Adaptive registration of diffusion tensor images on lie groups

Abstract

With diffusion tensor imaging (DTI), more exquisite information on tissue microstructure is provided for medical image processing. In this paper, we present a locally adaptive topology preserving method for DTI registration on Lie groups. The method aims to obtain more plausible diffeomorphisms for spatial transformations via accurate approximation for the local tangent space on the Lie group manifold. In order to capture an exact geometric structure of the Lie group, the local linear approximation is efficiently optimized by using the adaptive selection of the local neighborhood sizes on the given set of data points. Furthermore, numerical comparative experiments are conducted on both synthetic data and real DTI data to demonstrate that the proposed method yields a higher degree of topology preservation on a dense deformation tensor field while improving the registration accuracy.

Appendix

Solution for cost function

Given the current transformation $\varphi ^k$, compute a update $\varphi ^{(k+1)}$ by minimizing

$$\begin{aligned}&E(I_r,I_f,\varphi )\\ &\quad=\mathrm{Sim}(I_r,R(D_i)^\mathrm{T}\mathrm{log}(I_f\circ \varphi )R(D_i))+\mathrm{Reg}(\varphi )\\ & \quad =\Vert I_r-R(D_i)^\mathrm{T}\mathrm{log}(I_f\circ \varphi )R(D_i)\Vert ^2+\Vert \varphi \Vert ^2 \end{aligned}$$

(23)

For convenience, let $\Lambda (\varphi )=I_r-R(D_i)^\mathrm{T}\mathrm{log}(I_f\circ \varphi )R(D_i)$, we assume the linearization of $\Lambda (\varphi )$ at a given point $D_i$,

$$\begin{aligned} &\Lambda (\varphi (D_i\circ \mathrm{exp}(u)))\\ &\quad =I_r-R(D_i)^\mathrm{T}\mathrm{log}(I_f\circ \varphi (D_i\circ \mathrm{exp}(u)))R(D_i)\\ &\quad =I_r-R(D_i)^\mathrm{T}\mathrm{log}(I_f\circ \varphi (D_i\circ \mathrm{exp}(0)))R(D_i)+J(D_i).\varphi (D_i\circ \mathrm{exp}(u)) \end{aligned}$$

(24)

Equation (23) can be rewritten with the linearization in Eq. (24),

$$\begin{aligned} E(I_r,I_f,\varphi ) \approx \Bigg \Vert \begin{bmatrix} I_r-R(D_i)^\mathrm{T}log(I_f\circ \varphi ^k)R(D_i)\\ 0 \end{bmatrix} + \begin{bmatrix} J(D_i)\\ I \end{bmatrix} .\varphi ^{k+1} \Bigg \Vert ^2 \end{aligned}$$

(25)

the normal equation can be expressed in matrix form as follows,

$$\begin{aligned} \begin{bmatrix}J(D_i)^T\,I \end{bmatrix} . \begin{bmatrix} J(D_i)\\ I \end{bmatrix}.\varphi ^{k+1} =-\begin{bmatrix} J(D_i)^T&I \end{bmatrix}. \begin{bmatrix} I_r-R(D_i)^\mathrm{T}log(I_f\circ \varphi ^k)R(D_i)\\0 \end{bmatrix} \end{aligned}$$

(26)

which simplies into,

$$\begin{aligned} &(J(D_i)^T.J(D_i)+I).\varphi ^{k+1}\\ &\quad =-(I_r-R(D_i)^\mathrm{T}log(I_f\circ \varphi ^k)R(D_i)).J(D_i)^T\end{aligned}$$

(27)

solving for $\varphi ^{(k+1)}$, we achieve,

$$\begin{aligned} \varphi ^{k+1}=\frac{I_r-R(D_i)^\mathrm{T}\mathrm{log}(I_f\circ \varphi ^k)R(D_i)}{\Vert J(D_i)\Vert ^2+I}.J(D_i)^T \end{aligned}$$

(28)

where $J(D_i)=-\bigtriangledown R(D_i)^\mathrm{T}\mathrm{log}(I_f\circ \varphi ^k)R(D_i)$ and $R(D_i)=((J(D_i)J(D_i)^T)^{-\frac{1}{2}})J(D_i)$.

Finally, we can obtain the optimization solution by the Gauss–Newton iteration,