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

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