Fourier Coefficient Affine Superimposition

Green Open Access

Posted by Falk Mielke on 12th Jun 2021

Just before christmas 2018, I was a bit stuck with my research: I had analyzed kinematics, but only piglets. I had seen a talk about the work of Lees et al. (2016) at the SEB conference 2017. And so I sat down, coded a bit, and started tracking youtube videos.

Out came a new method to analyze cyclic joint angle profiles and a publication in the Zoological Journal of the Linnean Society (Mielke et al., 2019) together with my supervisors Chris and Peter. We called the method Fourier Coefficient Affine Superimposition. By analyzing the broad range of ungulates, I was able to give a phylogenetic frame to my study of piglets, and better anticipate what to expect in my own videos.

Yet in my opinion, the hidden gem of the paper is an equation in the Appendix (eqn. 6) which follows from the “delay theorem” of Fourier theory. This equation can be used to normalize the phase of a periodic signal. Thereby, I was able to align the joint profiles - which is convantionally done by superimposing footfall times - in a mathematically optimal way. Likewise, the other affine signal components (mean and amplitude) can be removed, and I argue that this makes signals dynamically similar sensu Alexander and Jayes (1983). More explanation can be found in my presentation at DZG annual meeting 2019.

In the meantime, the period for green open access has passed since online publication, and so I dug out the document version which went to final submission.

Here it is:

A master student recently applied the method to data from wild and domestic pigs, and we could identify the key differences in their forelimb kinematics. It turned out that the most pronounced changes appear in the elbow, which was also the focus of my publication above. I hope we will report more on that soon, but until then the code is already available here:

Of course I would like to see more researchers explore the possibilities of analyzing periodic phenomena in the frequency domain. An over the past year, I also received three requests by researchers to who were unable to get past the paywall which guards my article. I hope I could help that now, and I welcome further requests and discussions!

Enjoy reading!


  • Lees J, Gardiner J, Usherwood J, Nudds R. 2016. Locomotor preferences in terrestrial vertebrates: An online crowdsourcing approach to data collection. Scientific Reports, 6:28825.
  • Mielke F, Van Ginneken C, Aerts P. 2019. Quantifying intralimb coordination of terrestrial ungulates with Fourier coefficient affine superimposition. Zoological Journal of the Linnean Society, Volume 189, Issue 3, July 2020, Pages 1067–1083, doi:10.1093/zoolinnean/zlz135
  • Alexander RM, Jayes AS. 1983. A dynamic similarity hypothesis for the gaits of quadrupedal mammals. Journal of Zoology, 201(1):135–152.