Camera Models

A camera is a mapping between the 3D world and a 2D image

The principal camera of interest is central projection

Central Projection

• Cameras modeling Central projection are specialization of the general projective camera. It is examined using the tools of projective geometry.

• Specialized models fall into two classes:• (1) camera with a finite centre. (2) cameras with centre at infinity Affine camera is an important example

Finite cameras

• The basic pin hole model(most specialized)

Pinhole camera

• A point in space with coordinates

• X =(x,y,z)T is mapped to a point on the image plane.

• (x,y,z)T ____ ( f x/z , f y/z)T (5.1)

• The centre of projection is called the camera centre

Principal axis and principal point.

• The line from the camera centre perpendicular to the image plane is called the principal axis and point where it intersects the image plane is called the principal point.

• The plane parallel to the image plane and passing through the camera centre is called the principal plane

Central projection using homogenous coordinates

• If the world and image points are represented by homogenous vectors, then

• P = diag( f , f, 1) [I ! 0]

1

z

y

x

01

0

0

z

y

x

1

z

y

x

f

f

f

f

Image and camera coordinate systems

Principal point offset

• The expression (5.1) assumes that the origin of the coordinates in the image plane is the principal point. In practice, it may not be as follows.

• (x,y,z)T ____ ( f x/z + px , f y/z + py )T

• Where ( px, py) T are the coordinates of the principal point.

Principal point offset 2

• Now writing

(5.3)

1

z

y

x

01

0

0

z

y

x

1

z

y

x

f

f

f

f

1

pf

pf

K y

x

Principal point offset 3

• Then (5.3) has the concise form

• x = K [ I ! 0] xcam (5.5)

• (x, y,z, 1) as Xcam as the camera is assumed to be located at the origin of a Euclidean coordinate system with the principal axis of the camera pointing straight down the z-axis. K is called the camera calibration matrix.

The Euclidean transformation between the world and camera coordinate frame

Camera rotation and translation

• represents the coordinates of the camera centre in the world coordinate frame.

• is an inhomogeneous 3-vector in world coordinate frame

• R is 3x3 rotation axis

)C - X R( Xcam

C

X

Camera Rotation and translation

• • Putting (5.5) and (5.6) together leads to •

(5.6) X10

CR-R

1

Z

Y

X

10

CR-R Xcam

(5.7) X ]C - ! I [ KR x

Camera Rotation and translation 2

• The parameter contained in K are called internal camera parameters

• The parameter of R and which relate the orientation and position to a

world frame are called the external parameters. A more convenient form of the camera matrix is

• P = K [ R ! t] (5.8)•

C

CR- t

CCD Cameras

• The CCD camera may have rectangular pixels, where unit distances in x and y directions are mx and my, then

x0 = mx pxand y0 = my py

x = f mx , and y = f my

1

y

x

K 0y

0x

Finite projective camera

• S is the skew parameter• A camera

is called a finite projective camera. It has 11 degree of freedom, same as a 3x4 matrix defined up to an arbitrary scale

(5.11) ]C - ! I [ KR P

General projective Camera

The projective camera

A general projective camera P maps world point X to image points x according to

x = PX

Camera centre

Camera centre 2

Column vectors

• The columns of P are pi , i = 1, 2, 3, 4

• Then p1 , p2 , p3 are the vanishing points of the world coordinate x , y, z axes respectively.

• For example: x axis ahs direction D =(1,0,0,0), which is imaged at p1 = PD

• The column p4 is the image of the world origin

The three image points defined by the columns pi, i= 1, 2, 3 of the projection matrix are the vanishing points of the

directions of the world axes

Row vectors

Principal plane

Axis planes

Summary of the properties of a projective camera P=[M ! p4]

M

Summary of the properties of a projective camera 2

Principle point

Principle axis

Principle axis 2

Principle axis 3

Two of the three planes defined by the rows of the projection matrix

Action of a projective camera on points

Back projection of points to rays

Back projection of points to rays 2

Back projection of points to rays 3

Depth of points

Depth of points 2

Linear optics

Decomposition of the camera matrix

Finding camera orientation and internal parameters

Finding camera orientation and internal parameters 2

Example 5.2 The camera matrix

Euclidean vs Projective space

Euclidean and affine interpretation

Cameras at infinity

Affine cameras

Increasing focal length from left to right

Affine camera 2

Affine camera 3

Affine camera 4

Focal length increases as the object distance between the camera increases

The image remains the same size, but perspective effects diminish

Perspective vs weak perspective

projection

Orthographic projection

Orthographic projection 2

Scaled orthographic projection

Weak perspective projection

Affine camera

A general affine camera

A general affine camera 2

More properties of the affine camera

General cameras at infinity

Pushbroom camera

Pushbroom camera 2

Pushbroom camera 3

Pushbroom camera – mapping of line

Line camera

Line camera 2

Acquisition geometry of a pushbroom camera