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Herbert Hatze
05-28-1997, 02:44 AM
RE-INVENTING THE WHEEL IN BIOMECHANICAL DATA SMOOTHING
AND DERIVATIVE COMPUTATION


The recent summary posting on filtering techniques to be applied to
noisy data sequences of a running analysis has shown once again the
wide-spread and surprising unawareness of the existence of techniques
available to solve this problem. The methods suggested in the summary
ranged from high-order (collocating ?) polynomial fitting (Zlatnik)
to various techniques for estimating the cut-off frequency for a
low-pass Butterworth filter, empiric FFT-analysis of the signal,
visual signal inspection, and other techniques.

As has been pointed out correctly by Yu and Orendurff respectively,
Dave Winter's residual analysis for determining the cut-off frequency
is not an optimal procedure and may, in fact, lead to substantial
over-smoothing of the data with potentially unpleasant effects.

The subject of filtering optimally noise-contaminated biomechanical
data sequences, and of computing their optimally filtered first and
second time derivatives, is not a trivial one. Indeed, it partly
(numerical differentiation) belongs to the class of so-called
INCORRECTLY POSED PROBLEMS (Hatze, H. (1990)
Data Conditioning and Differentiation Techniques. In:
Biomechanics of Human Movement (N. Berme, A. Cappozzo, ed.),
pp. 237-248, Bertec, Worthington). These do not, in general,
possess unique solutions but may be converted into well-posed
problems by a procedure called REGULARIZATION. For biomechanical
data sequences, an algorithm implementing OPTIMAL REGULARIZATION
for Fourier series was devised by me in 1981 (Hatze, H. (1981) The use of optimally
regularized Fourier series for estimating higher-order derivatives of
noisy biomechanical data. J. Biomechanics 14, 13-18). This algorithm
(and the associated computer program) performs a complete analysis of the periodogram of
the signal and, based on this analysis, selects the OPTIMAL FILTERING
WINDOW (CUT-OFF FREQUENCY) automatically, for the signal itself
(smoothing), as well as for each of the derivatives.(It should be
noted though that optimally filtered derivatives can, in general, NOT be
obtained by simply differentiating the smoothed signal).

Soon after the publication of this method, Lanshammar (1982, J.
Biomech. 15, 459-470) performed a precision analysis of numerical
derivatives, while later Woltring (1985, Human Movement
Sci. 4, 229-245) used the work of Golub et al. (1979, Technometrics
21, 215-223) on Generalized Cross-Validation (a type of
regularization procedure) and the optimal spline smoothing technique
of Utreras (1981, SIAM J. Sci. Stat. Comp. 2, 349-362) to compile his
Fortran package for generalized cross-validatory spline smoothing and
differentiation, which is available on the net.

Both techniques, my method of optimally filtered Fourier series (now
improved by eliminating boundary effects) and Woltring's generalized
cross-validatory splines in essence yield an OPTIMAL CUT-OFF
FREQUENCY (or rather an optimal low-pass filter window) for smoothing
a noisy signal and obtaining its derivatives. So there is no need to
"guess" or otherwise "estimate" the cut-off frequency. For periodic
data such as result from gait, running, or other human motion
analysis, the Fourier series method is to be preferred.

Considering that these basic developments, including the underlying
theory, were completed by Woltring and myself in the 1980's, it is
most astonishing to observe the same fundamental questions being
asked again and again, mostly by students. Perhaps, there are certain
deficiencies in the teaching of biomechanics?

H. Hatze, Ph.D.
Professor of Biomechanics
University of Vienna
Auf der Schmelz 6
A-1150 WIEN
AUSTRIA