How to use deep learning to add local (e.g. repairing) transformations to images?
There is a paper called Spatial Transformer Networks written by Max Jaderberg et al. What it does is trying to find the canonical shape of its input by reducing transformations, like translation and rotation, or even diminishing the distortion of the inputs. It introduces a module which helps convolutional network to be spatial invariant. One of the significant achievements of this module is that it tries to enhance distorted inputs. Take a look at here.


I edit the answer because of the request of one of our friends. First I quote something from the popular book written by Pr. Gonzalez, I hope nothing goes wrong with copyright. Then I suggest my recommendation. 

Figure 2.40, the one that I've attached, shows an example of the steps in Fig. 2.39. In this case, the transform used was the Fourier transform, which we mention briefly later in this section and discuss in detail in Chapter 4. Figure 2.40(a) is an image corrupted by sinusoidal interference and Fig. 2.40(b) is the magnitude of its Fourier transform, which is the output of the first stage in Fig. 2.39. As you will learn in Chapter 4, sinusoidal interference in the spatial domain appears as bright bursts of intensity in the transform domain. In this case, the bursts are in a circular pattern that can be seen in Fig. 2.40(b). Figure 2.40(c) shows a mask
image (called a filter) with white and black representing 1 and 0, respectively.
For this example, the operation in the second box of Fig. 2.39 is to multiply the mask by the transform, thus eliminating the bursts responsible for the interference. Figure 2.40(d) shows the final result, obtained by computing the inverse of the modified transform. The interference is no longer visible, and important detail is quite clear. In fact, you can even see the fiducial marks (faint crosses) that are used for image alignment.



Okay! After those quotes, I refer to my suggestion. In the ST network, the authors have claimed that their differentiable module can learn different kinds of transformations other than affine transformation. The point about that is that we usually apply two kinds of transformations. One transformation is applied to the intensity of the image, which is popular by the means of filters. The other one is called image warping where we change the position of intensities and the intensity values do not change unless they locate between the discrete grids of image entries also called pixels, picture element. Spatial transformers are good for the second task but they also can be used for the first task. There are studies about using CNNs for evaluating the images in their frequency domain and not in the spatial domain. You can employ these differentiable modules in those nets. 

Finally about your specific task, what I'm seeing in your pictures, your noise has a same behaviour. I guess if that is true for all cases, your task is not learning and can be solved using usual image processing techniques.