Diffeomorphic Image Registration of Diffusion MRI Using Spherical Harmonics
ABSTRACT Nonrigid registration of diffusion magnetic resonance imaging (MRI) is crucial for group analyses and building white matter and fiber tract atlases. Most current diffusion MRI registration techniques are limited to the alignment of diffusion tensor imaging (DTI) data. We propose a novel diffeomorphic registration method for high angular resolution 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 intra-voxel fiber crossings. The registration is based on optimizing a diffeomorphic demons cost function. Unlike scalar images, deforming ODF maps requires ODF reorientation to maintain its consistency with the local fiber orientations. Our method simultaneously reorients the ODFs by computing a Wigner rotation matrix at each voxel, and applies it to the SH coefficients during registration. Rotation of the coefficients avoids the estimation of principal directions, which has no analytical solution and is time consuming. The proposed method was validated on both simulated and real data sets with various metrics, which include the distance between the estimated and simulated transformation fields, the standard deviation of the general fractional anisotropy and the directional consistency of the deformed and reference images. The registration performance using SHs with different maximum orders were compared using these metrics. Results show that the diffeomorphic registration improved the affine alignment, and registration using SHs with higher order SHs further improved the registration accuracy by reducing the shape difference and improving the directional consistency of the registered and reference ODF maps.
- SourceAvailable from: Julio Martin Duarte-Carvajalino[Show abstract] [Hide abstract]
ABSTRACT: Registration of diffusion-weighted magnetic resonance images (DW-MRIs) is a key step for population studies, or construction of brain atlases, among other important tasks. Given the high dimensionality of the data, registration is usually performed by relying on scalar representative images, such as the fractional anisotropy (FA) and non-diffusion-weighted (b0) images, thereby ignoring much of the directional information conveyed by DW-MR datasets itself. Alternatively, model-based registration algorithms have been proposed to exploit information on the preferred fiber orientation(s) at each voxel. Models such as the diffusion tensor or orientation distribution function (ODF) have been used for this purpose. Tensor-based registration methods rely on a model that does not completely capture the information contained in DW-MRIs, and largely depends on the accurate estimation of tensors. ODF-based approaches are more recent and computationally challenging, but also better describe complex fiber configurations thereby potentially improving the accuracy of DW-MRI registration. A new algorithm based on angular interpolation of the diffusion-weighted volumes was proposed for affine registration, and does not rely on any specific local diffusion model. In this work, we first extensively compare the performance of registration algorithms based on (i) angular interpolation, (ii) non-diffusion-weighted scalar volume (b0), and (iii) diffusion tensor image (DTI). Moreover, we generalize the concept of angular interpolation (AI) to non-linear image registration, and implement it in the FMRIB Software Library (FSL). We demonstrate that AI registration of DW-MRIs is a powerful alternative to volume and tensor-based approaches. In particular, we show that AI improves the registration accuracy in many cases over existing state-of-the-art algorithms, while providing registered raw DW-MRI data, which can be used for any subsequent analysis.Frontiers in Neuroscience 01/2013; 7:41.
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ABSTRACT: Diffusion MRI provides important information about the brain white matter structures and has opened new avenues for neuroscience and translational research. However, acquisition time needed for advanced applications can still be a challenge in clinical settings. There is consequently a need to accelerate diffusion MRI acquisitions. A multi-task Bayesian compressive sensing (MT-BCS) framework is proposed to directly estimate the constant solid angle orientation distribution function (CSA-ODF) from under-sampled (i.e., accelerated image acquisition) multi-shell high angular resolution diffusion imaging (HARDI) datasets, and accurately recover HARDI data at higher resolution in q-space. The proposed MT-BCS approach exploits the spatial redundancy of the data by modeling the statistical relationships within groups (clusters) of diffusion signal. This framework also provides uncertainty estimates of the computed CSA-ODF and diffusion signal, directly computed from the compressive measurements. Experiments validating the proposed framework are performed using realistic multi-shell synthetic images and in vivo multi-shell high angular resolution HARDI datasets. Results indicate a practical reduction in the number of required diffusion volumes (q-space samples) by at least a factor of four to estimate the CSA-ODF from multi-shell data. This work presents, for the first time, a multi-task Bayesian compressive sensing approach to simultaneously estimate the full posterior of the CSA-ODF and diffusion-weighted volumes from multi-shell HARDI acquisitions. It demonstrates improvement of the quality of acquired datasets by means of CS de-noising, and accurate estimation of the CSA-ODF, as well as enables a reduction in the acquisition time by a factor of two to four, especially when "staggered" q-space sampling schemes are used. The proposed MT-BCS framework can naturally be combined with parallel MR imaging to further accelerate HARDI acquisitions. Magn Reson Med, 2013. © 2013 Wiley Periodicals, Inc.Magnetic Resonance in Medicine 12/2013; · 3.27 Impact Factor
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ABSTRACT: We propose a large deformation diffeomorphic metric mapping algorithm to align multiple b-value diffusion weighted imaging (mDWI) data, specifically acquired via hybrid diffusion imaging (HYDI), denoted as LDDMM-HYDI. We then propose a Bayesian model for estimating the white matter atlas from HYDIs. We adopt the work given in Hosseinbor et al. (2012) and represent the q-space diffusion signal with the Bessel Fourier orientation reconstruction (BFOR) signal basis. The BFOR framework provides the representation of mDWI in the q-space and thus reduces memory requirement. In addition, since the BFOR signal basis is orthonormal, the L2 norm that quantifies the differences in the q-space signals of any two mDWI datasets can be easily computed as the sum of the squared differences in the BFOR expansion coefficients. In this work, we show that the reorientation of the $q$-space signal due to spatial transformation can be easily defined on the BFOR signal basis. We incorporate the BFOR signal basis into the LDDMM framework and derive the gradient descent algorithm for LDDMM-HYDI with explicit orientation optimization. Additionally, we extend the previous Bayesian atlas estimation framework for scalar-valued images to HYDIs and derive the expectation-maximization algorithm for solving the HYDI atlas estimation problem. Using real HYDI datasets, we show the Bayesian model generates the white matter atlas with anatomical details. Moreover, we show that it is important to consider the variation of mDWI reorientation due to a small change in diffeomorphic transformation in the LDDMM-HYDI optimization and to incorporate the full information of HYDI for aligning mDWI.Medical Image Analysis. 09/2013;
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) ))())()( ((
, 0∑ ∑
(2) || ||
even , 0
(5) ) Regρ
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).