6 Geometric Calibration of Digital Imaging Systems
6.5 Camera models
When optical systems are used for measurement, modeling the entire process of image formation is decisive in obtaining accuracy. Basically, the same ideas apply, for example, to projection systems for which models can be set up similarly to imaging systems.
Before we continue, we have to define animage-coordinate system KB in the image plane of the camera. In most electro-optical cameras, this image plane is defined by the sensor plane; only in special designs (e. g., in réseau scanning cameras [8]), is this plane defined differently.
While in the majority of analog cameras used for metrology purposes the image-coordinate system is defined by projected fiducial marks or réseau crosses, this definition is not required for digital cameras. Here
6.5 Camera models 159
Figure 6.3: Definition of image-coordinate system.
it is entirely sufficient to place the origin of image-coordinate system in the center of the digital images in the storage (Fig. 6.3). Because the pixel interval in column direction in the storage is equal to the in-terval of the corresponding sensor elements, the unit “pixel in column direction” may serve as a unit of measure in the image space. All pa-rameters of interior orientation can be directly computed in this unit, without conversion to metric values.
6.5.1 Calibrated focal length and principal-point location
The reference axis for the camera model is not the optical axis in its physical sense, but a principal ray, which on the object side is perpen-dicular to the image plane defined in the foregoing and intersects the latter at theprincipal point PH(xH, yH). The perspective centerOj is located at distancecK (also known ascalibrated focal length) perpen-dicularly in front of the principal point [14].
The original formulation of Eq. (6.1) is thus expanded as follows:
"
The following additional correction function can be applied to Eq. (6.3) for radial symmetrical, radial asymmetrical and tangential distortion:
" Here, dx and dy may now be defined differently, depending on the type of camera used, and are made up of the following different
com-ponents:
dx = dxsym+dxasy+dxaff
dy = dysym+dyasy+dyaff (6.5) Radial-symmetrical distortion. Theradial-symmetrical distortion typ-ical of a lens can generally be expressed with sufficient accuracy by a polynomial of odd powers of the image radius (xijandyijare hence-forth called x and y for the sake of simplicity):
drsym=A1(r3−r02r )+A2(r5−r04r )+A3(r7−r06r ) (6.6) where drsym is the radial-symmetrical distortion correction; r is the image radius fromr2 =x2+y2; A1,A2, A3 are the polynomial coef-ficients; andr0is the second zero crossing of the distortion curve, so that we obtain
dxsym= drsym
r x and dysym= drsym
r y (6.7)
A polynomial with two coefficients is generally sufficient to describe radial symmetrical distortion. Expanding this distortion model, it is possible to describe even lenses with pronounced departure from per-spective projection (e. g., fisheye lenses) with sufficient accuracy. In the case of very pronounced distortion it is advisable to introduce an additional point of symmetryPS(xS, yS). Figure6.4 shows a typical distortion curve.
For numerical stabilization and far-reaching avoidance of correla-tions between the coefficients of the distortion function and the cali-brated focal lengths, a linear component of the distortion curve is split off by specifying a second zero crossing [15].
Lenz [16] proposes a different formulation for determining radial-symmetrical distortion, which includes only one coefficient. We thus obtain the following equation:
drsym=r1−
1−4Kr2 1+
1−4Kr2 (6.8)
whereKis the distortion coefficient to be determined.
Radial-asymmetrical and tangential distortion. To handle radial-asymmetrical andtangential distortion, various different formulations are possible. Based on Conrady [17], these distortion components may be formulated as follows [2]:
dxasy = B1(r2+2x2)+2B2xy
dyasy = B2(r2+2y2)+2B1xy (6.9)
6.5 Camera models 161
Figure 6.4: Typical distortion curve of a lens.
In other words, these effects are always described with the two ad-ditional parametersB1 andB2.
This formulation is expanded by Brown [18], who adds parameters to describe overall image deformation or the lack of image-plane flat-ness:
dxasy = (D1(x2−y2)+D2x2y2+D3(x4−y4))x/cK
+ E1xy+E2y2+E3x2y+E4xy2+E5x2y2 dyasy = (D1(x2−y2)+D2x2y2+D3(x4−y4))y/cK
+ E6xy+E7x2+E8x2y+E9xy2+E10x2y2
(6.10)
In view of the large number of coefficients, however, this formula-tion implies a certain risk of too many parameters. Moreover, because this model was primarily developed for large-format analog imaging systems, some of the parameters cannot be directly interpreted for ap-plications using digital imaging systems. Equation (6.7) is generally sufficient to describe asymmetrical effects. Figure 6.5 shows typical effects for radial-symmetrical and tangential distortion.
Affinity and nonorthogonality. The differences in length and width of the pixels in the image storage caused by synchronization can be taken into account by anaffinity factor. In addition, an affinity direction may be determined, which primarily describes the orthogonality of the axes of the image-coordinate systemKB. An example may be a line scanner that does not move perpendicularly to the line direction. Allowance for
DCWS ASYMMETRIC EFFECTS RolleiMetric
Recording system: Grundig FA85 / Schneider Xenoplan 1.8/6.5
Unit of vectors: [µm] 0.8050
Figure 6.5: Radial symmetrical and tangential distortion.
DCWS EFFECTS OF AFFINITY RolleiMetric
Recording system: Grundig FA85 / Schneider Xenoplan 1.8/6.5
Unit of vectors: [µm] 100.72
Figure 6.6: Effects of affinity.
these two effects can be made as follows:
dxaff=C1x+C2x and dyaff=0(6.11) Figure6.6gives an example of the effect of affinity.
Additional parameters. The introduction of additional parameters may be of interest for special applications. Fryer [19] and Fraser and Shortis [20] describe formulations that also make allowance for distance-related components of distortion. However, these are primarily ef-fective with medium- and large-image formats and the corresponding
6.6 Calibration and orientation techniques 163 lenses and are of only minor importance for the wide field of digital uses.
Gerdes et al. [21] use a different camera model in which an additional two parameters have to be determined for the oblique position of the sensor.
6.6 Calibration and orientation techniques