Joint Image Reconstruction and Segmentation Using the Potts Model

We propose a new algorithmic approach to the non-smooth and non-convex Potts problem (also called piecewise-constant Mumford-Shah problem) for inverse imaging problems. We derive a suitable splitting into specific subproblems that can all be solved efficiently. Our method does not require a priori knowledge on the gray levels nor on the number of segments of the reconstruction. Further, it avoids anisotropic artifacts such as geometric staircasing. We demonstrate the suitability of our method for joint image reconstruction and segmentation. We focus on Radon data, where we in particular consider limited data situations. For instance, our method is able to recover all segments of the Shepp-Logan phantom from $7$ angular views only. We illustrate the practical applicability on a real PET dataset. As further applications, we consider spherical Radon data as well as blurred data

Dependent Nonparametric Bayesian Group Dictionary Learning for online reconstruction of Dynamic MR images

In this paper, we introduce a dictionary learning based approach applied to the problem of real-time reconstruction of MR image sequences that are highly undersampled in k-space. Unlike traditional dictionary learning, our method integrates both global and patch-wise (local) sparsity information and incorporates some priori information into the reconstruction process. Moreover, we use a Dependent Hierarchical Beta-process as the prior for the group-based dictionary learning, which adaptively infers the dictionary size and the sparsity of each patch; and also ensures that similar patches are manifested in terms of similar dictionary atoms. An efficient numerical algorithm based on the alternating direction method of multipliers (ADMM) is also presented. Through extensive experimental results we show that our proposed method achieves superior reconstruction quality, compared to the other state-of-the- art DL-based methods

Low-rank and sparse reconstruction in dynamic magnetic resonance imaging via proximal splitting methods

Dynamic magnetic resonance imaging (MRI) consists of collecting multiple MR images in time, resulting in a spatio-temporal signal. However, MRI intrinsically suffers from long acquisition times due to various constraints. This limits the full potential of dynamic MR imaging, such as obtaining high spatial and temporal resolutions which are crucial to observe dynamic phenomena. This dissertation addresses the problem of the reconstruction of dynamic MR images from a limited amount of samples arising from a nuclear magnetic resonance experiment. The term limited can be explained by the approach taken in this thesis to speed up scan time, which is based on violating the Nyquist criterion by skipping measurements that would be normally acquired in a standard MRI procedure. The resulting problem can be classified in the general framework of linear ill-posed inverse problems. This thesis shows how low-dimensional signal models, specifically lowrank and sparsity, can help in the reconstruction of dynamic images from partial measurements. The use of these models are justified by significant developments in signal recovery techniques from partial data that have emerged in recent years in signal processing. The major contributions of this thesis are the development and characterisation of fast and efficient computational tools using convex low-rank and sparse constraints via proximal gradient methods, the development and characterisation of a novel joint reconstruction–separation method via the low-rank plus sparse matrix decomposition technique, and the development and characterisation of low-rank based recovery methods in the context of dynamic parallel MRI. Finally, an additional contribution of this thesis is to formulate the various MR image reconstruction problems in the context of convex optimisation to develop algorithms based on proximal splitting methods

SPHR-SAR-Net: Superpixel High-resolution SAR Imaging Network Based on Nonlocal Total Variation

High-resolution is a key trend in the development of synthetic aperture radar (SAR), which enables the capture of fine details and accurate representation of backscattering properties. However, traditional high-resolution SAR imaging algorithms face several challenges. Firstly, these algorithms tend to focus on local information, neglecting non-local information between different pixel patches. Secondly, speckle is more pronounced and difficult to filter out in high-resolution SAR images. Thirdly, the process of high-resolution SAR imaging generally involves high time and computational complexity, making real-time imaging difficult to achieve. To address these issues, we propose a Superpixel High-Resolution SAR Imaging Network (SPHR-SAR-Net) for rapid despeckling in high-resolution SAR mode. Based on the concept of superpixel techniques, we initially combine non-convex and non-local total variation as compound regularization. This approach more effectively despeckles and manages the relationship between pixels while reducing bias effects caused by convex constraints. Subsequently, we solve the compound regularization model using the Alternating Direction Method of Multipliers (ADMM) algorithm and unfold it into a Deep Unfolded Network (DUN). The network's parameters are adaptively learned in a data-driven manner, and the learned network significantly increases imaging speed. Additionally, the Deep Unfolded Network is compatible with high-resolution imaging modes such as spotlight, staring spotlight, and sliding spotlight. In this paper, we demonstrate the superiority of SPHR-SAR-Net through experiments in both simulated and real SAR scenarios. The results indicate that SPHR-SAR-Net can rapidly perform high-resolution SAR imaging from raw echo data, producing accurate imaging results

Model-based T1, T2* and Proton Density Mapping Using a Bayesian Approach with Parameter Estimation and Complementary Undersampling Patterns

Purpose: To achieve automatic hyperparameter estimation for the joint recovery of quantitative MR images, we propose a Bayesian formulation of the reconstruction problem that incorporates the signal model. Additionally, we investigate the use of complementary undersampling patterns to determine optimal undersampling schemes for quantitative MRI. Theory: We introduce a novel nonlinear approximate message passing framework, referred to as ``AMP-PE'', that enables the simultaneous recovery of distribution parameters and quantitative maps. Methods: We employed the variable flip angle multi-echo (VFA-ME) method to acquire measurements. Both retrospective and prospective undersampling approaches were utilized to obtain Fourier measurements using variable-density and Poisson-disk patterns. Furthermore, we extensively explored various undersampling schemes, incorporating complementary patterns across different flip angles and/or echo times. Results: AMP-PE adopts a model-based joint recovery strategy, it outperforms the $l_1$-norm minimization approach that follows a decoupled recovery strategy. A comparison with an existing joint-recovery approach further demonstrates the advantageous outcomes of AMP-PE. For quantitative $T_1$ mapping using VFA-ME, employing identical k-space sampling patterns across different echo times produced the best performance. Whereas for $T_2^*$ and proton density mappings, using complementary sampling patterns across different flip angles yielded the best performance. Conclusion: AMP-PE is equipped with built-in parameter estimation, and works naturally in clinical settings with varying acquisition protocols and scanners. It also achieves improved performance by combining information from the MR signal model and the sparse prior on images