-handedorthonormal 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

PRP

zyx

zyx

OP

ABA

B

B

B

B

BBBA

A

A

AAA

kjikji

Coordinate Changes: Rigid Transformations

ABAB

AB OPRP

Block Matrix Multiplication

2221

1211

2221

1211

BBBB

BAAAA

A

What is AB ?

2222122121221121

2212121121121111

BABABABABABABABA

AB

Homogeneous Representation of Rigid Transformations

11111P

TOPRPORP A

BA

ABAB

AA

TA

BBA

B

0

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

Shear matrix

Horizontal shear

Vertical shear

2D planar transformations

2D planar transformations

2D planar transformations

3D transformation

Pinhole Perspective Equation

zyfy

zxfx

''

''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

Normalized ImageCoordinates

Physical Image Coordinates (f ≠1)

Units:k,l : pixel/m

f : m: pixel

The Intrinsic Parameters of a Camera

Calibration Matrix

The PerspectiveProjection Equation

TzyxP )1,,,(

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 anglesR can be written as a product of three elementary rotations, and described by three angles

M is 3 x 4 matrix with 11 parameters5 intrinsic parameters: α, β, u0, v0, θ6 extrinsic parameters: 3 angles defining R and 3 for t

TirT

ir

Explicit Form of the Projection Matrix

Note:

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

Tir : i-th row of R

Theorem (Faugeras, 1993)

Projection equation

• The projection matrix models the cumulative effect of all parameters• Useful to decompose into a series of operations

ΠXx

1************

ZYX

ssysx

11010000100001

100'0'0

31

1333

31

1333

x

xx

x

xxcy

cx

yfsxfs

000 TIRΠ

projectionintrinsics rotation translation

identity matrix

Camera parametersA 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