Diffeomorphic Metric Landmark Mapping Using Stationary Velocity Field Parameterization

Abstract

Large deformation diffeomorphic metric mapping (LDDMM) has been shown as an effective computational paradigm to measure anatomical variability. However, its time-varying vector field parameterization of diffeomorphism flow leads to computationally expensive implementation, as well as some theoretical issues in metric based shape analysis, e.g. high order metric approximation via Baker–Campbell–Hausdorff (BCH) formula. To address these problems, we study the role of stationary vector field parameterization in context of LDDMM. Under this setting registration is formulated as finding the Lie group exponential path with minimal energy in Riemannian manifold of diffeomorphisms bringing two shapes together. Accurate derivation of Euler–Lagrange equation shows that optimal vector field for landmark matching is associated with singular momenta at landmark trajectories in whole time domain, and a new momentum optimization scheme is proposed to solve the variational problem. Length of group exponential path is also proposed as an alternative shape metric to geodesic distance, and pair-wise metrics among a population are computed through an approximation method via BCH formula which only needs registrations to a template. The proposed methods have been tested on both synthesized data and real database. Compared to non-stationary parameterization, this method can achieve comparable registration accuracy in significantly reduced time. Second order metric approximation by this method also improves significantly over first order, which can not be achieved by non-stationary parameterization. Correlation between the two shape metrics is also investigated, and their statistical power in clinical study compared.

Appendix

In this Appendix, we compute the variation of transformation $\phi _1^v$ given variation direction $h$ for stationary vector field $v$. Variation ${\partial _h}\phi _1^v$ can be considered as the special case of the result for non-stationary parameterization, but for clarity its variational derivation will be given below. First we compute ${\partial _h}\phi _t^v$,

$$\begin{aligned} {\partial _h}\phi _t^v&= {\left. {\frac{{d\phi _t^{v + \varepsilon h}}}{{d\varepsilon }}} \right| _{\varepsilon = 0}}\\&= \frac{{d(\int _0^t {[v(\phi _t^{v + \varepsilon h}) + \varepsilon h(\phi _t^{v + \varepsilon h})]} \cdot dt)}}{{d\varepsilon }} \\&= \int _\mathrm{{0}}^t {[Dv(\phi _s^v){{\left. {\frac{{d\phi _s^{v + \varepsilon h}}}{{d\varepsilon }}} \right| }_{\varepsilon = 0}} + h(\phi _s^v)} ]ds \\ \end{aligned}$$

Differentiation of the above equation yields

$$\begin{aligned} \frac{{d({\partial _h}\phi _t^v)}}{{dt}} = Dv(\phi _t^v){\partial _h}\phi _t^v + h(\phi _t^v) \end{aligned}$$

(39)

The solution to the homogenous differential equations $\frac{{d({\partial _h}\phi _t^v)}}{{dt}} = Dv(\phi _t^v){\partial _h}\phi _t^v$ is $D\phi _t^v \cdot C$ ($C$ is a constant vector), because we have the following equation by differentiation of Eq. (6)

$$\begin{aligned} \frac{{d(D\phi _t^v)}}{{dt}} = Dv(\phi _t^v)D\phi _t^v \end{aligned}$$

It can be shown that the general solution to the non-homogenous differential equation of (39) has the form

$$\begin{aligned} {\partial _h}\phi _t^v = D\phi _t^v \cdot C + \int _0^t {D\phi _t^v{D^{ - 1}}\phi _s^vh(\phi _s^v)ds} \end{aligned}$$

Then from the initial condition ${\partial _h}\phi _\mathrm{{0}}^v = \mathrm{{0}}$, we get

$$\begin{aligned} {\partial _h}\phi _\mathrm{{1}}^v = \int _0^1 {D\phi _\mathrm{{1}}^v{D^{ - 1}}\phi _t^vh(\phi _t^v)dt} = \int _0^1 {D\phi _{t\mathrm{{1}}}^vh(\phi _t^v)dt} \end{aligned}$$