The TPS Direct Transport: A New Method for Transporting Deformations in the Size-and-Shape Space

Abstract

Modern shape analysis allows the fine comparison of shape changes occurring between different objects. Very often the classic machineries of generalized Procrustes analysis and principal component analysis are used in order to contrast the shape change occurring among configurations represented by homologous landmarks. However, if size and shape data are structured in different groups thus constituting different morphological trajectories, a data centering is needed if one wants to compare solely the deformation representing the trajectories. To do that, inter-individual variation must be filtered out. This maneuver is rarely applied in studies using simulated or real data. A geometrical procedure named parallel transport, that can be based on various connection types, is necessary to perform such kind of data centering. Usually, the Levi Civita connection is used for interpolation of curves in a Riemannian space. It can also be used to transport a deformation. We demonstrate that this procedure does not preserve some important characters of the deformation, even in the affine case. We propose a novel procedure called ‘TPS Direct Transport’ which is able to perfectly transport deformation in the affine case and to better approximate non affine deformation in comparison to existing tools. We recommend to center shape data using the methods described here when the differences in deformation rather than in shape are under study.

Appendix

In the following we derive the right value of $\nu$ coefficient used for quantifying the absolute value of the bending energy (Eqs. 5.8, 5.9).

We start from the Eq. (3.2) from Kent, J. T., & Mardia, K. V. (1994). The link between kriging and thin plate splines. In Probability, Statistics and Optimization, Ed. F. P. Kelly, pp. 324–339. New York: Wiley.

$$\begin{aligned} \sigma _\alpha (h)= & {} c_{\alpha ,d}|h|^{2\alpha } \alpha>0, \alpha ~ \text {not an integer}\\ \sigma _k(h)= & {} b_{k,d}|h|^{2k} \log |h| \alpha =k, k>0~ \text {an integer} \end{aligned}$$

where

$$\begin{aligned} c_{\alpha ,d}= & {} 2^{-2\alpha }\pi ^{d/2}\varGamma (-\alpha )/\varGamma (k+d/2)\\ b_{k,d}= & {} 2^{-2k+1}(-1)^{k-1}\pi ^{d/2}/\{\varGamma (k+d/2)k!\} \end{aligned}$$

In the same paper the Eq. (5.3) gives the expression for the Bending Energy:

$$\begin{aligned} J^d_{r+1}(y^*)=(2\pi )^2 y^TBy \end{aligned}$$

where $r=1=\alpha$.

1.1 Two Dimensional Case

In the two dimensional case $d=2,\alpha =1=k$ (Theorem 1 p. 333 Kent and Mardia 1994).

$$\begin{aligned} \sigma _1(h)= & {} b_{k,d}|h|^2 \log |h|\\ b_{k,d}= & {} b_{1,2}=2^{-1}(-1)^0\frac{\pi ^{2/2}}{\varGamma \left( 1+\frac{1}{2}\right) 1!} =\frac{1}{2}\frac{\pi }{\varGamma \left( 2\right) }=\frac{\pi }{2} \end{aligned}$$

Covariation function $\sigma (h)=\frac{\pi }{2}|h|^2 \log |h|$ leads to Bending Energy $J=(2\pi )^2 y^TBy$.

So, with $\sigma (h)=|h|^2 \log |h|$ we obtain a Bending Energy

$$\begin{aligned} J=(2\pi )^2 y^TBy \times \frac{2}{\pi }= 8\pi y^T{ By} \end{aligned}$$

If we use $\sigma (h)=|h|^2 \log |h|^2=2|h|^2 \log |h|$ we obtain a Bending Energy

$$\begin{aligned} J=(2\pi )^2 y^TBy \times \frac{4}{\pi }= 16\pi y^T{ By} \end{aligned}$$

1.2 Three Dimensional Case

In the three dimensional case $d=3,\alpha =1/2$ (Theorem 1 p. 333 Kent and Mardia 1994).

$$\begin{aligned} \sigma _{1/2}(h)= & {} c_{\alpha ,d}|h|^{1/2}\\ c_{1/2,3}= & {} 2^{-1}\frac{\pi ^{3/2}\varGamma (-1/2)}{\varGamma (1/2+3/2)}=\frac{1}{2}\frac{\pi ^{3/2}(-2)\sqrt{\pi }}{\varGamma (2)}=-\pi ^2 \end{aligned}$$

where we used the properties $\varGamma (2)=1$ and $\varGamma (-1/2)=-2\sqrt{\pi }$.

So $\sigma _{1/2}(h)=-\pi ^2|h|^{1/2}$ gives $J=(2\pi )^3 y^TBy$.

Hence $\sigma (h)=-|h|^{1/2}$ gives

$$\begin{aligned} J=\frac{8\pi ^3}{\pi ^2} y^TBy = 8\pi y^TBy. \end{aligned}$$