Biological Shape Analysis

The following sections are included:

• Preface

• List of Symposium Participants

• Contents

Variation in corolla shape has been widely observed at inter- and intra-specific levels in entomophilous plants. Many ecologists have focused on this variation in the context of co-evolution of wild plants and their pollinators. The corolla shape is considered to be an important visual cue that attracts their pollinators. However, the ability of pollinators to discriminate among the corolla shapes under natural conditions has not yet been supported by a large body of evidence. This study demonstrates the efficacy of combining a traditional behavioral experiment with computerized shape analysis of actual flowers. Elliptic Fourier descriptors were used to describe these shapes by transforming coordinate information of the contours into coefficients. Principal components analysis then transformed these coefficients to a smaller number of variables called principal components. Using these methods, artificial flowers were created based on the natural diversity of petal shape in Primula sieboldii and Brassica rapa. Dual-choice tests were then performed to investigate the ability of bumblebees (Bombus ignitus) to detect differences in the aspect ratio of petals. The insects showed significant ability to detect differences in the aspect ratio when the morphological distance increased. Moreover, the bumblebees showed a significant preference for narrow petals in P. sieboldii. The insects also detected differences in this parameter of the B. rapa flowers at close range. The effectiveness of this behavioral experiment was demonstrated using the actual variation present in the corolla shapes of the two plant species. Whether pollinators can discriminate among flowers based on the variation in corolla shape and whether the variation affects pollinator behavior remain fundamental questions regarding the mechanisms responsible for the formation and maintenance of diversity in these characteristics of entomophilous plants. The method proposed here could be useful in answering these questions for many other plants.

This research aims at deriving a well founded biologically quantitative method for evaluating citrus blade (part of the leaf) shapes. In this paper, attention has been paid to the blade's outline and other geometric features. The quantitative analysis was implemented using five typical varieties of citrus. The numerical evaluation of outline form was carried out using a methodology based on P-type Fourier Descriptors (PFDs) and Principal Component Analysis (PCA), as proposed in our previous research based on lotus petal shape. The average blade outline, which includes size information about each variety, can also be viewed as an inverse-PFD transformation. In contrast to the evaluation method based solely on PFDs, the blade's size information has now been included in the analysis. For the geometric evaluation of blade shape, three geometric variables were used: [1] Area, (the total pixels within each blade); [2] Circularity, (((Maximum length²)/Area) * π/4); and [3] Aspect Ratio, (Maximum length/Maximum width). The quantitative evaluation of blade shape consisted of the PCA scores calculated from the PFDs and the three geometric variables. An ANOVA was then computed and plot-box graphs drawn to investigate and display the changes of blade shape with variety. Classification of the varieties was determined from the PC scores derived from the PFDs. A linear Support Vector Machine (SVM) classification model was used in this experiment.

The method of combining P-type Fourier Descriptors (PFDs) and principal component analysis (PCA) was effective for the quantitative evaluation of ordinal "plant type" of rice. The evaluation method involved plant type outlines from the 2^nd, 3^rd and 4^th leaves of the 5^th leaf stage. These were extracted and described quantitatively with PFDs. In a previous study, it was possible to distinguish differences in plant type between rice cultivars "Koshihikari" and "Kasalath" with this method. In the current study, PFDs were applied to chromosome segment substitution lines (CSSLs), which have substituted a part of the Koshihikari chromosome with that of the Kasalath chromosome and allowed the estimation of chromosomal regions related to each rice plant type. The materials in this experiment consisted of 39 Koshihikari / Kasalath CSSLs from the Rice Genome Resource Center and the parental lines of "Koshihikari" and "Kasalath". Eight individuals per line were used for the analysis. The curves of the 2^nd, 3^rd and 4^th leaves at the 5^th leaf stage were extracted. The coefficients of the PFDs were calculated from the curves and summarized with PCA. The obtained PC scores were used to describe characteristics of plant type shape. To visualize the plant type shape, the curves were reconstructed from the PC scores using "Inverse Fourier Descriptors". Plant type traits evaluated by each PC were identified from the reconstructed outlines. The t-test was used to assess mean differences between the Koshihikari and the CSSL component identified from the PC scores. These CSSLs were found to be statistically significant and facilitated the estimation of those chromosomal regions that are related to plant type including the gene(s) that may be involved.

Plant shape is known as an important character in soybean (Glycine max) for its association with lodging resistance, efficient utilization of sunlight, and adaptability to machine harvesting. Although the plant shape is recognized as a typical quantitative trait, its evaluation is traditionally performed by visual inspection. The quantitative evaluation of plant shape based on changes during growth is important in relation to yield and understanding of the genotype x environment interaction. Soybean plant shape at maturity has been analyzed quantitatively by many researchers. However, there is no report available on the quantitative evaluation of changes in plant shape during the growth process. Quantification of change in soybean plant shape during growth was done by image analysis. In this study, it was attempted to ascertain which methods could be used to quantify the changes in plant shape during the growing stages. Seven soybean cultivars were cultivated and sequential photographs of were taken at six-day intervals. Binary images were obtained from threshold-based digital images. The binary images were then analyzed using two methods: (1) Area histogram and (2) gray level histogram analysis. In Area histogram method, the horizontal and vertical comparative distributions were calculated as quantitative values. Gray level histogram analysis treats the histogram as a first-order statistical image that provides the simplest way for describing the texture of images. As a result, the Area histogram method reflects the expansion and position of branches and leaves. The gray level histogram method explained the overlap of branches and leaves. The change of plant shape at various stages of development was well explained by uniting both methods.

Crop organ shape as well as size is an important trait for the genetic improvement of crops because it directly or indirectly influences the values of agricultural products. In general, crop organ shape shows continuous variation. In order to clarify its genetic structure it is necessary to measure this genetic variation with a quantitative method.

Principal component analysis (PCA) of a set of elliptic Fourier descriptors (EFDs) enables one to decompose the shape variation into statistically independent characteristics and to analyze these characteristics as ordinary quantitative traits. So far, crop organ shape has been analyzed with various quantitative genetic methods based on the principal components (PCs) of EFDs. Diallel analysis enabled the clarification of the mode of inheritance of commercially important shape characteristics in the radish root. Genome-wide association studies enabled the estimation of the positions and effects of quantitative trait loci (QTL) controlling rice grain shape variation observed in a germplasm collection. The results of these analyses were easy-to-understand because of the visualization of the results, and served as useful knowledge for plant breeding. In a series of analyses, different PCs of EFDs showed a different mode of inheritance or were controlled by different QTLs in general, indicating that PCA of EFDs unraveled the inheritance of shape controlled by a number of genes into simpler and more easily understood components. Recent advances in molecular biology technology allow one to utilize a more advanced method in quantitative genetics for the understanding of the biological processes controlling crop organ shape. For example, genomic selection may enable one to predict unobserved organ shapes of unknown genotypes and select desirable genotypes based on the predicted shape. Combining the PCA of EFDs with statistical genetic methods, may provide genetic improvement of crop organ shape. Moreover, this approach allowed for a more rapid and efficient procedure for the design of new and novel crop organ shapes.

Wheat is a major food source all over the world. In India three classes of wheat are cultivated: viz. bread wheat (Triticum aestivum), macaroni wheat (Triticum durum), and emmer wheat (Triticum dicoccum). Several varieties in each class are in use. Apart from the correct identification of wheat grains as starting material for research or for a breeding programme, correct identification is also important for end-use purposes. Bread and macaroni wheats are most appropriate for the production of leavened bread and pasta products respectively. Emmer wheat is used in snacks and recommended in special diets. Among the classes, varieties differ in their dough properties and hence the use of an appropriate class and variety is essential for obtaining a good quality end product. While it is possible for a trained eye to identify the class and variety by visual grain inspection, this method is subjective. The grains of the classes/varieties differ in morphology such as dimensions, shapes and colour. Computer-based shape analysis for distinguishing between varieties and classes was used. Images of grains were acquired by placing fifty grains of each sample for each wheat type on a flat bed scanner. In-house developed software was used to analyze these images and a modified Euclidean distance (ED) metric was the criterion for comparing the relationship between wheat grain samples. Earlier experiments using shape analysis showed that it was possible to distinguish between fifteen wheat varieties. In a previous study, a parent variety and three genetically closely-related lines derived from it were analyzed. Another study showed that the EDs for the same variety samples harvested in different environments were lower than between different variety samples indicating that in spite of an environmental effect, closeness among samples of the same variety was present. In this study, fifteen wheat varieties belonging to the three classes were subjected to analysis. Forty-five morphometric and 18 colour parameters were derived from the images. EDs correctly divided the varieties into three clusters with the exception of two of the 15 varieties. The results showed that computer-based shape analysis has potential for varietal identification.

The spread of invasive species is a rapidly increasing problem worldwide. In Japan, many lucanid (stag) beetle species have been imported from Southeast Asia as pets or collected as a leisure pursuit or hobby. Occasionally, these specimens are released into the wild by these owners. This has led to the accidental hybridization between some of the beetle species found in the wild. This suggests that the unique genetic variation of the populations of native lucanid beetles in Japan might be lost via genetic introgression. The body shapes of hybrids are influenced by the genetic architecture of the parental species and subspecies. In an attempt to understand the characteristics of the inheritance of the shape of body parts, an intercross experiment using three subspecies of stag beetles was conducted. Shape features of the mandibles of hybrids and their parental subspecies were extracted using two different morphometric tools, thin-plate splines (TPS) and elliptic Fourier analysis (EFA). Male mandibles were chosen as the test of body parts because the shape shows conspicuously large variation between sub-species. Intercross experiments and subsequent morphometric analyses demonstrated that the mandibular shape of male hybrids were similar to those of their maternal lineage. These findings suggest that the similarity of mandibular shape of the hybrid males to their maternal lineages could be attributed to maternal effects. Both methods of shape analysis were shown to be successful in classifying subspecies and cross-bred beetles with elliptic Fourier analysis giving completely correct results and thin-plate splines miss-classifying only one out of 55 individuals.

The identification of skeletons cannot always be achieved using DNA and/or dental records. This is especially true of remains thought to belong to US military personnel that were recovered from the Korean War. A solution to this problem may be found by matching ante-mortem and post-mortem radiographs since the Joint POW/MIA Accounting Command (JPAC) holds ante-mortem chest radiographs of 71% of the unaccounted for individuals from this conflict. Initial tests of visual methods revealed that seven skeletons of known identity (representing five true positive and two true negative identification scenarios) could be morphoscopically identified without errors from radiographs of up to 1000 subjects and from "open" ended sequential tests using claviculae and cervical/thoracic vertebrae anatomy. While visual radiographic comparison methods therefore hold promise, their quantification is paramount for efficient database searching. As radiographs for more than 6500 missing personnel exist, weeks would be required for human operators to undertake visual searches for any single individual. This paper, therefore, reports the results of Elliptical Fourier Analysis (EFA) to quantify clavicle shapes and to provide the basis for expedited electronic searches. To accommodate for radiographs that display only partial clavicles, outlines of clavicle shafts were used to generate four closed shapes for each individual (representing the superior and inferior borders of the left and right clavicles). Amplitudes, calculated from EFA of these shapes, served as the basis for post-mortem-ante-mortem image comparisons. Although some complicating factors existed, amplitude values provided a simple numerical coding system that enabled large proportions of the AM chest radiograph library to be quickly and correctly excluded, when searches for a correctly matching AM radiograph are made in reference to a single skeleton.

Quantitative descriptions of human facial form are valuable in forensic science. For example: classification (e.g. ethnic group recognition), individual identification and missing persons investigations. In medicine they can be applied for diagnosis, in treatment planning and for training purposes. The methods must capture biologically meaningful information in numerical form and must represent all relevant anatomical structures. The methods described here attempt to meet these requirements and are based on the creation of 3-D facial archetypes ("averages") and the modelling of population variances in the characteristics of interest. Two methods are described, one a very simple averaging of depth values; the other one uses dense correspondence models of the face followed by Principal Components Analysis to create multi-dimensional face-spaces. Examples are given of the use of these archetypes for the diagnosis of genetic syndromes that affect facial form.

The characterization and quantification of craniofacial form has been a primary interest of various researchers including clinicians and comparative anatomists. While modern geometric morphometric techniques are proving very successful at describing craniofacial shape, the dissection of the multifactorial determinants of that shape has proven more elusive. The current research program seeks to elucidate the genetic underpinnings of variation in the craniofacial and dentognathic complex in both human and a nonhuman primate model, specifically the baboon. Using statistical genetic approaches, the genetic architecture of the craniofacial complex has been examined using as "phenotypes", simple linear and angular metrics as well as landmarks and semi-landmarks in geometric morphometric analyses. When approaching the genetic analysis of a complex shape, such as the cranium, consideration must be given to the mode of acquisition of phenotypic characters with the ultimate goal of the genetic analyses. There is, currently, a vast toolkit available to the researcher for the description of two- and three-dimensional shapes. While simple methodologies, such as traditional morphometrics, may provide relatively weak description of shape compared to more sophisticated geometric techniques, the former may outperform the latter in elucidating the genetic underpinnings influencing trait variation. With proper consideration, the goals of both the maximal description of biological shape, and the dissection of the genetic architecture influencing that shape, can be accomplished.

The cranium is often regarded as a composite of different functional or developmental modules. Thus, morphological variation can be interpreted as a product of interactions of these modules during growth. This study aims to identify the degree and pattern of these interactions as seen in craniofacial traits among genetically distant populations. To assess this pattern of craniofacial integration, the skulls of males from Skolt Lapps in Finland and from modern Japanese were measured. Three manually-obtained indices of horizontal facial flatness were regressed on shape variation as measured from nine anatomical landmarks of cranial face and base digitized from lateral cephalograms. Extraction of this shape variation was attained through a process of geometric morphometrics; specifically, with the use of Procrustes superimposition and thin-plate spline transformations (TPS). The TPS-derived face-base variation displayed the inter-population differences between the Lapps and the Japanese populations. However, it did not co-vary with the degree of horizontal facial flatness. On the other hand, multiple regression analyses (MANCOVA) applied to the intra-population variation revealed significant correlations between the TPS-derived face-base shape variables and the facial flatness indices. Their covariate patterns were similar, and could be considered as a common trend among the two populations, while the three facial flatness indices exhibited different patterns with each other, indicating a different interaction of the three facial regions with the sagittal craniofacial profile. The frontal flatness is related to the combined position of the cranial base and the face, and also to the degree of cranial base flexion. The zygomaxillary flatness is related to the antero-posterior length of the palate. The results suggest that a common covariate pattern is detectable in the within-population variation and can be considered a consequence of developmental plasticity which can be controlled by both genes as well as functional demands.

The development of quantitative models is an essential first step in the elucidation of biological processes in terms of explanation and prediction. The conventional metrical approach (CMA), consisting of distances, angles and ratios, is one such model. CMA, however, is an insufficient model when applied to the shape of complex irregular forms. Shape is defined here as the boundary of a form in ℜ². Some of the insufficiencies of CMA include the: (1) difficulty of unambiguously removing the confounding effect of size; (2) distances and angles, as commonly used, are too sparse; and consequently, bear little relationship to the actual shape; (3) inability to precisely re-create the shape from the measurements; and (4) inevitable subjectivity inherent in the choice of measurements. Computational Shape Analysis (CSA) was developed, in part, to surmount the deficiencies in CMA. CSA is based on a combined approach: (1) elliptical Fourier functions (EFFs) to provide measures of global aspects and (2) continuous wavelet transforms (CWTs) to objectively identify localized features. To test the utility of this Fourier-wavelet model, a study utilizing the lateral view of the human cranial vault (CV) focused on shape changes during human evolution. The paucity of CMA landmarks makes the CV a good candidate for analysis with the Fourier-wavelet model. The fossil hominid study consisted of three samples: H. erectus, H. neanderthalensis and H. sapiens (total n=104). The fossil hominid CV results demonstrated statistically-significant differences between H. sapiens and both H. erectus and H. neanderthalensis. In contrast, there were no statistically-significant differences between H. erectus and H. neanderthalensis. The results suggest that H. erectus and H. neanderthalensis evolved as a single lineage, while H. sapiens represents a separate evolutionary development. The results demonstrate the usefulness of CSA as a method for eliciting significant information contained in the bounded outline of biological forms.

The shape of the orbital rim is one of the essential craniofacial features that play a crucial role in face perception, recognition, or personal identification. As a consequence, it continues to attract a large variety of researchers, e.g., skeletal or forensic anthropologists, anatomists, physicians, psychologists etc. Despite being a focus since the rise of traditional anthropometrics, the 3-D nature of the orbital rim has not been satisfactorily quantified. This paper presents an approach to numerically describe the within- and among-population shape variation of the orbital rim, its curviness and spatial orientation using elliptic Fourier analysis. The sample consisted of 336 crania from present-day skeletal collections (Czech Republic, Portugal). For each specimen, a pair of orbital outlines was recorded using the MicroScribe G2X, a portable digitizer. Regardless of the body side, the trace line started and ended at dacryon located on the inner margin of the orbital rim. Cartesian coordinates of three fixed points defining a 3-D Frankfurt Horizontal plane were also acquired. In order to align for spatial orientation of the orbits, each pair was rotated to the Frankfurt Horizontal. The outlines were divided into 150 evenly distributed points and subsequently processed using elliptic Fourier analysis with 20 harmonics. The results showed that the specimen's population affinity, sex, age and body size are factors responsible for the shape of the orbital rim. In contrast to the generally assumed pattern of sex differences, the shape of frontally-viewed orbits failed to act as a good sex predictor. Sex-related differences, as described by 3-D shape analysis, were linked mostly to a specific "curviness" and to spatial arrangement of orbits within the craniofacial skeleton. Whereas the male orbital outline is characteristically curved and placed in the frontal plane, the female outline is flat and diverged. The results demonstrate that elliptic Fourier analysis is a powerful methodological tool that can provide new insights into the structural relationships within biological objects.

Human orbit shape has been previously analyzed with a few non-specific landmarks or with subjective shape descriptions. This study uses elliptical Fourier analysis, a landmark-free method of measurement, to quantify the entire shape of the orbit and test the influence of sex, age, ancestry, and geographic location on orbit shape. Digital photographs were taken of 162 individuals of known sex, age and ancestry from the skeletal collections in Knoxville, Tennessee and Pretoria, South Africa. Orbit contours were traced from the photos and then analyzed in the elliptical Fourier program "SHAPE v. 1.3". Principal component (PC) values from SHAPE were analyzed for significant contributing variables using an ANCOVA. In the full sample, PCs 1,2, 4, and 5 were affected by at least one of the independent variables. Of these, geographical location figured prominently in 3 of the PCs. This underscores the importance of considering environmental effects over genetic ancestral affinities when analyzing human variation. Sex and the interaction between ancestry and continent of origin also significantly affected orbit shape. Although two PCs show some similarities to established descriptors of orbit shape, other components reveal more shape variation than have been previously prescribed. The corners of the orbits, features that are not captured in many current metric analyses, appear to play a significant role in shape variation. The technique used for this study proved to be effective in detecting significant contributing variables influencing human orbit shape. The study focused on descriptive analyses, but the quantitative data allows for the formulation of discriminant functions. These should not be hastily constructed as other variables associated with orbit shape remain unexplained.

Quantitative comparisons of long bone morphology between human and great apes are important to better understand the evolution of human bipedal walking. However, detailed quantitative knowledge about long bone morphology is relatively scarce. In this study, we present an integrated method to quantitatively analyze and visualize the morphological variation of the long bone diaphysis. Three-dimensional data of human femora are acquired using medical CT technology. Cross-sectional features (i.e., cortical bone thickness, external radius and surface curvature) can be measured along the entire diaphysis utilizing elliptical Fourier analysis and are visualized using the concept of morphometric mapping. Morphometric maps are further analyzed using a combination of 2-D Fourier transforms and multivariate analysis of shape to explore patterns of within- and between-taxon variation of long bone morphology. These methods are used here to analyze the morphological variability of the modern human femoral diaphysis. The analysis revealed complex patterns of intraspecific variation of cortical bone distribution in the human femoral diaphysis.

The description of natural shapes and the study of their development is based on our human perception and consequently the study of science is based on the notion of isotropy. The introduction of natural anisotropy through Gielis' curves and surfaces led to the notion of Universal Natural Shapes with a uniform description of shapes of flowers, molecules, space-time models and much more. They lead to a generalization of the Pythagorean Theorem.

The following sections are included:

• Index