Funk, P. (1916). “Uber eine geometrische anwendung der abelschen integralgleichung.” Math. Ann. 77(129-135).

Funk’s work is limited to integrals on great circles of the sphere. Is this more evidence for Arnold’s law: Discoveries are rarely attributed to the correct person?

I’d also like to mention that, as an engineer who does a lot of image processing, I suspect that real CT scan computations use some sort of least squares solution to the system of equations. I find this to be the most practical way of solving complex image transforms in the presence of noise and imperfections, where a real solution is often inconsistent.

2-D phase unwrapping, for example, doesn’t really work when you have missing data points, but a weighted least squares solution is indistinguishable from perfect in most cases.

There are a couple of caveats:

If a person has metal implants, the implants strongly scatter x-rays and ordinary CT won’t work. There are researchers actively seeking ways to account for this.

Ordinary CT neglects phase changes of the x-rays on propagation through tissue, but this influence is still there and leads to propagation changes far enough downstream. In recent years such ‘phase contrast tomography’ has become an important research topic.

You’re right that most inverse scattering problems are nonlinear and require techniques which are quite involved. CT is somewhat special because it uses high-energy photons (x-rays), which barrel through the body with little deviation. Most inverse scattering theory since then seems to have involved trying to achieve similar success with photons at lower energies, where multiple scattering and diffraction effects are significant.

I ask because the (nonlinear) inverse problems I have come across end up being formulated as an optimization (minimization) problem.

You’re welcome, I’ll send the bill at the end of the month!

I’ll just re-iterate what Mr Walker said and say thanks for some excellent post on scientific subjects. Living in a town that is one of the major centres in the world for CT production and being a historian of science myself I was fascinated by an obviously well researched and well-written piece on their origins.

Your posts do a great job of helping people understand that science isn’t a magical thing done only by geniuses. It’s a pragmatic activity done by real people for real, commonsense reasons.