International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B7. Istanbul 2004 
two angles (a and B) of a triangle relative to its base (baseline 
D) determines the dimensions of this triangle. The complete 
range equations are derived in (Blais, 2004). 
For an incremental change of distance, AZ, one measures the 
incremental angle shift AB. This laser spot sensor is in fact an 
angle sensor. The angular shift AB caused by the displacement 
of the surface is observed through a shift in laser spot position 
Ap=(p; — pz). For practical matters, the errors with a 
triangulation-based laser scanner come mainly from the 
estimate of p, through Sy: An error propagation computation 
gives the approximation of the uncertainty in Z, 
t2 
N 
(1) 
  
ó,2——Ó0, 
z fD I 
where f= effective position of laser spot sensor 
D = baseline 
5, = uncertainty in laser spot position 
Z = distance to object. 
From the equation above, one finds that the measurement 
uncertainty in Z is inversely proportional to both the camera 
baseline and the effective position of the angle sensor wrt the 
lens, but directly proportional to the square of the distance. 
Unfortunately, f and D cannot be made as large as desired. The 
baseline D is limited mainly by the mechanical structure of the 
optical set-up (stability of the whole system decreases as D 
increases) and by shadow effects (self occlusion problems 
increase with D). Rioux, 1984 presents an approach to 
triangulation-based range imaging that allows very large fields 
of view without compromising the performance of the system. 
The value of Ó, depends on the type of laser spot sensor used, 
the peak detector algorithm, the signal-to noise ratio (SNR) and 
the imaged laser beam shape. Each sensing method will 
perform differently (Blais ct al., 1986; Naidu et al., 1991). In 
the case of discrete response laser spot sensors (Beraldin et al., 
2003), assuming sub-pixel laser spot position estimation and 
high SNR, the limiting factor will be speckle noise (Baribeau et 
al., 1991; Dorsch et al., 1994; Jáhne et al., 1999; Amann et al., 
2001). The effect of speckle noise on spot position uncertainty 
is approximately given by 
= 1 ; 7 
$ dm (2) 
where fn = receiving lens f-number 
À = laser wavelength. 
For instance, A=0.68 um and fn=4, the laser sub-pixel 
uncertainty is about 1.4 pm. This estimate is for high SNR and 
for well-designed 3D systems. The SNR deteriorates rapidly 
with distance. The fact that the amount of light collected 
decreases with the distance squared and that the majority of 
triangulation-based systems don’t use optical sensors with a 
built-in current gain mechanism (like those used in time-of- 
flight systems) contribute to a deterioration of the SNR. 
Overall, the maximum range of triangulation-based laser 
scanners, even with a baseline of 1 m, does not exceed 10 m. A 
more detailed model of the spatial measurement uncertainty can 
be derived by computing the actual joint density function of the 
spatial error (probability distributions). The law of propagation 
of errors is only an approximation. The complete analysis can 
show that the spatial error distribution is skewed and oriented 
with the line of sight, ie. anisotropic and in-homogeous 
(Johnson et al., 1997). This is represented schematically on 
Figure 2. 
f. .Emor 
| 
Z 
g^ 
| 
i 
— 
X 
Figure 2. Schematic diagram showing a representation of the 
shape of the spatial error distribution for a laser scanner. 
2.4.2 Time delay systems 
A fundamental property of a light wave is its velocity of 
propagation. In a given medium, light waves travel with a finite 
and constant velocity. Thus, the measurement of time delays 
created by light traveling in a medium from a source to a 
reflective target surface and back to the source (round trip) 
offers a very convenient way to evaluate distance. The current 
accepted value for the speed of light in a vacuum is exactly ¢ = 
299,792,458 m/sec. If the light waves travel in air then a 
correction factor equal to the refraction index (depending on the 
air density) must be applied to ¢, n = 1.00025 (n-water = 1.33). 
Let us assume that the speed of light is 3% 10% m/sec. Different 
strategies have been devised to exploit this measurement 
principle: Time-Of-Flight (TOF) with pulsed lasers, Amplitude 
Modulation (AM), Frequency modulation (FM) systems with or 
without coherent detection (Koskinen et al., 1991; Baltsavias, 
E.P. 1999b.; Wehr et al., 1999; Amann et al., 2001). 
Long-range sensors (range exceeding 10 m) are usually based 
on the time-of-flight (TOF) technology (also known as laser 
radar or lidar for short). The camera to object distance Z is 
measured by sending a relatively short impulse of light on a 
reflective surface and measuring the round trip, 7, Z=c*1/2. The 
range uncertainty for a single pulse is approximately given by 
the following equation: 
so ST (3) 
where 7, - pulse rise time 
J, = uncertainty in range estimation pulse system. 
A round trip of T=1 microsecond corresponds to a distance of 
about 150 m. Assuming a SNR=100 and T,=1 nanosecond, the 
range uncertainty is close to 1.5 cm. Such a pulse rise time is 
equivalent to a system bandwidth of about 350 MHz (0.35/1 
nanoseconds). Most commercial systems based on TOF provide 
a range uncertainty in the range 1 cm to 10 cm. Averaging N 
measurements will reduce ó., by a factor proportional to square 
root of N. Expansion of the SNR can show that the range 
uncertainty depends on distance and the detection mechanism 
(avalanche photodiode, etc.). Incidentally, Eq. (3) is similar the 
result obtain from the estimation of arrival time for the radar 
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