International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B5. Istanbul 2004
D - hi oh; - 0
Gz(xo-Xm) *(Yo- Yan) *c?-R3-0
Taking into account that V, ^ [H,,, Ha, Hs" V2 =[H12, Ha, Hy]
with respect to the elements of homography H and substituting
them into G, it can be proved that G is equivalent to D only if
Hy Hs, # 0, which indicates that the first equation of plane-based
calibration — Eq.(9) — is that of the calibration sphere when both
vanishing points are finite.
3. THE CALIBRATION ALGORITHM
The developed calibration algorithm relies on the simultaneous
estimation of two orthogonal vanishing points on each view, to-
gether with their common interior orientation elements. The va-
nishing points are estimated from individual point observations
X;, yi on converging image lines. The fitted lines are constrained
to converge to a corresponding vanishing point V(xy, yy) accord-
ing to following observation equation, whereby t represents the
slope Ax/Ay of the converging line with respect to the y axis:
xi -xv -(yi - yv)t* 0 (11)
According to each line direction, Eq. (11) is also formulated in
terms of the slope t = Ay/Ax with respect to the x-axis. Finally,
introducing the coefficients k,, k, of the radial symmetric lens
distortion, Eq. (11) becomes (Gammatikopoulos et al., 2003):
xi - (xi - xo (kir? + kar* )- NV
12
[yi = (yi = Yo) (Kir2 + kar4) - yy Jt=0 a2)
The sum of squares of the image coordinate residuals are mini-
mised in the adjustment. Alternative formulations of these equa-
tions are also possible, in order to minimise the distances of the
points from the fitted line. Each individual point on a line of an
image offers one such condition, (11) or (12), to the adjustment.
Besides, every pair of vanishing points in orthogonal directions
gives rise to one Eq. (1) or, equivalently, Eq. (3). It is clear that,
if the normal camera is assumed, three pairs of vanishing points
(i.e. images) suffice for an estimation of the internal camera pa-
rameters. The image aspect ratio a can also be integrated, at the
expense of one additional pair of vanishing points (one image)
using Eq. (4) or (5) instead of (1) or (3). All equations above are
non-linear with respect to the unknowns, thus imposing a need
for initial values. Generally, the principal point may be assumed
at the image centre, the camera constant can then be approxima-
ted using a pair of vanishing points, line slope is estimated from
two line points, while vanishing point coordinates can be appro-
ximated using two converging lines. Finally, it is worth noting
that vanishing points might be weighted with the corresponding
elements of the covariance matrix from a previous solution.
4. PRACTICAL EVALUATION
4.1 Synthetic data
For the practical evaluation of the algorithm both simulated and
real data have been used. First, three sets consisting of 7, 13 and
22 synthetic images (1600x1200) of a 10x10 planar grid were
generated with random exterior orientation (Fig. 5). All interior
orientation parameters were kept invariant: c = 1600, x, = 802,
yo 7 604, k, = 2x I0 5 15s -3,. 5x19 (the aspect ratio has been
ignored). Gaussian noise with zero mean and three standard de-
viations 0 (+0.1, 0.5, +1 pixel) was added to the image points.
EHE n
Figure 5. Some of the synthetic images used in the evaluation.
For all image sets, results were compared with plane-based cali-
bration and self-calibrating bundle adjustment. Plane-based ca-
libration was carried using the ‘Calibration Toolbox for Matlab’
of Bouguet (2004), available on the Internet (the latest version
and complete documentation may be downloaded from the cited
web site). The bundle adjustment — in which no tie points were
used — has been carried out with the software ‘Basta’ (Kalispe-
rakis & Tzakos, 2001). All results are presented in Tables 1-3.
Table 1. Results from 7 simulated images (noise: £0, pixel)
Ac AX, Ayo Kk; k, So
S (969) | (pixel) | (pixel) | (x109) | (x10!) | (pixel)
CS | 026 | 0.591 -020] 195 | 340 | +£0.10
01 PB! O1S| 001} 002 | 195 | 345 | 40:10
BA | 0.20 | 0.03 | 0.16] 195 | —343 | £0.09
CS | 142 139 | —0.88 |. 2.02 ] -—3,66 | #31
os PB| 010] 036 | 008 | 1.99 | 350 | +£0.50
BA| 0.08] 0.55, -0.15]| 2.00 |] -3.49 | £0.50
CS| 199]-003] 290, 193 | 335 ] 1099
10] PR! 1.62 |—1357|1 253 | 1.70 | 323 ] +0,99
BA | 154 | —1 32 | 242 [| 170 | -323 |] £1.00
Table 2. Results from 13 simulated images (noise: ou pixel)
Ac AX, Ayo k, k, Go
(969) |(pixel) | (pixel) | (x105) | (x10'^) | (pixel)
CS | -0.28 | —0.15 | —0.13 | 2.03 | -3.55 | +0.10
0.1 | PB | -0.19 | -0.03 |] -0.17 | 2.02 | —3.54 | £0.10
BA | -0.18 | -0.05 | -0.11 | 2.02 | —3.55 | £0.10
CS | 0.80] -290|-157| 191 | -337 | 10.50
03] PBÍ-062] 1113] -093] 1.80 | 320 | +0.49
BA | -0.59 | —1.23 | -0.90 | 1.80 | -3.20 | £0.50
CS | 149 | -—241 | 0.77 | 2.03 | -3.46-| +097
1.0 |-PBR | 211 | -098] 122] 1.78 ] -328 4097
BA | 2.13 | -090] 1.31 | 1/79 | —3.29 :| +0,98
ON
Table 3. Results from 22 simulated images (noise: +oy pixel)
Ac Ax, Ay, k. ks Go
x (%0) | (pixel) | (pixel) | (<10°) | (<10'*) | (pixel)
CS | Q15. | 002] Q12 ; 1.99 |} —3.50 ] £0.10
01 | PR | 0111-002, QUS | 2.00 | -3.50 | +0,10
BA | 0.17 | —0.04 | 0.08 | 2.00 | -3.50 | £0.10
CS | 0.13] —1.43 1.—2.09 |. 2.00 | -3.50 |-+031
0.5 | PB | 0.26 | —0.64 | -0.18 | 1.98 | -3.49 | +0.50
BA | 0.29 | —0.74 | -0.12 | 1.98 | -3.49 | £0.50
CS 1.20. | -0.21 0.911 2:12: «3.56 4:30.99
1,0 | PR | 0.56 0201 0.817 204 | 51 1| 40.97
BA | -0.54 0.03 |: 0.51 1.04 | -3.51 | £0.98
102
In the above Tables, c is given as deviation per mil, x, and y, as
deviations in pixels, while k; and k; in true values. The symbols
Internation
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