How to find a noisy line in a picture
What about an algorithm in image domain?

For every pixel, check into the direction you desire to know (up-down in your images above, I guess) if there is a pixel with "the same" value (i.e. a value inside an acceptance band, of course. Let's say +/- 10%) as the pixel under investigation. If so, go further into that direction. Count every pixel that you find. If the number of counts exceeds a certain threshold (i.e. 1/4 FOV or so in your case), then there is a line.

I would first try to use something simple like that instead of using more complicated algorithms that might be suitable for different kinds of images. Yours pretty much look like MRI phase images that can be nasty ;) )