INTRODUCTION
The direct linear transformation (DLT) is a
technique commonly used to locate spatial points filmed with two or more cameras. (Marzan
and Karara, 1975). The DLT does not necessarily yield coefficients which correspond to an
orthogonal orientation matrix of the image to object coordinate system (Hatze 1988). Hatze
demonstrated that increased accuracy for the reconstruction of points is achieved by
imposing an orthogonality constraint upon the optimization procedure used to obtain the
DLT coefficients. Hatze called this DLT with orthogonality constraint the modified direct
linear transformation (MDLT).
The purpose of this paper is to present a technique
called physical parameter transformation (PPT) which allows the use of panning cameras.
The PPT is built upon the colinearity photogrammetric relations from which the DLT is
derived. Like the MDLT, PPT is implemented such that orthogonality of the orientation
matrix of the image to object coordinate system is guaranteed. PPT with panning will be
demonstrated to have greater accuracy than the DLT.
METHOD
The direct linear transformation (DLT) is a
technique commonly used to locate spatial points. The colinearity photogrammetric
relations provide the mapping from spatial coordinates to image coordinates. The mapping
is a function of 16 physical parameters which describe the central projection model of a
camera. In general the 16 parameters are not known; thus, they must be determined through
a calibration procedure. Calibration is implemented by minimizing the mapping error over a
set of control points whose spatial and digitizer coordinates are known. The minimization
for these physical parameters is nonlinear; therefore, this approach is not typically
used.
The DLT is obtained from the colinearity relations.
The colinearity conditions may be rearranged into a form requiring 11 coefficients. These
11 coefficients are functions of the 16 physical parameters. The minimization of residual
error with respect to these 11 coefficients is linear; thus, the calibration procedure is
simplified. The 11 parameters are the coefficients of the widely used DLT method.
The rotational orientation matrix of the camera with
respect to the spatial coordinate system provides 9 of the 16 physical parameters. The DLT
calibration procedure does not necessarily yield coefficients which correspond to an
orthogonal orientation matrix. This nonorthogonality increases error in spatial
reconstruction.
If the rotational orientation matrix of the camera
is expressed as a function of three suitable angles, the number of physical parameters
reduces to 10, minimization of mapping error with respect to these 10 physical parameters
automatically insures that the resulting orientation matrix is orthogonal. The
minimization is still nonlinear; thus, numerical optimization technique is required along
with an initial estimate for the 10 physical parameters.
The 10 physical parameters may be expressed as
functions of the 11 DLT coefficients; thus, the DLT provides a good initial estimate for
the 10 physical parameters. Newton's method was employed to iterate from the initial
estimate to the 10 optimum physical parameters. This photogrammetric procedure involving
10 physical parameters is called the physical parameter transformation (PPT). Like the
DLT, once the mapping parameters are known for two or more cameras, spatial locations of
points whose digitizer coordinates are known may be obtained by solution of a linear
system.
The PPT may easily accommodate panning cameras if
the displacement of the camera relative to its calibration position is known. In addition
to the cameras orientation matrix, the location of the projection center provides three
physical parameters which may vary with the panning angle. Both camera orientation and
projection center are transformed via the displacement yielding PPT coefficients for a
panned camera position.
In this study, single axis panning was considered.
The panning angle was provided by an optical encoder yielding 10 minutes of resolution.
Panning axis location and direction were determined by performing 2 normal PPT
calibrations corresponding to different panning angles. The two calibrations yield
positions of the camera which only differ by a rotational displacement about the panning
axis; thus, the calibrations provide enough information to determine panning axis location
and direction. Since the displacement may be expressed as a function of panning angle,
axis direction, and axis location, the PPT coefficients corresponding to any panning angle
about a single axis is determinable. Since the location and direction of the panning axis
relative to the panning camera is constant by construction, this special calibration
procedure needs to be performed only one time.
Accuracy of the DLT and PPT with panning were
investigated by filming two control objects each comprised of 15 symmetrically located
points. Each object was 1 cubic meter. Coordinates of all control points were located
within 3mm. The two control objects were horizontally translated 3 meters apart. The axis
location and direction calibration procedure was performed by calibrating each object.
Three cameras were used. A side panning view and a
front still view was used for the panning data. A side stiff view which was displaced far
enough away from the control objects such that all points were visible was used along with
the front still view for evaluating the DLT.
The object most positive in x henceforth referred to
as the right object was used for calibration in all methods. Accuracy of each method was
evaluated by calculating root mean squared error in reconstructing each object.
RESULTS AND DISCUSSION
Reconstruction error of the noncalibration object
(left object) was 29.8mm and 5mm for the DLT and PPT with panning respectively.
Reconstruction error for the right calibration object was 5.4mm and 4.6mm for the DLT and
PPT with panning respectively. Since the PPT with panning yields about one sixth the DLT
error in reconstructing the noncalibration object, PPT with panning has better
extrapolation characteristics than DLT for our test data.
CONCLUSION
A photogrammetric data processing technique (PPT)
was presented which allows the use of panning cameras. The PPT was developed such that the
rotational matrix of the camera is guaranteed to be orthogonal which is not the case with
DLT.
For our experimental data, the PPT with panning was
more accurate than the DLT. This improvement is probably due to the increased digitizing
resolution made possible with panning and the PPT's satisfaction of the orthogonality
condition.
REFERENCES
- Hatze, H. (1988) High-precision three-dimensional
photogrammetric calibration and object space reconstruction using a modified DLT-approach.
J Biomechmics 21, 553-538.
- Marzan, G. T. and Karara, H. M.
(1975). A computer program for direct linear transformation solution of the colinearity
condition, and some applications of it. Proceedings of the Symposium on Close-Range
Photogrammetric Systems, pp. 420-476. American Society of Photogrammetry, Falls
Church.
|
