Variational Modeling with Wavelets

Abstract:

In many geometric modeling paradigms the user sculpts a curve or
surface by dragging around some type of control points (eg. Bezier or
B-spline). A more intuitive modeling interface allows the user to
directly manipulate curves and surfaces. This manipulation defines
some set of constraints that the curve or surface must satisfy (such
as interpolation and tangent constraints). Direct manipulation,
however, usually leads to an underconstrained problem since there are,
in general, many possible surfaces meeting some set of constraints.
Finding the ``best'' solution requires solving a variational problem.
Unfortunately, this can be costly to compute. In particular,
iterative descent methods converge slowly when a finite element basis
such as B-splines is used.
This paper discusses how geometric variational modeling problems can
be solved more efficiently by using a wavelet basis. Because the
wavelet basis is hierarchical, the iterative methods converge rapidly.
And because the wavelet coefficients indicate the degree of detail in
the solution, they can be used to determine the number of basis
functions needed to express the variational minimum, thus avoiding
unnecessary computation. An implementation is discussed and
experimental results are reported.