We propose a linear-elastic registration method to register diffusion-weighted MRI (DW-MRI) data sets by mapping their diffusion orientation distribution functions (ODFs). The ODFs were reconstructed using a q-ball imaging (QBI) technique to resolve intravoxel fiber crossing. The registration method is based on mapping the ODF maps represented by spherical harmonics which yield analytic solutions and reduce the computational complexity. ODF reorientation is required to maintain the consistency with transformed local fiber directions. The reorientation matrices are extracted from the local Jacobian and directly applied to the coefficients of spherical harmonics. The similarity cost of the registration is defined by the ODF shape distance calculated from the spherical harmonic coefficients. The transformation fields are regularized by linear elastic constraints. The proposed method was validated using both synthetic and real data sets. Experimental results show that the elastic registration improved the affine alignment by further reducing the ODF shape difference; reorientation during the registration produced registered ODF maps with more consistent principle directions compared to registrations without reorientation or simultaneous reorientation.