Muon tomography is one of the most important spinoffs of fundamental research with particle detectors -if not the most important. 
It was realized already some sixty years ago that muons produced in the upper atmosphere by energetic cosmic radiation (protons or light nuclei) constituted a very useful probe of the interior of inaccessible volumes: since muons are very penetrating particles, and they withstand only minor Coulomb scattering as they pass through dense layers of matter (with a useful dependency of the scattering angle on the density of the material), by placing tracking detector before and after the interaction of muons with a unknown volume it is possible to infer the 3D map of the material in the volume; or, when it is impossible to have detectors tracking muons before they enter the volume, it is still possible to achieve that result by leveraging knowledge of the expected flux.

One of the earliest demonstrations of this principle was the scan of the interior of a pyramid in Giza, by a team led by Luis Alvarez in the 1960ies. Since then, muon tomography has become an incredibly versatile tool for scanning anything from archaeological structures, volcanos, mines, or even the interior of plasma reactors. Commercial applications also include border control devices, cultural heritage prospections, wear surveys in concrete bridges, etcetera. 

To long-time readers of this blog, though, the question is: what can be the interest in muon tomography of a particle physicist with a penchant for artificial intelligence? After all, the analytical problem of reconstructing densities from scattering angles is not too complicated, nor does it require the specific background that a particle physicist has; nor is the physics of the employed particle detectors particularly exciting - we know very well how to track muons and there is little to learn there.

The answer is that muon tomography lends itself excellently as a simple use case of the advanced technology I wish to develop for particle physics applications. This is the complete software modeling of particle interactions with detectors and data-generation devices, their reconstruction, and the inference that can be extracted from the resulting data, within a software pipeline that allows the automatic scanning of the parameter space of detector design choices. In other words, it is possible today to put together complex computer simulations that learn the optimal way of designing particle detectors, if one is capable of deciding what exactly is the objective that their instrument needs to achieve.

The field of differentiable programming - a new term that replaces and enhances the now aged concept of "deep learning" - has been booming in the past decade. It has been realized that the engine under the hood of neural networks - the back-propagation of the derivatives of a loss function through the network layers - can be used as the key ingredient in complex simulations that model not only the inference procedures (as is normally done in classification or regression tasks with neural networks, for instance), but the whole system under study, as well as the cost of the hardware, any constraints that prevent specific solutions, time considerations, and other boundary conditions.

Muon tomography is perfect as a test case because we only deal with one particle at a time there - as opposed to the thousands produced in a LHC collision, e.g. It is then possible to create a fully differentiable model of the interaction of the muon with the detectors and with the unknown volumes, and produce a fully differentiable pipeline where the utility function can be maximized by stochastic gradient descent.

It has taken us about three years to come up with a well-developed software package that does precisely that: it models a detector scanning a unknown volume, models muons interacting with the system, models the inference and reconstruction of the volume density, and the precision of the inference. This allows the system to search iteratively for the most promising configuration of the detector - the one that costs less money for the same result, or that minimize the scanning time for a fixed wanted precision, or a combination of these requirements.

The software is still not terribly useful for market applications as it is, because it does not handle the totality of possible applications of tomography: it is more like a demonstrative tool for now, with only a few tested use cases (one of which is however an important industrial one, the task of measuring the level of molten iron inside furnace ladles). Nevertheless, thanks to the wits of our main developer Giles Strong, it has been written in a way that is easily scalable. Anybody who has the competence to do so by him or herself can now download the full software package from GitHub and customize to the use case of interest; of course, the other way is to interact with us authors for an expert customization.

The TomOpt package is described in a recent preprint, now submitted for publication in IOP: Machine learning for Science and Technology. You can find the software in github here.

TomOpt is one of the projects we have been developing within the MODE Collaboration ( There are a large number of studies ongoing that all share the idea of end-to-end modeling. Some of them are described in these two publications:
- T. Dorigo et al., Toward the end-to-end optimization of particle physics instruments with differentiable programming, Rev. in Ph. 10 (2023) 100085
- M. Aehle et al., Progress in end-to-end optimization of detectors for fundamental physics with differentiable programming, arXiv:2310.05673