We mainly analyze and solve the overfitting problem of deep image prior (DIP). Deep image prior can solve inverse problems such as super-resolution, inpainting and denoising. The main advantage of DIP over other deep learning approaches is that it does not need access to a large dataset. However, due to the large number of parameters of the neural network and noisy data, DIP overfits to the noise in the image as the number of iterations grows. In the thesis, we use hybrid deep image priors to avoid overfitting. The hybrid priors are to combine DIP with an explicit prior such as total variation or with an implicit prior such as a denoising algorithm. We use the alternating direction method-of-multipliers (ADMM) to incorporate the new prior and try different forms of ADMM to avoid extra computation caused by the inner loop of ADMM steps. We also study the relation between the dynamics of gradient descent, and the overfitting phenomenon. The numerical results show the hybrid priors play an important role in preventing overfitting. Besides, we try to fit the image along some directions and find this method can reduce overfitting when the noise level is large. When the noise level is small, it does not considerably reduce the overfitting problem.

翻译：本文主要分析并解决深度图像先验（DIP）的过拟合问题。深度图像先验可解决超分辨率、图像修复和去噪等逆问题。与其他深度学习方法相比，DIP的主要优势在于无需访问大规模数据集。然而，由于神经网络参数数量庞大且数据包含噪声，随着迭代次数增加，DIP会过度拟合图像中的噪声。本文采用混合深度图像先验来避免过拟合。混合先验是将DIP与显式先验（如全变分）或隐式先验（如去噪算法）相结合。我们使用交替方向乘子法（ADMM）融合新先验，并尝试不同形式的ADMM以避免ADMM步骤内循环带来的额外计算开销。同时，我们研究了梯度下降动力学与过拟合现象之间的关系。数值结果表明，混合先验在防止过拟合方面具有重要作用。此外，我们尝试沿特定方向拟合图像，发现当噪声水平较大时该方法可减少过拟合；但当噪声水平较小时，其对过拟合问题的改善并不显著。