Inter-subject affine registration
Inter-subject diffeomorphic registration
norm stdev gFA
Tab.1. Normalized standard deviation of gFA and average DC in intra-subject (affine align the data in the 2nd session
to the 1st), inter-subject affine registration and diffeomorphic registration with different orders of SHs.
(6) ))())()( ((
12 12 12
, 0∑ ∑
(2) || ||
even , 0
(5) ) Regρ
) (v)(h (F D(R
Diffeomorphic Image Registration of Diffusion MRI Using Spherical Harmonics
X. Geng1, H. Gu1, W. Zhan2, W. Shin1, Y-P. Chao3, N. Schuff4, C-P. Lin5, and Y. Yang1
1Neuroimaging, National Institute on Drug Abuse, NIH, Baltimore, MD, United States, 2Department of Radiology, University of California, San Francisco, 3Department
of Electrical Engineering, National Taiwan University, Taiwan, 4Radiology, University of California, San Francisco, 5Institute of Brain Science, National Yang-Ming
INTRODUCTION: Nonlinear registration of diffusion MRI is crucial for group analyses and for building a white matter and fiber tract atlas. Most current diffusion
MRI registration techniques [1-2] are limited to the alignment of diffusion tensor imaging (DTI) data. Here we propose a novel diffeomorphic registration method for
diffusion images by mapping their orientation distribution functions (ODFs). ODFs can be reconstructed using q-ball imaging (QBI) techniques  and represented by
spherical harmonics (SHs) to resolve intravoxel fiber crossings. The registration is based on optimizing a diffeomorphic demons cost function. Unlike scalar images,
deforming ODF maps requires ODF reorientation to maintain the consistency with the local fiber orientations. Our method simultaneously reorients the ODFs by
computing a Wigner rotation matrix at each voxel, and applies it on the SH coefficients during registration. Rotations on the coefficients avoid the estimation of
principle directions, which has no analytical solution and is time consuming. The
performance of the proposed method was tested using different SH orders. Results show
that registration with higher orders (2nd-order is equivalent to tensor models ) improves
the performance in terms of smaller similarity error and higher directional consistency.
METHODS: Theory: The ODF, F(u), reconstructed by a great circle integration on the
sphere of the diffusion-attenuated signal in q-space , can be represented as a linear
combination of a set of real SH basis with order l and phase factor m (Eq.(1)), using only
even orders due to the antipodal symmetry. The invariant shape distance between two
ODFs can be defined as the Euclidean distance between their coefficients (Eq.(2)). A 3D
rotation can be decomposed to three Euler angles using z-y-z convention (Eq.(3)). The real
SH coefficients can be rotated in a similar way
as vectors with Wigner matrices (Eq.(4)) .
The registration was stated as an optimization
problem of finding spatial transformation h12
that minimizes a cost function defined in
Eq.(5). The rotation was extracted from the
inverse of the local Jacobian of h12 as in Eq.(6).
h12 was initialized to identity and the velocity
field υ12 was initialized to zeros and updated
iteratively by gradient decent optimization.
The updated υ12 was composed to h12 to obtain
the updated transformation.
Data acquisition: Human brain QBI data from
five healthy subjects were acquired at a 3T Siemens scanner. Each subject was scanned twice.
Isotropic axial images were obtained using a single-shot diffusion spin-echo EPI sequence with
TR/TE=8s/114 ms, FOV=195mm, matrix=78x78, yielding a 2.5mm image resolution. 162
diffusion encoding directions with a b-value of 3000 s/mm2 and one reference image were
acquired. 48 slices with slice thickness=2.5mm were obtained to cover the whole brain. The
total scan time was approximately 26 minutes.
Experimental design: Three mixtured zero-mean Gaussian diffusion tensors were
simulated with SNR=100 using “Camino”  to validate the rotation along y, z-axes. A
25o z-rotation of a real QBI data with and without ODF reorientation was simulated to
demonstrate the necessity of ODF reorientation during spatial alignment. To test the
proposed method, two metrics, generalized fractional anisotropy (gFA)  and directional
consistency (DC) were computed after affine and the proposed method with different
orders of SHs. DC was defined by the cosine value of the rotation angle that minimizes
two spherical shapes.
RESULTS & DISCUSSION: Fig.1(a-d) illustrate the ODF reorientation by rotating
coefficients with simulated 3-tensor data sets and (e-f) demonstrate that ODF reorientation
ensures the consistency with rotated fiber orientations. Fig.2 shows that diffeomorphic
registration aligns the macro structure of the target image to the reference more accurately
compared to affine registration; registration with higher orders maintains fiber crossings
and results in more precise ODF alignments. Fig.3 plots histograms of the normalized
standard deviation of gFA and the average DC after diffeomorphic registration with
different SH orders. With higher orders, gFA had smaller standard deviation and DC had
values closer to 1, indicating less shape similarity error and higher directional consistency.
Tab.1 summarizes the two measures for intra-subjects, and affine and diffeomorphic
registrations of inter-subjects. The diffeomorphic registration with higher SH orders
produced values closer to the intra-subject case.
CONCLUSIONS: We have developed a novel diffeomorphic diffusion MRI
registration method by aligning the SH coefficients of their ODF maps. ODF reorientation was done by applying rotation matrices over the coefficients. Higher order
SH registration provides better registration performance compared to 2nd order SH registration comparable to current available DTI registration techniques.
REFERENCES: 1. Alexander et al.,
IEEE TMI, 2001. 2. Zhang et al., IEEE TMI,
2007. 3. Hess et al. MRM, 2006. 4.
Barmpoutis et al., MICCAI, 2007. 5. Geng et
al., IPMI 2009. 6. Cook et al., ISMRM, 2006.
7. Tuch, MRM, 2007.
Fig.3. Comparison of diffeomorphic ODF registrations using SH
coefficients with different orders.
(a) ODF maps (using SH
coefficients at l=m=0) of
subject 1 and subject 2 after
registration in axial and
(b) ODF maps
of subject 1 at
(c) ODF maps
of subject 2
Fig.2. Representative registration results after affine and diffeomorphic
aligning subject 2 to 1.
Fig.1. Illustration of the ODF reorientation with simulated 3-tensor data
sets (a-d) and real QBI data (e-f). (a) A simulated 3-tensor ODF; (b) & (c)
rotated ODF along z & y-axis, (d) SH coefficients of the three ODFs; (e)
original and rotated ODFs; (f) enlarged ODFs of the original data (left
panel), rotated data without & with reorientation (middle & right panel).