>What I find fascinating is the application of information that is not present in the data but that can be confirmed by the data. I’m thinking specifically of deblurring algorithms people use to recover text.

I would argue that high frequency information is not missing in a blurred image, rather it's spread out spatially. Every sample in the blurred image is a convolution of multiple samples (i.e. high frequency data) from the originating source. Theoretically, it should be invertible, provided the exact function is known that does the convolution, but there are practical limitations, such as knowing how to apply the inverse operations at the edges and dealing with the extreme sensitivity to errors.