Background: In Diffusion Tensor Imaging (DTI), Riemannian framework
based on Information Geometry theory has been proposed for
processing tensors on estimation, interpolation, smoothing,
regularization, segmentation, statistical test and so on. Recently
Riemannian framework has been generalized to Orientation
Distribution Function (ODF) and it is applicable to any Probability
Density Function (PDF) under orthonormal basis representation.
Spherical Polar Fourier Imaging (SPFI) was proposed for ODF and
Ensemble Average Propagator (EAP) estimation from arbitrary sampled
signals without any assumption. Purpose: Tensors only can represent
Gaussian EAP and ODF is the radial integration of EAP, while EAP has
full information for diffusion process. To our knowledge, so far
there is no work on how to process EAP data. In this paper, we
present a Riemannian framework as a mathematical tool for such task.
Method: We propose a state-of-the-art Riemannian framework for EAPs
by representing the square root of EAP, called wavefunction based on
quantum mechanics, with the Fourier dual Spherical Polar Fourier
(dSPF) basis. In this framework, the exponential map, logarithmic
map and geodesic have closed forms, and weighted Riemannian mean and
median uniquely exist. We analyze theoretically the similarities and
differences between Riemannian frameworks for EAPs and for ODFs and
tensors. The Riemannian metric for EAPs is diffeomorphism invariant,
which is the natural extension of the affine-invariant metric for
tensors. We propose Log-Euclidean framework to fast process EAPs,
and Geodesic Anisotropy (GA) to measure the anisotropy of EAPs. With
this framework, many important data processing operations, such as
interpolation, smoothing, atlas estimation, Principal Geodesic
Analysis (PGA), can be performed on EAP data. Results and
Conclusions: The proposed Riemannian framework was validated in
synthetic data for interpolation, smoothing, PGA and in real data
for GA and atlas estimation. Riemannian median is much robust for
atlas estimation.