Shape-space smoothing splines for planar landmark data

Biometrika ◽  
2007 ◽  
Vol 94 (3) ◽  
pp. 513-528 ◽  
Author(s):  
A. Kume ◽  
I. L. Dryden ◽  
H. Le
Keyword(s):  
Smoothing Splines ◽  
Shape Space ◽  
Landmark Data

scholarly journals Centric Allometry: Studying Growth Using Landmark Data

2021 ◽  
Author(s):  
Fred L. Bookstein
Keyword(s):  
Principal Component ◽  
Shape Space ◽  
Allometric Growth ◽  
Differential Growth ◽  
Data Set ◽  
Geometric Morphometric ◽  
Franz Boas ◽  
Landmark Data ◽  
Per Se ◽  
Algebraic Constraints

AbstractThe geometric morphometric (GMM) construction of Procrustes shape coordinates from a data set of homologous landmark configurations puts exact algebraic constraints on position, orientation, and geometric scale. While position as digitized is not ordinarily a biologically meaningful quantity, and orientation is relevant mainly when some organismal function interacts with a Cartesian positional gradient such as horizontality, size per se is a crucially important biometric concept, especially in contexts like growth, biomechanics, or bioenergetics. “Normalizing” or “standardizing” size (usually by dividing the square root of the summed squared distances from the centroid out of all the Cartesian coordinates specimen by specimen), while associated with the elegant symmetries of the Mardia–Dryden distribution in shape space, nevertheless can substantially impeach the validity of any organismal inferences that ensue. This paper adapts two variants of standard morphometric least-squares, principal components and uniform strains, to circumvent size standardization while still accommodating an analytic toolkit for studies of differential growth that supports landmark-by-landmark graphics and thin-plate splines. Standardization of position and orientation but not size yields the coordinates Franz Boas first discussed in 1905. In studies of growth, a first principal component of these coordinates often appears to involve most landmarks shifting almost directly away from their centroid, hence the proposed model’s name, “centric allometry.” There is also a joint standardization of shear and dilation resulting in a variant of standard GMM’s “nonaffine shape coordinates” where scale information is subsumed in the affine term. Studies of growth allometry should go better in the Boas system than in the Procrustes shape space that is the current conventional workbench for GMM analyses. I demonstrate two examples of this revised approach (one developmental, one phylogenetic) that retrieve all the findings of a conventional shape-space-based approach while focusing much more closely on the phenomenon of allometric growth per se. A three-part Appendix provides an overview of the algebra, highlighting both similarities to the Procrustes approach and contrasts with it.

scholarly journals A New Method for Landmark-Based Studies of the Dynamic Stability of Growth, with Implications for Evolutionary Analyses

2021 ◽  
Author(s):  
Fred L. Bookstein
Keyword(s):  
Growth Regulation ◽  
Pattern Analysis ◽  
Developmental Time ◽  
Dynamical Stability ◽  
New Method ◽  
Data Sets ◽  
Data Set ◽  
Landmark Data ◽  
Study Designs ◽  
Changes Over Time

AbstractA matrix manipulation new to the quantitative study of develomental stability reveals unexpected morphometric patterns in a classic data set of landmark-based calvarial growth. There are implications for evolutionary studies. Among organismal biology’s fundamental postulates is the assumption that most aspects of any higher animal’s growth trajectories are dynamically stable, resilient against the types of small but functionally pertinent transient perturbations that may have originated in genotype, morphogenesis, or ecophenotypy. We need an operationalization of this axiom for landmark data sets arising from longitudinal data designs. The present paper introduces a multivariate approach toward that goal: a method for identification and interpretation of patterns of dynamical stability in longitudinally collected landmark data. The new method is based in an application of eigenanalysis unfamiliar to most organismal biologists: analysis of a covariance matrix of Boas coordinates (Procrustes coordinates without the size standardization) against their changes over time. These eigenanalyses may yield complex eigenvalues and eigenvectors (terms involving $$i=\sqrt{-1}$$ i = - 1 ); the paper carefully explains how these are to be scattered, gridded, and interpreted by their real and imaginary canonical vectors. For the Vilmann neurocranial octagons, the classic morphometric data set used as the running example here, there result new empirical findings that offer a pattern analysis of the ways perturbations of growth are attenuated or otherwise modified over the course of developmental time. The main finding, dominance of a generalized version of dynamical stability (negative autoregressions, as announced by the negative real parts of their eigenvalues, often combined with shearing and rotation in a helpful canonical plane), is surprising in its strength and consistency. A closing discussion explores some implications of this novel pattern analysis of growth regulation. It differs in many respects from the usual way covariance matrices are wielded in geometric morphometrics, differences relevant to a variety of study designs for comparisons of development across species.

Methods for Studying Morphological Integration and Modularity

2010 ◽  
Vol 16 ◽  
pp. 213-243 ◽  
Author(s):  
Anjali Goswami ◽  
P. David Polly
Keyword(s):  
Distance Matrix ◽  
Japanese Macaques ◽  
Morphological Integration ◽  
Matrix Analysis ◽  
Euclidean Distance Matrix ◽  
Phenotypic Data ◽  
Graphical Modelling ◽  
Landmark Data ◽  
Competing Models ◽  
Matrix Correlation

Morphological integration and modularity are closely related concepts about how different traits of an organism are correlated. Integration is the overall pattern of intercorrelation; modularity is the partitioning of integration into evolutionarily or developmentally independent blocks of traits. Modularity and integration are usually studied using quantitative phenotypic data, which can be obtained either from extant or fossil organisms. Many methods are now available to study integration and modularity, all of which involve the analysis of patterns found in trait correlation or covariance matrices. We review matrix correlation, random skewers, fluctuating asymmetry, cluster analysis, Euclidean distance matrix analysis (EDMA), graphical modelling, two-block partial least squares, RV coefficients, and theoretical matrix modelling and discuss their similarities and differences. We also review different coefficients that are used to measure correlations. We apply all the methods to cranial landmark data from and ontogenetic series of Japanese macaques,Macaca fuscatato illustrate the methods and their individual strengths and weaknesses. We conclude that the exploratory approaches (cluster analyses of various sorts) were less informative and less consistent with one another than were the results of model testing or comparative approaches. Nevertheless, we found that competing models of modularity and integration are often similar enough that they are not statistically distinguishable; we expect, therefore, that several models will often be significantly correlated with observed data.

Simple shape space for 3D face registration

2009 ◽  
Author(s):  
Andrej Košir ◽  
Igor Perkon ◽  
Drago Bracun ◽  
Jurij Tasic ◽  
Janez Mozina
Keyword(s):  
Shape Space ◽  
Simple Shape ◽  
3D Face

scholarly journals Apolar and polar transitions drive the conversion between amoeboid and mesenchymal shapes in melanoma cells

2015 ◽  
Vol 26 (22) ◽  
pp. 4163-4170 ◽  
Author(s):  
Sam Cooper ◽  
Amine Sadok ◽  
Vicky Bousgouni ◽  
Chris Bakal
Keyword(s):  
Quantitative Imaging ◽  
Three Dimensional ◽  
Shape Space ◽  
Melanoma Cells ◽  
Collagen Matrices ◽  
Morphological Heterogeneity ◽  
Rho Family Gtpases ◽  
Metastatic Process ◽  
Rho Family ◽  
Single Living

Melanoma cells can adopt two functionally distinct forms, amoeboid and mesenchymal, which facilitates their ability to invade and colonize diverse environments during the metastatic process. Using quantitative imaging of single living tumor cells invading three-dimensional collagen matrices, in tandem with unsupervised computational analysis, we found that melanoma cells can switch between amoeboid and mesenchymal forms via two different routes in shape space—an apolar and polar route. We show that whereas particular Rho-family GTPases are required for the morphogenesis of amoeboid and mesenchymal forms, others are required for transitions via the apolar or polar route and not amoeboid or mesenchymal morphogenesis per se. Altering the transition rates between particular routes by depleting Rho-family GTPases can change the morphological heterogeneity of cell populations. The apolar and polar routes may have evolved in order to facilitate conversion between amoeboid and mesenchymal forms, as cells are either searching for, or attracted to, particular migratory cues, respectively.

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