Gregory M. Nielson

The asymptotic decider: resolving the ambiguity in marching cubes

A method for computing isovalue or contour surfaces of a trivariate function is discussed. The input data are values of the trivariate function, F/sub ijk/, at the cuberille grid points (x/sub i/, y/sub j/, z/sub k/), and the output of a collection of triangles representing the surface consisting of all points where F(x,y,z) is a constant value. The method is a modification that is intended to correct a problem with a previous method.<<ETX>>

Scattered data modeling

A variety of methods for modeling scattered data are discussed, with an emphasis on two types of data: volumetric and spherical. To demonstrate the performance of the various methods, results from an empirical study using trivariate scattered data are presented. The authors objective is to provide information to aid in selecting a type of method or to form the basis for customizing a method for a particular application.<<ETX>>

Visualizing functions over a sphere

Techniques for the visualization of a scalar-valued function defined over a spherical domain are discussed. Functions of this type arise, for example, in the fitting of measured data on the surface of the earth. Methods for contouring and for displaying the graph of a function are presented, along with several examples. The projected graph technique can also be considered as a method for modeling closed surfaces.<<ETX>>

On marching cubes

A characterization and classification of the isosurfaces of trilinear functions is presented. Based upon these results, a new algorithm for computing a triangular mesh approximation to isosurfaces for data given on a 3D rectilinear grid is presented. The original marching cubes algorithm is based upon linear interpolation along edges of the voxels. The asymptotic decider method is based upon bilinear interpolation on faces of the voxels. The algorithm of this paper carries this theme forward to using trilinear interpolation on the interior of voxels. The algorithm described here will produce a triangular mesh surface approximation to an isosurface which preserves the same connectivity/separation of vertices as given by the isosurface of trilinear interpolation.

SOME PIECEWISE POLYNOMIAL ALTERNATIVES TO SPLINES UNDER TENSION

Publisher Summary Splines under tension were first introduced by Schweikert. The motivation was to be able to imitate the behavior of cubic interpolating splines yet avoid the extraneous inflection points. This chapter discusses the characterization of splines under tension and presents some piecewise polynomial alternatives to splines under tension. The splines are characterized under tension as having a certain minimal property. This motivates the choice for an analogous minimization problem that has piecewise polynomials as its solution. These functions are referred to as ν-splines. Many of the properties of splines under tension are shared by this class of interpolants. The chapter also illustrates the effect of the tension for both ν-splines and splines in tension and describes the advantages that ν-splines have compared to the normal splines in tension.

Scattered Data Interpolation and Applications: A Tutorial and Survey

The multivariate scattered data interpolation problem is introduced and the reasons for the difficulty of the problem compared to the one dimensional case are discussed. Basic ideas for interpolation (or approximation) of scattered data are introduced. Various types of data sets and some strategies for dealing with some of them are given. Readily available algorithms for the solution of the problem are discussed and suitability for various types of data, along with discussion of situations where they have been useful is given. Some related ideas are briefly mentioned. Throughout there are bountiful references to the existing literature.

The side-vertex method for interpolation in triangles☆

Abstract Interpolation schemes which assume prescribed values on the boundary of a triangle are presented. The development of these interpolants is based upon univariate interpolation along line segments joining a vertex and a side. Initially, methods which only interpolate to function values on the boundary are described. This is followed by the application of several techniques which extend these methods so as to include interpolation to first order derivatives on the boundary.

Direct manipulation techniques for 3D objects using 2D locator devices

Three dimensional input techniques have long remained a problem in computer graphics because of the two dimensional nature of our display and interaction media. The most successful approaches to date are based upon specialized input hardware ranging from special rooms in which three dimensional manipulations can be performed [Suth 681 to simple multiaxis joysticks with an additional joint to support an additional degree of freedom [Brit 781. As well as a host of others.

Terrain simulation using a model of stream erosion

The major process affecting the configuration and evolution of terrain is erosion by flowing water. Landscapes thus reflect the branching patterns of river and stream networks. The network patterns contain information that is characteristic of the landscapes topographic features. It is therefore possible to create an approximation to natural terrain by simulating the erosion of stream networks on an initially uneroded surface. Empirical models of stream erosion were used as a basis for the model presented here. Stream networks of various sizes and shapes are created by the model from a small number of initial parameters. The eroded surface is represented as a surface under tension, using the tension parameter to shape the profiles of valleys created by the stream networks. The model can be used to generate terrain databases for flight simulation and computer animation applications.

Dual Marching Cubes

We present the definition and computational algorithms for a new class of surfaces which are dual to the isosurface produced by the widely used marching cubes (MC) algorithm. These new isosurfaces have the same separating properties as the MC surfaces but they are comprised of quad patches that tend to eliminate the common negative aspect of poorly shaped triangles of the MC isosurfaces. Based upon the concept of this new dual operator, we describe a simple, but rather effective iterative scheme for producing smooth separating surfaces for binary, enumerated volumes which are often produced by segmentation algorithms. Both the dual surface algorithm and the iterative smoothing scheme are easily implemented.