To $e$ or not to $e$ in Poisson Image Reconstruction and Minimax Rates for Poisson Inverse Problems with Physical Constraints

In photon-limited image reconstruction, observations can be modeled as y~Poisson(f), where f:=exp(g) is the intensity of interest and g is the log-intensity. Previous work in this area has considered applying regularizers such as the total variation semi-norm to either f or to g:=log(f). The former is less stable at very low intensity levels and makes selecting tuning parameters challenging. The latter is more amenable to tuning via cross-validation, particularly at low intensity levels, but exhibits numerical instabilities and slow convergence at high intensity levels. This paper describes a novel hybrid approach in which the regularization mode is locally adapted to the signal intensity level. The resulting method yields strong empirical performance relative to previous approaches.

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We consider fundamental limits for solving sparse inverse problems in the presence of Poisson noise with physical constraints. Such problems arise in a variety of applications, including photon-limited imaging systems based on compressed sensing. Our minimax lower and upper bounds reveal that due to the interplay between the Poisson noise model, the sparsity constraint and the physical constraints: (i) the mean-squared error does not depend on the sample size $n$ other than to ensure the sensing matrix satisfies RIP-like conditions and the intensity $T$ of the input signal plays a critical role; and (ii) the mean-squared error has two distinct regimes, a low-intensity and a high-intensity regime and the transition point from the low-intensity to high-intensity regime depends on the input signal $f^*$.