X-Ray Transform, the Legendre Transform, and Envelopes

We consider the X-ray transform, defined as an integral over affine subspaces, of piecewise-smooth functions, and describe the behaviour of the X-ray transform in a neighbourhood of its singular locus. We consider the problem of reconstruction of the singular locus of the original function given the singular locus of the X-ray transform. The latter problem is studied also under the condition that only part of the data is known, the so called X-ray transform with sources on a curve. The technical tools are different generalizations of the Legendre transform. The results provide new procedures for constructing envelopes for families of affine subspaces of Rn.