Analytical methods for polynomial weighted convolution surfaces with various kernels

doi:10.1016/S0097-8493(02)00087-0

Computers & Graphics

Volume 26, Issue 3, June 2002, Pages 437-447

https://doi.org/10.1016/S0097-8493(02)00087-0 Get rights and content

Abstract

Convolution surface has the advantage of being crease-free and bulge-free over other kinds of implicit surfaces. Among the various types of skeletal elements, line segments can be considered one of the most fundamental as they can approximate curve skeletons. This paper presents analytical solutions for convolving line segments with varying kernels modulated by polynomial weighted functions. We derive the closed-form formulae for most classical kernel functions, namely Gaussian, inverse linear, inverse squared, Cauchy, and quartic functions, and compare their computational complexity. These analytical solutions can be incorporated into existing implicit surface modeling systems for more convenient modeling of generalized cylindrical shapes. We demonstrate their high potentials for modeling and animating branching and tubular organic objects with some examples. We also propose a new competitive kernel function that has a smoothness control parameter.

Introduction

Geometric objects are often modeled using parametric surfaces and polygon meshes; however, many smooth deformable objects with complex and time-varying topology are more conveniently modeled as field-based implicit surfaces. Examples of such objects are liquids, clouds, tree branches and other organic shapes. Consequently, implicit surfaces have gained acceptance in shape morphing, surface reconstruction, natural phenomena simulation, and space deformation [1], [2], [3], [4], [5], [6], [7].

Metaballs, distance surfaces, convolution surfaces, and variational surfaces are some well-known types of implicit surfaces. Metaballs (or blobs, soft objects) [8], [9], [10], [11] are defined in terms of point fields, and are widely implemented in commercial modeling packages (e.g., Softimage, 3D Studio Max) and supported by many ray-tracing software (e.g., Rayshade, POV-Ray). Nevertheless, point-based field surfaces suffer from some drawbacks: flat surfaces have to be approximated by many closely packed metaballs to avoid bumps, causing expensive computation; curve skeletons, which naturally abstract many shapes, have to be converted to points, causing incompact representations. Distance surface offers a solution to this problem by generalizing point-based skeletons to higher dimensional ones [12], [13]. Unfortunately, distance surfaces have a major weakness; wherever a skeleton is not convex, the surface may have bulges, creases and curvature discontinuity [14].

Bloomenthal and Shoemake presented convolution surfaces [15] (Fig. 1) as a natural and powerful extension to point-based field surfaces to include line segments, curves, polygons as skeletal elements. By convolving these skeletons with a three-dimensional (3D) low-pass Gaussian filter kernel, convolution surfaces overcome the problem of bulges and curvature discontinuity in distance surfaces. Their other desirable advantages include intuitive shape design, well-behaved blending and fluid topology changes in accordance with the underlying skeleton. Computer vision research has shown that any 3D object can be defined entirely from a geometric skeleton [16]; that is, skeletons are natural abstractions for 3D objects. Convolution surfaces also offer a means of controlling the shape of an underlying modeling object by controlling its skeleton, which is easier to manipulate due to its lower dimension.

While the modeling potentials of convolution surfaces are very attractive, their mathematical formulations still pose some open problems, stemming from the fact that convolution integrals seldom yield closed-form solutions. There are limited choices of kernel functions and skeletal primitives that can be convolved together analytically. By using the superposition property of convolution surface and separable property of the Gaussian filter, Bloomenthal and Shoemake calculated the field function numerically based on point-sampling method [15], which unfortunately suffers from potential under-sampling artifacts and large storage. The existence of closed-form solutions depends on both the skeletal element and the kernel function. By employing a kernel function called Cauchy function McCormack and Sherstyuk deduced analytical solutions for points, line segments, polygons, arcs and planes [17], [18], [19].

In convolution surface modeling, line segments can be viewed as one of the most fundamental ones because many objects can be abstracted into curve skeletons, and curve skeletons can in turn be subdivided into line segments. The analytical model for line-segment primitive derived by McCormack and Sherstyuk treats the weight distribution along the skeleton uniformly, thus modeling tapering or generalized cylindrical shapes requires specifying multiple line segments. Unlike generalized cylindrical distance surfaces, which can be produced by simply varying the distance in the field computation, this approach fails for generalized cylindrical convolution surfaces [20].

In an earlier work, we have presented an analytical solution for line-segment skeletons convolved with the Cauchy function modulated by polynomial weighted distributions [21]. This paper further proves that analytical solutions for line-segment skeletons with polynomial weighted distributions can be derived for most other classical kernel functions, namely Gaussian, inverse linear, inverse squared, inverse cubic, inverse quintic, and quartic polynomial functions. In addition, we propose a new competitive kernel function that has a smoothness control parameter and derive the corresponding analytical solutions. Our model can also be applied to curve skeletons by subdividing the skeletons into polylines as well as subdividing the given polynomial weighted distribution. With the analytical formulae derived for all the commonly used kernel functions of implicit surfaces, our method can be incorporated into any software.

Section snippets

Convolution surfaces

A convolution surface is determined by a skeleton consisting of 3D points, each of which contributes to the field function according to its distance to a space point in question. Let P(x,y,z) be a space point in R³, and let g:R³→R be a tri-variate geometry function that represents a modeling skeleton V: $g(P)= 1, P ∈ skeleton V, 0 otherwise .$ Let f:R³→R be a potential function that describes the field generated by a single point in the skeleton, and Q be a point in the skeleton V, then the total field

Polynomial field computation for line segments with various kernel functions

A line segment L(t) of length l, with start point b and unit direction n, can be represented parametrically as $L (t)= b +t n, 0⩽t⩽l.$ The squared distance from a point P to a point on the line L(t) is then $r^{2} (t)=d^{2} +t^{2} −2t d n =(t−h)^{2} +(d^{2} −h^{2}),$ where d=P−L(0), d=||d||, and h=d radical dot n.

The idea of using weight functions in defining convolution surfaces was introduced in [15], but no closed-form solutions were developed. Our primary purpose is to develop analytical convolution surface solutions for line-segment

Weight distribution control with cubic profile curves

As it is difficult to imagine the shape of a polynomial curve only from its coefficients, we present an intuitive interface for defining a cubic spline curve to control the weight distribution along a line-segment skeleton. The idea of this cubic control curve is similar to that of Kochanek et al's interpolating splines [25], which are widely used for designing keyframe animation in commercial animation software such as Softimage, Alias|Wavefront, and Maya.

A degree n Bezier curve q(u) with

NURBS curve skeletons

Many natural objects can be abstracted as curves. Since NURBS curve is a standard for representing freeform curves, we now consider generating convolution surfaces from NURBS curve skeletons. The NURBS curves are assumed to be cubic with clamped knot vectors (i.e., the curves interpolate the two end control points). By specifying a dense parameter that controls the number of points into which each polynomial segment is to be divided [26], and applying the deBoor algorithm, we can approximate

Computational efficiency and results

We have implemented our algorithm on a Pentium III 400E PC with 128M main memory. To analyze the computational complexity of the various kernels, we use a skeleton consisting of two line segments: from (−4,0,0) to (4,0,0), and from (0,−4,0) to (0,4,0). The space volume is organized into a uniform grid of 150×150×150 nodes, yielding 3.375 million function evaluations. The time taken to evaluate the skeleton with cubic polynomial distribution and the various kernels is given in Table 1. When

Conclusions

Convolution surfaces have the advantage of producing pleasing crease-free and bulge-free features over other kinds of implicit surfaces. One drawback that prevents its more extensive use is the expensive computation and high memory cost incurred by numerical methods if closed-form solution does not exist for calculating the field function. In this paper, we present the closed-form solutions for line-segment skeletons with polynomial weighted distributions for most of the commonly used kernel

Acknowledgements

Part of this research work was conducted while the first author was a visiting researcher at the Hong Kong University of Science and Technology. We are grateful to Andrei Sherstyuk for his help on convolution surfaces and to anonymous reviewers for their constructive suggestions. This work received support from Hong Kong Research Grant Council (HKUST6215/99E), National Natural Science Foundation of China (Grant No. 69973040), Zhejiang Provincial Natural Science Foundation (Grant No. 698022),

Xiaogang Jin is a professor of the State Key Lab of CAD&CG, Zhejiang University, People’s Republic of China. He received his B.Sc. degree in Computer Science in 1989, M.Sc. and Ph.D. degrees in Applied Mathematics in 1992 and 1995, all from Zhejiang University. His research interests include implicit surface modeling, space deformation, computer animation and realistic image synthesis.

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Chiew-Lan Tai is an Assistant Professor in the Department of Computer Science, Hong Kong University of Science & Technology. She received her B.Sc. and M.Sc. in Mathematics from the University of Malaya, and her M.Sc. in Computer and Information Sciences from the National University of Singapore. She earned her D.Sc. in Information Science from the University of Tokyo in 1997. Her research interests include geometric modeling, computer graphics, digital Chinese art, and interpretation of engineering drawings.

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Technical SectionAnalytical methods for polynomial weighted convolution surfaces with various kernels

Abstract

Introduction

Section snippets

Convolution surfaces

Polynomial field computation for line segments with various kernel functions

Weight distribution control with cubic profile curves

NURBS curve skeletons

Computational efficiency and results

Conclusions

Acknowledgements

Computers & Graphics

Computer Vision and Image Understanding

Computer Aided Geometric Design

An introduction to implicit surfaces

Animation of deformable models using implicit surfaces

IEEE Transactions on Visualization and Computer Graphics

A modeling and rendering method for snow by using metaballs

Computer Graphics Forum

Function representation of solids reconstructed from scattered surface points and contours

Computer Graphics Forum

A generalization of algebraic surface drawing

ACM Transactions on Graphics

Object modeling by distribution function and a method of image generation

Transactions on IECE

Data structure for soft objects

The Visual Computer

Field functions for implicit surfaces

The Visual Computer

Technical Section
Analytical methods for polynomial weighted convolution surfaces with various kernels