  In this work, we study the problem of reconstructing shapes from simple
nonasymptotic densities measured only along shape boundaries. The particular
density we study is also known as the integral area invariant and corresponds
to the area of a disk centered on the boundary that is also inside the shape.
It is easy to show uniqueness when these densities are known for all radii in a
neighborhood of r = 0, but much less straightforward when we assume that we
only know the area invariant and its derivatives for only one r > 0. We present
variations of uniqueness results for reconstruction (modulo translation and
rotation) of polygons and (a dense set of) smooth curves under certain
regularity conditions.
