Guohui Song

Guohui Song, Anne Gelb

This investigation seeks to establish the practicality of numerical frame approximations. Specifically, it develops a new method to approximate the inverse frame operator and analyzes its convergence properties. It is established that sampling with {\em well-localized frames} improves both the accuracy of the numerical frame approximation as well as the robustness and efficiency of the (finite) frame operator inversion. Moreover, in applications such as magnetic resonance imaging, where the given data often may not constitute a well-localized frame, a technique is devised to project the corresponding frame data onto a more suitable frame. As a result, the target function may be approximated as a finite expansion with its asymptotic convergence solely dependent on its smoothness. Numerical examples are provided.

Yao Xiao, Anne Gelb, Guohui Song

This paper introduces a new sparse Bayesian learning (SBL) algorithm that jointly recovers a temporal sequence of edge maps from noisy and under-sampled Fourier data. The new method is cast in a Bayesian framework and uses a prior that simultaneously incorporates intra-image information to promote sparsity in each individual edge map with inter-image information to promote similarities in any unchanged regions. By treating both the edges as well as the similarity between adjacent images as random variables, there is no need to separately form regions of change. Thus we avoid both additional computational cost as well as any information loss resulting from pre-processing the image. Our numerical examples demonstrate that our new method compares favorably with more standard SBL approaches.

Guohui Song, Jacqueline Davis, Anne Gelb

The ability to efficiently and accurately construct an inverse frame operator is critical for establishing the utility of numerical frame approximations. Recently, the admissible frame method was developed to approximate inverse frame operators for one-dimensional problems. Using the admissible frame approach, it is possible to project the corresponding frame data onto a more suitable (admissible) frame, even when the sampling frame is only weakly localized. As a result, a target function may be approximated as a finite frame expansion with its asymptotic convergence solely dependent on its smoothness. In this investigation, we seek to expand the admissible frame approach to two dimensions, which requires some additional constraints. We prove that the admissible frame technique converges in two dimensions and then demonstrate its usefulness with some numerical experiments that use sampling patterns inspired by applications that sample data non-uniformly in the Fourier domain.

Feng Yu, Lixin Shen, Guohui Song

Sparse Bayesian Learning (SBL) models are extensively used in signal processing and machine learning for promoting sparsity through hierarchical priors. The hyperparameters in SBL models are crucial for the model's performance, but they are often difficult to estimate due to the non-convexity and the high-dimensionality of the associated objective function. This paper presents a comprehensive framework for hyperparameter estimation in SBL models, encompassing well-known algorithms such as the expectation-maximization (EM), MacKay, and convex bounding (CB) algorithms. These algorithms are cohesively interpreted within an alternating minimization and linearization (AML) paradigm, distinguished by their unique linearized surrogate functions. Additionally, a novel algorithm within the AML framework is introduced, showing enhanced efficiency, especially under low signal noise ratios. This is further improved by a new alternating minimization and quadratic approximation (AMQ) paradigm, which includes a proximal regularization term. The paper substantiates these advancements with thorough convergence analysis and numerical experiments, demonstrating the algorithm's effectiveness in various noise conditions and signal-to-noise ratios.

Weihong Guo, Guohui Song, Yue Zhang

We propose in this paper a novel two-stage Projection Correction Modeling (PCM) framework for image reconstruction from (non-uniform) Fourier measurements. PCM consists of a projection stage (P-stage) motivated by the multi-scale Galerkin method and a correction stage (C-stage) with an edge guided regularity fusing together the advantages of total variation (TV) and total fractional variation (TFV). The P-stage allows for continuous modeling of the underlying image of interest. The given measurements are projected onto a space in which the image is well represented. We then enhance the reconstruction result at the C-stage that minimizes an energy functional consisting of a fidelity in the transformed domain and a novel edge guided regularity. We further develop efficient proximal algorithms to solve the corresponding optimization problem. Various numerical results in both 1D signals and 2D images have also been presented to demonstrate the superior performance of the proposed two-stage method to other classical one-stage methods.