  Diffusion models have been demonstrated as powerful deep learning tools for
image generation in CT reconstruction and restoration. Recently, diffusion
posterior sampling, where a score-based diffusion prior is combined with a
likelihood model, has been used to produce high quality CT images given
low-quality measurements. This technique is attractive since it permits a
one-time, unsupervised training of a CT prior; which can then be incorporated
with an arbitrary data model. However, current methods rely on a linear model
of x-ray CT physics to reconstruct or restore images. While it is common to
linearize the transmission tomography reconstruction problem, this is an
approximation to the true and inherently nonlinear forward model. We propose a
new method that solves the inverse problem of nonlinear CT image reconstruction
via diffusion posterior sampling. We implement a traditional unconditional
diffusion model by training a prior score function estimator, and apply Bayes
rule to combine this prior with a measurement likelihood score function derived
from the nonlinear physical model to arrive at a posterior score function that
can be used to sample the reverse-time diffusion process. This plug-and-play
method allows incorporation of a diffusion-based prior with generalized
nonlinear CT image reconstruction into multiple CT system designs with
different forward models, without the need for any additional training. We
develop the algorithm that performs this reconstruction, including an
ordered-subsets variant for accelerated processing and demonstrate the
technique in both fully sampled low dose data and sparse-view geometries using
a single unsupervised training of the prior.
