Figure 5. Left image : corners of sub images are projected on the predefined plane and the enclosed rectangle in object space is
found. Right image: backprojections of the corners of the rectangle define the size of the virtual image.
enclosed rectangle of the images is defined. Finally, all pixels
are resampled and the virtual image is created. Since demand-
ing accuracy is less important for archaeological documentation
and the terrain height differences are not large, the influences in
planimetry and in height are generally small.
The reference plane may be defined and approximated in two
different ways: in the first case, a given horizontal plane at a
specified height or a best fitted plane from a group of 3D points.
The equation of the plane is given as a,*x a,*y + a; = z and
with least squares adjustment the best fitting plane is calculated.
In the second case the intersection of the ray with the plane is
calculated through Eq. 1. For nonplanar surfaces, a higher order
polynomial surface may be fitted e.g. a; + a,*x + a*x? t a4
*x*y + as *y” +a = z and the intersection point of the ray with
the plane may be found by iterative methods.
X Xo a,
Y |=|Yo |+t-|b, (D
Z Zo C,
where ,. ap(x, — Xo) * b. (y, — Yo) + cp (z, — Zo)
apa, +b,b, +b,b,
(a,,b,,c;) » 1=Lp direction vector of ray and normal vector of
plane
Xo,Yo,Zo camera position
XprVpaZp Point that lies on the plane
The version using a best fitted plane was selected as a compro-
mise between accuracy and computation time. Compared to the
horizontal plane the surface was better approximated and com-
pared to the higher polynomial surface it was faster. The plane
has been selected, so that the residuals of all 3D points were
minimum (Fig. 6).
0.8 -
0.6
0.4
CR
*
0.2
103 102 ” 106
y
Figure 6. Plane fitted over the 3D points.
When generating the virtual image the corrections resulting
from lens distortion have to be calculated. The Brown model
(Eq. 2), modeling radial and descentering distortion gives the
correction from distorted to the undistorted image, as the input
are the coordinates of the distorted image. The inverse trans-
formation corrections have to be computed by solving a two
dimensional non-linear system of equations with an iterative
method, e.g Newton Raphson. The corrections are given regard-
ing to the distorted coordinates, therefore undistorted coordi-
nated can not be used directly as input in the equation (see be-
low).
Ax =Xr"K, +xr"K, + Xr°K, X r°)P +2xyP, (2)
Ay 2 yr' K, - yr'K, 4 yrK, * 2xyP. - Qy^ 4 Y^P
where x- x—x, and y 2 y - y,
rz [52 my.
x,,y, Principal point coordinates
Both x,, y, undistorted coordinates have to be corrected by Ax
and Ay after projecting from ground to image, give distorted
coordinates and resample the grey value from that position. If
the distortions of the camera are small the corrections can be
approximated by using Eq. 2 with opposite signs in the parame-
ters, but in case of the Mavica, the distortions are large and
therefore a more accurate computation is needed. In Fig. 7 the
influence of lens distortions for Mavica FD81 is shown. The
two-dimensional system of equations (Eq. 3) is solved by calcu-
lating the Jacobian matrix, inverting and multiplying with the
observation vector. The solution vector updated and the itera-
tions continue till the convergence of the solution. As initial ap-
proximation the undistorted coordinates are used.
f y) 20 (3)
f, y) 20
where
f,G,y) - x «xr' K,exr! Ke xr'K, (n -2x?)B
*T2XPP, —x 4,
f505y) S y * yr K,* yr! K, c yr'K, e (n? 2y?)P,
*2xyP, - y, * y,
With x y the undistorted coordinates
—442-
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Figure 6.
