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Compressive sensing leverages the compressibility of natural signals to trade off the convenience of data acquisition against computational complexity of data reconstruction. Thus, CS appears to be an excellent technique for data acquisition and reconstruction in a wireless sensor network (WSN) which typically employs a smart fusion center (FC) with a high computational capability and several dumb front-end sensors having limited energy storage. This paper presents a novel signal reconstruction method based on CS principles for applications in WSNs. The proposed method exploits both the intra-sensor and inter-sensor correlation to reduce the number of samples required for reconstruction of the original signals. The novelty of the method relates to the use of the Frechet mean of the signals as an estimate of their sparse representations in some basis. This crude estimate of the sparse representation is then utilized in an enhanced data recovering convex algorithm, i.e., the penalized ℓ1 minimization, and an enhanced data recovering greedy algorithm, i.e., the precognition matching pursuit (PMP). The superior reconstruction quality of the proposed method is demonstrated by using data gathered by a WSN located in the Intel Berkeley Research lab.