  Inferring brain connectivity and structure \textit{in-vivo} requires accurate
estimation of the orientation distribution function (ODF), which encodes key
local tissue properties. However, estimating the ODF from diffusion MRI (dMRI)
signals is a challenging inverse problem due to obstacles such as significant
noise, high-dimensional parameter spaces, and sparse angular measurements. In
this paper, we address these challenges by proposing a novel deep-learning
based methodology for continuous estimation and uncertainty quantification of
the spatially varying ODF field. We use a neural field (NF) to parameterize a
random series representation of the latent ODFs, implicitly modeling the often
ignored but valuable spatial correlation structures in the data, and thereby
improving efficiency in sparse and noisy regimes. An analytic approximation to
the posterior predictive distribution is derived which can be used to quantify
the uncertainty in the ODF estimate at any spatial location, avoiding the need
for expensive resampling-based approaches that are typically employed for this
purpose. We present empirical evaluations on both synthetic and real in-vivo
diffusion data, demonstrating the advantages of our method over existing
approaches.
