  A novel algorithm is proposed for quantitative comparisons between compact
surfaces embedded in the three-dimensional Euclidian space. The key idea is to
identify those objects with the associated surface measures and compute a weak
distance between them using the Fourier transform on the ambient space. In
particular, the inhomogeneous Sobolev norm of negative order for a difference
between two surface measures is evaluated via the Plancherel theorem, which
amounts to approximating a weighted integral norm of smooth data on the
frequency space. This approach allows several advantages including high
accuracy due to fast-converging numerical quadrature rules, acceleration by the
nonuniform fast Fourier transform, and parallelization on many-core processors.
In numerical experiments, the 2-sphere, which is an example whose Fourier
transform is explicitly known, is compared with its icosahedral discretization,
and it is observed that the piecewise linear approximations converge to the
smooth object at the quadratic rate up to small truncation.
