The Measurement of Local Variation in Shape

Abstract

Geometric morphometrics comprises tools for measuring and analyzing shape as captured by an entire set of landmark configurations. Many interesting questions in evolutionary, genetic, and developmental research, however, are only meaningful at a local level, where a focus on “parts” or “traits” takes priority over properties of wholes. To study variational properties of such traits, current approaches partition configurations into subsets of landmarks which are then studied separately. This approach is unable to fully capture both variational and spatial characteristics of these subsets because interpretability of shape differences is context-dependent. Landmarks omitted from a partition usually contain information about that partition’s shape. We present an interpolation-based approach that can be used to model shape differences at a local, infinitesimal level as a function of information available globally. This approach belongs in a large family of methods that see shape differences as continuous “fields” spanning an entire structure, for which landmarks serve as reference parameters rather than as data. We show, via analyses of simulated and real data, how interpolation models provide a more accurate representation of regional shapes than partitioned data. A key difference of this interpolation approach from current morphometric practice is that one must assume an explicit interpolation model, which in turn implies a particular kind of behavior of the regions between landmarks. This choice presents novel methodological challenges, but also an opportunity to incorporate and test biomechanical models that have sought to explain tissue-level processes underlying the generation of morphological shape.

Appendix

The following sections provide mathematical details for the interpolation approaches discussed in this work, emphasizing common principles shared by these methods.

Interpolating Splines

Interpolating spline functions use information contained in landmark-wise differences between two configurations to map every location in a reference configuration onto a target form (Woods 2003). In general, m-dimensional interpolant functions are optimized by minimizing a roughness penalty $\rho \in \mathbb{R}^{m}$ (m = 1, 2, 3), defined in terms of the second order partial derivatives of the interpolant function ${{\Upphi}}( {\mathbf{p}} )$

$$ \rho \left( {{{\Upphi}}_{x,y} } \right) = \int {\int_{R^{2}}} {\left( {\frac{{\partial^{2} {{\Upphi}}}}{{\partial x^{2} }}} \right)^{2} + \left( {\frac{{\partial^{2} {{\Upphi}}}}{{\partial y^{2} }}} \right)^{2} + 2\left( {\frac{{\partial^{2} {{\Upphi}}}}{\partial x\partial y}} \right)^{2} {\text{d}}x{\text{d}}y} $$

(3)

for the 2-D case (Dryden and Mardia 1998), and

$$ \rho \left( {{{\Upphi}}_{x,y,z} } \right) = \int\int{\int_{R^{3}} {\left( {\frac{{\partial^{2} {{\Upphi}}}}{{\partial x^{2} }}} \right)^{2} + \left( {\frac{{\partial^{2} {{\Upphi}}}}{{\partial y^{2} }}} \right)^{2} + \left( {\frac{{\partial^{2} {{\Upphi}}}}{{\partial z^{2} }}} \right)^{2} + 2\left( {\frac{{\partial^{2} {{\Upphi}}}}{\partial x\partial y}} \right)^{2} + 2\left( {\frac{{\partial^{2} {{\Upphi}}}}{\partial x\partial z}} \right)^{2} + 2\left( {\frac{{\partial^{2} {{\Upphi}}}}{\partial y\partial z}} \right)^{2} {\text{d}}x{\text{d}}y{\text{d}}z} } $$

(4)

for the 3-D case (Meinguet 1979). The minimized sum of $\rho \left( {{\Upphi}} \right)$ over splines is the minimum penalty incurred from deforming a set of fixed set of landmarks into another set of equivalent (e.g., homologous) landmarks (Bookstein 1992), and it is termed the total bending energy. Minimization of Eqns. 3 and 4 leads to the smooth interpolation of deformation values at arbitrary locations of the form.

The interpolating splines emphasized in this study, namely TPS, belongs to a class of interpolants termed radial basis functions (RBF; Rohr 2001; Holden 2008), with general form

$$ {{\Upphi}}( {\mathbf{p}} ) = \mathop \sum \limits_{j}^{h} \beta_{j} U_{j} ({\mathbf{p}}) + \mathop \sum \limits_{i}^{k} \alpha_{i} R\left( {\left\| {{\mathbf{p}} - {\mathbf{q}}_{i} } \right\|} \right) $$

(5)

where ${{\Upphi}}( {\mathbf{p}} )$ represents the interpolation function at the 2- or 3-D point p (Holden 2008). In this formalism, a deformation is modeled as the sum of an affine component, given by $U({\mathbf{p}})$, and a function R that provides the non-affine component of the deformation, which smoothes the propagation of deformations throughout an object as a function of the distance between the interpolation point p and sampled landmarks q _i (i = 1, …, k). Thus, the influence of a landmark on the non-affine component of an interpolated site decays with the distance between the two locations, i.e., $\left\| {{\mathbf{p}} - {\mathbf{q}}_{i} } \right\|$. Finally, β _j and α _i are weights for the affine and non-affine components, respectively. Eqns. 3 and 4 can be solved as a set of linear equations with general form:

$$ \left[ {\begin{array}{*{20}c} {\mathbf{R}} & {\mathbf{Q}} \\ {\mathbf{Q}} & 0 \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} \alpha \\ \beta \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {\mathbf{P}} \\ 0 \\ \end{array} } \right] $$

(6)

which describes a deformation of a reference configuration Q onto a known target configuration P (Holden 2008). Deformation functions $R_{i} = R\left( {\left\| {{\mathbf{p}}-{\mathbf{q}}_{i} } \right\|} \right)$ are collected in matrix R, and we solve for vectors of coefficients α and β.

A set of thin-plate splines is given by the function

$$ \left[ {{{\Upphi}}_{1} ( {\mathbf{p}} ), \ldots ,{{\Upphi}}_{m} ( {\mathbf{p}} )} \right]^{T} = {\mathbf{b}} + {\mathbf{Ap}} + {\mathbf{W}}^{T} r( {\mathbf{p}} ) $$

(7)

where ${\mathbf{b}} + {\mathbf{Ap}}$ defines the affine component of the deformation, b (m × 1), A (m × m), and W (k × m) are represented by coefficients α and β in Eqns. 5 and 6, and $r( {\mathbf{p}} ) = \left( {R\left( {{\mathbf{p}} - {\mathbf{q}}_{1} } \right), \ldots ,R\left( {{\mathbf{p}} - {\mathbf{q}}_{k} } \right)} \right)^{T}$ is computed using the logarithmic interpolants $R_{i} = r_{i}^{2} \log r_{i}$ (for 2-D) and $R_{i} = r_{i}$ (for 3-D), where r _i represent the Euclidean distances between each landmark and the interpolation point in the reference (Bookstein 1992; Dryden and Mardia 1998). Unknown coefficients in Eqn. 7 are estimated using the set of linear equations

$$ \left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\mathbf{R}} \\ {{\mathbf{{1}}_{k}^{T}} } \\ {{\mathbf{Q}}^{T} } \\ \end{array} } & {\begin{array}{*{20}c} {{\mathbf{1}_{k}} } \\ 0 \\ 0 \\ \end{array} } & {\begin{array}{*{20}c} {\mathbf{Q}} \\ 0 \\ 0 \\ \end{array} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\mathbf{W}} \\ {{\mathbf{b}}^{T} } \\ {{\mathbf{A}}^{T} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {\mathbf{P}} \\ 0 \\ 0 \\ \end{array} } \right] $$

(8)

where $R_{ij}$ elements from R (k × k) are computed using the interpolation function between all possible pairs of landmarks in the reference (using $R_{ij} = 0$ for i = j), 1 _k (k × 1) is a vector of ones, and Q and P (k × m) are the reference and target configurations being compared, respectively (Dryden and Mardia 1998).

Smoothing Splines

One generalization of the spline methodology treats the choice and adjustment of interpolant functions as an estimation problem, where landmarks are assumed to be measured with error that is minimized prior to computation of the splines (Hutchinson and Gessler 1994). In this approach, sampled landmarks q are modeled as instances of an intrinsic function $g\left( {\mathbf{q}} \right)$, so that $q_{i} = g\left( {q_{i} } \right) + \varepsilon_{i} \left( {q_{i} } \right)$, where $\varepsilon_{i}$ denotes an error function (Hutchinson and Gessler 1994; Laslett 1994). The idea is to choose the function g that best predicts the values at sampled landmarks (Laslett 1994). The optimal function is the one that minimizes the quantity

$$ \mathop \sum \limits_{i = 1}^{n} \left[ {q_{i} - g\left( {q_{i} } \right)} \right]^{2} + {{\uplambda}}\rho \left( {g\left( {\mathbf{q}} \right)} \right) $$

where λ > 0 is a smoothing parameter, estimable through cross-validation (Wahba 1990; Hutchinson and Gessler 1994), and ρ is as defined in Eqns. 3 and 4, so that λρ can be treated as a roughness penalty (Dryden and Mardia 1998). When TPS are used as the g function, the system in Eqn. 8 becomes

$$ \left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {{\mathbf{R}} + {{\uplambda}}{\mathbf{I}}_{k} } \\ {{\mathbf{{1}}_{k}^{T}} } \\ {{\mathbf{Q}}^{T} } \\ \end{array} } & {\begin{array}{*{20}c} {{\mathbf{1}}_{k} } \\ 0 \\ 0 \\ \end{array} } & {\begin{array}{*{20}c} {\mathbf{Q}} \\ 0 \\ 0 \\ \end{array} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\mathbf{W}} \\ {{\mathbf{b}}^{T} } \\ {{\mathbf{A}}^{T} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {\mathbf{P}} \\ 0 \\ 0 \\ \end{array} } \right] $$

(9)

which also can be used for interpolation. Smoothing splines could be used to test among alternative spatial patterns of roughness penalties ($\lambda$) given an interpolation function, as well as for choosing among alternative g functions. Simulations have shown, however, that the use of cross-validation to optimally fit splines for spatial prediction often leads to over-smoothing and a consequent loss of valuable spatial information, an issue that seems related to fluctuations in the spatial density of the samples used for interpolation (Laslett 1994).

Kriging

Kriging is a method for linear prediction of the value of a variable using the information contained in a set of neighboring samples (Laslett 1994). Kriging presupposes the existence of a random field $\left\{ {Z( {\mathbf{p}} ) :{\mathbf{p}} \in D \subset \mathbb{R}^{m} } \right\}$ spanning a continuous domain D and observed at sampled points q, which determines the value of the variable of interest at each location (Matheron 1973; Schabenberger and Gotway 2005). The value at any arbitrary site p is given by the linear predictor $\hat{Z}( {\mathbf{p}} )$, defined by

$$ \hat{Z}( {\mathbf{p}} ) = \mathop \sum \limits_{i = 1}^{k} {{\uplambda}}_{i} Z\left( {{\mathbf{q}}_{i} } \right) = {{\Uplambda}}Z\left( {\mathbf{q}} \right) $$

where ${{\Uplambda}} = \left[ {{{\uplambda}}_{1} , \ldots ,{{\uplambda}}_{k} } \right]$ for k landmarks, and λ _i are weights for the q landmarks, which are computed based on the information content at each landmark about the true value of the random field at the estimation site $Z( {\mathbf{p}} )$ (Goovaerts 1997). The goal of kriging is to find the weights λ _i that minimize the mean square prediction error ${\text{MSPE}}( {\mathbf{p}} ) = E\left[ {\left( {Z( {\mathbf{p}} ) - \hat{Z}( {\mathbf{p}} )} \right)^{2} } \right]$, under the unbiasedness constraint $E\left[ {\hat{Z}( {\mathbf{p}} ) - Z( {\mathbf{p}} )} \right] = 0$ (Goovaerts 1997).

The central element of kriging prediction is the covariance function $\sigma \left( {t - s} \right)$, which describes the relationship between the difference between points t and s, and their values in the random field, i.e., $Z\left( t \right) - Z\left( s \right)$ (Cressie 1993). A common approach in spatial analysis consists of fitting an arbitrary continuous function to an empirical (discrete) variogram and using this function to derive a covariance function (Goovaerts 1997; Cressie 1993). More generally, kriging estimates the covariance function underlying the residual portion $\varepsilon ( {\mathbf{p}} )$ of the random field Z given the linear model

$$ Z( {\mathbf{p}} ) = \mathop \sum \limits_{j = 1}^{h} \beta_{j} g_{j} ( {\mathbf{p}} ) + \mathop \sum \limits_{i = 1}^{k} \alpha_{i} K\left( {\left\| {{\mathbf{p}} - {\mathbf{q}}_{i} } \right\|} \right) $$

(10)

where $\sum\nolimits_{i = 1}^{k} {\alpha_{i} K\left( {{\mathbf{p}}_{i} } \right)} = \varepsilon ( {\mathbf{p}} )$ defines the residual and $\sum\nolimits_{i = 1}^{h} {\beta_{i} g_{i} ( {\mathbf{p}} )} = E\left[ {Z( {\mathbf{p}} )} \right]$ the trend of the field, with $g( {\mathbf{p}} ) = \left[ {g_{1} ( {\mathbf{p}} ), \ldots ,g_{h} ( {\mathbf{p}} )} \right]$ a linear combination of known functions (Matheron 1973; Hutchinson and Gessler 1994). A comparison between Eqns. 5 and 10 suggests similarities between the trend of the random field in kriging and the uniform component of a deformation in terms of splines, and between the unique residuals at each point estimated using kriging with the non-affine component of the interpolating splines. In fact, when the rank of the function monomials $g( {\mathbf{p}} )$ equals the degree of the derivative used to estimate the roughness parameters in splines (see Eqns. 3 and 4) minus one, kriging and interpolating splines are identical methods (Hutchinson and Gessler 1994).

Alternative kriging strategies differ in their treatment of the trend as a universal constant, as a local constant, or as a function, such as an interpolant function, upon which locally predicted mean values are modeled (Goovaerts 1997). The latter, termed universal kriging has been developed in the context of shape deformations by Mardia et al. (1996), whose work has demonstrated that 2-D TPS are in fact universal kriging predictors with covariance function given by $\sigma \left( r \right) = r_{i}^{2} \log r_{i}$.

Finite Elements

Finite element methods (FEM) extract local deformation information by generalization or interpolation of the displacement at landmarks forming the vertices of rigid polygons or polyhedrons (Lewis et al. 1980). Two types of FEM have been applied in morphometric studies, namely homogeneous FE, in which shape changes within an element are assumed to be uniform throughout (Bookstein 1986; Moss et al. 1987), and non-homogeneous FE, whereby interpolation is used to model the deformation as a non-linear function of the changes observed at landmarks (Lewis et al. 1980; Cheverud et al. 1983; McAlarney 1995; McAlarney and Chiu 1997).

In homogeneous FEM, a set of normalized landmark displacements of an individual with respect to a reference configuration, A, are used to estimate tensors and principal strains via eigendecomposition of the sum of squares and cross-products matrix A ^T A (Bookstein 1986; Moss et al. 1987). The resulting eigenvectors can be envisaged as the principal axes of the ellipse that results from deforming the circumference that passes through all of the vertices of an element in the reference configuration to match the target configuration (Dryden and Mardia 1998). Non-homogeneous FEM, on the other hand, use normalized landmark displacements between reference and target configuration to parameterize a system of continuous functions whose purpose is to find a common set of coefficients that explain both these displacements and all the deformations inside the element boundaries (Lewis et al. 1980). If we denote a shape transformation at the ith vertex of a triangle as $T_{i} = Q_{i} + R_{i} ,\;\;\left( {i = { 1},{ 2},{ 3}} \right)$, where T and Q represent the target and reference configurations, respectively, and R represents a deformation, then for any arbitrary point p within Q, we can write the interpolated value of the deformation at this point as

$$ r_{p} = \mathop \sum \limits_{i = 1}^{3} {{\uplambda}}_{i} R_{i} $$

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(Lewis et al. 1980) where λ denotes vertex-specific weights derived from an interpolation function. In practice, Eqn. 11 can be parameterized using polynomials (Pepper and Heinrich 1992), or an interpolator, such as TPS (Bookstein 1986).

Both types of FEM require a priori choices regarding the shape and location of finite elements, and the absence of unambiguous criteria to make this choice (Cheverud and Richtsmeier 1986) is generally perceived a weakness of the method (McAlarney and Chiu 1997; Dryden and Mardia 1998). In the case of homogeneous FEM, the assumption of spatial uniformity of interpolated values suggests a preference for elements of small size, the optimal choice being to use triangles (2-D) and tetrahedrons (3-D), compiled together as a mesh (Pepper and Heinrich 1992). Even though this choice should lead to relatively precise estimates, it also imposes undesirable restrictions, namely the fact that triangles can only undergo affine deformations (Dryden and Mardia 1998), limiting their usefulness as shape descriptors. Non-homogeneous FEM, on the other hand, can be applied to more complex volumes (e.g., Cheverud et al. 1983), which in turn requires making ad hoc choices regarding which landmarks to use as element vertices, once again incorporating arbitrary elements in the measurement of shape (Zienkiewicz 1971; McAlarney and Chiu 1997; Dryden and Mardia 1998). Additional issues with FEM are the lack of a standard approach for computation of a mean and registration (Cheverud et al. 1983), and the absence of a clear-cut separation of global and local components of a deformation, due to the fact that both components are defined only within elements.

Comparison Among Alternative Methods

The common purpose of shape analysis by interpolation, smoothing, or kriging methods is the estimation or prediction of an unknown local deformation value Z(p) as a function of its spatial location, based on observed landmark deformations, Z(q). Measurements Z(q) are used to infer or parameterize an underlying function g[Z(p)], which is then applied throughout the whole structure under analysis. An analogous function is also derived in non-homogeneous FEM to estimate values within FE.

The function g[Z(p)] comprises two parts: an affine component that captures the portion of the total shape deformation that is shared throughout the whole structure under analysis, and a non-affine component that captures local deviations from the global trend, modeled using a covariance function tracking the influence of sampled points as a function of the distances amongst them. Splining methods select the function g among the universe of possible functions that minimizes the deformations required to explain the differences between two shapes (Bookstein 1986), according to an a priori deformation model. In kriging, on the other hand, g defines a random field whereby local deformations, whether measured or predicted, are treated as stochastic realizations of an intrinsic process (Matheron 1973). In practical applications, however, the two methods largely overlap (Hutchinson and Gessler 1994).

Thus, interpolation-based shape analyses by smoothing, kriging or finite elements methods are all closely related, and become indistinguishable from interpolating splines under the circumstances we have outlined.