  Topological transforms have been very useful in statistical analysis of
shapes or surfaces without restrictions that the shapes are diffeomorphic and
requiring the estimation of correspondence maps. In this paper we introduce two
topological transforms that generalize from shapes to fields, $f:\mathbf{R}^3
\rightarrow \mathbf{R}$. Both transforms take a field and associate to each
direction $v\in S^{d-1}$ a summary obtained by scanning the field in the
direction $v$. The transforms we introduce are of interest for both
applications as well as their theoretical properties. The topological
transforms for shapes are based on an Euler calculus on sets. A key insight in
this paper is that via a lifting argument one can develop an Euler calculus on
real valued functions from the standard Euler calculus on sets, this idea is at
the heart of the two transforms we introduce. We prove the transforms are
injective maps. We show for particular moduli spaces of functions we can upper
bound the number of directions needed determine any particular function.
