Towards a Sampling Theory for Implicit Neural Representations
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.
Code (0)
등록된 구현이 없습니다.
Tasks
Super-ResolutionMethods 이 논문이 사용한 방법론
Similar Papers 제목 키워드 기반
A Sampling Theory Perspective on Activations for Implicit Neural Representations
Implicit Neural Representations (INRs) have gained popularity for encoding signals as compact, differentiable entities. While commonly using techniques like Fourier positional encodings or non-traditional activation func…
Sampling Theory for Super-Resolution with Implicit Neural Representations
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 …
Super-ResolutionA Level Set Theory for Neural Implicit Evolution under Explicit Flows
Coordinate-based neural networks parameterizing implicit surfaces have emerged as efficient representations of geometry. They effectively act as parametric level sets with the zero-level set defining the surface of inter…
Inverse RenderingEnhancing Surface Neural Implicits with Curvature-Guided Sampling and Uncertainty-Augmented Representations
Neural implicit representations have become a popular choice for modeling surfaces due to their adaptability in resolution and support for complex topology. While previous works have achieved impressive reconstruction qu…
3D ReconstructionSurface ReconstructionInference, Sampling, and Learning in Copula Cumulative Distribution Networks
The cumulative distribution network (CDN) is a recently developed class of probabilistic graphical models (PGMs) permitting a copula factorization, in which the CDF, rather than the density, is factored. Despite there be…