## 2 Related Works  
In many data collection scenarios in WSNs, such as environmental monitoring [17], to fully sense physical quantities efficiently, it is necessary to transmit all sensing data to the sink node correctly and accurately. Doing so would consume too much energy in the networks. However, it is unnecessary to get the entire sensing data, given that there exists high correlation in the sensing data. Hence, the best strategy is to explore energy efficient data collection methods which approximately get the sensing data with a given accuracy. The conventional approximate data collection methods include data aggregation techniques, distributed source coding and collaborative in-network compression methods. Data aggregation techniques [8, 9, 10, 11] formulate the network sensor nodes into a hierarchical structure like trees or clusters. Generally the sensing nodes send the sensing data to the middle aggregator nodes. These aggregators collect data from multiple sensor nodes and aggregate the sensing data by aggregation functions, and then send the aggregated results to upper aggregators or sink nodes. Only a subset of nodes is involved in the communication with an aggregator and thus the overall energy consumption of the network during data collection is reduced. Distributed source coding schemes are based on the Slepian-Wolf coding theory [3, 4, 5]. In these methods, the sensor nodes are modeled as correlated sources of information and the dependency between the measured data are exploited to obtain data compression. Collaborative in-network compression methods can be viewed as an extension of traditional transforms in signal processing to sensor networks with irregular topology. Nodes can communicate with their neighbors and spatial correlation is exploited by collaborative transforms such as the distributed wavelet transform [6] and the graph wavelet transform [7]. Although the above mentioned conventional data collection methods are considered to be energy efficient and widely used in WSNs applications, they have their respective advantages in different aspects and the energy efficiency of these methods is relative to specific application scenarios. For example, the aggregation methods generally depend on a particular routing protocol, and the source coding methods typically require exact knowledge of the correlation between the measurements of different sensor nodes. It is highly desired to investigate the generic data collection method with high energy efficiency. The typical representatives are the recently proposed compressive sensing (CS) based methods [18, 19, 20, 21, 22] and the LRMA based methods [14, 15, 16]. The CS based methods provide two features of universal sampling and decentralized encoding, which make it a new paradigm for data collection in WSNs. In a single-hop network, a universal compressive wireless sensing method is proposed to deliver the sensing data by synchronized amplitude-modulated analog transmissions to the fusion center [23]. The sparsity of the sensing data and the CS decoding algorithms are discussed for data collection application in [24]. The first complete scheme to apply compressive sensing theory to data collection in large scale sensor networks is presented in [18]. Then an adaptive compressive sensing based data collection method is proposed in [19]. By maximizing information gain per energy cost during each measurement, the method adaptively collects data in an energy efficient way. However, during each measurement, the projected vector is obtained by solving an NP-hard optimization problem, which brings in relatively high computational and communication overhead. Energy efficiency of applying CS to data collection was investigated in [20], aiming at minimizing energy consumption through combining compressed aggregation with routing protocol. The optimization problem is proved to be of NP-completeness and both optimized and nearly-optimized solutions are given.