Bayesian Experimental Design for Compressed Sensing

Compressed sensing (CS) can be addressed as instance of Bayesian experimental design, which is based on similar models than common estimation (or energy minimisation) methods, but fundamentally differs from the latter, in that estimates of uncertainty and correlations of latent variables are maintained. The Bayesian computations are approximated by the expectation propagation fixed point alg orithm, which is scaled up to the application of interest here through a novel scheduling mechanism, exploiting the fact that marginal uncertainty estimates are available at all times. An important application of CS is the optimisation of architectures for measuring natural images. In a large study, we compare various CS reconstruction methods utilising random measurement filters from different ensembles to a number of techniques which sequentially search for these filters, including our own, and Bayesian projection optimisation (1). We find that a simple heuristic of measuring wavelet coefficients in a fixed, top-down ordering significantly outperforms CS methods using random measurements; the approach of (1) performs even worse. In contrast, our Bayesian design method learns filters that o utperform the wavelet heuristic. Our results show that the property of incoherence of a measurement design, which plays a central role in the "unstructured except for random sparsity" theoretical CS setting, bears no significa nce for measuring real natural images. Our framework is not restricted to sparse signals, other notions of signal or noise structure can easily be accommodated. We give concrete ideas how our method can be scaled up to large signal representations. Part of this work has been presented at a conference (2).