Local sparsity and recovery of fusion frame structured signals

Abstract

The problem of recovering signals of high complexity from low quality sensing devices is analyzed via a combination of tools from signal processing and harmonic analysis. By exploiting fusion frames, we introduce a compressed sensing framework in which we split the dense information into subchannels and fuse the local estimations. Each piece of information is measured by linear, potentially low quality sensors, and recovered via compressed sensing. Finally, by a fusion process within the fusion frames, we are able to recover accurately the original signal. We illustrate our findings with numerical experiments, first consider various artificial setups in which we show that splitting a signal via local projections allows for accurate, stable, and robust estimation. We verify that by increasing the size of the fusion frame, a certain robustness to noise is also achieved. While the computational complexity remains relatively low, we achieve stronger recovery performance compared to usual single-device compressed sensing systems. We finally show how our techniques can be applied in various signal processing tasks such as Doppler signal denoising, natural scene scanning and reconstruction, and MR Image reconstruction. In these examples, we empirically verify that visually good reconstructions are obtained, even in highly undersampled and noisy regimes.