An Integrated Approach for Landmark-Based Resistant Shape Analysis in 3D

Abstract

The study of shape changes in morphology has seen a significant renovation in the last 20 years, particularly as a consequence of the development of geometric morphometric methods based on Cartesian coordinates of points. In order to extract information about shape differences when Cartesian coordinates are used, it is necessary to establish a common reference frame or system for all specimens to be compared. Therefore, a central issue in coordinate-based methods is which criterion should be used to align these configurations of points, since shape differences highly depend on those alignments. This is usually accomplished by aligning the configurations in a way that the sum of squared distances between coordinates of homologous points (landmarks) is minimized: the least-squares superimposition method. However, it is widely recognized that this method has some limitations when shape differences are not homogeneous across landmarks. Here we present an integrated approach for the resistant shape comparison of 3D landmark sets. It includes a new ordinary resistant Procrustes superimposition and its corresponding generalized resistant Procrustes version. In addition, they are combined with existing resistant multivariate statistical techniques for depicting the results. We demonstrate, by using both simulated and real datasets, that resistant Procrustes better detects and measures localized shape variation whenever present in up to half but one of the landmarks. The resistant Procrustes results are highly concordant with a priori biological information, and might dramatically improve the quality of inferences on patterns of shape variation.

Appendix: Optimality of the Presented Resistant Method (Overall Performance When Shape Variation is Located in Half but One of the Landmarks)

The 3D version of the resistant method presented in this work achieves the best possible superimposition whenever shape differences are located in up to 50 % of the landmarks. This is proved next.

Theorem

Suppose there exist parameters ρ > 0 (an isotropic scaling), R _{(v , θ)} (a 3D rotation matrix with associated rotation angle θ and rotation axis v , respectively) and t (a translation vector) such that the equation:

$$ {\mathbf{l}}_{i}^{(2)} = \rho {\mathbf{l}}_{i}^{(1)} {\mathbf{R}}_{{({\mathbf{v}},\theta )}}\, { + } \, {\mathbf{t}} $$

(14)

holds for more than $\frac{{{n} + 1}}{2}$ different landmarks, the repeated-medians estimates (6), (7), (9) and the single median (11) are exactly those values; that is,

$$ \mathop {\text{med} }\limits_{i} \,(\mathop {\text{med} }\limits_{{{j} \ne {i}}} \, \rho_{ij} ) = \widetilde{\rho }, $$

$$ \mathop {\text{med} }\limits_{i}\, (\mathop {\text{med} }\limits_{{{j} \ne {i}}} \, {\mathbf{v}}_{ij} ) = \widetilde{{\mathbf{v}}}, $$

$$ \mathop {\text{med} }\limits_{i} \,(\mathop {\text{med} }\limits_{{{j} \ne {i}}} \, \theta_{ij} ) = \tilde{\theta }, $$

and

$$ \mathop {\text{med} }\limits_{i} \, {\mathbf{t}}_{i} = \widetilde{{\mathbf{t}}}. $$

Proof

The result is showed only for the rotation matrix parameters, because they use the specific formulation for the 3D case. The proof for the remaining parameters is analogous.

Whenever more than $\frac{{{n} + 1}}{2}$ in a set of values are the same, the median is that repeated value. Then, when landmarks l ^(k) _i and l ^(k) _j satisfy (14):

$$ {\mathbf{l}}_{j}^{(2)} - {\mathbf{l}}_{i}^{(2)} = \rho ({\mathbf{l}}_{j}^{(2)} - {\mathbf{l}}_{i}^{(2)} ){\mathbf{R}}_{{({\mathbf{v}},\theta )}}\, { + }\,{\mathbf{t}}, $$

(15)

and setting for the moment ρ = 1 (the scale factor can be estimated afterwards, without loss of generality) the rotation matrix R _ij satisfies equations (1), (2), (3) and consequently (4), producing:

$$ {\mathbf{R}}_{ij} = {\mathbf{R}}_{{({\mathbf{v}}_{ij} ,\theta_{ij} )}} = {\mathbf{R}}_{{({\mathbf{v}},\theta )}}. $$

From this, whenever landmark l ^(k) _i satisfies (14), more than half of the remaining landmarks l ^(k) _j (j ≠ i) also satisfy (14) and therefore:

$$ \mathop {\text{med} }\limits_{{{j} \ne {i}}} \, {\mathbf{v}}_{ij} = {\mathbf{v}},\,\left( {\text{componentwise median}} \right) $$

$$ \mathop {\text{med} }\limits_{{{j} \ne {i}}} \, \theta_{ij} = \theta , $$

which leads to:

$$ \mathop {\text{med} }\limits_{i}\, (\mathop {\text{med} }\limits_{{{j} \ne {i}}} \, {\mathbf{v}}_{ij} ) = \widetilde{{\mathbf{v}}} = {\mathbf{v}}, $$

$$ \mathop {\text{med} }\limits_{i} \,(\mathop {\text{med} }\limits_{{{j} \ne {i}}} \, \theta_{ij} ) = \widetilde{\theta } = \theta, $$

and finally:

$$ \tilde{\bf{R}}_{{(\tilde{\bf{v}},\tilde{\theta })}} \, = \,\bf{R}_{(\bf{v},\theta )}, $$

which concludes the proof.