提出隐式神经表示在超分辨率中的采样理论，证明了低通傅里叶采样可精确重建图像。
Sampling Theory for Super-Resolution with Implicit Neural Representations
- 通过带正则化的单层ReLU网络建模连续图像，将非凸优化转为测度空间的凸问题
- 理论证明：当傅里叶采样数≥20时，可精确恢复低宽网络生成的图像
- 适用于图像重建、超分辨等需连续域建模的任务，适合研究逆问题的学者
隐式神经表示（INRs）已成为计算机视觉与计算成像中求解逆问题的强大工具。INRs 将图像表示为以空间坐标为输入的神经网络所实现的连续域函数。然而，与传统的像素表示不同，关于使用 INRs 在线性逆问题中估计图像的采样复杂度，目前了解甚少。为此，我们研究了通过拟合具有 ReLU 激活和傅里叶特征层的单隐藏层 INR，并采用广义权重衰减正则化，从其低通傅里叶样本中恢复连续域图像所需的采样要求。我们的关键洞察是将该非凸参数空间优化问题的最小值，关联到无限维测度空间上一个凸惩罚问题的最小值。我们确定了足够数量的傅里叶样本，使得由 INR 实现的图像可通过求解 INR 训练问题被精确恢复。为验证该理论，我们实证评估了低宽度单隐藏层 INRs 实现图像的精确恢复概率，并展示了 INRs 在连续域幻影图像超分辨率恢复中的性能。
原文摘要 · Abstract (English)
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.
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