Risk-based approaches for Poisson image denoising with neural networks

In this work, we address the problem of restoring images corrupted by Poisson noise using a Neural Network (NN). Following the Deep Image Prior (DIP) approach, we optimize the parameters of the NN to produce a denoised version of a single noisy input image. This is achieved by minimizing the Kullback-Leibler (KL) divergence between the noisy image and the network’s prediction. To regularize this approach, we compute the value of the Poisson Asymptotically Unbiased Kullback-Leibler risk estimator (PAUKL) across the iterations of the optimization process, and we stop the process once the minimum of the estimator is attained. In fact, PAUKL is an estimator of the expected value of the KL divergence between the (unknown) noise-free image and the NN prediction. Moreover, we propose to optimize the NN parameters by minimizing the PAUKL estimator itself. We prove that Multi-Layer Perceptrons (MLPs) with arbitrary number of layers and sigmoid activation functions satisfy the hypotheses which guarantee that PAUKL is asymptotically unbiased. Finally, we assess the performance of the proposed techniques with numerical tests carried out in both simulated and real-case scenarios.