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Morphological quantification of outlines
250
Morphological
Biology of the Cell
quantification
(1998190, 247-290
of outlines
Mathieu Schmittbuhl Fbdiration de Recherche Odontologique, RSERM U 424, 11, rue Humann, F-67085 Strasbourg, France, and lnstitut de Gkologie - Centre de Gkochimie de la Surface, Universitg Louis Pasteur, f-67084 Strasbourg, France Morphological quantification of outlines is an indispensable prerequisite for characterization and comparison of biological structures. Among the multivariate morphometric methods used to quantify shapes, Fourier methods constitute an useful approach when a few or no landmarks are available on the outline. The expansion in Fourier series of a simple outline is the subject of new development in order to decompose the shape in a series of simple geometrical concepts (Schmittbuhl M et al. (1998) American Journal of Physical Anthropology, in press). The shape of simple outlines is quantified by an ordered series of harmonics ; each harmonic being characterized by an amplitude and a phase. Amplitude and phase correspond to the Fourier descriptors, and provide the direct representation of the geometrical contribution of the Fourier harmonic. This geometrical interpretation is then used to quantify the more or less marked elongation (2nd harmonic), triangularity (3rd harmonic), quadrangularity (4th harmonic), and n-angularity (nth harmonic) of the shape studied. A new methodological approach of Fourier analysis allows to compare geometrically and quantitatively simple shapes, in order to provide a direct geometrical interpretation of morphological differences which would have been difficult to give using the Fourier coefficients classically used. The shape of complex outlines can be quantified by a more elaborated method called the elliptical Fourier analysis. This elliptical Fourier analysis has been rarely applied at this present time, probably
because of its mathematical complexity and difficulty to translate the Fourier coefficients into simple geometrical concepts. From the theoretical existing development, a new geometrical approach of the Fourier development of a complex outline is proposed in order to define new parameters for which the connection with the geometry is direct (Schmittbuhl M et al. (1995) In Infonnatique et Imagerie Medicale (eds A. Wackenheim & G. Zdllner), pp. 85-99. Paris : Masson). It can be demonstrated that the geometrical locus of the points associated to each harmonic used in the decomposition of the Fourier series is an ellipse. The contribution of each harmonic is then characterized by four geometrical parameters corresponding respectively, for each ellipse, to its size,its shape, its orientation, and to the ordination of the points on the ellipse. The quantification of a complex shape then consists in providing the list of the elliptical descriptors characteristic of each Fourier harmonic ; these elliptical descriptors allowing a direct geometrical interpretation of the contributions of the Fourier harmonics. From the list of the elliptical descriptors, the outline can be reconstructed whatever its morphological complexity: A new methodological approach allows to compare geometrically and quantitatively the shape of the outlines using elliptical descriptors, in order to provides a direct geometrical interpretation of morphological differences from the comparison, harmonic by harmonic, of the elliptical descriptors of same order.
Morphological
Biology of the Cell
quantification
(1998190, 247-290
of outlines
Mathieu Schmittbuhl Fbdiration de Recherche Odontologique, RSERM U 424, 11, rue Humann, F-67085 Strasbourg, France, and lnstitut de Gkologie - Centre de Gkochimie de la Surface, Universitg Louis Pasteur, f-67084 Strasbourg, France Morphological quantification of outlines is an indispensable prerequisite for characterization and comparison of biological structures. Among the multivariate morphometric methods used to quantify shapes, Fourier methods constitute an useful approach when a few or no landmarks are available on the outline. The expansion in Fourier series of a simple outline is the subject of new development in order to decompose the shape in a series of simple geometrical concepts (Schmittbuhl M et al. (1998) American Journal of Physical Anthropology, in press). The shape of simple outlines is quantified by an ordered series of harmonics ; each harmonic being characterized by an amplitude and a phase. Amplitude and phase correspond to the Fourier descriptors, and provide the direct representation of the geometrical contribution of the Fourier harmonic. This geometrical interpretation is then used to quantify the more or less marked elongation (2nd harmonic), triangularity (3rd harmonic), quadrangularity (4th harmonic), and n-angularity (nth harmonic) of the shape studied. A new methodological approach of Fourier analysis allows to compare geometrically and quantitatively simple shapes, in order to provide a direct geometrical interpretation of morphological differences which would have been difficult to give using the Fourier coefficients classically used. The shape of complex outlines can be quantified by a more elaborated method called the elliptical Fourier analysis. This elliptical Fourier analysis has been rarely applied at this present time, probably
because of its mathematical complexity and difficulty to translate the Fourier coefficients into simple geometrical concepts. From the theoretical existing development, a new geometrical approach of the Fourier development of a complex outline is proposed in order to define new parameters for which the connection with the geometry is direct (Schmittbuhl M et al. (1995) In Infonnatique et Imagerie Medicale (eds A. Wackenheim & G. Zdllner), pp. 85-99. Paris : Masson). It can be demonstrated that the geometrical locus of the points associated to each harmonic used in the decomposition of the Fourier series is an ellipse. The contribution of each harmonic is then characterized by four geometrical parameters corresponding respectively, for each ellipse, to its size,its shape, its orientation, and to the ordination of the points on the ellipse. The quantification of a complex shape then consists in providing the list of the elliptical descriptors characteristic of each Fourier harmonic ; these elliptical descriptors allowing a direct geometrical interpretation of the contributions of the Fourier harmonics. From the list of the elliptical descriptors, the outline can be reconstructed whatever its morphological complexity: A new methodological approach allows to compare geometrically and quantitatively the shape of the outlines using elliptical descriptors, in order to provides a direct geometrical interpretation of morphological differences from the comparison, harmonic by harmonic, of the elliptical descriptors of same order.