# Two-and-a-half Order Score-based Model for Solving 3D Ill-posed Inverse Problems

arXiv: https://arxiv.org/abs/2308.08511 · HTML: https://mlsanity.com/paper/2308.08511/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Placed in Medical imaging and segmentation, one of 14 directions in Diffusion models.
- Builds on 23 of its 70 references that are in the map.
- First posted on arXiv 16 Aug 2023.

## Placed in

- Diffusion models / [Medical imaging and segmentation](https://mlsanity.com/diffusion/medical-imaging-and-segmentation/)

## Builds on (most cited first, 23 of 23)

- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Improved Techniques for Training Score-Based Generative Models](https://mlsanity.com/paper/2006.09011/) (2020)
- [Diffusion Probabilistic Models for 3D Point Cloud Generation](https://mlsanity.com/paper/2103.01458/) (2021)
- [LION: Latent Point Diffusion Models for 3D Shape Generation](https://mlsanity.com/paper/2210.06978/) (2022)
- [Solving Inverse Problems in Medical Imaging with Score-Based Generative Models](https://mlsanity.com/paper/2111.08005/) (2021)
- [fastMRI: An Open Dataset and Benchmarks for Accelerated MRI](https://mlsanity.com/paper/1811.08839/) (2018)
- [Deep Convolutional Neural Network for Inverse Problems in Imaging](https://mlsanity.com/paper/1611.03679/) (2016)
- [Point-Voxel CNN for Efficient 3D Deep Learning](https://mlsanity.com/paper/1907.03739/) (2019)
- [Score-based diffusion models for accelerated MRI](https://mlsanity.com/paper/2110.05243/) (2021)
- [Low-Dose CT with a Residual Encoder-Decoder Convolutional Neural Network (RED-CNN)](https://mlsanity.com/paper/1702.00288/) (2017)
- [Solving 3D Inverse Problems using Pre-trained 2D Diffusion Models](https://mlsanity.com/paper/2211.10655/) (2022)
- [Deep learning for undersampled MRI reconstruction](https://mlsanity.com/paper/1709.02576/) (2017)
- [X2CT-GAN: Reconstructing CT from Biplanar X-Rays with Generative Adversarial Networks](https://mlsanity.com/paper/1905.06902/) (2019)
- [CNN-Based Projected Gradient Descent for Consistent Image Reconstruction](https://mlsanity.com/paper/1709.01809/) (2017)
- [Improving 3D Imaging with Pre-Trained Perpendicular 2D Diffusion Models](https://mlsanity.com/paper/2303.08440/) (2023)
- [Generative Modeling in Sinogram Domain for Sparse-view CT Reconstruction](https://mlsanity.com/paper/2211.13926/) (2022)
- [PWLS-ULTRA: An Efficient Clustering and Learning-Based Approach for Low-Dose 3D CT Image Reconstruction](https://mlsanity.com/paper/1703.09165/) (2017)
- [Multi-domain Integrative Swin Transformer network for Sparse-View Tomographic Reconstruction](https://mlsanity.com/paper/2111.14831/) (2021)
- [Low-dose spectral CT reconstruction using L0 image gradient and tensor dictionary](https://mlsanity.com/paper/1801.01452/) (2017)
- [Deep Learning based Spectral CT Imaging](https://mlsanity.com/paper/2008.13570/) (2020)
- [A deep cascade of ensemble of dual domain networks with gradient-based T1 assistance and perceptual refinement for fast MRI reconstruction](https://mlsanity.com/paper/2207.01791/) (2022)
- [Spectral2Spectral: Image-spectral Similarity Assisted Spectral CT Deep Reconstruction without Reference](https://mlsanity.com/paper/2210.01125/) (2022)

## Built on by (most cited first, 5 of 5)

- [Deep Generative Models for 3D Medical Image Synthesis](https://mlsanity.com/paper/2410.17664/) (2024)
- [Data-iterative Optimization Score Model for Stable Ultra-Sparse-View CT Reconstruction](https://mlsanity.com/paper/2308.14437/) (2023)
- [Generative Drifting for Conditional Medical Image Generation](https://mlsanity.com/paper/2604.19736/) (2026)
- [MInDI-3D: Iterative Deep Learning in 3D for Sparse-view Cone Beam Computed Tomography](https://mlsanity.com/paper/2508.09616/) (2025)
- [Wavelet-Optimized Pseudo-3D Accelerated Diffusion Model for Truncated Computed Laminography](https://mlsanity.com/paper/2606.31318/) (2026)

## Abstract (arXiv)

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.