Richard H. Bartels

Hierarchical B-spline refinement

Refinement is usually advocated as a means of gaining finer control over a spline curve or surface during editing. For curves, refinement is a local process. It permits the change of control vertices and subsequent editing of the detail in one region of the curve while leaving control vertices in other regions unaffected. For tensor-product surfaces, however, refinement is not local in the sense that it causes control vertices far from a region of interest to change as well as changing the control vertices that influence the region. However, with some care and understanding it is possible to restrict the influence of refinement to the locality at which an editing effect is desired. We present a method of localizing the effect of refinement through the use of overlays, which are hierarchically controlled subdivisions. We also introduce two editing techniques that are effective when using overlays: one is direct surface manipulation through the use of edit points and the other is offset referencing of control vertices.

Interpolating splines with local tension, continuity, and bias control

This paper presents a new method for using cubic interpolating splines in a key frame animation system. Three control parameters allow the animator to change the tension, continuity, and bias of the splines. Each of these three parameters can be used for either local or global control. Our technique produces a very general class of interpolating cubic splines which includes the cardinal splines as a proper subset.

Surface fitting with hierarchical splines

We consider the fitting of tensor product parametric spline surfaces to gridded data. The continuity of the surface is provided by the basis chosen. When tensor product splines are used with gridded data, the surface fitting problem decomposes into a sequence of curve fitting processes, making the computations particularly efficient. The use of a hierarchical representation for the surface adds further efficiency by adaptively decomposing the fitting process into subproblems involving only a portion of the data. Hierarchy also provides a means of storing the resulting surface in a compressed format. Our approach is compared to multiresolution analysis and the use of wavelets.

Minimization Techniques for Piecewise Differentiable Functions: The

A new algorithm is presented for computing a vector x which satisfies a given m by

l_1

n(m > n \geqq 2)

Solution to an Overdetermined Linear System

linear system in the sense that the

Constraint-based curve manipulation

l_1

Ray Tracing Free-Form B-Spline Surfaces

norm is minimized. That is, if A is a matrix having m columns

Reversing subdivision rules: local linear conditions and observations on inner products

a_1 , \cdots ,a_m

Multiresolution Curve and Surface Representation: Reversing Subdivision Rules by Least-Squares Data Fitting

each of length n, and b is a vector with components