Generative Modeling in Sinogram Domain for Sparse-view CT Reconstruction

The radiation dose in computed tomography (CT) examinations is harmful for patients but can be significantly reduced by intuitively decreasing the number of projection views. Reducing projection views usually leads to severe aliasing artifacts in reconstructed images. Previous deep learning (DL) techniques with sparse-view data require sparse-view/full-view CT image pairs to train the network with supervised manners. When the number of projection view changes, the DL network should be retrained with updated sparse-view/full-view CT image pairs. To relieve this limitation, we present a fully unsupervised score-based generative model in sinogram domain for sparse-view CT reconstruction. Specifically, we first train a score-based generative model on full-view sinogram data and use multi-channel strategy to form highdimensional tensor as the network input to capture their prior distribution. Then, at the inference stage, the stochastic differential equation (SDE) solver and data-consistency step were performed iteratively to achieve fullview projection. Filtered back-projection (FBP) algorithm was used to achieve the final image reconstruction. Qualitative and quantitative studies were implemented to evaluate the presented method with several CT data. Experimental results demonstrated that our method achieved comparable or better performance than the supervised learning counterparts.Comment: 11 pages, 12 figure

Stage-by-stage Wavelet Optimization Refinement Diffusion Model for Sparse-View CT Reconstruction

Diffusion models have emerged as potential tools to tackle the challenge of sparse-view CT reconstruction, displaying superior performance compared to conventional methods. Nevertheless, these prevailing diffusion models predominantly focus on the sinogram or image domains, which can lead to instability during model training, potentially culminating in convergence towards local minimal solutions. The wavelet trans-form serves to disentangle image contents and features into distinct frequency-component bands at varying scales, adeptly capturing diverse directional structures. Employing the Wavelet transform as a guiding sparsity prior significantly enhances the robustness of diffusion models. In this study, we present an innovative approach named the Stage-by-stage Wavelet Optimization Refinement Diffusion (SWORD) model for sparse-view CT reconstruction. Specifically, we establish a unified mathematical model integrating low-frequency and high-frequency generative models, achieving the solution with optimization procedure. Furthermore, we perform the low-frequency and high-frequency generative models on wavelet's decomposed components rather than sinogram or image domains, ensuring the stability of model training. Our method rooted in established optimization theory, comprising three distinct stages, including low-frequency generation, high-frequency refinement and domain transform. Our experimental results demonstrate that the proposed method outperforms existing state-of-the-art methods both quantitatively and qualitatively

Directional Sinogram Inpainting for Limited Angle Tomography

In this paper we propose a new joint model for the reconstruction of tomography data under limited angle sampling regimes. In many applications of Tomography, e.g. Electron Microscopy and Mammography, physical limitations on acquisition lead to regions of data which cannot be sampled. Depending on the severity of the restriction, reconstructions can contain severe, characteristic, artefacts. Our model aims to address these artefacts by inpainting the missing data simultaneously with the reconstruction. Numerically, this problem naturally evolves to require the minimisation of a non-convex and non-smooth functional so we review recent work in this topic and extend results to fit an alternating (block) descent framework. \oldtext{We illustrate the effectiveness of this approach with numerical experiments on two synthetic datasets and one Electron Microscopy dataset.} \newtext{We perform numerical experiments on two synthetic datasets and one Electron Microscopy dataset. Our results show consistently that the joint inpainting and reconstruction framework can recover cleaner and more accurate structural information than the current state of the art methods

Directional sinogram inpainting for limited angle tomography

In this paper we propose a new joint model for the reconstruction of tomography data under limited angle sampling regimes. In many applications of Tomography, e.g. Electron Microscopy and Mammography, physical limitations on acquisition lead to regions of data which cannot be sampled. Depending on the severity of the restriction, reconstructions can contain severe, characteristic, artefacts. Our model aims to address these artefacts by inpainting the missing data simultaneously with the reconstruction. Numerically, this problem naturally evolves to require the minimisation of a non-convex and non-smooth functional so we review recent work in this topic and extend results to fit an alternating (block) descent framework. We illustrate the effectiveness of this approach with numerical experiments on two synthetic datasets and one Electron Microscopy dataset.Cantab Capital Institute for the Mathematics of Information PIHC innovation fund of the Technical Medical Centre of UT Dutch 4TU programme Precision Medicine Netherlands Organization for Scientific Research (NWO), project 639.073.506 Henslow Research Fellowship at Girton College, Cambridge Clare College Junior Research Fellowshi

DOLCE: A Model-Based Probabilistic Diffusion Framework for Limited-Angle CT Reconstruction

Limited-Angle Computed Tomography (LACT) is a non-destructive evaluation technique used in a variety of applications ranging from security to medicine. The limited angle coverage in LACT is often a dominant source of severe artifacts in the reconstructed images, making it a challenging inverse problem. We present DOLCE, a new deep model-based framework for LACT that uses a conditional diffusion model as an image prior. Diffusion models are a recent class of deep generative models that are relatively easy to train due to their implementation as image denoisers. DOLCE can form high-quality images from severely under-sampled data by integrating data-consistency updates with the sampling updates of a diffusion model, which is conditioned on the transformed limited-angle data. We show through extensive experimentation on several challenging real LACT datasets that, the same pre-trained DOLCE model achieves the SOTA performance on drastically different types of images. Additionally, we show that, unlike standard LACT reconstruction methods, DOLCE naturally enables the quantification of the reconstruction uncertainty by generating multiple samples consistent with the measured data.Comment: 29 pages, 21 figure

Mathematical Challenges in Electron Microscopy

Development of electron microscopes first started nearly 100 years ago and they are now a mature imaging modality with many applications and vast potential for the future. The principal feature of electron microscopes is their resolution; they can be up to 1000 times more powerful than a visible light microscope and resolve even the smallest atoms. Furthermore, electron microscopes are also sensitive to many material properties due to the very rich interactions between electrons and other matter. Because of these capabilities, electron microscopy is used in applications as diverse as drug discovery, computer chip manufacture, and the development of solar cells. In parallel to this, the mathematical field of inverse problems has also evolved dramatically. Many new methods have been introduced to improve the recovery of unknown structures from indirect data, typically an ill-posed problem. In particular, sparsity promoting functionals such as the total variation and its extensions have been shown to be very powerful for recovering accurate physical quantities from very little and/or poor quality data. While sparsity-promoting reconstruction methods are powerful, they can also be slow, especially in a big-data setting. This trade-off forms an eternal cycle as new numerical tools are found and more powerful models are developed. The work presented in this thesis aims to marry the tools of inverse problems with the problems of electron microscopy: bringing state-of-the-art image processing techniques to bear on challenges specific to electron microscopy, developing new optimisation methods for these problems, and modelling new inverse problems to extend the capabilities of existing microscopes. One focus is the application of a directional total variation to overcome the limited angle problem in electron tomography, another is the proposal of a new inverse problem for the reconstruction of 3D strain tensor fields from electron microscopy diffraction data. The remaining contributions target numerical aspects of inverse problems, from new algorithms for non-convex problems to convex optimisation with adaptive meshes.Cantab Capital Institute for Mathematics of Informatio