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Herman J. Woltring
08-14-1991, 06:25 AM
Dear Biomch-L readers,

One of the peculiarities of electronic publication is the simplicity of
revision and modification. While this entails the risk of posterior
censorship and premature publication, it is a great advantage that useful
modifications in `objective' material can easily be implemented.

Following last week's posting on the constrained DLT, some modifiations
have been studied and implemented in the published DLTDSP routines .
Since the constraints are nonlinear, the routines are based on iterative
linearization and adjustment calculus until convergence, and finding
suitable initial estimates is rather critical if the control point
distribution is shallow (i.e., relatively small in one dimension) or
small in all three dimensions (i.e., the camera is relatively far away
/ the solid angle subtended by the control object on the camera is
rather small).

In this case, it has been observed that initial estimates can reliably
be found by the (incorrect) assumption that perspective imaging is an
equiform transformation [1] (i.e. a shape-invariant transformation
described by a position vector, an attitude matrix, and a scaling factor)
between the object coordinates {XPi,YPi,ZPi, i=1,n} and the corresponding
image co-ordinates {XIi,YIi, i=1,n}, under the assumptions of *constant*
image Z-coordinates ZIi = C and given prior estimates of the camera's
principal point (Xo,Yo) and principal distance C. If the principal
distances (Cx,Cy) per image axis are strongly different, prior scaling
per image axis is advisable. Because of the erroneous model and of
perspective compensation between various camera parameters, the estimated
values will be different from the true values, but they will act as
appropriate, inital values for the constrained DLT.

In an old study [2], a general 3-D calibration procedure was described
without the need for a voluminous 3-D calibration object as required under
the `classical' DLT; instead, a rather large *plane* of calibration points
was found to be sufficient. At ISB 1989, a modified approach was reported
emphasizing that a very small control point distribution was, in principle,
sufficient [3]; thus, landmark clusters affixed to moving body segments
could be used not only for reconstructing 3-D kinematics but also for calib-
rating the camera configuration, *without* the need to know the positions
and attitudes of those clusters (except for one defined reference) since
these parameters are calibrated *with* the unknown camera parameters.
However, initialization of the nonlinear procedure proved quite critical.

>From some simulation studies, the present approach to obtain constrained
DLT parameters (with subsequent conversion to the equivalent, conventional
internal/external camera parameters) has been found to allow rather robust
and accurate camera calibration. Now, it is merely necessary to observe a
number of planar or spatial landmark clusters (whose local coordinates must
be accurately known), and to combine measured image data on these clusters
throughout the field of view with limited-accuracy prior estimates of the
cameras' internal parameters. The latter are improved during the final,
iterative procedure of [2], together with the external camera parameters
of position and attitude.

In principle, separate calibration program runs can now be completely
avoided, by viewing the unknown and/or prior estimated camera parameters
as *nuisance* parameters to be estimated and eliminated during the
3-D kinematics reconstruction process. However, this is numerically
inefficient, and it is more advisable to select a number of time samples
from a movement record which are used for estimating the camera parameters.

In summary, the careful and expensive machining and cumbersome manipulation
of voluminous 2-D or 3-D calibration objects can be avoided by appropriate
modelling approaches in a PC environment. Both for laboratory and field
studies, this is thought a considerable advantage. However, some additional
control (e.g., at least three non-collinear points at the edges of a force
plate or of the field of view) are recommended in order to ensure a stable,
global frame of reference.

Herman J. Woltring, Eindhoven/NL

References:

[1] Journal of Biomechanics 21(1), 45-54, 1988
[2] Journal of Biomechanics 13(1), 39-48, 1980
[3] Paper #197, Proc. ISB XII, UCLA 1989

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