A canonical form-based approach to affine registration of DTI

Abstract

Due to the orientation feature of diffusion tensor images (DTI), tensors need to be reoriented during an affine registration. There exists two active reorientation schemes: finite strain (FS) and preserving principal direction (PPD). However, FS scheme limits its application on rigid deformation and PPD scheme suffers from computation load caused by the iteration. In order to overcome these shortcomings, we propose a canonical form-based affine registration of DTI, named as CFARD. We transform voxel sets into canonical forms where an affine registration is simplified as a rigid registration, while still preserves the effects of non-rigid components. This transforming thus extends the application of FS scheme to affine deformation. Furthermore, to reduce computation load, the quaternion technique is skillfully employed to seek a closed-form solution of the optimal rotation where no iteration is required. Extensive experiments are conducted on synthetic and real DTI data from the human brain. In contrast to four existing algorithms, the proposed CFARD improves the consistency between tensor orientation and the anatomical structures after deformation, and performs a better balance between accuracy and computational complexity.

Appendix

Let rotation matrix $R$ transform $S_{f}^{(k)}$ to $S_{r}^{(k)}$,

$$ S_{r}^{(k)}=RS_{f}^{(k)}. $$

(32)

Consider rotation by the unit quaternion $q$.

$$ S_{r}^{(k)}=qS_{f}^{(k)}q^{*}. $$

(33)

We try to seek $q$ by maximizing,

$$ \sum\limits_{k = 1}^{n}(qS_{r}^{(k)}q^{*})S_{f}^{(k)}. $$

(34)

According to the property of products of quaternions in Section 2.2, (33) can be rewritten as the following form,

$$ \sum\limits_{k = 1}^{n}(qS_{r}^{(k)})\dot(S_{f}^{(k)}q). $$

(35)

Assume $S_{r}^=({S_{x}^{r}}, {S_{y}^{r}}, {S_{z}^{r}})$, $S_{f}^=({S_{f}^{r}}, {S_{f}^{r}}, {S_{f}^{r}})$, then use multiplication of quaternions, we get,

$$ qS_{r}^{(k)}=\left[ \begin{array}{llll} 0&-{S_{x}^{r}}&-{S_{y}^{r}}&-{S_{z}^{r}}\\ {S_{x}^{r}}&0&{S_{z}^{r}}&-{S_{y}^{r}}\\ {S_{y}^{r}}&-{S_{z}^{r}}&0&{S_{x}^{r}}\\ {S_{z}^{r}}&{S_{y}^{r}}&-{S_{x}^{r}}&0 \end{array} \right]^{(k)}q=R_{r}q, $$

(36)

and

$$ S_{f}^{(k)}q=\left[ \begin{array}{llll} 0&-{S_{x}^{f}}&-{S_{y}^{f}}&-{S_{z}^{f}}\\ {S_{x}^{f}}&0&-{S_{z}^{f}}&{S_{y}^{f}}\\ {S_{y}^{f}}&-{S_{z}^{f}}&0&{S_{x}^{f}}\\ {S_{z}^{f}}&-{S_{y}^{f}}&{S_{x}^{f}}&0 \end{array} \right]^{(k)}q=R_{f}q. $$

(37)

thus, the problem of maximizing (34) is equivalent to,

$$ \sum\limits_{k = 1}^{n}(R_{r}q)\dot(R_{f}q)=\sum\limits_{k = 1}^{n}q^{\mathrm{T}}(R_{r}^{\mathrm{T}}\dot R_{f})q. $$

(38)

finally, we have,

$$ \sum\limits_{k = 1}^{n}q^{\mathrm{T}}N_{i}q=q^{\mathrm{T}}Nq.. $$

(39)

According to the method of [12], the optimal rotation $q$ corresponds to the eigenvector of the symmetric matrix $N$ associated with the most positive eigenvalue.