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Initial Guess Problem
The filter function of the Butterworth low-pass filter has
the following general form:
 ,
[1]
where ak & bk =
filter coefficients, x[i] = raw coordinate of frame i, and x'[i]
= filtered coordinate. A second order filter, thus, can be expressed as
 .
[2]
[2] shown above has two main
problems:
1. It requires the raw and filtered coordinates of the
two previous frames. Thus this equation can not be used in the first two
frames.
2. Since it can not be used for the first two frames,
the filtered coordinates of the first two frames are not available and,
therefore, it can not be used for the third frame, either.
In other words, certain initial guess is necessary to use [2]
in filtering the position data in motion analysis, such as
 .
[3]
It was assumed that the filtered coordinates are equal to
the raw coordinates in the first two frames. Since the filtered coordinates of
the first two frames are now available, [2] can be
used for the third frame, etc. Generalizing [3]:
 ,
[4]
where k = 0, ..., N - 1, and N =
order of the filter.
One problem of this approach is the error propagation from
the first N frames to the rest of the frames. Since a Butterworth
low-pass filter is an IIR (infinite impulse response) filter, the
coordinates of the first N frames are continuously involved in the
filtering of the coordinates of the rest of the frames. Thus, errors included in
the filtered coordinates of the first N frames will be propagated to the
rest of the frames. The effects of the first N frames gradually diminish
as the filtering progresses from one frame to the next.
There are three possible ways to reduce the error
propagation from the first N frames to the rest of the frames:
1. Digitize additional frames before and after the
target frame range to be analyzed.
2. Pad additional frames numerically before and after
the digitized frames.
3. Combine both methods 1 and 2.
Both methods 1 and 2 above are based on the fact that the
effects of the first N frames gradually diminish as the filtering
progresses from one frame to the next. In either case, it is important to
digitized the frames accurately at the beginning of the trial since the amount
of error in the first N frames will affect the rest of the frames and the
data padding is based on these frames.
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Data Padding
Data padding is the process of generating additional frame
data numerically at the boundary of the frame range, based on the digitized
coordinates. The typical data padding methods include the point-symmetric
padding method and the cubic padding method.
Point-symmetric padding was originally called as reflection padding
but it is not an accurate term.
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References & Related
Literature
Smith, G.
Giakas, G
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