  Compression of point clouds has so far been confined to coding the positions
of a discrete set of points in space and the attributes of those discrete
points. We introduce an alternative approach based on volumetric functions,
which are functions defined not just on a finite set of points, but throughout
space. As in regression analysis, volumetric functions are continuous functions
that are able to interpolate values on a finite set of points as linear
combinations of continuous basis functions. Using a B-spline wavelet basis, we
are able to code volumetric functions representing both geometry and
attributes. Geometry is represented implicitly as the level set of a volumetric
function (the signed distance function or similar). Attributes are represented
by a volumetric function whose coefficients can be regarded as a critically
sampled orthonormal transform that generalizes the recent successful
region-adaptive hierarchical (or Haar) transform to higher orders. Experimental
results show that both geometry and attribute compression using volumetric
functions improve over those used in the emerging MPEG Point Cloud Compression
standard.
