  This thesis analyzes the challenging problem of Image Deblurring based on
classical theorems and state-of-art methods proposed in recent years. By
spectral analysis we mathematically show the effective of spectral
regularization methods, and point out the linking between the spectral
filtering result and the solution of the regularization optimization objective.
For ill-posed problems like image deblurring, the optimization objective
contains a regularization term (also called the regularization functional) that
encodes our prior knowledge into the solution. We demonstrate how to craft a
regularization term by hand using the idea of maximum a posterior estimation.
Then, we point out the limitations of such regularization-based methods, and
step into the neural-network based methods.
  Based on the idea of Wasserstein generative adversarial models, we can train
a CNN to learn the regularization functional. Such data-driven approaches are
able to capture the complexity, which may not be analytically modellable.
Besides, in recent years with the improvement of architectures, the network has
been able to output an image closely approximating the ground truth given the
blurry observation. The Generative Adversarial Network (GAN) works on this
Image-to-Image translation idea. We analyze the DeblurGAN-v2 method proposed by
Orest Kupyn et al. [14] in 2019 based on numerical tests. And, based on the
experimental results and our knowledge, we put forward some suggestions for
improvement on this method.
