Master Thesis - Fachbereich Informatik

62 CHAPTER 4. LENGTH MEASUREMENT APPROACH
Figure 4.10: Visualization of the resulting radial distortion model. The computed center of
distortion indicated by the ‘◦’ is slightly displaced from the optical center (‘×’). The image
area of interest considered in this application lies in between the red lines.
Perspective Warping One possibility to compute a synthetic fronto-orthogonal view of
an image is based on the extrinsic relationship of the camera plane and a particular
world plane (e.g. conveyor plane) that can be extracted in a calibration step. With the
extrinsic parameters it is possible to describe the position and orientation of the world
plane in the camera reference frame. Finally, one can compute a transformation that
maps the world plane into a plane parallel to the image plane or vice versa, and warp the
image to a synthetic fronto-orthogonal view. This approach has a significant drawback.
First of all, the accuracy of the results is closely related to the calibration accuracy.
Furthermore, the extrinsic parameters of a camera change if the camera is moved even
slightly compared to the intrinsic parameters that can be assumed constant as long as
the focus is not changed. Thus, one has to recalibrate the extrinsic parameters as well as
the transformation parameters every time the camera is moved, which seemed to be not
practicable in this particular application.
There are other methods that can be used to compute a fronto-orthogonal view of an
perspective image, which are based on characteristic image features such as parallel or
orthogonal lines, angles, or point correspondences and do not need any knowledge on the
interior or exterior camera parameters [30]. One common approach is based on point
correspondences of at least 4 points xi and x ′ i with x′ i = Hxi (1 ≤ i ≤ 4) and
⎡
H = ⎣
h1 h2 h3
h4 h5 h6
h7 h8 h9
⎤
⎦ (4.2)
the projective transformation matrix representing the 2D homography.
The unknown parameters of H can be computed in terms of the vector cross product
x ′ i × Hxi = 0 using a Direct Linear Transformation (DLT) [30]. To correct the perspective
of an image one has to find four points in the image that lie on the corners of a rectangle
in the real world, but are perspectively distorted in the image. These points xi have to be
mapped to points x ′ i that represent the corners of a rectangle in the image. Then, after H