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The circuit board inspection camera can measure a volume of 50 mm in width, 50 mm in depth, and 50 mm in 
height at a resolution of nine bits. The light source in the system is a semiconductor laser diode with a wavelength 
of 820 nm and an average power on the scene of 100 mW. The laser beam is scanned horizontally by a 24-facet 
polygonal mirror rotating at a speed 656.25 revolutions per second corresponding to a rate of 15,750 lines per second, 
while a galvanometer driven mirror imparts a vertical motion to the laser beam at a rate of 60 Hz. 
The real-time space camera, developed in collaboration with the Canadian Space Agency, has an angular field of view 
of 30? x 30? and is capable of producing a range image precision of 0.2 mm at 0.5m and 2.0 mm at a maximum 
distance of 1.0 m corresponding to 12 bits of resolution. The source is an Nd-YAG laser emitting a laser beam with 
a wavelength of 1,060 nm and a maximum power on the scene of 2 W. 
Both cameras have two video channels coded in RS-170 video standard composed of 483 lines, each having 512 pixels 
at 30 Hz interlace. One channel is the intensity obtained from the scattered light from the scene and the second, 
the corresponding range information. Using this video standard allows easy interface to an image processor through 
an analog or digital video frame grabber. 
3. DIRECT ESTIMATION OF RIGID MOTION PARAMETERS 
3.1 The Range Flow Equation 
Let the coordinate system of the scene be a Cartesian (z,y, z) one. The output of the range sensor is a height 
measurement z of points on the object surface located at (z, y) for each video frame interval. The shape of the 
surface at time t is written as z = F(x,y,t). For each measurement point, one can locally approximate the surface 
by a set of small tangent patches. One can define the center point and the surface normal vector of the patch by 
(zo, yo, Zo), where zo — F(zo, yo, t), and 5 2 (n;, ny,nz)! . When this patch moves with an increment between two 
frames of Ap — (Az, Ay, Az)T in each direction during a time interval At, then the movement of the patch in the 
observation direction of a unit vector rà — (m;,my,, m;)7 is d. The distance between the patches before and after 
the motion is 7 - Ap. If one defines the angle between M and m as y, cosy is equal to Ÿ - 7. Therefore, 
n-Apzdn.. (1) 
If one can measure d, ri and ri, then Equation (1) is a linear equation with unknowns Ap. Let us suppose m to be 
aligned with the z axis, that is, à — (0,0, 1)7. Then, d 2 F(z,y,t - At) — F(z,y,t). Defining d as F;, Equation 
(1) can be rewritten as 
ne Az + nyAy+n,Az=n,F;. (2) 
On the other hand, the equation of the patch is 
nz(z — zo) - ny(y — yo) - n;(z — zo) 2 0. (3) 
Differentiating Equation (3) with respect to x and y, one obtains 
n, -n,F,-—0 (4) 
ny -n,F,-0 (5) 
where (Fy, Fy) is a tangent gradient of the surface F'(z,y,t) at (zo,y0). Substituting Equations (4) and (5) into 
Equation (2), one can obtain : 
F,-Ap+F,=0 (6) 
where É, = (Fy, Fy, —1)T. Note that F,, F, and F; can be easily calculated from the range image sequence. 
Equation (6) is a fundamental constraint of motion in range sequence analysis. 
3.2 Estimation of 3-D Rigid Motion Parameters 
Range flow Ap = (Az, Ay, Az)T of a point vector p — (z,y,2)T on arigid object is given along with the translation 
velocity vector T = (Tz, Ty, T:)T and the angular velocity vector R = (E. Hy R;)T at the origin as parameters 
— 
p=Rxp+T. (7) 
IAPRS, Vol. 30, Part 5W1, ISPRS Intercommission Workshop “From Pixels to Sequences”, Zurich, March 22-24 1995 
  
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