Henrik Weimer

A subdivision scheme for surfaces of revolution

This paper describes a simple and efficient non-stationary subdivision scheme of order 4. This curve scheme unifies known subdivision rules for cubic B-splines, splines-in-tension and a certain class of trigonometric splines capable of reproducing circles. The curves generated by this unified subdivision scheme are C^2 splines whose segments are either polynomial, hyperbolic or trigonometric functions, depending on a single tension parameter. This curve scheme easily generalizes to a surface scheme over quadrilateral meshes. The authors hypothesize that this surface scheme produces limit surfaces that are C^2 continuous everywhere except at extraordinary vertices where the surfaces are C^1 continuous. In the particular case where the tension parameters are all set to 1, the scheme reproduces a variant of the Catmull-Clark subdivision scheme. As an application, this scheme is used to generate surfaces of revolution from a given profile curve.

Subdivision schemes for fluid flow

The motion of fluids has been a topic of study for hundreds of years. In its most general setting, fluid flow is governed by a system of non-linear partial differential equations known as the Navier-Stokes equations. However, in several important settings, these equations degenerate into simpler systems of linear partial differential equations. This paper will show that flows corresponding to these linear equations can be modeled using subdivision schemes for vector fields. Given an initial, coarse vector field, these schemes generate an increasingly dense sequence of vector fields. The limit of this sequence is a continuous vector field defining a flow that follows the initial vector field. The beauty of this approach is that realistic flows can now be modeled and manipulated in real time using their associated subdivision schemes. CR Categories: I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling— Physically based modeling; I.6.5 [Simulation and Modeling]: Model Development—Modeling Methodologies

Efficient co-triangulation of large data sets

Presents an efficient algorithm for the reconstruction of a multivariate function from multiple sets of scattered data. Given N sets of scattered data representing N distinct dependent variables that have been sampled independently over a common domain and N error tolerance values, the algorithm constructs a triangulation of the domain of the data and associates multivariate values with the vertices of the triangulation. The resulting linear interpolation of these multivariate values yields a multivariate function, called a co-triangulation, that represents all of the dependent data up to the given error tolerance. A simple iterative algorithm for the construction of a co-triangulation from any number of data sets is presented and analyzed. The main contribution of this paper lies in the description of a highly efficient framework for the realization of this approximation algorithm. While the asymptotic time complexity of the algorithm certainly remains within the theoretical bounds, we demonstrate that it is possible to achieve running times that depend only linearly on the number of data even for very large problems with more than two million samples. This efficient realization of the algorithm uses adapted dynamic data structures and careful caching in an integrated framework.

Subdivision: Functions as Fractals

This chapter reviews some of the basic ideas behind modeling shapes using both functions and fractals. A subdivision may be viewed as the synthesis of two previously distinct approaches to modeling shape: functions and fractals. One interesting property of the subdivision scheme for linear splines is that it makes no explicit use of the piecewise polynomial definition of the unit hat function n[x]. The scheme uses only the subdivision matrix S. For piecewise linear splines, the subdivision matrix S defined a convergent process. Thus, the scaling functions N [2 k x] associated with the subdivision scheme could be computed entirely in terms of the matrix S. Two popular techniques for representing complex shapes have been described— explicit functions and fractal procedures. When describing shapes with functions, piecewise polynomials are a particularly common and practical choice. Piecewise polynomial functions provide a convenient and efficient method of defining smooth curved shapes. Using subdivision, a curve is defined as the limit of an increasingly faceted sequence of polygons. Each polygon is related to a successor by a simple linear transformation that provides an increasingly accurate approximation to the final limit curve. Many of the traditional shapes used in geometric design can be defined using subdivision. The chapter concludes with a brief introduction to the fundamentals of subdivision.

An Integral Approach to Uniform Subdivision

This chapter introduces a general method for creating scaling functions that may be refined—directional integration. This integral approach is the standard technique used in constructing subdivision schemes for B-splines and box splines and leads directly to a simple repeated averaging algorithm for subdividing these splines. For example, the B-spline basis functions have a simple recursive definition in terms of integration. Based on this definition, the refinement relation for B-splines is derived and the associated subdivision scheme is constructed. . The chapter presents an equivalent geometric method for constructing the B-spline basis functions as the cross-sectional volumes of high dimensional hypercubes. This construction easily generalizes from the univariate case to the bivariate case and allows creation of refineable scaling functions in two variables. The splines associated with these scaling functions, box splines, also have an elegant subdivision algorithm. The chapter concludes with a simple method for constructing explicit piecewise polynomial expressions for both B-splines and box splines. Again, these constructions have an interesting geometric interpretation in terms of high-dimensional geometry.

A Differential Approach to Uniform Subdivision

This chapter introduces a differential method for constructing subdivision schemes. Given a differential equation whose solutions form the segments of the spline, this method involves deriving a finite difference equation that characterizes a sequence of solutions to the differential equation on increasingly fine grids. If the right-hand side of this difference equation is chosen appropriately, successive solutions to this finite difference equation, p k-1 [x] and p k [x], can be related via a subdivision recurrence of the form p k [x] = s k−1 [x]p k-1 [x 2 ]. The primary advantage of this approach is that it can reproduce all of the schemes resulting from the integral approach, such as B-splines and box splines, while yielding new schemes, such as those for thin plates and slow flows that do not possess any equivalent integral formulation. The chapter considers the two examples of B-splines and box splines, from a differential point of view. In each case, the differential approach yields the same subdivision mask as the integral approach. By taking a slightly more abstract, operator-based view, the relation between these two approaches can be captured with beautiful precision. Starting from the differential approach, the subdivision scheme for exponential B-splines is derived. The chapter concludes with a discussion on the useful variant of exponential splines.

Chapter 7 – Averaging Schemes for Polyhedral Meshes

This chapter discusses a method of subdividing surface meshes using repeated averaging, including an extension to the case of surfaces of revolution. A more flexible representation for surfaces—in terms of topological meshes capable of representing surfaces of arbitrary topology—avoids this problem. The chapter also includes a simple method for handling creases and boundaries of subdivision surfaces. Some of the basic properties of topological meshes are introduced and two subdivision schemes for such meshes: linear subdivision for triangle meshes and bilinear subdivision for quadrilateral meshes are investigated. A smooth variant of bilinear subdivision along with its several extensions is presented. It concludes with a brief survey of the wealth of current work on subdivision schemes for polyhedral meshes. The main application of constant subdivision lies in serving as a preprocessor for higher-order dual schemes. The method for generating these schemes is the same as in the primal case: apply an appropriate averaging operator to the results of some type of simple subdivision operation. For primal meshes, applying quad averaging to the result of linear subdivision leads to smooth primal schemes. For dual meshes, applying quad averaging to the result of constant subdivision leads to smooth dual schemes.

Chapter 5 – Local Approximation of Global Differential Schemes

This chapter derives an infinitely supported subdivision scheme for solutions to the harmonic equation via the differential approach and discusses building finite approximations to this scheme. Splines that are solutions to the polyharmonic equation are considered. These polyharmonic splines possess a globally supported, but highly localized, bell shaped basis that can be expressed as a difference of the integer translates of the traditional radial basis function. This bell-shaped basis function possesses a refinement relation that itself has a globally supported, but highly localized, subdivision mask. Three techniques for computing finite approximations to this infinite mask have been considered — Laurent series expansion, Jacobi iteration, and linear programming. Each method yields a locally supported subdivision scheme that approximates the polyharmonic splines. The chapter discusses the problem of generating a subdivision scheme for simple types of linear flows. As is the case of polyharmonic splines, the subdivision mask for these flows follows directly from the discretization of the differential operators governing these flows.

Chapter 6 – Variational Schemes for Bounded Domains

This chapter considers variational extensions of the differential method for the problem of generating subdivision rules along the boundaries of finite domains. The key to this approach is to develop a variational version of the original differential problem. Given this variational formulation, we construct an inner product matrix, E k that plays a role analogous to d k [x] in the differential case. Remarkably, such inner products can be computed exactly for limit functions defined via subdivision, even if these functions are not piecewise polynomial. A matrix relation of the form E k S k-1 == U k-1 E k-1 is derived, where U k-1 is an upsampling matrix. This new relation is a matrix version of the finite difference relation d k [x]S k-1 [x] == 2d k-1 [x 2 ] that characterized subdivision in the differential case. Solutions S k-1 to this matrix equation yields subdivision schemes that converge to minimizers of the original variational problem. To illustrate these ideas, the problem of constructing subdivision schemes for two types of splines with simple variational definitions: natural cubic splines and bounded harmonic splines are considered. For bounded harmonic splines, no choice of finite element basis leads to locally supported subdivision rules. Instead, the heuristic of choosing the finite element basis with appropriate smoothness and support as small as possible is followed. In practice, this rule appears to lead to a variational scheme whose subdivision rules are highly localized.

Convergence Analysis for Uniform Subdivision Schemes

This chapter considers the question of convergence and smoothness. The key to answering this question is developing a method for subdividing various differences associated with the vector p k . For many subdivision schemes, especially interpolatory ones, there is no known piecewise representation; the only information known about the scheme is the subdivision mask s[x]. In this case, it is necessary to understand the behavior of the associated subdivision process in terms of the structure of the subdivision mask s[x] alone. Given a subdivision mask s[x] and an initial set of coefficients p 0 , the subdivision process p k [x] = s[x] p k-1 [x 2 ] defines an infinite sequence of generating functions. The basic approach of this chapter is to associate a piecewise linear function p k [X] with the coefficients of the generating function p k [x] and to analyze the behavior of this sequence of functions. The chapter reviews some mathematical basics necessary to understand convergence for a sequence of functions. Sufficient conditions on the mask s[x] for the functions p k [x] to converge to a continuous function p ∞ [x] are derived. The chapter considers the bivariate version of this question and derives sufficient conditions on the mask s[x, y] for the functions p k [x, y] to converge to a continuous function p ∞ [x, y].