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Elliptical Fourier analysis: fundamentals, applications, and value for forensic anthropology

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Abstract

The numerical description of skeletal morphology enables forensic anthropologists to conduct objective, reproducible, and structured tests, with the added capability of verifying morphoscopic-based analyses. One technique that permits comprehensive quantification of outline shape is elliptical Fourier analysis. This curve fitting technique allows a form’s outline to be approximated via the sum of multiple sine and cosine waves, permitting the profile perimeter of an object to be described in a dense (continuous) manner at a user-defined level of precision. A large amount of shape information (the entire perimeter) can thereby be collected in contrast to other methods relying on sparsely located landmarks where information falling in between the landmarks fails to be acquired. First published in 1982, elliptical Fourier analysis employment in forensic anthropology from 2000 onwards reflects a slow uptake despite large computing power that makes its calculations easy to conduct. Without hurdles arising from calculation speed or quantity, the slow uptake may partly reside with the underlying mathematics that on first glance is extensive and potentially intimidating. In this paper, we aim to bridge this gap by pictorially illustrating how elliptical Fourier harmonics work in a simple step-by-step visual fashion to facilitate universal understanding and as geared towards increased use in forensic anthropology. We additionally provide a short review of the method’s utility for osteology, a summary of past uses in forensic anthropology, and software options for calculations that largely save the user the trouble of coding customized routines.

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Authors and Affiliations

  1. Laboratory for Human Craniofacial and Skeletal Identification (HuCS-ID Lab), School of Biomedical Sciences, The University of Queensland, Brisbane, QLD, 4072, Australia

    Jodi Caple & Carl N. Stephan

  2. Defense Prisoner of War/Missing in Action Accounting Agency, 590 Moffet St., Building 4077, Joint Base Pearl Harbor-Hickam, Oahu, HI, 96853, USA

    John Byrd

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  1. Jodi Caple
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  2. John Byrd
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Correspondence to Jodi Caple.

Electronic supplementary material

Animation depicting how the harmonics relate in an epicyclic manner in Fourier outline analysis using three harmonics. A comparison between traditional a) and elliptical b) Fourier analysis methods. Outline extracted and reconstructed using 500 equally spaced semi-landmarks. Bold line indicates reconstructed outline, dotted lines are phasors. Images created using R [44] (MP4 3022 kb)

Animation depicting interlinked movement of the harmonics in elliptical Fourier outline analysis using three harmonics. Outline extracted and reconstructed using 1000 equally spaced semi-landmarks. Bold line indicates reconstructed outline, dotted lines are phasors. Images created using R [44] (MP4 3531 kb)

Animation depicting interlinked movement of the harmonics in elliptical Fourier outline analysis using 40 harmonics. Outline extracted and reconstructed using 1000 equally spaced semi-landmarks. Bold line indicates reconstructed outline, dotted lines are phasors. Images created using R [44] (MP4 3872 kb)

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Caple, J., Byrd, J. & Stephan, C.N. Elliptical Fourier analysis: fundamentals, applications, and value for forensic anthropology. Int J Legal Med 131, 1675–1690 (2017). https://doi.org/10.1007/s00414-017-1555-0

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