International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B5. Istanbul 2004
to setup error equations.
S
p
Figure 6. Coplanarity between space and image lines
The coplanar equation among p, S, P and Q is:
Up HP We ;
Xe ecran Vs Om (
~ x =
X mg. any TEE
where bar, w,) is the model coordinate of image point p,
fr Y. 2) the coordinate of S, (x Y Ze) and
(x, y zy the coordinates of part points. Error equation of
line photogrammetry can be written as:
A,dx , * Ady , + A;dp + Ado + Adi + Ag dX 5 +
A,dY + A,dZ, * Ado" + Apdo’ + A, dk? (2)
A, dAX + A dAY® + A, dAZ" + A dX) + A, dr) +
AZ} + A dX 9 + A dF] + Ay dZ] + Fy = 0
where 4, — 45, are the partial derivatives of unknowns and F,
the constant item. Besides the coplanar equation among p, S, P
and Q, there exits another equation among q, S. P and Q. The
linearised form is similar to that of equation (2).
For parts that are very simple or there are a few line segments,
the geometric configuration is very poor. Grid points should be
combined into the adjusmtent model to ensure the reliability of
reconstruction. Error equations of grid points are:
v. 2B dp+B,do+B.dr + B,dX + B.dY, +
BdZ + B,d% +B, dY + BAZ — I. 3)
v, 2 Cido * C,do * C,dk * C,dX , * C,dY, *
Cds + Cdt v CuY + Cod? — 1
where / ool , are constant items, B, >
Brion Ve
coefficients of unknowns. Please refer to (Kraus 1993) for more
detail about the coefficients of error equations. If the
coordinates of grid points can be treated as known, terms of (dX,
dY, dZ) should be removed from the error equations. The
692
model of hybrid point-line photogrammetry is composed of
equation (2) and equation (3). It can be used to reconstruct the
wire frame model of parts.
For lots of board-like industrial parts, the reconstruction of
complex shapes is also very important but hard to deal with in
. practice. An effective approach to reconstruct circles, connected
arcs and lines based on one-dimensional point template
matching and direct object space solution will be presented in
the following.
Camera parameters, which can be obtained with hybrid point-
line photogrammetry, are treated as known. Since the end
points of small line segments are results of template matching
and also functions of space circles or lines, the parameters of
space circles or lines and images are related directly. Thus
parameters of circles, arcs and lines can be obtained directly
from several images by least squares template matching.
Suppose the plane where circle or arc lies in is known. This is
true since the plane can be determined by the reconstructed wire
frame CAD model. The camera parameters of the images can be
rotated to level the plane. So the circle equation in the level
plane is very simple:
X = X, +R-cos0
: (4)
YzY,t-R:snO
where X,Y, and R are the center and radius of circle or arc,
@ varies from 0 degree to 360 degree for circle, and from start
angle to end angle for arc. In this paper, circles and arcs are
represented by a number of points with certain intervals of
different angle @. The top of Figure 7 is a space circle, and the
bottom is the projected ellipse with known camera parameters.
Each point A on the space circle defined by Ó has its
corresponding point a in the image.
Projection
Figure 7. Projected image point of circle
If equation (4) is substituted into collinearity equations, the
unknowns are the center and radius of circle or arc since Z is a
fixed value after rotation. For certain angle &, the object point
is projected onto image and the tangential vector with angle 0
can be easily determined. A small window with length of 2-5
pixels taken the projected image point as center and OU as
tangential vector is rotated into horizontal, followed by one-
dimensional point template matching. The displacement dr
determined by template matching can be rotated back to image
I SN ul PE LR AEST end.
