Introduction to Sports Biomechanics: Analysing Human Movement ...

QUANTITATIVE ANALYSIS OF MOVEMENT
cannot be genlocked, event and time synchronisation can be achieved by placing a
timing device, such as a digital clock, in the fields of view of all cameras. Time
synchronisation must then be performed mathematically at a later stage; obviously
some error is involved in this process. It may also not be possible, particularly in
competition, to include a timing device in the fields of view of the cameras. Some
other event synchronisation must then be used, based on information available from
the recorded sports movement, such as the instant of take-off in a jump.
From two or more sets of image coordinates, some method is needed to reconstruct
the three-dimensional movement-space coordinates. Several algorithms can be used
for this purpose and the choice of the algorithm may have some procedural implications.
Most of these algorithms involve the explicit or implicit reconstruction of the
line (or ray) from each camera that is directed towards the point of interest, such as a
skin marker. The location of that point is then estimated as that which is closest to
the intersection of the rays from the two or more cameras.
The simplest algorithm requires two cameras to be aligned with their optical axes
perpendicular to each other. The cameras are then largely independent and the
depth information from each camera is used to correct for perspective error for
the other. The alignment of the cameras in this technique is difficult, although the
reconstruction equations are very simple. This technique is generally too restrictive
for use in sports competitions, where flexibility in camera placements is beneficial
and sometimes essential.
Flexible camera positions can be achieved with the most commonly used reconstruction
algorithm, the ‘direct linear transformation (DLT)’. This transforms the
video image coordinates to movement-space coordinates by camera calibration
involving independently treated transformation parameters for each camera. The
algorithm requires a minimum of six calibration points with known threedimensional
coordinates and measured image coordinates to establish the DLT
(transformation) parameters, or coefficients, for each camera independently.
The DLT parameters incorporate the optical parameters of the camera and
linear lens distortion factors. Because of the errors in sports biomechanical data,
the DLT equations also incorporate residual error terms. The equations can
then be solved directly by minimisation of the sum of the squares of the residuals.
Once the DLT parameters have been established for each camera, the unknown
movement-space coordinates of other points, such as skin markers, can then
be reconstructed using the DLT parameters and the image coordinates for all
cameras. Additional DLT parameters can also be included, if necessary, to allow
for symmetrical lens distortion and asymmetrical lens distortions caused by
decentring of the lens elements. No improvements in accuracy are usually achieved
by incorporating non-linear lens distortions. The DLT algorithms impose several
experimental restrictions.
� An array of calibration (or control) points is needed, the coordinates of which are
accurately known with respect to three mutually perpendicular axes. This is
usually provided by some form of calibration frame (for example, Figures 4.4
and 4.7) or similar structure. The accuracy of the calibration coordinates
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