  Modern geometric measure theory, developed largely to solve the Plateau
problem, has generated a great deal of technical machinery which is
unfortunately regarded as inaccessible by outsiders. Some of its tools (e.g.,
flat norm distance and decomposition in generalized surface space) hold
interest from a theoretical perspective but computational infeasibility
prevented practical use. Others, like nonasymptotic densities as shape
signatures, have been developed independently for data analysis (e.g., the
integral area invariant).
  The flat norm measures distance between currents (generalized surfaces) by
decomposing them in a way that is robust to noise. The simplicial deformation
theorem shows currents can be approximated on a simplicial complex,
generalizing the classical cubical deformation theorem and proving sharper
bounds than Sullivan's convex cellular deformation theorem.
  Computationally, the discretized flat norm can be expressed as a linear
programming problem and solved in polynomial time. Furthermore, the solution is
guaranteed to be integral for integral input if the complex satisfies a simple
topological condition (absence of relative torsion). This discretized
integrality result yields a similar statement for the continuous case: the flat
norm decomposition of an integral 1-current in the plane can be taken to be
integral, something previously unknown for 1-currents which are not boundaries
of 2-currents.
  Nonasymptotic densities (integral area invariants) taken along the boundary
of a shape are often enough to reconstruct the shape. This result is easy when
the densities are known for arbitrarily small radii but that is not generally
possible in practice. When only a single radius is used, variations on
reconstruction results (modulo translation and rotation) of polygons and (a
dense set of) smooth curves are presented.
