  Here we present a new non-parametric approach to density estimation and
classification derived from theory in Radon transforms and image
reconstruction. We start by constructing a "forward problem" in which the
unknown density is mapped to a set of one dimensional empirical distribution
functions computed from the raw input data. Interpreting this mapping in terms
of Radon-type projections provides an analytical connection between the data
and the density with many very useful properties including stable
invertibility, fast computation, and significant theoretical grounding. Using
results from the literature in geometric inverse problems we give uniqueness
results and stability estimates for our methods. We subsequently extend the
ideas to address problems in manifold learning and density estimation on
manifolds. We introduce two new algorithms which can be readily applied to
implement density estimation using Radon transforms in low dimensions or on low
dimensional manifolds embedded in $\mathbb{R}^d$. We test our algorithms
performance on a range of synthetic 2-D density estimation problems, designed
with a mixture of sharp edges and smooth features. We show that our algorithm
can offer a consistently competitive performance when compared to the
state-of-the-art density estimation methods from the literature.
