  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.
