"In 1975, the Jet Propulsion Laboratory at Caltech called Krantz with a problem. Pictures of Venus and Mars transmitted by the Voyager spacecraft had been taken in the dark while the craft was moving, and were, as was expected, horribly blurred. The Jet Propulsion Laboratory had a deblurring technique using an on-board computer that logged the spacecraft's motion, but the technique took a month and Congress wanted to see the pictures now. A researcher discovered a way to unblur the pictures more quickly if he could factor polynomials of several variables in a certain way. He called Krantz and asked him how to do it. "I thought about it for a while, and then I said, 'It can't be done.' Like most people, he just assumed that he had called the wrong person."
Funny--I came here for the same reason. My best guess is that they were trying to solve a deconvolution problem the naive way (by solving a huge linear system). I think the reference to polynomial factorization might have something to do with solving the problem by taking the Z-transform of both sides, and then having to divide the resulting polynomials and invert the transform.
For the demonstration at the end that shows that it is harder to perturb an almost circle into a circle than doing the reverse I would use the perfectly circular rim of a bicycle wheel.
It is easy to make such a wheel "almost round", but virtually impossible to revert that action.
"In 1975, the Jet Propulsion Laboratory at Caltech called Krantz with a problem. Pictures of Venus and Mars transmitted by the Voyager spacecraft had been taken in the dark while the craft was moving, and were, as was expected, horribly blurred. The Jet Propulsion Laboratory had a deblurring technique using an on-board computer that logged the spacecraft's motion, but the technique took a month and Congress wanted to see the pictures now. A researcher discovered a way to unblur the pictures more quickly if he could factor polynomials of several variables in a certain way. He called Krantz and asked him how to do it. "I thought about it for a while, and then I said, 'It can't be done.' Like most people, he just assumed that he had called the wrong person."