  The purpose of this paper is twofold. In the first part we concentrate on
hyperplane sections of algebraic schemes, and present results for determining
when Gr\"obner bases pass to the quotient and when they can be lifted. The main
difficulty to overcome is the fact that we deal with non-homogeneous ideals. As
a by-product we hint at a promising technique for computing implicitization
efficiently.
  In the second part of the paper we deal with families of algebraic schemes
and the Hough transforms, in particular we compute their dimension, and show
that in some interesting cases it is zero. Then we concentrate on their
hyperplane sections. Some results and examples hint at the possibility of
reconstructing external and internal surfaces of human organs from the parallel
cross-sections obtained by tomography.
