Can we build continuous generative models which generalize across scales, can be evaluated at any coordinate, admit calculation of exact derivatives, and are conceptually simple? Existing MLP-based architectures generate worse samples than the grid-based generators with favorable convolutional inductive biases. Models that focus on generating images at different scales do better, but employ complex architectures not designed for continuous evaluation of images and derivatives. We take a signal-processing perspective and treat continuous image generation as interpolation from samples. Indeed, correctly sampled discrete images contain all information about the low spatial frequencies. The question is then how to extrapolate the spectrum in a data-driven way while meeting the above design criteria. Our answer is FunkNN -- a new convolutional network which learns how to reconstruct continuous images at arbitrary coordinates and can be applied to any image dataset. Combined with a discrete generative model it becomes a functional generator which can act as a prior in continuous ill-posed inverse problems. We show that FunkNN generates high-quality continuous images and exhibits strong out-of-distribution performance thanks to its patch-based design. We further showcase its performance in several stylized inverse problems with exact spatial derivatives.

翻译：能否构建一类连续生成模型，使其能跨尺度泛化、在任意坐标处求值、支持精确导数计算，且概念上简洁？现有基于多层感知机的架构生成的样本质量，劣于借助有利卷积归纳偏置的网格基生成器。侧重多尺度图像生成的模型虽表现更优，但其复杂架构并非为图像与导数的连续求值而设计。我们从信号处理视角出发，将连续图像生成视为基于样本的插值问题。事实上，正确采样的离散图像已包含低空间频率的全部信息。问题在于：如何在满足上述设计准则的前提下，以数据驱动方式外推频谱？我们的答案是FunkNN——一种新型卷积网络，它能学习在任意坐标处重建连续图像，并可适用于任何图像数据集。与离散生成模型结合后，它即成为函数生成器，可作为连续病态逆问题的先验。实验表明，得益于基于块的设计，FunkNN能生成高质量连续图像，并在分布外场景中展现出强泛化性能。我们进一步在若干涉及精确空间导数的风格化逆问题中展示了其性能。