Muon tomography is an application of particle detectors where we exploit the peculiar properties of muons to create three-dimensional images of the interior of unknown, inaccessible volumes. You might also want to be reminded that muons are unstable elementary particles; they are higher-mass versions of electrons which can be found in cosmic ray showers or produced in particle collisions.
I wrote a couple of posts on muon tomography in this blog in the recent past, where I proposed a novel technique to leverage a not-well-studied physical phenomenon (K capture) that takes place when negatively-charged muons stop inside matter.  This generated some interest from a few students (I don't even know from which country) who contacted me for discussions. It turns out that these students (a pack of high-school blokes who evidently are too bright to spend their time on videogames) want to participate in a call that CERN and DESY laboratories are making, offering their accelerators for innovative or didactical projects. This is called "Beamline for Schools", and it is a great opportunities for would-be-physicists! I wish I could participate in something like that when I was 17.

So, being the outreach-enthusiast that I am, I could not shy away from supporting them, and I am providing some advisory. On the other hand, during our discussions it occurred to me that there is a further method to exploit the physical processes that take place when muons traverse materials, beyond the well-used scattering and absorption techniques. Unlike k-capture, this is actually not difficult to study and test.

The idea is to exploit the decay of both negative and positive muons inside the volume of interest. Muons who are not captured by nuclei decay by producing an electron (with the same charge of the muon) plus two neutrinos. The electron is emitted with an energy ranging from a few to 50 MeV, and therefore it cannot traverse large amounts of materials - it soon gets absorbed. The change of making it through a certain thickness of material will thus depend on the initial energy as well as the specific material type. 

Suppose you want to extract information on the material traversed by an electron in emerging from the decay point where the muon stopped in the volume. What do you do? Here is a plan.

1) surround the volume of interest with tracking detectors;
2) record the direction of the incident muons (that come from cosmic ray interactions in the upper atmosphere);
3) record the direction of arrival of electrons exiting the volume.

The above recipe generates a dataset where for each muon (an event) you have information about its decay point (which you can fit from the two straight lines obtained by the measurements). What do you do with it? Here is the key plan:

1) Assume that the volume has a uniform material composition, and define what that is. This could be the atomic species with solid-state density closest to the observed ratio of total weight divided by volume (if you have access to weight), or only a wild guess;
2) Virtually divide the volume into voxels - small 3D elements like cubes; then compute an estimate for the number of muons stopped in each voxel, by using the model of the material composition and a model of the incident momentum of the flux of muons. Note that if you surrounded the volume with detectors, you have a rough estimate of which muons stopped inside (those for which no exiting signal was seen).
3) Compute the probability for each observed electron to have made it through the volume and into your detector, based on the assumed material model and the actual voxels electrons traversed;
4) Compare your data to the rate of decays you should have reconstructed, given the assumed material composition; find the derivative of the difference between the two quantities with respect to the material hypothesis in each voxel; update the material model using those derivatives; iterate to convergence.

The above is a classical use of gradient descent for the solution of the problem; however, note that in principle the problem per se is not intractable - it can be handled by a maximum likelihood method, if you parametrize all the ingredients (the physics of muon incidence, energy loss, decay, and the physics of electron energy loss).

I do not have time to put together a simulated proof of principle of this method, but I suspect it has been done before. So the next step is to check the literature. See, I have no shame - if I feared to show my ignorance I would have first checked the literature; but then, upon finding out that my ideas, put in a much more refined form, had been already had by others before me, all the fun would be gone, and you would not be reading this post!

If you are interested in the topic, however, I can promise I will get back to it at some point.