A key search term is *geometric tomography*.
The problem posed is a version of Hammer’s X-ray reconstruction problem:

> Gardner, Richard J., and Markus Kiderlen. "A solution to Hammer's X-ray reconstruction problem." *Advances in Mathematics* 214.1 (2007): 323-343.
[PDF download](http://home.math.au.dk/kiderlen/preprints/gard:kide07.pdf)

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[![Xrays][1]][1]
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Two shapes with the same parallel X-rays.
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  [1]: https://i.sstatic.net/0qPjh.jpg