Reconstruction in compressed sensing relies on knowledge of a sparsifying transform. In a setting where a sink reconstructs a field based on measurements from a wireless sensor network, this transform is tied to the locations of the individual sensors, which may not be available to the sink during reconstruction. In contrast to previous works, we do not assume that the sink knows the position of each sensor to build up the sparsifying basis. Instead, we propose the use of spatial interpolation based on a predetermined sparsifying transform, followed by random linear projections and ratio consensus using local communication between sensors. For this proposed architecture, we upper bound the reconstruction error induced by spatial interpolation, as well as the reconstruction error induced by distributed compression. These upper bounds are then utilized to analyze the communication cost tradeoff between communication to the sink and sensor-to-sensor communication
