Master Thesis - Fachbereich Informatik

14 CHAPTER 2. TECHNICAL BACKGROUND
physical parameters of the camera. At this stage, one can distinguish between intrinsic
and extrinsic parameters [24].
Intrinsic Parameters The intrinsic parameters describe the projection of a point in the
camera frame onto the image plane, i.e. the transformation of camera coordinates into
image coordinates. This transformation extends the ideal perspective camera model introduced
in the previous section with respect to properties of real CCD cameras. One can
derive the following projection matrix Mi:
⎛
⎞
−f/sx k ox
Mi = ⎝ 0 −f/sy oy ⎠ (2.6)
0 0 1
where f represents the focal length, sx and sy theeffectivepixelsizeinxand y direction
respectively, k the skew coefficient, and (ox,oy) the coordinates of the image center. α = sy
is the aspect ratio of the camera. If α = 1, the sensors of the CCD array are ideally square.
The skew coefficient k determines the angle between the pixel axis and is usually zero, i.e.
the x- and y axis are perpendicular. (ox,oy) can be seen as an offset that translates the
projection of the camera origin onto the image origin in pixel dimensions. If sx = sy =1
and ox = oy = k =0,Mi represents an ideal pinhole perspective camera.
Extrinsic Parameters The extrinsic parameters take the transformation between a fixed
world coordinate system (or object coordinate system) and the camera coordinate system
into account. This includes the translation and rotation of the coordinate axis [65], i.e. a
translation vector T =(Tx Ty Tz) T and a 3 × 3 rotation matrix R such as:
⎛
Me = ⎝
r11 r12 r13 −R T 1 T
r21 r22 r23 −R T 2 T
r31 r32 r33 −R T 3 T
⎞
sx
⎠ (2.7)
where rij (i, j ∈{1, 2, 3}) are the matrix elements of R at (i, j) andRi indicates the
ith row of R.
Thus, the relationship between world and image coordinates can be written in terms of
two matrix multiplications [65]:
⎛
⎝
x1
x2
x3
⎞
⎠ = Mi Me
⎛
⎜
⎝
X
Y
Z
1
⎞
⎟
⎠
(2.8)
with (X, Y, Z, 1) T representing a 3D world point in homogeneous coordinates, and image
coordinates can be computed as x = x1/x3 andy = x2/x3 respectively. M = MiMe is
denoted as projection matrix in the following.
Image Distortion The resulting image coordinates may be distorted by the lens, i.e.
linear projection is not guaranteed. If high accuracy and precision is required, the simple
mathematical relationships introduced before are not sufficient.
To overcome this effect, a model of the distortion has to be defined. A common radial
distortion model [30] can be written as: