Elliptic Fourier Descriptors Research Articles

Description of the morphology of roots of Chicorium intybus L. partim by means of image analysis: Comparison of elliptic Fourier descriptors and classical parameters

In the framework of DUS testing (distinctness, uniformity and stability) of new varieties of Chicorium intybus L. partim, experts are manually measuring or scoring the shape of the roots. This way of working has several disadvantages in comparison to image analysis techniques. In this study two methods based on image analysis are statistically compared and evaluated: elliptic Fourier descriptors and classical parameters. From the study, we can conclude that the former can be used to analyze and explore the variance available, whereas the latter can be used to improve interpretability and the accessibility of the data. As such, both approaches are complementary. Also new characteristics for DUS testing are suggested.

Reslicing axially sampled 3D shapes using elliptic Fourier descriptors

We propose a new method that interpolates between parallel slices from a 3D shape for the purposes of reslicing and putting into correspondence organ shapes acquired from volumetric medical imagery. By interpolating the coefficients of elliptic Fourier descriptors for a set of parallel contours, a new set of slices can be directly generated at desired axial locations. Neither an explicit correspondence between points on adjacent contours nor a 3D interpolating surface needs to be obtained. We apply the proposed reslicing method to experimental datasets of both synthetic 3D shapes and real prostate contours, and demonstrate that it performs as well as a common method based on variational implicit surfaces, for a much lower computational cost. We also show that reslicing and putting into correspondence an ensemble of axially sampled 3D organs enables the construction of shape models for accurate 3D segmentation.

Characterization of local populations of the common sole Solea solea (Pisces, Soleidae) in the NW Mediterranean through otolith morphometrics and shape analysis

Shape analyses were carried out on otoliths of the common sole, Solea solea (Linnaeus, 1758), in order to discriminate local populations in the North-Western Mediterranean Sea. Samples were collected in various environments like coastal lagoons, the outlet of the Rhone River as well as other marine sites. Morphological analyses highlighted a significant asymmetry between the left and right otoliths. This character could be acquired during or shortly after settlement on soft-bottoms when the individuals really become flatfishes. The otolith shape was described by seven harmonics from elliptic Fourier descriptors and by five indices of shape (coefficient of form, roundness, circularity, rectangularity, and ellipticity). The existence of several local populations of common sole in the NW Mediterranean was demonstrated. In particular, discriminant analyses highlighted significant differences in otolith shape according both to fish size and to the types of environment in which the fish live, i.e. coastal lagoons vs. marine sites, but also between sites belonging to the same type (lagoons, and marine sites). The differences in shape could be linked (1) to the particular environmental conditions of each site, and (2) to changes in metabolic and/or physiological conditions according to the stage of development of the fish which most likely influences the otolith growth.

Quantitative evaluation of characteristics of an auditorium shape in horizontal cross section

The room shape of an auditorium is often classified as a ‘‘shoe-box type,’’ ‘‘fan-shape type,’’ ‘‘round type,’’ ‘‘surrounding (vineyard) type,’’ or other shape. The surface diffusivity index (SDI), which is an evaluation index of surface irregularities, has been proposed by Haan and Fricke, but the SDI is based on visual surveys of photographs or drawings [Appl. Acoust. 51(1), 53–69 (1997)]. Clarification of the relationship between an auditorium’s shape and its acoustic characteristics would be simplified if the auditorium shape were understood objectively rather than subjectively. Therefore, in this study, we tried to objectively obtain characteristics of a room shape in horizontal cross section using an elliptic Fourier descriptor, which can detect the periodicity of a two-dimensional contour. Room shapes were expressed as spatial frequency characteristics calculated from their Fourier descriptors. Based on them, we can understand the amplitude of irregularities and their scale. In addition, by using principal component analysis, it is possible to understand, quantitatively, the main differences of room shapes in horizontal cross section.

Pipe eccentricity measurement using laser triangulation

A pipe eccentricity measurement system based on laser triangulation is presented. The custom measurement head implements the Scheimpflug condition to ensure maximum working depth and integrates an FPGA to perform real time image processing at a rate of 200 frames per second. The eccentricity of the pipe is estimated by approximating the position of the upper tangent point of the pipe and used as a measure of its straightness. The correlation coefficients for the parameters r, x 0 and y 0 estimating from a least square approximation of a circle via the algebraic distance are theoretically derived, demonstrated using Monte-Carlo simulation and experimentally verified. It is shown that perturbation of the measurement, leads to less uncertainty in the estimated position of the tangent point than for the centre point. Harmonic filtering based on Elliptical Fourier Descriptors is used to filter the measurement data. Laboratory measurements show that the repeatability is in the order of 1 μm. The accuracy of industrial measurements exhibits a systematic bias of −4 μm and has a standard deviation of 2.6 μm. Selected measurements from the production plant are presented to show that the deviation of the cross-section of the pipe from a perfect circle can be determined from the measured radius of curvature.

Genetic Combining Ability of Petal Shape in Garden Pansy (Viola × wittrockiana Gams) based on Image Analysis

Flower appearance is an important target for improvement in garden pansy (Viola × wittrockiana Gams). Flowers of this species consist of three petal types: inferior, lateral, and superior. By means of principal-components (PC) analysis of elliptic Fourier descriptors (EFDs), we estimated the general and specific combining abilities (GCA and SCA) of the floral characteristics of F1 progeny derived from diallel crosses of four inbred lines. The greatest variation in petal shape was explained by the aspect ratio (the first PC) in each petal type. The second or third PC of each petal type was associated with the curvatures of the various parts of the petal. The highly significant GCA effects indicate the importance of additive genetic variance in the transmission of parental petal characteristics to the progeny. The fact that the SCA mean squares were not significant for aspect ratio and petal area indicates that these characteristics of a single-cross progeny can be sufficiently predicted on the basis of GCA. Significant SCA effects were observed in the curvatures of the distal and proximal parts of lateral and superior petals. Correlation analyses indicated several associations between the shape elements of lateral and superior petals, suggesting that genes for these shape elements may be associated, linked, or pleiotropic in the parents used in this study. We successfully demonstrated the use of EFD–PCA to evaluate the petal shape of garden pansy, and of analyzing combining ability and correlations based on the PC scores of EFDs.

Quantitative Evaluation of Petal Shape and Picotee Color Pattern in Lisianthus by Image Analysis

Petal shape and picotee colour pattern of lisianthus [Eustoma grandiflorum (Raf.) Shinn.] were qualitatively evaluated by means of personal computer-based methods. In lisianthus, many cultivars have been improved to obtain various floral characteristics. Picotee color patterns and flower shape are commercially important in this species and the availability of an objective and quantitative evaluation method is of vital importance for investigations related to the genetic and physiological aspects of these characteristics. Our objectives were to evaluate petal shape variation quantitatively and to establish a new quantitative evaluation method for picotee color patterns. We succeeded in quantitatively evaluating petal shape variation by means of elliptic Fourier descriptors and principal-components analysis, and in evaluating picotee color patterns by a newly developed procedure based on comparative marginal distribution. Petal shape variation was divided into symmetrical and asymmetrical elements of the entire shape variation. Both groups were additionally divided into several components. The variations in picotee color pattern were effectively described by the first through fourth principal components. Comparing the varietal effect of these components, nested analyses of variance showed that the differences between cultivars in picotee color pattern were smaller than those of the symmetrical shape elements. In addition, the environmental effects on picotee color formation were greater than those of symmetrical shape formation. The evaluation methods described in this study are effective for further investigations, and are applicable to other floricultural crops as well as lisianthus.

Morphometric Analysis of Intraspecific Shape Variation in Male and Female Genitalia of <I>Phyllophaga hirticula</I> (Coleoptera: Scarabaeidae: Melolonthinae)

The diversity of male genitalia observed among many disparate arthropod taxa has stimulated a number of studies investigating which evolutionary factors contribute to this phenomenon. Much of this research, however, is focused on testing patterns of selection on male genitalia only. Because copulatory processes are a joint act between males and females, detailed study of the selection pressures on female morphology seems equally appropriate. In the current analysis, I used geometric morphometrics to quantify intraspecific shape variation of male and female genitalia in the scarab beetle species Phyllophaga hirticula (Knoch). Characters under sexual selection are predicted to be more phenotypically variable than nonsexual characters because of the persistent tendency of sexual selection to eliminate bouts of stabilizing selection. Thus, this analysis tested whether phenotypic variation of male and female genitalia is equivalently higher than shape variation of a nonsexual character, the right elytron. Shape variation was analyzed in SHAPE, a program that transforms two-dimensional outline data into elliptic Fourier descriptors to be analyzed in a principal components analysis. Data from this study supported the hypothesis that male and female genitalia were significantly more variable than the right elytron. In addition, P. hirticula female genitalia were significantly more variable than male genitalia. Comparable phenotypic variation of male and female genitalia suggests that like male genitalia, female genitalia also may be subject to sexual selection and should be considered when testing hypotheses of genitalic evolution.

An improved numerical method of calculating the striation thinning in static mixers

An improved method of calculating the rate of striation thinning for laminar flow in a static mixer has been developed. The shapes of the objects studied are reconstructed using elliptic Fourier descriptors, allowing discrimination between true changes in shape and noise due to discretization errors. The method also allows one to choose the level of complexity in the objects to be reconstructed which faciliates optimization of the computations without loss of information. The evaluation of the rate of striation thinning is based on local velocities giving a result closely related to the volumetric flow in different regions of the static mixer.

Diversity of Landraces of the White-flowered Gourd (Lagenaria siceraria) and its Wild Relatives in Kenya: Fruit and Seed Morphology

Kenyan landraces of the cultivated white-flowered gourd Lagenaria siceraria have highly variable morphology. In order to reveal the inter- and intra-specific variation in fruit and seed morphology in L. siceraria and its wild relatives in Kenya, various traits were examined in a total of 425 strains from L. siceraria (269), Lagenaria sphaerica (124), Lagenaria abyssinica (27) and Lagenaria breviflora (5). Data analysis revealed the following patterns: (1) L. siceraria is more diverse than its wild relatives in both qualitative and quantitative and fruit traits are more variable than seed traits within each species. (2) Principal component analysis of L. siceraria with 15 quantitative traits showed a continuous variation among strains, in which general size factor of fruit and seed, shape factor of fruit and shape factor of seed were the principal causes of variation. (3) No correlation was found between fruit and seed shape, or between size and shape. (4) Image analysis with elliptic Fourier descriptors revealed continuous shape variation in the landraces of L. siceraria. Fruit shape features such as the contrast between a wide base with a distinct handle, and a slender base with an indistinct handle and the degree of bulge of the elongated handle (bilobate shape) were evaluated quantitatively. (5) Analysis of variance of 12 quantitative traits based on the progeny test demonstrated that the degree of heterozygosity is considerably low in the white-flowered gourd existing in the natural environment in Kenya. (6) The quantitative evaluation of the intra-specific variation in fruit and seed in L. siceraria was possible, but it was difficult to classify the landraces into distinct groups. Most of the variation observed in the cultivated L. siceraria, including differences in fruit size and shape, shell thickness and handle development, probably resulted from selection by the local human population.

Fruit shape variation in Fraxinus mandshurica var. japonica characterized using elliptic Fourier descriptors and the effect on flight duration

AbstractFruit shape variation in Fraxinus mandshurica var. japonica was characterized using principal component analysis of elliptic Fourier descriptors (EFDs), and experimental release of the fruits from a 8.3‐m tower was conducted to examine the effect of fruit shape on flight ability. The differences in the first five components of symmetrical variation in fruit shape, area and weight were highly significant among individual trees. Experimental release of fruits indicated that three principal components of fruit shape, area and weight were significantly related to flight duration. Thus, intraspecific morphological variation of fruits will affect the dispersion potential of individual trees.

Quantitative evaluation of the petal shape variation in Primula sieboldii caused by breeding process in the last 300 years

Primula sieboldii: (E. Morren) has been a popular garden plant at least since the Edo period, about 300 years ago. We compared petal form between cultivars and wild populations in order to characterise the changes that have occurred during domestication. The comparison was made using EF-PCA analysis, which describes overall petal shape mathematically by transforming petal contour coordinates into elliptic Fourier descriptors; it subsequently summarises these descriptors by principal component analysis (PCA). Rearing cultivars in a common-garden experiment identified the PCs with a substantial genetic element. A clear heritable component was detected for the PCs characterising symmetrical variation in flower shape, but not the asymmetrical variation. Wild populations of this species have become endangered owing to habitat destruction by human activity, and many lowland floodplain habitats have been lost. Variation within the remaining wild populations was significantly lower than in the cultivars for PC1 (aspect ratio), PC3 (curvature of proximal and distal parts) and petal area; but not for PC2 (depth of head notch) and PC4 (position of the centre of gravity). The shifts in petal form from the wild populations to the cultivars parallel those seen in other crop-types following domestication, including an increase in size and diversity of forms: cultivars have shallower head notches, more fan-shaped petals and larger petals than do wild P. sieboldii.

Identification of Rhus succedanea L. Cultivars Using Elliptic Fourier Descriptors Based on Fruit Shape

Abstract We have developed a method to identify cultivars of Rhus succedanea L. based on their fruit contour shape. For this, we collected fruits of five cultivars from three different environments (differing in site and/or year of collection) and the horizontal contour shape of each fruit was expressed by 37 elliptic Fourier descriptors, normalized in terms of size, rotation, shift and starting point of contour tracing. The first six components derived from a principal component analysis of the elliptic Fourier descriptors explained 89% of the variance. The differences among cultivars, environments and the cultivar x environment interaction were significant at the 0.01% probability level for all six principal components according to ANOVA. UPGMA cluster analysis based on the six principal components showed a high degree of clustering and most (but not all) ramets from the same cultivar clustered together. However, results of a UPGMA cluster analysis of Mahalanobis’ generalized distances among cultivars and environments, based on the 37 elliptic Fourier descriptors, showed that samples from the same cultivars clustered together, regardless of the environmental factors. We then applied a ‘similarity probability’ test, based on Mahalanobis’ generalized distances and a randomization test. The similarity probabilities between descriptors in the database and sampled fruits, when the cultivars they represented were included in the database, were >97%. In contrast, for samples representing cultivars that were not included in the database, the probabilities were <46%. These figures also apply to pairs of samples included in the database that represented the same cultivar, and different cultivars, respectively. These results suggest that it is possible to identify R. succedanea cultivars based on fruit contour shape using elliptic Fourier descriptors and similarity probability analysis.

Silhouette encoding and synthesis using elliptic Fourier descriptors, and applications to videoconferencing

This paper investigates the use of elliptic Fourier descriptors as a shape descriptor for encoding the silhouette of a person. Shape descriptors are here used for predicting the shape of silhouettes in missing frames within a sequence. This prediction scheme is applied to the case of generating in-between images in a low frame rate videoconferencing system, where the reconstructed silhouette is used as a binary mask for reducing the computational time for the frame reconstruction.

Genetic Control of Root Shape at Different Growth Stages in Radish (Raphanus sativus L.)

Since the changes in the root shape of radish (Raphanus sativus L.) during development cannot be easily observed, it is often difficult to determine the suitable time after sowing for the selection of genotypes with desirable root shapes. To determine the optimum time for selection of radish root in breeding, we conducted diallel analyses of the root shape at different growth stages and investigated the changes in the mode of inheritance of the root shape. A set of diallel crosses of 6 inbred lines was harvested 6 times from 40 to 75 days after sowing. The major shape characteristics, associated with principal components of elliptic Fourier descriptors (EFDs), included the ratio of length to width (AP1), the bluntness of the distal part (AP2), and the curvature of the middle part (BP1). ANOVA indicated that the growth stages, genetic effects, and their interaction were significant in all 3 characteristics. The interaction did not affect appreciably the difference in the mode of inheritance of AP1 among growth stages, suggesting that the time of harvest did not affect the selection of this characteristic. The interaction caused a certain degree of heterogeneity in the ordering of genotypes and in the mode of inheritance of AP2 and BP1 at different growth stages. In AP2, the degree of dominance increased at the later growth stages. In BP1, the genetic parameters varied, but showed no apparent trend with the growth process. The results suggest that AP1 can be efficiently selected at an early stage, i.e., at the beginning of thickening growth. For AP2 and BP1, the selection at a stage around the normal harvest time is recommended. However, the fluctuations in the time of selection within a 2-week period around the normal harvest time did not affect the selection of the characteristics, because the interaction was not significant within this period.

Interaction between Genetic Effects and Soil Type in Diallel Analysis of Root Shape and Size of Japanese Radish (Raphanus sativus L.)

We conducted diallel analyses of root shape and size in Japanese radish to determine the interaction between genetic effects and soil type in the selection for these traits in a breeding program. A set of diallel crosses of six inbred lines was grown in an andosol, a clayey soil and a sandy soil. The major shape characteristics accounted for by the principal components of elliptic Fourier descriptors were the ratio of length to width, the bluntness of the distal part, the curvature of the central part, and the curvature of the proximal and distal parts. Analysis of variance of diallel tables showed that the genetic effects were highly significant for all the characteristics. In contrast, the influence of the soil type was markedly different among the characteristics. Soil type effect was significant for size characteristics and the bluntness of the distal part. In the bluntness of the distal part, the interaction between genetic effects and soil type was significant, and the heritability and the degree of dominance varied between the soil types. The results indicated that the differences in the soil types did not influence the selection for the root size or shape in most characteristics. However, the interaction between genetic effects and soil type should be taken into account in the selection for the bluntness of the distal part.

SHAPE: a computer program package for quantitative evaluation of biological shapes based on elliptic Fourier descriptors.

Quantitative evaluation of the shapes of biological organs is often required in various research fields, such as agronomy, medicine, genetics, ecology, and taxonomy. Elliptic Fourier descriptors (EFDs), proposed by Kuhl and Giardina (1982), can delineate any type of shape with a closed two-dimensional contour and have been effectively applied to the evaluation of various biological shapes in animals (Bierbaum and Ferson 1986; Diaz et al. 1989; Ferson et al. 1985; Rohlf and Archie 1984) and plants (Furuta et al. 1995; Iwata et al. 1998; McLellan 1993; Ohsawa et al. 1998; White et al. 1988). Quantization of shapes is a prerequisite for evaluating the inheritance of morphological traits in quantitative genetics. There are many reports showing that measurements based on EFDs are helpful for such quantization of the shapes of plant and animal organs. For instance, Iwata et al. (2000) conducted a diallele analysis of the shape of Japanese radish (Raphanus sativus L.) roots, using the principal component scores of the EFDs as shape characteristics. Quantitative trait loci (QTL) analysis has also been conducted using the principal component scores of EFDs concerning the shape of the male genitalia of Drosophila species (Laurie et al. 1997; Liu et al. 1996). The shape evaluation method based on EFDs can be a powerful tool for analyzing biological shapes, but it is not easy for a researcher to use this method because it involves several complex procedures, such as image processing, contour recording, derivation of the descriptors, and multivariate analysis of the descriptors. In this article we present SHAPE, a package of programs for evaluating biological contour shapes based on EFDs. This package contains programs for image processing, contour recording, derivation of EFDs, principal component analysis of EFDs, and visualization of shape variations estimated by the principal components. With the aid of this package, a researcher can easily analyze shapes on a personal computer without special knowledge about the procedures related to the method. The principal component scores obtained by the procedures can be used directly as observed values of shape characteristics for the subsequent analyses. SHAPE is characterized by the following features: (1) The packaged programs are easily operated with the aid of a graphical user interface (GUI); (2) No special computer devices for image processing are required; (3) A large number of samples (say 1,000) can be treated; (4) The scores of principal components are stored in tabbed text format files and can be easily exported for analysis by other software; and (5) The variations in shape accounted for by the principal components can be visualized and printed out.

The human mandible in lateral view: elliptical fourier descriptors of the outline and their morphological analysis

An original procedure based on new developments in elliptical Fourier methods associated with image analysis technics was applied to 117 human mandibles in lateral view (69 males, 48 females) in order to analyze interindividual variability and sexual dimorphism. Original parameters, called elliptical Fourier descriptors, allowed for the quantification of the shape of an outline irrespective of its morphological complexity, and provided a precise individual characterization. The use of step by step reconstructions with an increasing number of harmonics allowed for demonstration of the morphological contributions of the elliptical Fourier descriptors, and relationships to precise anatomical features were established. The first harmonics (1st to 8th) described the general mandibular shape, and the later harmonics (8th to 14th) characterized accessory and finer morphological features. The proportion of individuals presenting significant sexual dimorphism was 97.1% in males and 91.7% in females, and after size normalization 84.1% in males and 81.2% in females. Even after removal of the size component, these results in regard to mandibular dimorphism remained better than those found by classical approaches; this could be due to the use of elliptical Fourier analysis which takes into account all the information of the mandibular outline. Compared to the size component, the shape component seemed to be more important in the sexual dimorphism of the human mandible than previously recognized. The present approach opens interesting anthropological and clinical perspectives for the characterization of mandibular morphology.

Diallel Analysis of Leaf Shape Variations of Citrus Varieties Based on Elliptic Fourier Descriptors.

The inheritance of citrus leaf shape was investigated by analyzing a diallel set of crosses of five citrus varieties/selections. Leaf contours were extracted by image processing, and described by elliptic Fourier descriptors. Mathematically independent shape characteristics were then identified by principal component analysis of the descriptors. The 1st, 2nd, 3rd and 4th components accounted for 51%, 20%, 10% and 6% of the total shape variation, and were good measures of the ratio of length to width, the position of the centroid, the curvature and the leaf wing size, respectively. Diallel analyses indicated that these components differed substantially in their mode of inheritance. The broad sense heritability was fairly high (>0.90) in the 2nd and 4th components and high (>0.70) in the 1st component, whereas the narrow sense heritability was high only in the 2nd component. There was no significant genetic effect in the 3rd component. The 1st, 2nd and 4th components showed over-, incomplete- and complete- dominance, respectively. The results indicate that elliptic Fourier descriptors can be successfully applied to the quantitative genetic analysis of citrus leaf shape.

The Evaluation of Genotype * Environment Interactions of Citrus Leaf Morphology Using Image Analysis and Elliptic Fourier Descriptors.

Variations of leaf form, i.e., differences in shape and size, among citrus varieties and the genotype × environment (GE) interactions influencing the variations were examined. Leaf shape was quantitatively measured by the score of the principal components of elliptic Fourier descriptors (EFDs). Leaf size was measured in terms of area and perimeter. The first four principal components of EFDs, which could explain over 90 % of the shape variations, were good measures of the length to width ratio, the position of the center of gravity, the curvature, and the degree of roundness. Nested ANOVA for the leaf form variations in nine citrus varieties indicated that the variations accounted for by the 1st, 2nd and 4th components were inherited. GE interaction of the leaf form was investigated using the data appertaining to seven genotypes from eight locations; the interaction was significant for all the principal components examined except the third. Although the interaction fitted very poorly to a joint linear regression analysis model developed from the data, it was effectively explained by an additive main effect and multiplicative interaction model where the score of the interaction principal component was highly correlated with the stability indices. Our results suggest that the genotype was the main source of variation in leaf shape, but not in size, and that the contribution of GE interaction was minor to both shape and size, although statistically significant at the 1 % probability level.