# Camera Models and Parameters - University of .Camera Models and Parameters ... coordinate system

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Camera Models and Parameters

We will discuss camera geometry in more detail.Particularly, we will outline what parameters are importantwithin the model. These parameters are important to severalkey computer vision tasks and must be computed(calibrated) using approaches we will discuss in laterlectures.

Important Definitions Frame of reference: a measurements are made with respect to a

particular coordinate system called the frame of reference.

World Frame: a fixed coordinate system for representing objects(points, lines, surfaces, etc.) in the world.

Camera Frame: coordinate system that uses the camera center as itsorigin (and the optic axis as the Z-axis)

Image or retinal plane: plane on which the image is formed, note thatthe image plane is measured in camera frame coordinates (mm)

Image Frame: coordinate system that measures pixel locations in theimage plane.

Intrinsic Parameters: Camera parameters that are internal and fixed toa particular camera/digitization setup

Extrinsic Parameters: Camera parameters that are external to thecamera and may change with respect to the world frame.

Camera Models Overview

Extrinsic Parameters: define the location and orientation of the camerawith respect to the world frame.

Intrinsic Parameters: allow a mapping between camera coordinatesand pixel coordinates in the image frame.

Camera model in general is a mapping from world to imagecoordinates.

This is a 3D to 2D transform and is dependent upon a number ofindependent parameters.

Pinhole Model Revisited Select a coordinate system (,x,y,z) for the three-dimensional space to

be imaged

Let (u,v) be the retinal plane Then, the two are related by:

y

v

x

u

z

f==

Which is written linearly in homogeneous coordinates as:

=

10100

000

000

z

y

x

f

f

S

V

U

The Retinal Plane

Mxyz

vuy

xF

m uv

f

Camera orientation in the world

The position of the camera in the world must be recovered rotational component

translation component

Describes absolute position of the focal plane in the world coordinatesystem

This is a Euclidean transform from one coordinate system to another

-translation, rotation

F

x

y

z

Translation

rA = rB + tA

where ta representsthe puretranslation fromframe A to frameB written in frameA coordinates

AB

yA

xA

yB

xB

tA

rA

rB

Frames A and B are related througha pure translation

RotationsFrames A and B are related through

a pure rotation

yA

yB

xA

xB

r

A position vector r, inframe B, can beexpressed in the Acoordinate frame byemploying the 3 X 3transformation matrix

ARB . . .

Rotation Matrix

This projection of frame B onto frame A clearly convertsa position vector, , written in frame B, into thecorresponding coordinates in frame A, .

rB

rA

A A B B

A B A A BBA B

B BA BA A B A

A BA B A A BB

r R r

i i i j i krx rx

ry j i j j j k ry

rz rzk i k j k k

=

=

v v

)) ) ) ) )

)) ) ) ) )

) ) ) )) )

Interpreting the Rotation Matrix

To interpret the rotation matrix for thistransformation:

the rows of ARB represent the projection ofthe basis vectors for frame A onto the basisvectors of frame B

the columns of ARB represent the basisvectors of frame B projected onto the basisvectors of frame A

Rotations

One way to specify the rotation matrix ARB isto write the base vectors

in frame A coordinates and to enter the resultinto the columns of ARB

( ), ,B

i j k)) )

If is a column vector representing theaxis of frame B written in frame Acoordinates, then

ABx

)x

cos( ) -sin( ) 0

R sin( ) cos( ) 0

0 0 1

A BA A Ax y zB B B

= =

) ) )

Rotations

Rotations: x

For completeness, we will look at the rotationmatrix for rotations about all three axes:

1 0 0

( , ) 0 cos( ) sin( )

0 sin( ) cos( )

rot x

=

)

Rotations: y

For completeness, we will look at the rotationmatrix for rotations about all three axes:

cos( ) 0 sin( )

( , ) 0 1 0

sin( ) 0 cos( )

rot y

=

)

Rotations: z

For completeness, we will look at the rotationmatrix for rotations about all three axes:

cos( ) sin( ) 0

( , ) sin( ) cos( ) 0

0 0 1

rot z

=

)

Extrinsic Parameters

Recall the fundamental equations of perspective projection assumed the orientation of the camera and world frame known

this is actually a difficult problem known as extrinsic pose problem

using only image information recover the relative position andorientation of the camera and world frames

This transformation is typically defined by: 3-D translation vector T=[x,y,z]T

defines relative positions of each frame

3x3 rotation matrix, R rotates corresponding axes of each frame into each other

R is orthogonal: (RTR = RRT =I)

Extrinsic Parameters

R T

PXW

YW

ZWZC

XC

YC

Note: we write R=r11 r12 r13r21 r22 r23r31 r32 r33

( )How to write both rotation and translation as a single,composed transform?

We consider the general case where frame 0and frame 2 are related to one anotherthrough both rotation and translation

0y)

0x)

0rv

0tv

1x)

2x)

1y)

2y) 2r

v

1

20

Homogeneous Transformations

0y)

0rv

Homogeneous Transformations

The homogeneous transform is amechanism for expressing this form of

compound transformation

0y)

0x)

0rv

0tv

1x)

2x)

1y)

2y) 2r

v

1

20

Homogeneous Transforms

Expand the dimensionality of the domain space

Same transformation now can be expressed in a linearfashion

Linear transforms can be easily composed and written as asingle matrix multiply

Vectors, in homoeneous space take on a new parameter r.This is the scale of the vector along the new axis and isarbitrary: [x y z r]

Normalization, after the transform has been applied isaccomplished simply by dividing each vector componentby r [x y z 1] = [x/r y/r z/r r/r]

Homogeneous Transformations

Let 0T2 be the compoundtransformation consisting of atranslation from 0 to 1, followed bya rotation from 1 to 2

Homogeneous Transformations

In vector notation, this homogeneoustransformation and correspondinghomogeneous position vectors are written:

0 2T =1 2R

0 0 0

0tr

1[ ]

2

21

x

y

z

r

rr

r

=

r

0 0 2 2

1 2 2 0

T

R

r r

r t

=

= +

r

rrThen,

Composing Transformations

The homogeneous transformprovides a convenient means of

constructing compoundtransformations

Composing Transformations

Example: Suppose

0 1 0

1 2 1

2 3 2

3 4 3

where:

T ( , 1.0)

T ( , 1.0)

T ( , 1.0)

T ( , / 4)

translation x

translation y

translation z

rotation y

====

)

)

)

)

0 4 0 1 1 2 2 3 3 4T T T T T=

Composing Transformations

Example: 0 4 0 1 1 2 2 3 3 4T T T T T=

0x)

0 y)

0z)

1x)

1y)

1z)

2x)

2y)

2z)

3x)

3, 4y y) )

3z)

4x)

4z)

4

3

2

10

/4

The resulting compound transformation is

Composing Transformations

0 4

0.707 0 0.707 1

0 1 0 1T

0.707 0 0.707 1

0 0 0 1

=

Extrinsic ParametersR T

P

XW

YW

ZWZC

XC

YC

R=r11 r12 r13r21 r22 r23r31 r32 r33

( )wc pp P=

=

1000

TRP [ ]Tzyx TTT=T

Intrinsic Parameters Characterize the optical, geometric, and digital

characteristics of the camera Defined by:

perspective projection: focal length f

transformation between camera frame and pixel coordinates

geometric distortion introduced by the lens

Transform between camera frame and pixels:

x = -(xim - ox)sxy = -(yim - oy)sy

(ox,oy) image center (principle point)

(sx,sy) effective size of pixels in mm in horizontal andvertical directions

Camera Lens Distortion Optical system itself a source of distortions

evident at the image periphery

worsened by large field of view

Modeled accurately as radial distortion

x = xd (1 + k1r2 + k2r4)

y = yd (1 + k1r2 + k2r4)

(xd,yd) distorted points, and r2= x2d + y2d note: this is a radial displacement of the image points

because k2

Camera models (again)

Recall

Using the equations from previous slides, we are able to transformpixel coordinates to world points this is our camera model.

-(xim - ox)sx = fRT1(Pw - T)

RT3(Pw - T)

-(xim - ox)sx = fRT2(Pw - T)

RT3(Pw - T)

Ri, i=1,2,3 is a 3D vector formed by the I-th row of R

x = f y = f X

Z

Y

Z

Linear Perspective Projection Equations:

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