Sampling Theory for Super-Resolution with Implicit Neural Representations

TL;DR

This paper investigates the sample complexity for super-resolution image recovery using implicit neural representations, establishing theoretical recovery guarantees and validating them through empirical experiments.

Contribution

It introduces a theoretical framework linking INR minimizers to convex measures, providing sample complexity bounds for exact image recovery.

Findings

Identifies sufficient Fourier samples for exact INR-based image recovery.

Empirically demonstrates high probability of exact recovery with low-width INRs.

Shows effectiveness of INRs in super-resolution of phantom images.

Abstract

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 samples 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 Fourier samples for which an image realized by an INR is exactly recoverable by solving the INR training problem. 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 INRs on super-resolution recovery of continuous domain phantom images.