  The deep image prior showed that a randomly initialized network with a
suitable architecture can be trained to solve inverse imaging problems by
simply optimizing it's parameters to reconstruct a single degraded image.
However, it suffers from two practical limitations. First, it remains unclear
how to control the prior beyond the choice of the network architecture. Second,
training requires an oracle stopping criterion as during the optimization the
performance degrades after reaching an optimum value. To address these
challenges we introduce a frequency-band correspondence measure to characterize
the spectral bias of the deep image prior, where low-frequency image signals
are learned faster and better than high-frequency counterparts. Based on our
observations, we propose techniques to prevent the eventual performance
degradation and accelerate convergence. We introduce a Lipschitz-controlled
convolution layer and a Gaussian-controlled upsampling layer as plug-in
replacements for layers used in the deep architectures. The experiments show
that with these changes the performance does not degrade during optimization,
relieving us from the need for an oracle stopping criterion. We further outline
a stopping criterion to avoid superfluous computation. Finally, we show that
our approach obtains favorable results compared to current approaches across
various denoising, deblocking, inpainting, super-resolution and detail
enhancement tasks. Code is available at
\url{https://github.com/shizenglin/Measure-and-Control-Spectral-Bias}.
