What is a periodic signal in image processing?
You can think of the periodic signals like sine and cosine are basis for building other signal. See this figure for a 1-D example.

For images, sine and cosine extend to 2-D patterns as something like in here. As in 1-D signal, any images can be represented as a weighted sum of these basis images.  And roughly speaking, DFT are just weights in the weighted sum above. 

Convolution connects with DFT by the nice convolution theorem. Namely, if you have two images (one of them is typically a filter, but you can consider any filter as an image patch as well) A &amp; B, F(A * B)=F(A)F(B). That is, the Fourier transform of the convolved image of A and B is equal to the product of the Fourier transform of image A and the Fourier transform of image B.