Sensing by Random Convolution

Several recent results in compressive sampling (CS) show that a sparse signal (i.e. one which can be compressed in a known orthobasis) can be efficiently acquired by taking linear measurements against random test functions. In practice, however, it is difficult to build sensing devices which take these types of measurements. In this paper, we will show how to extend some of the results in CS to measurement systems which are more amenable to real-world implementation. In particular, we will show that taking measurements by subsampling a convolution with a random pulse is in some sense a universal compressive sampling strategy. We finish by briefly discussing how these results suggest a novel imaging architecture.