CNN Denoisers as Non-Local Filters: The Neural Tangent Denoiser

Convolutional Neural Networks (CNNs) are now a well-established tool for solving computational imaging problems. Modern CNN-based algorithms obtain state-of-the-art performance in diverse image restoration problems. Furthermore, it has been recently shown that, despite being highly overparametrized, networks trained with a single corrupted image can still perform as well as fully trained networks, a phenomenon encapsulated in the deep image prior. We introduce a novel interpretation of denoising networks with no clean training data in the context of the neural tangent kernel (NTK), elucidating the strong links with well-known non-local filtering techniques, such as non-local means or BM3D. The filtering function associated with a given network architecture can be obtained in closed form without need to train the network, being fully characterized by the random initialization of the network weights. While the NTK theory accurately predicts the filter associated with networks trained using standard gradient descent, our analysis shows that it falls short to explain the behaviour of networks trained using the popular Adam optimizer. The latter achieves a larger change of weights in hidden layers, adapting the non-local filtering function during training. We evaluate our findings via extensive image denoising experiments.