If you do something really simple like a Gaussian blur (which is a type of convolution), it might be possible to find the inverse convolution (de-convolution) and restore the original image with some accuracy.
One method is the Lucy-Richardson deconvolution [1], which is an iterative algorithm, and here [2] is the best practical example I could find right away. Unfortunately the text is not in English, but the illustrations and formulae might be enough to give some intuition of the process.
[1] https://en.wikipedia.org/wiki/Richardson%E2%80%93Lucy_deconv...