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Survey Method
In the DLT method, the accuracy of the camera calibration is
primarily determined by the accuracy of the object-space (3-D) or object-plane (2-D)
coordinates of the control points and the digitizing errors. Control points are normally
fixed to a rigid frame: the calibration frame. The size of the control volume one wants to
use is basically limited by the size of the calibration frame. It is possible to
reconstruct the object space/plane coordinates outside the control volume but
extrapolation is not recommended due to the intrinsic problem of the DLT algorithm. See
the Modified DLT page for the details.
When a huge control volume/area is needed (e.g., in long jump), the
typical rigid-frame approach does not work. Kwon3D provides a
survey method as an alternative to the typical rigid-frame approach. In this method, one
uses a set of range poles. The control points are marked on the range poles. By varying
the number and the size of the range poles, one can construct control volumes/areas of
different size. Since the shape of the control volume is not fixed, it is necessary to
compute the object-space/plane coordinates of the control points each time, case by case.
It is required to measure the horizontal angular positions of the poles and the vertical
angular positions of the control points marked on the poles. The object-space/plane
coordinates of the control points are computed from the angle data. This method was
originally developed at the Biomechanics Lab, Penn State University.
Here are the advantages of the survey method:
 | High flexibility. It is possible to change the number, size, and
location of the poles freely. |
| Large control volume. It is possible to construct a large (huge)
control volume/area. This is especially important in panning videography since one can
define single control volume/area. |
| Easy axis aligning. It is easy to define the direction of the X-axis
by using two poles (the origin pole & the X-axis pole). |
| Easy digitizing. Since the control points are all marked on the
poles, it is easy to identify them in digitizing. |
| Easy handling. It is easy to assemble, dissemble, and carry the
poles. |
| Flexible software supports. It is possible to incorporate useful
options into the survey program: repetition of the angle measurements, missing-point
option, axis-aligning, etc. |
The survey method also has few disadvantages:
| The object-space/plane coordinates of the control points must be
computed in each experiment. |
| A theodolite should be used in measuring the angular positions of the
poles and the control points. Since the accuracy of the object-space/plane coordinates
depends on the accuracy of the angle data, the theodolite must be handled carefully.
Surveying is a big burden in the competition situations. |
| Control points may not be distributed uniformly throughout the
control volume/area. |
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Procedures
The coordinates of the control points are computed based on the
following assumptions:
| All poles are set vertically. |
| The distance between the adjacent points is constant. |
The horizontal angular positions of the poles are used in computing
the X and Y coordinates of the control points while the vertical angular positions of the
control points are used in computing the Z coordinates.
Computation of the Distance & Height Factors of
the Poles
From the vertical angular positions of the control points marked on
a pole (Figure 1), the distance & height factors (D
& H) of the pole can be computed. For any two-point combination, the
following relationships hold:
 Figure 1
[1]
where Dij & Hij = the
distance & height factors obtained from combination i-j. Since each 2-point
combination provides a Dij & Hij pair, one can
obtain at most (n-1)! pairs from n control points marked on a pole.
Thus:
[2]
where D & H = mean distance & height
factors, respectively, and N = total number of 2-point combinations.
Computation of the 3-D Coordinates
The 3-D coordinates of control point i can be obtained as
follows:
[3]
where [xi, yi, zi]
= the 3-D coordinates of a control point, and H1 = the height factor
of pole 1. [3] takes into accounts the difference in the
height factors of the poles. Note that the origin of the coordinate system, at this point,
is located right below the theodolite at the height of the first control point of the
first pole and that the X-axis is aligned along the direction of the horizontal 0°.
Now, the 3-D coordinates need to be expressed in the user's
coordinate system. Let the coordinates of the first point of the origin pole be [xo,
yo, zo] and the horizontal coordinates of the
X-axis pole be [xx, yx]. Therefore:
[4]
where
[5]
and [xi,
yi,
zi] = the coordinates of point i
described in the user coordinate system.
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Kwon3D
The calibration program in Kwon3D provides
the following additional options:
- Repetition of the angular position measurement. One can repeat
measurement of each angle item (the horizontal angular position of a pole or the vertical
angular position of a point) up to three times. The program computes the mean angular
positions before the distance & angle factor computation.
- Missing points. One can omit any control point in the angular
position measurement. The program generates the 3-D coordinates of the missing control
points as long as at least two points are measured in each pole. This option is extremely
useful in a situation like competition where the survey must be done as quickly as
possible. One can measure the first and last points only, but still generates coordinates
for all control points on the pole. In some cases, one may intentionally use this option:
if the distance between the adjacent points is too short or if the theodolite is set too
far away from the poles, the parallax angle (qij
shown in Figure 1) gets small
and a small error in the angle measurement then can severely affect the distance &
height factors (see [1]). One may measure every second point
in this case.
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Application
See Kwon (1996) for an exemplar
application of the survey method.
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References & Related-Literature
Kwon, Y.-H. (1996). Effects of the method of body
segment parameter estimation on airborne angular momentum. Journal of Applied
Biomechanics, 12, 413-430.
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