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Both noise and deformations are defined in relation to a model - the perfect picture. Of course, when you take a picture, there is no accompanying 'standard' to allow you to compare. So, NO, there is no way to define noise or distortions in an image.
However, this subject is a really interesting field, and many scientists and students tried to find ways to approximate the noise in an image. So far, there is no accepted or perfect way to do it, and the best you can is to google and read papers on the subject.
As for geometrical deformations, Wiki is your friend. There is plenty of info out there, no need to paste it here. Again, you will not find any way to miraculously measure it.
By the way, I do not understand why professors set this as homework - it's a very popular topic among students.