Application of harmonized elliptic Fourier transform coefficients for comparing the shapes of biological structures (on the example of the attachment organs of monogenea)

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Abstract

Elliptic Fourier transform is a common method of describing the shape of objects by an unique sequence of coefficients that allow comparing the shapes by mathematical methods. However, raw coefficients contain unnecessary data unrelated to the shape, which does not provide a correct comparison. For this reason the coefficients are normalised. This removes some of the superfluous data, but leaves information about mirror symmetry and the order in which the contour vertices are declared, that are encoded in the signs of the coefficients. This also interfere with shape comparison. The paper describes an algorithm for harmonizing the coefficients, leveling the influence of the mentioned information. On the example of attachment organs of monogeneas, the advantages of using harmonized coefficients for comparing the shapes of biological structures are shown.

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A. M. Lyakh

A.O. Kovalevsky Institute of Biology of the Southern Seas of RAS

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Email: me@antonlyakh.ru
Russian Federation, Nakhimov av., 2, Sevastopol, 299011

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Supplementary files

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2. Fig. 1. Contours of the attachment organs of monogenes. The fill indicates the order of traversing the vertices of the contours (orientation of the contours): The points of a shaded contour go counterclockwise, and those of an unpainted one go clockwise.

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3. Fig. 2. Contours of the attachment organs of species of the genus Ligophorus used in the work. The contours are aligned vertically and adjusted to the same size using the normalization procedure, and using the matching algorithm described in the article, they are geometrically consistent with each other.

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4. Fig. 3. The values of 60 coefficients of the first 15 harmonics describing the shape of the four attachment hooks. It can be seen that with an increase in the harmonic number, the values of the normalized EPF coefficients decrease rapidly.

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5. Fig. 4. Rules for replacing signs and the action of matching operators: a – Ø – zero operator that does not affect the contour, • – operator for shifting the starting point of the contour to the opposite part, ϲ – operator for inverting the direction of traversal, ― and | – mirror reflection operators relative to the horizontal and vertical axes; b – rules for replacing signs the rule is written for the first two harmonics (eight coefficients), but it should be applied for each pair of harmonics, starting with the first; the following notation is used in the recording: × – inversion of the sign, × – the sign remains the same; c, d – masks for searching for operators corresponding to the sign difference of the matched contours, with vertical (c) and horizontal (d) alignment of contours; e – the result of the action of operators on contours.

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6. Fig. 5. Comparison using tanglegrams and the Baker gamma association index (the number in the middle of the tanglegram) of dendrograms constructed using normalized EPF coefficients of three sets of contours – A, B and C.

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7. Fig. 6. The matching algorithm automatically transformed chaotically arranged contours into ordered ones. In rare cases, the position of some contours (circled in red) differed from the rest, but this did not prevent using their numerical descriptions to compare shapes.

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8. Fig. 7. Comparison of dendrograms constructed according to agreed (cont.) and absolute (abs.) the values of the coefficients; the numbers show the values of the Baker gamma Association index.

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9. Fig. 8. The average shapes (black) of the attachment structures of representatives of five species of the genus Ligophorus (dark red), constructed by averaging the values of the agreed EPF coefficients.

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10. Fig. 9. A tree of medium shapes of monogenean attachment hooks, which can be used to identify shapes. Only two levels of the tree are shown. At the first level of the hierarchy, the average forms are constructed by averaging the agreed EPF coefficients belonging to one cluster; at the second level, by averaging the coefficients belonging to neighboring clusters.e

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