3. Biomechonics
Vol.22,No.s/9,pp.967-971,
1989.
Printedin GreatBritain
co21-9290/89
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C 1989Pergamon Press plc
TECHNICAL
PADDING
NOTE
POINT EXTRAPOLATION
TECHNIQUES
BUTTERWORTH
DIGITAL FILTER
FOR THE
GERALD SMITH
Biomechanics Laboratory, Pennsylvania State University, University Park, PA 16802, U.S.A.
INTRODUCTION
Biomechanical studies of movement kinematics and kinetics
often involve the measurement of analog quantities through
discrete sampling of the signal at some regular interval. The
digital data which result can, in most cases, adequately
represent the original signal, providing the sampling frequency is high enough to respond to the high frequency
components of that signal (Winter, 1979, p. 28). The ease of
computer processing of digital data is a distinct advantage
over analog data. However, in either case some signal
conditioning is usually required before any quantitative
analysis is performed. In the case of digital data, various
smoothing routines have been developed for dealing with the
random errors which are introduced in the measurement
process. The most effective of these methods (spline functions,
Fourier series and digital filters) have been described in detail
elsewhere (see Wood, 1982 for a general review).
The Buttenvorth digital filter is a recursive routine which
uses prior data points (both smoothed and raw data) to
predict subsequent smoothed data points (Gold and Rader,
1969). For biomechanical application, Winter (1979) suggested the use of a second order filter such as the following:
Yi=a,Xi+alXi-l+alX,_2+blYi_1+b,Yi_,
,
where Xi is the raw data value for time i, Yi is the smoothed data
value for time i, and the a and b coefficients are weighting
values based on the sampling and cutoff frequencies desired.
The recursive nature of the algorithm leads to a problem
inherent in this approach to data smoothing. Because each
smoothed data point requires two previous points as input
into the algorithm, the process cannot be started with the first
raw data point. Furthermore, the smoothing process requires
several iterations to become effective as prior smoothed
points also contribute to the value of each point. Therefore,
typical applications of digital filtering minimize these ‘startup’ problems with the filter by beginning the processing at a
point well ahead of the true region of interest. These extra
‘padding points’ would be expected to contain more noise
than the remainder of the data set. In practice, digital fi!ters
are commonly applied to data sets in both a forward and
reverse direction. This double filtering removes a time shift
which occurs with a single pass through the filter (Winter,
1979, pp. 36-37). As a result, padding points are necessary not
only at the beginning but also at the end of the data set. The
inclusion of padding points is not a serious problem when
collecting data by some electronic means; however, when
manual operations such as film digitizing are involved, the
extra padding point data may greatly increase the data
collection task. As a means of avoiding the extra digitizing
required for the creation of genuine raw data to be used as
Received
in finalfirm 21 November 1988.
padding points for a digital filter, several alternatives will be
presented in the following analysis.
In kinematic and kinetic studies it is often necessary to
determine velocities and accelerations from displacement
data. This process is somewhat problematic because of the
high frequency noise amplification which results. Therefore
prior to differentiation of displacement data some smoothing
of the raw data is usually required. The acceleration curves
which result from double differentiation of displacement-time data are quite sensitive to small displacement
differences. Thus, acceleration curves provide a good instrument for comparison of the subtle differences between various data smoothing routines. In particular, acceleration
curves will be used to distinguish the differences between the
methods for generating padding points.
METHODS
Several routines were created for generating padding
points at the beginning and end of a raw data set. A method
which has occasionally been used in digitizing when the
appropriate extra frames were not available (subject moved
out of the held of view, for example) is to redigitize or
duplicate the coordinates from the last available frame. Other
methods for artificially calculating padding point values
could involve extrapolations from the first (or last) few real
raw data points. For example, a linear extrapolation based
on the slope between the first (or last) two points could easily
be used to calculate the extra points. Alternatively a ‘reflection’ could be made about the end point, determining the
(E-i) point value from the (E + i) point value (where E is the
end point and i is the number of padding points). Each of
these techniques is illustrated in Fig. 1.
To assess the effect that each padding method had on the
resultant smoothed data produced by a Butterworth second
order digital filter (with forward and reverse passes through
the filter) a standard data set was used for comparison.
Pezzack et al. (1977) reported a comparison of accelerations
from several data smoothing methods with an independent
analog measure of acceleration. The data set they included
was used as the base for this study. However, the Pezzack et
al. data are somewhat unrepresentative of real biomechanical
data in their relative cleanness or absence of noise. To
produce a more realistic data set, random noise was introduced into the Pezzack et al. data (in a manner similar to that
used by Hatze, 1981; Lanshammer, 1982; and Wood, 1982).
The original data ranged from 0.1517 to 2.2047 rad. At each
data point a noise component was added of at maximum
_+O.l rad. The actual noise magnitude was determined by
computer generated random numbers. This noisy data set
was then used as the raw data to be digitally filtered under
various conditions.
To simulate the situation in actual digitizing, a ‘window’
was taken from the middle of the data set. The original data
967
968
METHODS
Technical Note
FOR GENERATING
PADDING
POINTS
FROM RAW DATA
ANGLE VS TIME: COMPARISON
OF ORIGINAL
TO NOISY DATA
1
04
0.0
I
I
9
I
,
0.5
1.0
1.5
2.0
2.5
SAMPLE NUMBER
Fig. 1. Padding points were created using three methods.
The duplication method simply reproduced the last data
point value. The linear extrapolation method used the slope
between the last two points to determine the padding points.
The reflection method used the E (end point) value as a
reference level, determining the E-i padding point value
based on the E + i point value.
0
TIME (WC)
Fig. 2. Comparison of the Pezzack et al. (1977) angle-time
data with the noisy data set used in this study. The noisy data
have been shifted upward in the graph for ease of comparison.
ANGULAR
ACCELERATION
VS TIME:
COMPARISON
OF ORIGINAL AND FILTERED DATA
set involved 142 points; a subset of 80 points was designated
the region of interest (from point 31 to point 110). The three
methods of padding point generation were applied to this 80
point region (with the number of padding points ranging
from 0 to 30) and then digitally filtered (cutoff frequency of
4 Hz). From each of the smoothed data sets, velocity vs time
and acceleration vs time functions were calculated using first
central difference methods (Miller and Nelson, 1976). These
acceleration curves were also compared with those obtained
in smoothing the corresponding range of points from the
noisy data set (as if the padding points were digitized).
The original Pezzack et al. (1977) data accelerations (obtained by smoothing and differentiating the ‘clean’ data) were
used as an estimate of the ‘actual’ acceleration curve (a cutoff
frequency of 9 Hz was used). The acceleration curve obtained
in this manner was illustrated in the Pezzack et al. paper and
provided an excellent approximation to the accelerometer
measurements. The beginning and end region acceleration
deviations from the ‘actual’ acceleration were represented by
a mean residual over the first (or last) 10 points. This value
was defined as the mean of the absolute value of the difference
between actual accelerations and the filtered noisy accelerations.
0.0
0.5
1.0
1.5
2.0
2.5
a 0
TIME (sac)
Fig. 3. Comparison of the angular accelerations from the
original angular data and the acceleration curve from a
corresponding smoothed angle-time data set. The fluctuations of the original accelerations suggest that some noise
was included in the sample. The solid curve illustrates the
acceleration function after smoothing; this was used as an
estimate of the ‘actual’ acceleration.
RESULTS
The original (‘clean’) angle vs time data (Pezzack et al.,
1977) describe a smoothly varying angular position. The
noisy data set was created by randomly introducing deviations added to the original data. The resulting angular data
(Fig. 2) are clearly noisy (probably exceeding the noise levels
of typical biomechanical data). The angular accelerations of
the original data (Fig. 3) illustrate the fluctuation one normally encounters after differentiating even relatively clean
raw displacement data. Differentiating the noisy data would
result in velocity and acceleration curves which obscure the
actual angular velocity and acceleration values. Signal processing of some type would clearly be a prerequisite to
determining angular accelerations from such noisy data.
The acceleration vs time curves for several conditions are
illustrated in Fig. 4. Except for the immediate vicinity of the
end points the curves are nearly identical. However, near the
end points, considerable variation in acceleration was observed between padding methods and also the number of
padding points included. Tables 1 and 2 summarize the mean
residuals observed for the start and end point deviations from
the actual acceleration in those regions. The starting region
was an area of small negative angular acceleration. The
residuals observed were similar for the reflection and the
digitize padding point procedures, while the linear extrapolation and duplicate end point methods were consistently
more deviant from actual accelerations. The end region was
an area of large negative acceleration. The large negative
peak acceleration was consistently attenuated by each
filtering procedure. The curves returned to closer proximity
near the end point. Values in Table 2 suggest a pattern
somewhat similar to that observed in the beginning region.
The reflection and digitize padding points methods were
similar in approximating the actual acceleration; however,
the linear extrapolation method was as good an estimator of
this acceleration condition.
The relationship of mean residual to the number of
padding points for each padding method is illustrated in
Technical Note
ANGULAR
ACCELERATION
969
ANGULAR
VS TIME
-100
i
0.5
1.0
ACCELERATION
ANGULAR
VS TIME
ACCELERATION
2.0
ACCELERATION
VS TIME
nsrz (SOC)
(4
ANGULAR
1.5
VS TIME
rlME(sec,
@I
ANGULAR
ACCELERATION
VS TIME
Fig. 4. Comparison of angular accelerations under various padding point conditions with the actual
acceleration. (a) No padding points;(b) duplicating end point (10 points);(c) linear extrapolation (10 points);
(d) reflection (10 points) and (e) reflection (30 points).
Fig. 5. The curves reach a plateau beyond 10 padding points,
with little improvement exhibited.
DISCUSSION
The data values used in creating padding points around a
data set for digital filtering purposes have typically been
assumed to derive from actual measurements. In some cases
such as film digitizing, the extra digitizing required may
substantially increase the data collection time. Furthermore,
allowing for the required padding points in field of view may
often result in smaller image size from which to digitize and a
resulting decrease in coordinate precision. Thus, artificial
means of generating padding points which would not
severely distort the smoothed data would be desirable.
970
Technical Note
Table 1
I
MEAN RESIDUALS
(ACTUAL
POINTS:
DUPLICATE
LINEAR
ACCELERATION):
NUMBER OF PADDING POINTS:
0
5
10
15
PADDING
METHOD:
DIGITIZE
- CALCULATED
1ST POINT:
EXTRAPOLATION:
REFLECTION:
FRAMES
1 - 10
20
26
30
9.4
2.3
2.4
2.2
2.2
2.3
2.3
9.4
6.3
6.3
6.3
6.3
6.3
6.3
9.4
3.6
4.5
4.2
4.2
4.2
4.2
9.4
2.4
2.2
2.2
2.2
2.2
2.2
Table 2
MEAN RESIDUALS
(ACTUAL
PADDING
METHOD:
DIGITIZE
POINTS:
DUPLICATE
LINEAR
1 ST POINT:
EXTRAPOLATION:
REFLECTION:
- CALCULATED
ACCELERATION):
NUMBER OF PADDING POINTS:
0
5
10
16
FRAMES 71 - 60
20
25
30
17.4
9.5
7.0
7.2
7.1
7.1
7.1
17.4
11.7
10.7
10.7
10.7
10.7
10.7
17.4
6.7
7.0
6.9
6.6
6.9
6.9
17.4
10.7
7.3
6.6
6.9
7.0
7.0
Each of the methods described above was superior to not
padding the filter at all. The simplest method (duplication of
the end point) was the least effective procedure tested. Both of
the extrapolation procedures (linear and reflection) resulted
in acceleration curves closely approximating those obtained
from actually digitizing data. However, the reflection method
produced slightly better end point accelerations under the
conditions of the test. The ease with which either technique
could be implemented within a computer program provides
little discrimination between methods. The reflection method
is perhaps a more aesthetically appealing choice. Its use is
somewhat reminiscent of the techniques involved in Fourier
analysis of functions over a finite time interval: The
function is simply extended through a point order reversal,
creating a periodic waveform in the process. In this case a
slightly more complex reflection is involved. The reflection
method is perhaps more justifiable on theoretical grounds as
well. The linear extrapolation and duplication methods
create regions of zero acceleration over the padding points. It
would be expected that accelerations near the boundary of
tHe data set would be influenced by the padding point
derivatives even though the padding point values were not
used in computing the accelerations. On the other hand, the
padding point derivatives for the reflection method would
more nearly match the actual boundary accelerations and
thus affect them minimally.
It is clear that the number of padding points used need not
be extended beyond 20 points. If digitized data are to be used
for padding, 10 points are probably sufficient. The effort
involved in additional digitizing may not improve the accelerations obtained from the smoothed data. With artificially
generated padding points there is a marginal improvement in
acceleration estimation in going to 15 or 20 points of
padding. In either case, there is no real penalty for extending
the padding further; the computer processing time for the
additional points is usually negligible.
The acceleration curves from the Pezzack et al. (1977) data
suggest that the end point values are influenced by the nature
of the underlying acceleration function. When the acceleration is changing rapidly it appears that a good approximation to the actual acceleration is likely to result from the
filtering/differentiating procedure. When the actual acceleration is fluctuating through minor oscillations, the calculated
acceleration may not closely follow it, particularly near the
end points. Thus some care is advisable in using derived
accelerations for the further prediction of kinetic variables,
particularly near the boundary values of the digitized
interval.
CONCLUSIONS
The calculation of first and second order derivatives from
digital data is a process which selectively amplifies the high
frequency noise components over the lower frequency signal
components characteristic of biomechanical data. To mini-
Technical Note
MEAN
04
0
(4
RESIDUALS
OF ACCELERATION:
FRAMES
1 - 10
I
10
30
20
NUMBER OF PADDIND POINT3
MEAN
RESIDUALS
OF ACCELERATION:
10
FRAMES
2'0
71-
.
DUPLCATE
.
UN-
.
REREClm4
80
971
mize the et&t of the noise components, smoothing routines
have been devised which filter out the high frequency noise.
One of the most easily implementable of these filtering
procedures is the Butterworth digital filter: a recursive
algorithm which uses both raw and smooth data values to
determine subsequent smoothed values. The use of this
digital tilter is dependent on ‘padding’ a data set with extra
points at both the beginning and end of the data. These
padding data are used to get the fitter running smoothly
before getting to the data in the region of interest. The use of
actual raw data for these padding points produces satisfactory filtering results but requires a significant additional
effort in measurement. Several artificial means were created
for producing padding points. The most effective of these
methods (reflection around the boundary point) resulted in
acceleration curves which closely matched those produced
with digitized padding data. The number of padding points
was found to affect the acceleration curves derived from the
smoothed data. Between 10 and 20 padding points were
found to be sufficient to produce a smooth transition from
padding to real data points with good approximation to
actual accelerations.
REFERENCES
3’0
NUMBER OF PADDING POINT3
Fig. 5. Mean residuals for the beginning (a) and ending
(b)portions of the 80 point region analyzed. The mean
residual was the average difference between acceleration
calculated and actual acceleration (for the first/last
10 points).
Gold, B. and Rader, C. M. (1969) Digital Processing of
Signals. McGraw-Hill, New York.
Hatze, H. (1981) The use of optimally regularized Fourier
series for estimating higher-order derivatives of noisy
biomechanical data. J. Biomechanics 14, 13-18.
Lanshammer, H. (1982) On practical evaluation of differentiation techniques for human gait analysis. J. Biomechanics 15, 99-105.
Miller, D. I. and Nelson, R. C. (1976) Biomechanics of Sport.
Lea 8r Febiger, Philadelphia.
Perzack, J. C., Norman, R. W. and Winter, D. A. (1977) On
assessment of derivative determining techniques used for
motion analysis. J. Biomechanics 10, 377-382.
Winter, D. A. (1979) Biomechanics ofHuman Mooement. John
Wiley, New York.
Wood, G. A. (1982) Data smoothing and differentiation
procedures in biomechanics. Exercise and Sport Science
Reviews (Edited by Terjung, R. L.), Vol. 10, pp. 308-362.
Franklin Institute Press, Philadelphia, PA.