Published on *Computer Science Department at Princeton University* (http://www.cs.princeton.edu)

Report ID:

TR-456-94

Authors:

Date:

March 1994

Pages:

11

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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.