Compressed Motion Sensing

We consider the problem of sparse signal recovery in dynamic sensing scenarios. Specifically, we study the recovery of a sparse time-varying signal from linear measurements of a single static sensor that are taken at two different points in time. This setup can be modelled as observing a single signal using two different sensors - a real one and a virtual one induced by signal motion, and we examine the recovery properties of the resulting combined sensor. We show that not only can the signal be uniquely recovered with overwhelming probability by linear programming, but also the correspondence of signal values (signal motion) can be established between the two points in time. In particular, we show that in our scenario the performance of an undersampling static sensor is doubled or, equivalently, that the number of sufficient measurements of a static sensor is halved.