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Performance bounds for Poisson compressed sensing using Variance Stabilization Transforms

The analysis of reconstruction errors for compressed sensing under Poisson noise is challenging due to the signal dependent nature of the noise, and also because the Poisson negative log-likelihood is not a metric. In this paper, we present error bounds for reconstruction of signals which are sparse or compressible under any given orthonormal basis, given compressed measurements corrupted by Poisson noise and acquired in a realistic physical system. The concerned optimization problem is framed based on the well-known Variance Stabilization Transforms which transform the noise to (approximately) Gaussian with a fixed variance. This problem also turns out to be convex. We demonstrate promising numerical results on signals with different sparsity, intensity levels and given different numbers of compressed measurements.