Abstract Tensor singular value decomposition (t‐SVD) provides a novel way to decompose a tensor. It has been employed mostly in recovering missing tensor entries from the observed tensor entries. The problem of applying t‐SVD to recover tensors from limited coefficients in any given ortho‐normal basis is addressed. We prove that an n × n × n 3 tensor with tubal‐rank r can be efficiently reconstructed by minimising its tubal nuclear norm from its O ( rn 3 n log 2 ( n 3 n )) randomly sampled coefficients w.r.t any given ortho‐normal basis. In our proof, we extend the matrix coherent conditions to tensor coherent conditions. We first prove the theorem belonging to the case of Fourier‐type basis under certain coherent conditions. Then, we prove that our results hold for any ortho‐normal basis meeting the conditions. Our work covers the existing t‐SVD‐based tensor completion problem as a special case. We conduct numerical experiments on random tensors and dynamic magnetic resonance images (d‐MRI) to demonstrate the performance of the proposed methods.

It has always been challenging to reconstruct magnetic resonance (MR) images from a limited set of k ‐space data due to the ill‐posed nature. An effective way to compensate for the data incompleteness is through the use of the sparsity‐based regularisation. Recent work in image processing suggests that exploiting structured sparsity may lead to improved results. In this study, this idea is explored in combination with additional support prior of the MR images in the analysis context. Put differently, the authors propose a highly effective regulariser constraining group sparsity of the analysis coefficients within the pre‐estimated cosupport. A two‐stage iterative algorithm is developed and proceeds by alternatively calling its two key components: image reconstruction and cosupport detection. The feasibility of the proposed method is demonstrated for individual and multiple T1/T2‐weighted MR images. Simulation results show considerable improvement of their method compared with the methods using structured sparsity and support knowledge in the synthesis context and other related reconstruction methods.

The problem of estimating rapidly time‐varying channels is considered to be one of the key challenges in high mobility orthogonal frequency division multiplexing (OFDM) systems. In such scenarios, fast time variation within OFDM symbol duration requires overloaded measurements for estimation. By exploiting the inherent sparsity of wireless channels, the authors cast the channel estimation as a compressive sensing (CS) problem to reduce the required pilot symbols. Different from the existing CS‐based estimators which mainly focus on the diagonal matrix model and treat the inter‐carrier interference as additive noise, the proposed methods are designed for the non‐diagonal matrix, a more precise representation of fast fading channels. To handle this more complex channel model, an iterative estimation scheme is presented, which adopts the recently introduced modified‐CS algorithm. In addition, a more simplified scheme is also designed by utilising a reasonable approximation of the system model. Compared to the former method, it has a reduced computational complexity with limited performance degradation. The simulation results demonstrate that the two proposed CS‐based methods are robust to large Doppler spreading and have better performance than conventional CS‐based estimators for fast time‐varying channels in OFDM systems.

Compressive sampling (CS) is a novel data collection and coding theory which allows us to recover sparse or compressible signals from a small set of measurements. This paper presents a new model for natural image recovery, in which the smooth l0norm and the approximate total-variation (TV) norm are adopted simultaneously. By using one-order gradient decrease, the speed of algorithm for this new model can be guaranteed. Experimental results demonstrate that the principle of the model is correct and the performance is as good as that based on TV model. The computing speed of the proposed method is two orders of magnitude faster than that of interior point method and two times faster than that of the Nesta optimization based on TV model.