The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part B5. Beijing 2008 
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In addition, spatial distance observations between object points 
(4) m and n 
s_ - V(X. - X J + (Y, - Yj + (Z„ - Z J (11) 
and (x, y, p)ij are the observables; (X, Y, Z\ are the object- 
space co-ordinates of point i; the parameters (X c , Y c , Z c , to, ((), 
k)j comprise the exterior orientation (EO) elements of image j; 
Ri, R 2 , R3 are the fundamental rotation matrices; (x p , y p , c)j are 
the interior orientation elements (10) of image j; and (Ax, Ay, 
Ap) represent the correction models for systematic errors in 
each observable. 
2.2 Systematic Error Models 
The camera-lens system error model used for LRC calibration is 
the standard model for digital cameras (e.g., Fraser, 1997), 
Ax = x(k,r 2 + k 2 r 4 + k 3 r 6 )+ p, (r 2 + 2x 2 )+ 2p 2 xy (5) 
+ a,x + a 2 y 
Ay = y(k,r 2 +k 2 r 4 + k 3 r 6 )-i-p 2 ( r2 + 2y 2 )+2p,xy (6) 
where (k 1? k 2 , k 3 ) are the radial lens distortion coefficients; (p t , 
p 2 ), are the decentring distortion terms; (a t , a 2 ) are the 
electronic biases and 
X = x - x„ 
y = y.j-y Pj 
2 —2 —2 
r = x + y 
(7) 
(8) 
(9) 
The rangefinder model comprises terms that have physical 
explanation (the d-terms and the first two e-terms) as well as 
empirical terms 
Ap = d 0 + d,p +J 
. (2 k n Ì f2 k n 
d - sin — P + d 2k + .cos — p 
(10) 
+ e,x + e 2 y + e 3 r + e 4 r 2 + X Z e 3m +n -i xm_n y n 
are included to allow estimation of the scale error. 
3. EXPERIMENT DESCRIPTION 
3.1 Hardware 
The subject of this study was the SwissRanger SR-3000 LRC 
system pictured in Figure 1. The principles of 3D range camera 
technology can be found in Lange and Seitz (2001), for 
example. The SR-3000 features a 176 pixel x 144 pixel array 
for which the pixel size and spacing are both 40 pm. The 
nominal principal distance of the lens is 8 mm. Several 
rangefinder system parameters such as the integration time and 
modulation frequency can be set by the user. For the 
experiments described herein, the former was set to the highest 
possible value of 51.2 ms so as to maximise the signal-to-noise 
ratio and the latter was 20 MHz, for which the corresponding 
maximum unambiguous range and, therefore, the unit length, is 
7.5 m. 
Figure 1. The SR-3000. 
where d 0 is the rangefinder offset; di is the scale error; d 2 to d 7 
are the cyclic error terms; U is the unit wavelength; ei and e 2 
are the clock skew errors (Du et al., 2005); and e 3 to e n 
represent empirical terms. In contrast to Lindner and Kolb 
(2006), who use B-splines to model the cyclic errors, the 
modelling approach chosen here is primarily driven by the 
known physical causes of these periodic effects (e.g., Riieger, 
1990). 
Hereafter the IO shall be understood to comprise the principal 
point, principal distance plus this set of additional parameters 
(APs; Equations 5, 6 and 10). The IO shall be considered 
network-invariant for a given sensor. 
2.3 Self-Calibration Solution and Spatial Distances 
For the integrated self-calibration all model terms (EO, IO, and 
object points) are simultaneously estimated in a free-network 
adjustment with inner constraints imposed on the object points. 
3.2 Target Field 
The integrated calibration approach required special 
considerations in terms of both geometric network design and 
target design. A purpose-built, multi-resolution field of high- 
contrast (black on white) targets measuring 3.6 m x 2.0 m was 
constructed. An SR-3000 intensity image of the target field is 
shown in Figure 3. The targets were mounted on two planar 
surfaces separated by 0.8 m to provide depth relief. Several 
sizes of rectangular targets were used since the network 
comprised images captured at multiple ranges, a requirement 
for estimation of the rangefinder APs. 
Several factors motivated this design. First, as the comers of the 
black rectangles constitute the targets, it was easy to measure 
spatial distances between targets. Second, no eccentricity 
correction was needed as is the case when circular targets are 
used (e.g., Ahn et al., 1999). Third, the materials were readily 
available. The disadvantage of this design stems from range 
biases that exist as a function of surface reflectivity (i.e. colour).