Local sparsity and recovery of fusion frames structured signals

The problem of recovering complex signals from low quality sensing devices is analyzed via a combination of tools from signal processing and harmonic analysis. By using the rich structure offered by the recent development in fusion frames, we introduce a framework for splitting the complex information into local pieces. Each piece of information is measured by potentially very low quality sensors, modeled by linear matrices and recovered via compressed sensing – when necessary. Finally, by a fusion process within the fusion frames, we are able to recover accurately the original high-dimensional signal. Using our new method, we show, and illustrate on simple numerical examples, that it is possible, and sometimes necessary, to split a signal via local projections for accurate, stable, and robust estimation. In particular, we show that by increasing the size of the fusion frame, we encourage a certain robustness to noise. While the complexity remains relatively low, we achieve stronger recovery performance compared to usual single-device compressed sensing. 1. Problem statement In a traditional sampling and reconstruction system, the sensors are designed so that the recovery of the signal(s) of interest is possible. For instance, when considering the sparse recovery problem, one tries to find the sparsest solution x̂ ∈ K from the noisy measurements y = Ax + e ∈ K and m N . Here K denotes the field R or C. This is done by solving the mathematical program x̂ := argmin ‖z‖0, subject to ‖Az− y‖2 ≤ η. (`-min) This problem being NP-Hard, it is usually approximated, for instance by solving its convex relaxation, known as the Basis Pursuit DeNoising x̂ := argmin ‖z‖1, subject to ‖Az− y‖2 ≤ η. (BPDN) It is known that for a given complexity s of the signal x, the number of random subgaussian linear measurements needs to grow as m & s log(N/s) for x̂ to be a good enough approximation to x. Said differently, if the design of a sensor can be made at will, then knowing the complexity of the signal, here characterized by the sparsity, is sufficient for a stable and robust recovery. This paper looks at the problem of sampling potentially complex signals when the quality of sensors is constrained. We investigate problems where the number of measurements m cannot be chosen based on the complexity of the signals to recover. In the context described above, one would have a limit on the sparsity of the vectors that can be recovered given by s . m/ log(N/m) (which is a very pessimistic bound). These constraints can be due to many reasons such as cost – e.g. using 10 sensors at a coarser resolution is cheaper than one at the finest –, frequency rate – sensors at 2000 THz might not exists for a while –, legal regulation – e.g. in nuclear medicine where one should not expose a patient to too high radiations at once. Problems arise when the signals being sampled are too complex for the usual mathematical theories. When one thinks about compressed sensing, the size of the sensor required is driven by a certain measure of complexity of the signals considered. Allowing for the recovery of signals with higher level of complexity entails the use of better sensors. Here we look at the problem differently: first, we assume constraints on the sensor design which are fixed due to some outside reasons. Under these assumptions, we take on the following challenge: split the information carried by the signal in a clever way so that a mathematical recovery is possible. This paper revisits the theory of fusion frames and applies it to the signal recovery problem. We show that by using advanced mathematical techniques stemming from applied harmonic analysis, it is possible to handle very complex signals in an efficient and stable manner. Before we dig into the more technical details, we present some real-world scenarios where our framework appears useful, if not essential.