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Y|=|Y [7% AY (1)
2} ed
OX oY
where P is the considered point, x and oz are the
components of the gradient vector in the same point and
AX, AY are the parameters that can be interpreted as
local planimetric coordinates.
The variation of AX and AY within properly choosen
intervals and with a certain incremental step define a set
of gridded points around P.
For each of these points one or more observation
equation can be written.
An indipendent equation can be written for each image in
the form:
&(x.)=G(X.Y) 2)
where
i = index of image ( (1, 2) or (1, 2, 3) in our
implementation);
X;,y; * image coordinates in the ;? image, related to (X,
Y, Z) by the collinearity equation;
G(X,Y) is the function that represent the gray value of
the imaged surface (a very simplified radiometric model
is considered here, in practice a linear radiometric
trasformation has been implemented).
The first step for the practical use of the equation (2) is to
eliminate the unknown function G( X, Y).
We have, with two images:
GERE M EC (3)
while for three images we can write
4816 3) [8 2) 50. ,)] =0 (4.b)
432, (x,.»,) - 438, (x,,,) 2 0 (4.8)
that are observation equations with incorrelated known
term of the same weight.
These equations must be linearized and then they can be
used in a standard least square adjustment.
There are two possible way for choosing the point to be
plotted:
e fix X,,Y,; in this case Z, and the components of
the gradient are the unknowns.
e. fix (3,3), the point must be determined along the
line of sight defined in this manner: again we have 1
degree of freedom for the position of the point and
two more unknowns to model the slope.
All this unknowns are determined as solutions of the
normal equations generated by the observation equation
(3) or alternatively (4.a) and (4.b).
421
It must be remarked that the unknowns are implicitly
contained in the (xy) image coordinates by means of
the collinearity equations and the parametric equation
(1).
5. TESTS
The system has been tested both with simulated and real
images.
5.1 Tests on simulated images
A specific program allow the simulation of images taking
with a given color pattern, surface height and camera
orientation. These tests allow an immediate checking of
the results and the consequent judgment of the system
working.
The grey function that describes the object pattern in the
simulate images is:
8(x,y)=|X- Xo|+|Y = ¥,| +20 sin(X)sin(Y) +5 (5)
where (X, Y) are the planimetric ground point
coordinates, (Xo, Yo) are the coordinates of the centre of
simmetry of the artificial pattern. The surface shape and
the cameras' orientation are different in the various tests,
however they are choosen so that images deformations
are small and the overlapping is more than 6096 in both
directions.
In the main tests the object surface is a plane, with
equations, in the different cases, Z=0,
Z — 0.005. X +0.005-Y and Z = 0.03- X +0.03-Y.
Least square matching has given good results, as shown
in the following table.
Test Std Dev max|S. D. x max [pixel]|S. D. y max [pixel]
Slope 0 1073 «103 «103
Slope0.005| | 2:10? 0.007 0.007
Slope 0.03 | 51073 0.042 0.038
Table 1: least square matching results.
The higher values for the last two trial can be explained
with the deformations of the images due to the slope.
The next step is images' orientation with bundle
adjustment procedure, the reference system is fixed by
constrains on the coordinates of the taking points of the
first and the third images and on the o angle of the
second image.
The results of the orientation procedure are shown in the
following table, the max error indicate the maximum error
of the coordinates of the points on the images, that are
known in the simulated tests.
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B2. Vienna 1996
RO ER enr mr rer
