Towards Globally Predictable k-Space Interpolation: A White-box Transformer Approach

Interpolating missing data in k-space is essential for accelerating imaging. However, existing methods, including convolutional neural network-based deep learning, primarily exploit local predictability while overlooking the inherent global dependencies in k-space. Recently, Transformers have demonstrated remarkable success in natural language processing and image analysis due to their ability to capture long-range dependencies. This inspires the use of Transformers for k-space interpolation to better exploit its global structure. However, their lack of interpretability raises concerns regarding the reliability of interpolated data. To address this limitation, we propose GPI-WT, a white-box Transformer framework based on Globally Predictable Interpolation (GPI) for k-space. Specifically, we formulate GPI from the perspective of annihilation as a novel k-space structured low-rank (SLR) model. The global annihilation filters in the SLR model are treated as learnable parameters, and the subgradients of the SLR model naturally induce a learnable attention mechanism. By unfolding the subgradient-based optimization algorithm of SLR into a cascaded network, we construct the first white-box Transformer specifically designed for accelerated MRI. Experimental results demonstrate that the proposed method significantly outperforms state-of-the-art approaches in k-space interpolation accuracy while providing superior interpretability.

Generalized Low-Rank Matrix Completion Model with Overlapping Group Error Representation

The low-rank matrix completion (LRMC) technology has achieved remarkable results in low-level visual tasks. There is an underlying assumption that the real-world matrix data is low-rank in LRMC. However, the real matrix data does not satisfy the strict low-rank property, which undoubtedly present serious challenges for the above-mentioned matrix recovery methods. Fortunately, there are feasible schemes that devise appropriate and effective priori representations for describing the intrinsic information of real data. In this paper, we firstly model the matrix data ${\bf{Y}}$ as the sum of a low-rank approximation component $\bf{X}$ and an approximation error component $\cal{E}$. This finer-grained data decomposition architecture enables each component of information to be portrayed more precisely. Further, we design an overlapping group error representation (OGER) function to characterize the above error structure and propose a generalized low-rank matrix completion model based on OGER. Specifically, the low-rank component describes the global structure information of matrix data, while the OGER component not only compensates for the approximation error between the low-rank component and the real data but also better captures the local block sparsity information of matrix data. Finally, we develop an alternating direction method of multipliers (ADMM) that integrates the majorization-minimization (MM) algorithm, which enables the efficient solution of the proposed model. And we analyze the convergence of the algorithm in detail both theoretically and experimentally. In addition, the results of numerical experiments demonstrate that the proposed model outperforms existing competing models in performance.

ERD: Exponential Retinex decomposition based on weak space and hybrid nonconvex regularization and its denoising application

The Retinex theory models the image as a product of illumination and reflection components, which has received extensive attention and is widely used in image enhancement, segmentation and color restoration. However, it has been rarely used in additive noise removal due to the inclusion of both multiplication and addition operations in the Retinex noisy image modeling. In this paper, we propose an exponential Retinex decomposition model based on hybrid non-convex regularization and weak space oscillation-modeling for image denoising. The proposed model utilizes non-convex first-order total variation (TV) and non-convex second-order TV to regularize the reflection component and the illumination component, respectively, and employs weak $H^{-1}$ norm to measure the residual component. By utilizing different regularizers, the proposed model effectively decomposes the image into reflection, illumination, and noise components. An alternating direction multipliers method (ADMM) combined with the Majorize-Minimization (MM) algorithm is developed to solve the proposed model. Furthermore, we provide a detailed proof of the convergence property of the algorithm. Numerical experiments validate both the proposed model and algorithm. Compared with several state-of-the-art denoising models, the proposed model exhibits superior performance in terms of peak signal-to-noise ratio (PSNR) and mean structural similarity (MSSIM).