Implicit neural representations (INRs) have emerged as a powerful tool for
solving inverse problems in computer vision and computational imaging. INRs
represent images as continuous domain functions realized by a neural network
taking spatial coordinates as inputs. However, unlike traditional pixel
representations, little is known about the sample complexity of estimating
images using INRs in the context of linear inverse problems. Towards this end,
we study the sampling requirements for recovery of a continuous domain image
from its low-pass Fourier coefficients by fitting a single hidden-layer INR
with ReLU activation and a Fourier features layer using a generalized form of
weight decay regularization. Our key insight is to relate minimizers of this
non-convex parameter space optimization problem to minimizers of a convex
penalty defined over an infinite-dimensional space of measures. We identify a
sufficient number of samples for which an image realized by a width-1 INR is
exactly recoverable by solving the INR training problem, and give a conjecture
for the general width-$W$ case. To validate our theory, we empirically assess
the probability of achieving exact recovery of images realized by low-width
single hidden-layer INRs, and illustrate the performance of INR on
super-resolution recovery of more realistic continuous domain phantom images.

隐式神经表示（INR）已成为计算机视觉和计算成像中解决反问题的强大工具。在线性反问题背景下，我们研究了使用单隐藏层 ReLU 激活和傅里叶特征层的广义权重衰减正则化方法，来从低通傅里叶系数恢复连续域图像的采样要求，验证了我们的理论，并展示了 INR 在超分辨恢复更实际的连续域幻影图像上的性能。