CT Reconstruction using Diffusion Posterior Sampling conditioned on a Nonlinear Measurement Model

TL;DR

The paper introduces Diffusion Posterior Sampling conditioned on a nonlinear physical model (DPS Nonlinear) for CT reconstruction, pairing an unsupervised diffusion prior with a nonlinear Poisson likelihood to respect realistic CT physics. It derives a conditional reverse-time SDE, trains a score-based prior on a large CT dataset, and implements a plug-and-play reconstruction workflow with an ordered-subsets acceleration variant (OS-DPS Nonlinear). Across fully sampled low-dose and sparse-view configurations, DPS Nonlinear demonstrates competitive PSNR/SSIM relative to MBIR and DOLCE, while OS-DPS Nonlinear offers meaningful runtime gains with modest accuracy trade-offs. The work highlights the potential and challenges of diffusion-based CT methods under nonlinear forward models, including hallucinations at low fluence and opportunities for extending to larger-scale 3D and spectral CT applications.

Abstract

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.

Figures (12)

• Figure 1: The workflow of the proposed DPS Nonlinear method. The blue arrow represents a forward stochastic process defined by the equation in the top wherein $\mathbf{x_0}$, a "clean" ground truth CT image is successively degraded in intermediate noisy images, $\mathbf{x_t}$, indexed by time $t$, to an image of Gaussian noise, $\mathbf{x_T}$,. A reverse stochastic process is defined at the bottom that is conditioned on measurement data, $\mathbf{y}$. The orange arrow represents this conditioning and is based on a general nonlinear physical measurement model.
• Figure 2: Low-dose CT reconstructions comparing FBP, MBIR (without and with TV regularization), DPS Linear, DOLCE, and the proposed DPS Nonlinear approach. All images share a common colormap with W/L = 2000/0 HU.
• Figure 3: Bias and standard deviation comparison for low-dose CT reconstructions comparing the proposed DPS Nonlinear and DPS Linear approach. Root-mean-squared bias and standard deviation are reported in Hounsfield units (HU) based on 16 generated samples from diffusion posterior sampling for each weighting coefficient $k$ value. Error bars ($\pm~1.0$ standard deviation) are shown based on the variability over samples. The best weighting coefficient $k$ was $310$ based on the lowest root-mean-squared bias as well as the lowest standard deviation.
• Figure 4: Additional DPS sample reconstructions and standard deviation image from the same noisy measurement illustrating variability in the outputs. Colormap: W/L = 2000/0 HU for sample image and W/L = 500/250 HU for standard deviation image.
• Figure 5: Sparse-view CT reconstructions comparing FBP, MBIR (without and with TV regularization), DPS Linear, DOLCE, and the proposed DPS Nonlinear approach. All images share a common colormap with W/L = 2000/0 HU.