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).
K.A. Stivers, G.B. Ariel, A. Vorobiev, M.A. Penny, A. Gouskov, N. Yakunin
international Center for Biomechanical Research, La Jolla, USA
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 .kith panning will be demonstrated to have
greator accuracy than the DLT.
The colinearity photograrnmetric 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 I I coefficients. These I I
coefficients are functions of the 16 physical parameters. The minimization of residual
error with respect to these I I coefficients is linear; thus, the calibration procedure is
simplified. The I I 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 I I 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 it's calibration position is known. In addition to the Camera's
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
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.
objects each comprised of 15 symmetrically located points. Each object
was I cubic meter. Coordinates of 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.
Accuracy of the DLT and PPT with panning were investigated by filming
Three cameras were used. A side panning view and a front still view was
used for the panning data. A side still view which was displaced far enough away from the
control objects such that
a 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.
Reconstruction error of the non calibration object (left object) was
29.8mm and 5mm for the DLT and PPT with panning respectively. Reconstruction error for the
right calibration object was 5.4mrn and 4.6mm for the DLT and PPT with panning
respectively. Since the PPT with panning yeilds about one sixth the DLT error in
reconstructing the noncalibration object, PPT with panning has better extrapolation
characteristics than DLT for our test data.
RESULTS AND DISCUSSION
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.
Hatze, H. (1988) High-precision three-dimensional photograrnmetric
calibration and object space reconstruction using a modified DLT-approach. J
of Photogrammetry, Falls Church. Proceedings
of the Symposium on Close-Range Photogrammetric Systems, pp. 420476. Marzan, G. T. and Karara, H. M. (1975). A computer program for direct
linear transformation solution of the colineatity condition, and some applications of it.
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