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Surface Representation and Morphometric Analysis Based on Discrete Cosine Transform

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Abstract

Fourier series are usually employed to describe closed or open, 2D or 3D outlines of biological samples. Landmark-based morphometric methods are widely used in the analysis of 3D surfaces. There are few investigations on the representation and morphometric analysis of 3D biological sample surfaces with methods relating to Fourier series. In this paper, we firstly extend discrete cosine transform (DCT), a Fourier-related method classically used to describe 2D open curves, but here to 3D surfaces. Surfaces are transformed into 3D curves with a path connecting all points. The path can be determined manually by an analyst or by algorithms. Before being represented with DCT, non-shape effects should be eliminated. A strategy to improve the selection of coefficients to approximate surfaces is also presented. As a result, the mathematical homology of the coefficients is preserved while fast convergence of the approximation is ensured. Three 3D surface examples are transformed into 3D curves and represented with DCT. The first example is four groups of 120 simulated surfaces generated with equations, and the other two examples are 3D surfaces extracted from aligned 3D human skulls with four types of diagnoses of coronal craniosynostosis. Principal component analysis, one-way analysis of similarity, and one-way permutational multivariate analysis of variance are utilized to analyze the coefficients obtained. The results of statistical analyses suggest that DCT is an effective and stable tool in describing 3D surfaces.

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Data Availability

The data and code that support this study are available in the supplementary materials.

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Acknowledgements

We want to acknowledge Emeric Gioan for sharing the 3D skull data of different diagnoses. This work is supported by the National Natural Science Foundation of China under Grant No. 51805080.

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Authors and Affiliations

  1. School of Mechanical Engineering and Jiangsu Key Laboratory for Design and Manufacture of Micro/Nano Biomedical Instruments, School of Mechanical Engineering, Southeast University, Nanjing, 211189, Jiangsu, China

    Bingjue Li & Heng Nie

  2. OPPO Mobile Telecommunications Co., Ltd., No. 158, Tianfu Fourth Street, Chengdu High-Tech Zone, Chengdu, 610000, Sichuan, China

    Shengmin Zhou

Authors
  1. Bingjue Li
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  2. Shengmin Zhou
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  3. Heng Nie
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Contributions

BL: conceptualization, methodology, validation, resources, writing—original draft, writing—review & editing, visualization, supervision, funding acquisition. SZ: methodology, software, investigation, writing—original draft, writing—review & editing, visualization. HN: software, investigation, data curation, writing—original draft, writing—review & editing, visualization.

Corresponding authors

Correspondence to Bingjue Li or Shengmin Zhou.

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The authors have no conflicts of interest on this research.

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Appendices

Appendix

Nomenclature

This appendix presents the definition of the major terms/notation used in this paper and their equivalents in (Zhou et al., 2021).

Notation

Equivalent in Zhou et al. (2021)

Definition

c[k]

w(k)

Scaling factor of cosine polynomial

CM

C

Matrix of coefficients generated from DCT

m

m

Number of harmonics with large amplitudes selected for surface reconstruction

N

N

Number of points that define a surface

n

 

Number of samples/surfaces

p

 

Number of harmonics whose amplitudes account for at least 99% (or other appropriate value) of the total signal energy for a surface

re

E

Mean residual between the fitting points and the original ones

q

 

Number of the sorted harmonics with larger amplitudes which contribute at least 99.5% (or other appropriate value) of the signal energy corresponding to the p selected harmonics for a surface

S[k]

S(k)

Spectrum of input signal

s[n]

s(n)

Input signal

SI

 

Union of the sequential index of the m selected harmonics for all surfaces

SIj

 

Sequential index of the selected harmonics for the jth surface

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Li, B., Zhou, S. & Nie, H. Surface Representation and Morphometric Analysis Based on Discrete Cosine Transform. Evol Biol 49, 102–122 (2022). https://doi.org/10.1007/s11692-021-09558-6

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