Procrustes analysis involves finding the optimal superposition of two or more "forms" via rotations, translations, and scalings. Procrustes problems arise in a wide range of scientific disciplines, ...especially when the geometrical shapes of objects are compared, contrasted, and analyzed. Classically, the optimal transformations are found by minimizing the sum of the squared distances between corresponding points in the forms. Despite its widespread use, the ordinary unweighted least-squares (LS) criterion can give erroneous solutions when the errors have heterogeneous variances (heteroscedasticity) or the errors are correlated, both common occurrences with real data. In contrast, maximum likelihood (ML) estimation can provide accurate and consistent statistical estimates in the presence of both heteroscedasticity and correlation. Here we provide a complete solution to the nonisotropic ML Procrustes problem assuming a matrix Gaussian distribution with factored covariances. Our analysis generalizes, simplifies, and extends results from previous discussions of the ML Procrustes problem. An iterative algorithm is presented for the simultaneous, numerical determination of the ML solutions.
Three experiments provide evidence that the perceived similarity between two images is systematically affected by the inherent direction of a transformation that links the two. Participants were ...shown short animations morphing one object into another from the same basic category. They were then asked to make directional similarity judgments ("How similar is object A to object B?") for two stationary images drawn from the morph continuum. Across three experiments, similarity ratings for identical comparisons were higher when the reference object, B, had appeared before the comparison object, A, in the preceding morph sequence. This response to dynamic transformational sequences is in accordance with the view that similarity depends on the ease of transformation between object representations and that transformations between objects in categorization and object recognition are psychologically real.
Exploring how many pattern blocks will completely fill the Rocket Ship puzzle, students are challenged to use the most and fewest number of blocks possible. They have the opportunity to explore the ...composition and decomposition of shapes and generalize ideas about the relationship between the size of the pieces and the number of pieces. Each month, elementary school teachers are presented with a problem along with suggested instructional notes; asked to use the problem in their own classrooms; and encouraged to report solutions, strategies, reflections, and misconceptions to the journal audience.
Three experiments examine 18-to 24-month-old (N = 78) toddlers' ability to spatially orient objects by their major axes for insertion into a slot. This is a simplified version of the posting task ...that is commonly used to measure dorsal stream functioning. The experiments identify marked developmental changes in children's ability to preorient objects for insertion, with 18-month-olds failing completely and 24-month-olds succeeding easily. In marked contrast, 18-month-olds preorient their empty hands for insertion into the same slots. This developmental dissociation between aligning hands and aligning objects to slots suggests that the key developmental change is in action with the goal of object-to-object alignment versus action on an object.
Morphology reflects ecological pressures, phylogeny, and genetic biophysical constraints. Disentangling their influence is fundamental to understanding selection and trait evolution. Here, we assess ...the contributions of function, phylogeny, and habitat to patterns of plastrom (ventral shell) shape variation in emydine turtles. We quantify shape variation using geometric morphometrics, and determine the influence of several variables on shape using path analysis. Factors influencing plastron shape variation are similar between emydine turtles and the more inclusive Testudinoidea. We evaluate the fit of various evolutionary models to the shape data to investigate the selective landscape responsible for the observed morphological patterns. The presence of a hinge on the plastron accounts for most morphological variance, but phylogeny and habitat also correlate with shape. The distribution of the shape variance across emydine phylogeny is more consistent with an evolutionary model containing two adaptive zones—one for turtles wiht kinetic plastra, and one for turtles with rigid plastra. Models with more complex adaptive landscapes often fit the data only as well as the null model (purely stochastic evolution). The adaptive landscape of plastron shape in Emydinae may be relatively simple because plastral kinesis imposes overriding mechanical constraints on the evolution of form.
Visual pattern processing becomes increasingly complex along the ventral pathway, from the low-level coding of local orientation in the primary visual cortex to the high-level coding of face identity ...in temporal visual areas. Previous research using pattern aftereffects as a psychophysical tool to measure activation of adaptive feature coding has suggested that awareness is relatively unimportant for the coding of orientation, but awareness is crucial for the coding of face identity. We investigated where along the ventral visual pathway awareness becomes crucial for pattern coding. Monoptic masking, which interferes with neural spiking activity in low-level processing while preserving awareness of the adaptor, eliminated open-curvature aftereffects but preserved closed-curvature aftereffects. In contrast, dichoptic masking, which spares spiking activity in low-level processing while wiping out awareness, preserved open-curvature aftereffects but eliminated closed-curvature aftereffects. This double dissociation suggests that adaptive coding of open and closed curvatures straddles the divide between weakly and strongly awareness-dependent pattern coding.
Proving an existence theorem is less intuitive than proving other theorems. This article presents a semiotic analysis of significant fragments of classroom meaning-making which took place during the ...class-session in which the existence of the midpoint of a line-segment was proven. The purpose of the analysis is twofold. First follow the evolution of students' conceptualization when constructing a geometric object that has to satisfy two conditions to guarantee its existence within the Euclidean geometric system. An object must be created satisfying one condition that should lead to the fulfillment of the other. Since the construction is not intuitive it generates a dilemma as to which condition can be validly assigned initially. Usually, the students' spontaneous procedure is to force the conditions on a randomly chosen object. Thus, the second goal is to highlight the need for the teacher's mediation so the students understand the strategy to prove existence theorems. In the analysis, we use a model of conceptualization and interpretation based on the Peircean triadic SIGN.
We present the development, convergence analysis, and numerical tests of the mimetic finite difference method for the Stokes problem on two-dimensional polygonal and three-dimensional polyhedral ...meshes.
Arthropods are known to display a variable number of eye lenses and this number mostly increases during their development. In trilobites, most species possessing schizochroal eyes exhibit a notable ...intraspecific variation in the number of dorso ventral files of eye lenses that can be age related (i.e., growth) or not (i.e., living environment). Several previous studies have shown that some trilobite groups (e.g. phacopids) tend to have fewer lenses/files in representatives from the deeper habitats than those from shallower habitats. In this study, we analyzed the pattern of variation in the number of dorso ventral files of eye lenses in two Devonian phacopid trilobites from the Prague Basin of the Czech Republic. We quantified their intraspecific variability. To better understand the patterning, we compared more than 120 individuals. Data first reveal evidence of a bimodal distribution of lens/file number without intermediate forms among each of two studied populations of Prokops prokopi (Chlupáč, 1971) and throughout the ontogeny of Pedinopariops insequens (Chlupáč, 1977). Our results indicate that caution must be taken for taxonomical affiliation and biodiversity analyses of taxa in which the intraspecific variability is unclear. Additionally, we investigated possible relations of these bimodalities to the stratigraphical position of studied populations and to the paleoenvironment. In Prokops prokopi, a slightly different age of both populations, together with supposed differences in the local environments can be responsible for observed variability. In Pedinopariops insequens, stress conditions possibly related to the approaching onset of the Basal Choteč Event can be responsible for surprising intrapopulation variability. We speculate that the stress conditions could cause a bimodal selection and possibly also the change of ontogenetic trajectory within this species. Pedinopariops insequens was the only phacopid in the Prague Basin that crosses the Lower/Middle Devonian boundary and survived also the onset of Basal Choteč Event.
Butterflies possess attributes that are sensitive to gradual environmental changes. Recently, the effects of environmental factors on the shapes of organisms, as well as the interactions of these ...elements, have been extensively examined, i.e., effects of seasonal changes on the colors of butterfly wings, and effects of landscape structure on butterfly distribution and morphology. However, few studies have dealt with variations in butterfly shapes in response to varying environmental conditions. Here we aimed to determine how body size and shape variations in butterflies are correlated to environmental heterogeneity. We used geometric morphometrics to quantify Pieris rapae wing shape variations. Results showed that forewing and hind wing sizes were significantly different among the 15 populations. P. rapae individuals with larger wing sizes were mainly distributed in mountainous areas, whereas those with smaller-sized wings were found on the plains. Canonical variate analysis was employed to examine the patterns of variation in wing shapes among and within the populations. Significant differences in shape were revealed in the forewings and the hind wings of P. rapae populations. All populations were divided into 2 groups on the first canonical variate axis (CV1), which followed the Qinling Mountains as an important boundary between the Palearctic and Oriental Realms in zoogeographical division of the world. The unweighted pair group method with arithmetic mean (UPGMA) clustered the 15 populations into 4 groups by forewing and hind wing shape in response to the 4 environment types in Qinling Mountains. We suggest that wing shapes of P.rapae are sensitive to environmental heterogeneity. The isolating effect of the Qinling Mountains on P. rapae population interactions was apparent.
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