Variational Modeling with Wavelets

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.