In this paper, we introduce a multi-dimensional approach to the problem of reconstruction of MR image sequences that are highly undersampled in k-space. By formulating the reconstruction as a high-order low-rank plus sparse tensor decomposition problem, we propose an efficient numerical algorithm based on the alternating direction method of multipliers (ADMM) to solve the optimization. Through extensive experimental results we show that our proposed method achieves superior reconstruction quality, compared to the state-of-the-art reconstruction methods.

It has been shown recently that incorporating priori knowledge into the basic compressive sensing results in significant improvement of its performance. This has motivated us to extend the incorporation of partial known support into the problem of Robust Principal Component Analysis (RPCA) from compressive measurements. Our proposed algorithm utilizes the known part of the support to recover a matrix as the sum of a low-rank matrix and a sparse component and is tested on the problem of surveillance video reconstruction from compressive measurements. Our experimental results show that the incorporation of partial known support, can significantly improve the reconstruction performance of video sequences.

Concerning medical images, which are known to have sparsity in either the spatial (or its derivative), DFT, DCT or curvelet domain, we propose a new approach for reconstruction from sparse samples, based on Simultaneous Perturbation Stochastic Optimization (SPSA) to minimize a nonconvex ℓp-norm for 00-norm as is computationally feasible. This approach is distinct from the homotopy-theoretic and hard-thresholding techniques of recent literature for ℓ0- and ℓp-norm minimization. For lack of space, our illustrations are limited to only one each of synthetic and real images.

A novel segmentation-based image approximation and coding technique is proposed. A hybrid quad-binary (QB) tree structure is utilized to efficiently model and code geometrical information within images. Compared to other tree-based representation such as wedgelets, the proposed QB-tree based method is more efficient for a wide range of contour features such as junctions, corners and ridges, especially at low bit rates.