Mathematical Problems in Engineering

Volume 2016 (2016), Article ID 2189563, 7 pages

http://dx.doi.org/10.1155/2016/2189563

## Compressive Sensing of Multichannel EEG Signals via Norm and Schatten- Norm Regularization

^{1}School of Computer Engineering, Jinling Institute of Technology, Nanjing 211169, China^{2}College of Computer and Information Engineering, Nanjing Xiaozhuang University, Nanjing 210017, China^{3}Jinling Institute of Technology, Nanjing, China

Received 11 June 2016; Revised 25 September 2016; Accepted 19 October 2016

Academic Editor: Raffaele Solimene

Copyright © 2016 Jun Zhu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In Wireless Body Area Networks (WBAN) the energy consumption is dominated by sensing and communication. Recently, a simultaneous cosparsity and low-rank (SCLR) optimization model has shown the state-of-the-art performance in compressive sensing (CS) recovery of multichannel EEG signals. How to solve the resulting regularization problem, involving norm and rank function which is known as an NP-hard problem, is critical to the recovery results. SCLR takes use of norm and nuclear norm as a convex surrogate function for norm and rank function. However, norm and nuclear norm cannot well approximate the norm and rank because there exist irreparable gaps between them. In this paper, an optimization model with norm and schatten- norm is proposed to enforce cosparsity and low-rank property in the reconstructed multichannel EEG signals. An efficient iterative scheme is used to solve the resulting nonconvex optimization problem. Experimental results have demonstrated that the proposed algorithm can significantly outperform existing state-of-the-art CS methods for compressive sensing of multichannel EEG channels.

#### 1. Introduction

The electroencephalogram (EEG) signal is one of the most frequently used biomedical signals [1, 2]. It is known that EEG signals are important health indicators for stroke and trauma; recent studies also indicate that EEG signals can be used for studying dementia and Alzheimer disease. Therefore the monitoring of these signals is of utmost importance. However, continuous EEG monitoring usually records a large number of data which is too large to be sampled and transmitted in many applications [3–5]. To overcome this issue, prior studies have proposed compressive sensing (CS) [6, 7].

From fewer measurements than suggested by the Nyquist theory, compressive sensing (CS) proves that a signal can be recovered when it is sparse in a transform domain. The sampling model is formulated as follows:where is the random measurement and is the sampling matrix. CS assumes that the signal can be represented as , where is the transform domain and only contains a small number of nonzero elements. Then the synthesis based -minimization model is formulated aswhere counts the number of nonzero elements in . Many methods are proposed to solve problem (2), such as BP [6], OMP [8], and IHT [9].

Different from the traditional sparse or block sparse signal model, the cosparse signal model [10, 11] assumes that a signal multiplied by an analysis operator results in a sparse vector. The analysis based -minimization problem can be formulated:where is the analysis operator and is the cosparse vector. The above minimization problem can be efficiently solved by many methods, including GAP [10], ABS [12], AIHT [13], and ACoSaMP [14].

The cosparse analysis method has a number of advantages for multichannel EEG signals, which has been demonstrated in [15]. First, compared with the sparse synthesis model which limits the incoherence of the sampling matrix, the cosparse analysis model allows the columns of the analysis operator to be coherent, which can obtain better recovery results. Second, the sparse synthesis model firstly estimates the sparse vector and then estimates the signal, but the cosparse analysis model directly estimates the EEG signal. In a word, the cosparse analysis method is more suitable than sparse synthesis approach for CS recovery of multichannel EEG signals.

Since the EEG signals from multiple channels are correlated with each other, they motivate us to recover multichannel EEG signals via low-rank regularization [16–18]. Recently, a simultaneous cosparsity and low-rank (SCLR) optimization model [15] has shown the state-of-the-art performance in CS recovery of multichannel EEG signals. SCLR chooses the second-order difference matrix as the analysis operator to enforce the approximate piecewise linear structure, and it takes use of norm and nuclear norm as a convex surrogate function for norm and rank function. However, SCLR approach may obtain suboptimal results in real application since the norm and nuclear norm may not be good surrogate functions for norm and rank. There exist irreparable gaps between norm, the real rank and norm, and nuclear norm, respectively. The optimization results based on convex surrogate functions essentially deviate from the real solution of original minimization problem.

Motivated by the fact that norm can obtain a more accurate result in sparse synthesis model [19, 20], schatten- norm can efficiently recover low-rank matrix in image denoising [21, 22]. They have been proved rigorously in theory that norm and schatten- norm are equivalent to norm and rank function, respectively, when and are tend to be 0. So it is desirable to take them together to better exploit cosparsity and low-rank property of multichannel EEG signals.

In this paper, a novel CS model based on norm and schatten- norm (LQSP) is proposed for the compressive sensing recovery of multichannel EEG signals reconstruction. We take use of norm for the norm to enforce cosparsity prior and employ schatten- norm for the matrix rank to enforce low-rank property prior. In addition, the alternating direction method of multipliers (ADMM) is used to efficiently solve the resulting nonconvex optimization problem.

The rest of the paper is organized as follows. In Section 2, we present our proposed LQSP in detail to exploit the cosparsity and low-rank property. In Section 3, we show that the optimization problems can be solved efficiently by the alternating direction multiplier method. Then we present the numerical experiments in Section 4. Section 5 provides some concluding remarks.

#### 2. Norm and Schatten- Norm for CS Recovery of Multichannel EEG Signals

The cosparse recovery model for multichannel EEG signals can be represented as [10]where and is the number of the channels. puts all the columns of into the column vector sequentially.

In Figure 1, we select chb01_31.edf which is used in our experiments as the test data and take the second-order difference matrix as the cosparse operator. From Figure 1, we can see that most entries of the cosparse vector are nearly zero and many singular values are close to 0, which have shown that our test data naturally have both cosparsity and low-rank property. So we simultaneously exploit these two useful priors in multichannel EEG signal recovery form the compressed measurement. Then the optimization model can be reformulated as [15]