Vittorio Casella
3 BASIC FORMULAS OF OUR ESTIMATION TECHNIQUE
This is the core of our paper, where we describe the method we are using and tuning. Let’s define, first of all, what we
mean by ramp. A ramp is a planar object whose projection on the X-Y plane is a rectangle. Besides, its height changes
along the longitudinal axis, but it doesn’t change along the transverse axis. Let’s now define the equation of a ramp.
Provided that the object coordinate system is (X,Y,Z), we will associate an intrinsic coordinate system, called,
(X',Y',Z') , to each ramp. Its origin coincides with the mid point of the lower edge; the Z' axis is parallel to the object
Z axis. The Y' axis is parallel to the longitudinal ramp axis and it grows along the direction that makes the ramp's height
grow. In such a system, the equation of the plane containing the ramp is
Z'-pY' (4)
where p is the slope. To express the same equation in respect to the object coordinate system, we have to take into ac-
count an offset, (X,.Y,.Z,) , which represents the position ofthe origin of the intrinsic system with respect to the object
system, and a planar rotation angle a . We have chosen to use the clockwise angle formed by the Y' axis with respect to
the Y axis, so that the angle represents the azimuth of the longitudinal axis of the ramp. Therefore the equation of the
plane containing the ramp is
Z = psina(X — X,)+ pcosa (Y —Y,) +2
5
= X psina +Y pcosa + Z, —(X,psina +Y,p cosa ) 6»
It is also possible to give a parametric description of the ramp in the following way
X = X, +tcosa +u sina
Y - Y, -tsina * ucosa (6)
Z-Z, tup
where u and 1 span the width and the length of the ramp, respectively. Now, let's come to the problem of the estimation
of the planimetric precision. Let's form the difference between the height measured by the scanner, Z, and the height of
the ramp in the position (X, Y) supplied by the scanner
A —Z —(X psina +Y pcosa +c) (7)
where we have synthesized all the non-essential parameters into c. This random variable can be investigated experimen-
tally and its mean and variance estimated: we have as many extractions as the number of the laser points which hit the
ramp. We will assume at the moment that ramp parameters, p and a are constant, while they should be more properly
treated as random variables themselves. Our assumption is justified because those parameters will be estimated with
GPS, with a much higher precision than laser points. In any case the formulas we are giving can be easily generalized to
consider the randomness contained in those parameters. Instead, the randomness contained in the laser-measured coor-
dinates will be taken into account, assuming that X,Y, and Z are random variables having respectively mean values
(X,.Y,y.Z,) and variances (s;,s;,s2). It’s easy to calculate mean and variance of the A variable
m, =Z,—(X,psina +Y,pcosa +c)=0 (8)
9 > . pi 2 ^
S1-s7-8s;p'sin'ats] p'cos?a
160 International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B3. Amsterdam 2000.
