## 1 Introduction  
Wireless Sensor Networks (WSNs) generally consist of a large number of sensor nodes and a sink node deployed in the detected environment to monitor various physical characteristics of the real world, such as temperature, voltage, wind direction, and so on. Furthermore, WSNs should have a long enough lifetime to successfully fulfill the monitoring task. However, sensor nodes are limited in terms of computational ability, communication bandwidth, and energy availability. In many cases, such as where there is an inaccessible or hostile field, a battery is difficult to recharge [1]. Therefore, how to gather sensor network data in an energy-efficient way becomes the crucial problem in practical applications. For instance, [2] indicates the importance of promoting the efficiency of intra-car multihop WSNs. A great many research works have addressed energy-efficient challenges for WSNs from the point of view of sleep schedule [3], topology control [4], area coverage [5], mobile sink [6], and data gathering [7]. However, the core of the study is to take advantage of data-gathering and routing algorithms in WSNs to promote network performance and balanced load. We can leverage the spatial characteristics in sensor node readings from real deployments, which is an essential technique to decrease data transmission costs while preserving relatively high recovery accuracy in the sink node [1]. In other words, the spatial correlation property of a sensor node leads to inherent data sparsity in some areas, such as wavelet domain and DCT domain. In order to solve the sparsity of such signals, compressive sensing (CS) is exploited as a novel signal-processing paradigm that provides an efficient compressive method and recovers sparse or compressible signals [8; 9; 10; 11]. In recent years, a large number of CS-based techniques have been investigated to gather data on WSNs [12; 13; 14; 15; 16; 17; 18; 19]. However, most of the current CS-based methods neglect the topology structure or property of sensor node readings, such as spatial characteristics. Therefore, the performance of the algorithm is limited. In this paper, the spatial property of sensor node readings is exploited to strengthen the performance of networks, considering the topology structure and sensor nodes' distance. Thus, we take advantage of spatial correlations of sensor node readings to further promote the efficiency of the data-gathering algorithm. So far, various CS-based data-gathering schemes have been presented to promote the networks' efficiency for data gathering in WSNs [12; 13; 14; 15]. (See Section 2 for more details.) For instance, [12] proposes a data-gathering scheme that diminishes the bottleneck of the sink in WSNs. In [13], the spatial correlation of sensor node readings is leveraged to improve the performance of WSNs. [PERSON] et al. propose a sparsest representation basis, which makes full use of the spatial property of sensor node readings [14]. [PERSON] et al. introduce a framework for data gathering considering CS and principal component analysis (PCA) simultaneously. In this scheme, the spatial characteristics of sensor node readings are utilized to enhance the recovery accuracy [15]. The routing protocol in data-gathering schemes has also attracted much attention in the past few years [16; 17; 18; 19]. In [16], multi-chain CS-based data gathering is presented by considering the routing protocol from the sensor nodes to the sink of WSNs. A special routing algorithm corresponding to the topology structure is given in [17]. In [18], a random walk routing approach is introduced to reduce the energy consumption of the whole network. In [19], [PERSON] et al. propose a cluster-based routing algorithm for data gathering in WSNs. However, these approaches do not consider the property of sensor node readings. Additionally, the optimal routing algorithm is not considered. In this paper, a sparser basis based on improved diffusion wavelets is provided, aiming to exploit the sensor node's spatial correlation features.