  Implicit Neural Representations (INRs) encoding continuous multi-media data
via multi-layer perceptrons has shown undebatable promise in various computer
vision tasks. Despite many successful applications, editing and processing an
INR remains intractable as signals are represented by latent parameters of a
neural network. Existing works manipulate such continuous representations via
processing on their discretized instance, which breaks down the compactness and
continuous nature of INR. In this work, we present a pilot study on the
question: how to directly modify an INR without explicit decoding? We answer
this question by proposing an implicit neural signal processing network, dubbed
INSP-Net, via differential operators on INR. Our key insight is that spatial
gradients of neural networks can be computed analytically and are invariant to
translation, while mathematically we show that any continuous convolution
filter can be uniformly approximated by a linear combination of high-order
differential operators. With these two knobs, INSP-Net instantiates the signal
processing operator as a weighted composition of computational graphs
corresponding to the high-order derivatives of INRs, where the weighting
parameters can be data-driven learned. Based on our proposed INSP-Net, we
further build the first Convolutional Neural Network (CNN) that implicitly runs
on INRs, named INSP-ConvNet. Our experiments validate the expressiveness of
INSP-Net and INSP-ConvNet in fitting low-level image and geometry processing
kernels (e.g. blurring, deblurring, denoising, inpainting, and smoothening) as
well as for high-level tasks on implicit fields such as image classification.
