EECS 274 Computer Vision

Geometric Camera Models

Geometric Camera Models

• Elements of Euclidean geometry• Intrinsic camera parameters• Extrinsic camera parameters• General form of perspective projection

• Reading: Chapter 1 of FP, Chapter 2 of S

Euclidean Geometry

Geometric camera calibration

z

y

x

zyxOP

OPz

OPy

OPx

Pkji

k

j

i

.

.

.

Euclidean coordinate system

1

and where

00

,],,[,],,[

00

z

y

x

d

c

b

a

dczbyax

dOAcbazyxP

OAOPAPTT

PΠ

PΠ

nn

nnn

homogenous coordinate

Planes

OBP = OBOA + OAP , BP = BOA+ APAP: point P in frame A

Pure translation

BABABA

BABABA

BABABABA R

kkkjki

jkjjji

ikijii

...

...

...

AB

AB

AB kji

TB

A

TB

A

TB

A

k

j

i

1st column:iA in the basis of (iB, jB, kB)

3rd row:kB in the basis of (iA, jA, kA)

Pure rotation

100

0cossin

0sincos

RBA

Rotation about z axis

Rotation matrix

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

Elementary rotation

• 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.

Properties of rotation matrix

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

PRP

z

y

x

z

y

x

OP

ABA

B

B

B

B

BBBA

A

A

AAA

kjikji

Pure rotations

ABAB

AB OPRP

Rigid transformation

2221

1211

2221

1211

BB

BBB

AA

AAA

What is AB ?

2222122121221121

2212121121121111

BABABABA

BABABABAAB

Homogeneous Representation of Rigid Transformations

11111

PT

OPRPORP ABA

ABAB

AA

TA

BBA

B

0

Block matrix manipulation

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

– Ratio of distance along a line• |p2-p1|/|p3-p2| is preserved

Shear matrix

Horizontal shear

Vertical shear

2D planar transformations

See Szeliski Chapter 2

2D planar transformations

2D planar transformations

3D transformation

Pinhole Perspective Equation

z

yfy

z

xfx

''

''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

Normalized ImageCoordinates

Physical Image Coordinates (f ≠1)

Units:

k,l : pixel/m

f : m(See EXIF tags)pixel

Intrinsic camera parameters

Scale parameters: k, l (image sensor may not be square)Offset: u0, v0

Manufacturing error: θ

Calibration matrix κ

The perspectiveprojection Equation

TzyxP )1,,,(

Intrinsic camera parameters

In reality

• Physical size of pixel and skew are always fixed for a given camera, and in principal known during manufacturing

• Some parameters often available in EXIF tag• Focal length may vary for zoom lenses when

optical axis is not perpendicular to image plane

• Change focus affects the magnification factor• From now on, assume camera is focused at

infinity

Extrinsic camera parameters

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 × 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 projection Matrix

Note:

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

Tir : i-th row of R

Explicit form of projection Matrix

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****

****

****

Z

Y

X

s

sy

sx

110100

0010

0001

100

'0

'0

31

1333

31

1333

x

xx

x

xxcy

cx

yfs

xfs

00

0 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

Camera calibration toolbox

• Matlab toolbox by Jean-Yves Bouguethttp://www.vision.caltech.edu/bouguetj/calib_doc/

• Extract corner points from checkerboard