Smoothing filter
Butterworth low-pass
The convention in human-movement research
A low-pass filter with the flattest possible passband, run forwards and then backwards so it introduces no lag. If a paper you are reading filtered its kinematics, this is almost certainly what it used.
- −3 dB point
- Lands on the stated cutoff
- Phase lag
- 0 frames
- Peak flattening
- 0.00°
- Single-frame spike
- 66° survives
Measured on our own test signals at 24fps: a clean sine wave, that wave with ±4° of random noise, and that wave with one frame knocked 108° out of place. These describe how the filter behaves — they are not a validation of markerless measurement against marker-based capture.
How it works
A low-pass filter keeps slow changes and removes fast ones. Real joint movement is slow — a knee does not change direction thirty times a second — while tracking jitter is fast, so a cutoff placed between the two keeps the movement and drops the noise.
The Butterworth is the low-pass whose passband is as flat as a filter of its order can be, which means it does not distort the amplitudes of what it keeps. That is the property it is named for and the reason it became the default.
Running it once would shift the whole signal later in time. Running it a second time backwards shifts it back by exactly the same amount, so the result is zero-lag and a peak stays on the frame it happened on.
Reach for it when
- You want the thing the literature does, so your numbers are comparable with published work.
- The noise is general jitter across the whole clip rather than a few wild frames.
- You care about the peaks — this changed the range of a clean test signal by 0.00°.
What it costs
It needs to know the frame rate, because a cutoff is measured in hertz. It also cannot help with a single wild frame: a spike is spread across its neighbours rather than removed.
What we actually run
So the page and the code say the same thing.
- 2nd-order, applied forwards and backwards, which gives 4th-order attenuation with no phase shift.
- Coefficients from the bilinear transform with pre-warping, so the cutoff lands where it is asked for rather than drifting with the frame rate.
- The cutoff is raised by 1/0.802 before use — the standard correction for a dual pass. Without it, filtering twice cuts well below the frequency you asked for.
- Default 6 Hz, the conventional choice for whole-body movement; 9 Hz and 3 Hz at the other two strengths, each held to a third of the frame rate so the setting still means something on a 24fps clip.
- Odd reflection padding at both ends, so the filter starts from a continuation of the signal rather than from a step at frame zero.
The research behind it
Butterworth, S. (1930). On the theory of filter amplifiers. Experimental Wireless and the Wireless Engineer, 7, 536–541.
The original: a filter whose passband is as flat as a filter of that order can be, which is the property the whole family is named for.
Winter, D. A., Sidwall, H. G., & Hobson, D. A. (1974). Measurement and reduction of noise in kinematics of locomotion. Journal of Biomechanics, 7(2), 157–159.
The paper that established low-pass filtering of human kinematics, and the residual analysis still used to choose a cutoff.
doi:10.1016/0021-9290(74)90056-6
Winter, D. A. (2009). Biomechanics and Motor Control of Human Movement (4th ed.). John Wiley & Sons.
The standard text. Chapter 3 sets out the forwards-and-backwards procedure and the conventional cutoffs for whole-body movement.
doi:10.1002/9780470549148
Robertson, D. G. E., & Dowling, J. J. (2003). Design and responses of Butterworth and critically damped digital filters. Journal of Electromyography and Kinesiology, 13(6), 569–573.
Where the correction for running a filter twice comes from — without it a dual pass cuts well below the cutoff you asked for.
doi:10.1016/S1050-6411(03)00080-4
Challis, J. H. (1999). A procedure for the automatic determination of filter cutoff frequency for the processing of biomechanical data. Journal of Applied Biomechanics, 15(3), 303–317.
On choosing the cutoff from the data instead of by convention — worth reading before trusting any default, ours included.
doi:10.1123/jab.15.3.303