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International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B5. Istanbul 2004
coordinate system according to OL. Error equations of circle or
arc reconstruction can be written as:
y = A| dX 9 45: dYg A4- dR — dx
v, 7 Bj-dXg 4 B5-dYg * B3: dR — dy
where A, A.» 4 ; Bi, B,, B, are the coefficients of
unknowns, dx, dy constant items. Thus the parameters of space
circles or arcs can be obtained directly from one or several
images by LSTM with initial values obtained from CAD data
and known camera parameters. The obtained parameters of
circles or arcs should be rotated back to world coordinate
system according to camera parameters.
Figure 8. Connected arcs and lines
In modern industry, arcs in parts are usually connected to lines.
As shown in Figure 8, two arcs c;, c; and three line segments /;,
l; and /; are connected to each other. Generally, they are very
difficult to reconstruct precisely. The model of obtaining
uniform solution of arcs and lines will be addressed. For
convenience of reconstruction, line segments are also rotated
into a level plane, and represented as follows:
X=X +i -AL-cosp
Y=Y -i-AL snp
(6)
where X XY. is the start point of line segment, p denotes the
direction of the line, AL is the length of small segment
approximately equal to the length of point window in circle and
arc matching. Error equations of line reconstruction are:
=
ll
mM dE. + M, dY, +M. 4-4 +
=NdX,+ N, dy, + N, dP - dv
=
|
where M,,M,, M; and N,,N,,N, are coefficients of
unknowns, dx, dy constant items. The circle or arc
reconstruction equation (5) can be combined with line
reconstruction equation (7) to get an uniform solution of
connected arcs and lines. To ensure the stability of
reconstruction, several geometric constrains should be added,
such as the center of arc c, should lies on bisector of line /;and
l, the center of arc c; should lies on bisector of line /;and /;, the
three lines should be tangential to the two arcs etc.
4. EXPERIMENTS
4.1 Overview of the System
To reduce the cost of measurement, only one non-metric CCD
camera is used. Figure 9 shows the hardware configuration of
the system. A planar grid is fixed on the rotation table for
camera calibration and offering initial values of camera
parameters during reconstruction. The part to be reconstructed
is put on the grid. Image sequence is obtained while the table
rotating under computer control. As mentioned in section 3,
world coordinate system is chosen the same as that of grid.
Figure 9. Hardware configuration of the measurement system
The developed software runs fully automatically. It can be used
to reconstruct and measure industrial parts mainly composed of
lines, circles, connected arcs and lines. The system is composed
of 4 steps. Firstly, Image sequence is acquired by CCD digital
camera automatically while the table turns around its center
controlled by computer. Image points and lines are obtained by
LSTM simultaneously with image acquiring. Then 3D wire
frame model of the part is reconstructed accurately with hybrid
point-line photogrammetry. Afterwards, circles, connected arcs
and lines are reconstructed by direct object space solution.
Finally, measurement can be done automatically or
interactively.
4.2 Real Data Experiments
The measurement system has been tested with real image data
of several parts taken by a pre-calibrated CCD camera (Zhang
et al 2003). Experiments of two parts both with dimension of
about 150mm will be presented in the following. The part to be
reconstructed is put on the planar grid, which is fixed on the
turntable. The CCD camera is fixed on a tripod with distance of
about 600mm to the part. One image of each part is shown in
Figure 10. A sequence of 25 images for each part is taken with
equal angle intervals while the table turns around. Image
matching is made simultaneously with image acquiring. Grid
points are detected as the intersection of two line segments
fitted to each corner. Lines are obtained by LSTM with initial
values projected by the CAD-designed data and the camera
parameters provided by the grid. Occlusions are detected before
template matching to reduce mismatches.
White lines in Figure 10 are the matched lines of parts. There is
nearly no mismatch for points of planar grid. But for parts that
are very thin, there maybe some mismatched lines. Most of
them can be removed successfully with Trifocal tensor (Hartley
et al 2000) computed with camera parameters provided by the
planar grid. The remained mismatches can be eliminated during
