Abstract
We introduce Deep Spectral Prior (DSP), a new formulation of Deep Image Prior (DIP) that redefines image reconstruction as a frequency-domain alignment problem. Unlike traditional DIP, which relies on pixel-wise loss and early stopping to mitigate overfitting, DSP directly matches Fourier coefficients between the network output and observed measurements. This shift introduces an explicit inductive bias towards spectral coherence, aligning with the known frequency structure of images and the spectral bias of convolutional neural networks. We provide a rigorous theoretical framework demonstrating that DSP acts as an implicit spectral regulariser, suppressing high-frequency noise by design and eliminating the need for early stopping. Our analysis spans four core dimensions establishing smooth convergence dynamics, local stability, and favourable bias-variance tradeoffs. We further show that DSP naturally projects reconstructions onto a frequency-consistent manifold, enhancing interpretability and robustness. These theoretical guarantees are supported by empirical results across denoising, inpainting, and super-resolution tasks, where DSP consistently outperforms classical DIP and other unsupervised baselines.
Abstract (translated)
我们介绍了深度频谱先验（DSP），这是一种新的深度图像先验（DIP）公式，它将图像重建定义为一个频率域对齐问题。与传统的DIP依赖于像素级损失和提前停止以减少过拟合不同，DSP直接匹配网络输出和观测测量之间的傅里叶系数。这种转变引入了明确的归纳偏差，倾向于频谱一致性，这符合已知的图像频率结构以及卷积神经网络的频谱偏置。我们提供了一个严格的理论框架，证明DSP作为隐式频谱正则化器的作用，在设计上抑制高频噪声，并消除了提前停止的需求。我们的分析涵盖了四个核心维度，建立了平滑收敛动力学、局部稳定性及有利偏差-方差权衡。此外，我们还展示了DSP自然地将重建投影到一致的频率流形上，增强了可解释性和鲁棒性。这些理论保证得到了去噪、修复和超分辨率任务中经验结果的支持，在这些任务中，DSP始终优于传统的DIP和其他无监督基准方法。
URL
https://arxiv.org/abs/2505.19873
