  Computed Tomography (CT) and Magnetic Resonance Imaging (MRI) are crucial
technologies in the field of medical imaging. Score-based models have proven to
be effective in addressing different inverse problems encountered in CT and
MRI, such as sparse-view CT and fast MRI reconstruction. However, these models
face challenges in achieving accurate three dimensional (3D) volumetric
reconstruction. The existing score-based models primarily focus on
reconstructing two dimensional (2D) data distribution, leading to
inconsistencies between adjacent slices in the reconstructed 3D volumetric
images. To overcome this limitation, we propose a novel two-and-a-half order
score-based model (TOSM). During the training phase, our TOSM learns data
distributions in 2D space, which reduces the complexity of training compared to
directly working on 3D volumes. However, in the reconstruction phase, the TOSM
updates the data distribution in 3D space, utilizing complementary scores along
three directions (sagittal, coronal, and transaxial) to achieve a more precise
reconstruction. The development of TOSM is built on robust theoretical
principles, ensuring its reliability and efficacy. Through extensive
experimentation on large-scale sparse-view CT and fast MRI datasets, our method
demonstrates remarkable advancements and attains state-of-the-art results in
solving 3D ill-posed inverse problems. Notably, the proposed TOSM effectively
addresses the inter-slice inconsistency issue, resulting in high-quality 3D
volumetric reconstruction.
