We consider the joint optimization of sensor placement and transmission structure for data gathering, where a given number of nodes need to be placed in a field such that the sensed data can be reconstructed at a sink within specified distortion bounds while minimizing the energy consumed for communication. We assume that the nodes use joint entropy coding based on explicit communication between sensor nodes, and consider both maximum and average distortion bounds. The optimization is complex since it involves an interplay between the spaces of possible transmission structures given radio reachability limitations, and feasible placements satisfying distortion bounds.

We address this problem by first looking at the simplified problem of optimal placement in the one-dimensional case. An analytical solution is derived for the case when there is a simple aggregation scheme, and numerical results are provided for the cases when joint entropy encoding is used. We use the insight from our 1-D analysis to extend our results to the 2-D case, and show that our algorithm for two-dimensional placement and transmission structure provides significant power benefit over a commonly used combination of uniformly random placement and shortest path trees.