Implicit modeling from polygon soup using convolution

We present a novel method for creating implicit surfaces from polygonal models. The implicit function is defined by convolving a kernel with the triangles in the polygonal model. By adopting a piecewise quartic polynomial kernel function with a finite support, we derive a convolution model that has a closed-form solution, and thus can be efficiently evaluated. The user only needs to specify an effective radius of influence to generate an implicit surface of desired closeness to the polygonal model. The resulting implicit surface is fast to evaluate, not requiring accumulating evaluation results using any hierarchical data structure, and can be efficiently ray-traced to reveal the detailed features.     

Convolution surfaces for arcs and quadratic curves with a varying kernel

A convolution surface is an isosurface in a scalar field defined by convolving a skeleton, comprising of points, curves, surfaces, or volumes, with a potential function. While convolution surfaces are attractive for modeling natural phenomena and objects of complex evolving topology, the analytical evaluation of integrals of convolution models still poses some open problems. This paper presents some novel analytical convolution solutions for arcs and quadratic spline curves with a varying kernel. In addition, we approximate planar higher-degree polynomial spline curves by optimal arc splines within a prescribed tolerance and sum the potential functions of all the arc primitives to approximate the field for the entire spline curve. Published online: November 20, 2002     

Shape and tone depiction for implicit surfaces

We present techniques for rendering implicit surfaces in different pen-and-ink styles. The implicit models are rendered using point-based primitives to depict shape and tone using silhouettes with hidden-line attenuation, drawing directions, and stippling. We present sample renderings obtained for a variety of models. Furthermore, we describe simple and novel methods to control point placement and rendering style. Our approach is implemented using HRBF Implicits, a simple and compact representation, that has three fundamental qualities: a small number of point-normal samples as input for surface reconstruction, good projection of points near the surface, and smoothness of the gradient field. These qualities of HRBF Implicits are used to generate a robust distribution of points to position the drawing primitives.     

Adaptive Sampling of Implicit Surfaces for Interactive Modelling and Animation

This paper presents a new adaptive sampling method for implicit surfaces that can be used in both interactive modelling and animation. The algorithm samples implicit objects composed of blending primitives and efficiently maintains this sampling over time, even when their topology changes (during fractures and fusions). It provides two complementary modes of immediate visualization: displaying “scales” lying on the surface, or a “primitive-wise” polygonization. The sampling method efficiently avoids unwanted blending between different parts of an object. Moreover, it can be used for partitioning an implicit surface into local bounding boxes that will accelerate collision detection during animation and ray-intersections during final rendering.     

SCALe‐invariant Integral Surfaces

Extraction of skeletons from solid shapes has attracted quite a lot of attention, but less attention was paid so far to the reverse operation: generating smooth surfaces from skeletons and local radius information. Convolution surfaces, i.e. implicit surfaces generated by integrating a smoothing kernel along a skeleton, were developed to do so. However, they failed to reconstruct prescribed radii and were unable to model large shapes with fine details. This work introduces SCALe‐invariant Integral Surfaces (SCALIS), a new paradigm for implicit modelling from skeleton graphs. Similarly to convolution surfaces, our new surfaces still smoothly blend when field contributions from new skeleton parts are added. However, in contrast with convolution surfaces, blending properties are scale‐invariant. This brings three major benefits: the radius of the surface around a skeleton can be explicitly controlled, shapes generated in blending regions are self‐similar regardless of the scale of the model and thin shape components are not excessively smoothed out when blended into larger ones.     

基于任意骨架的隐式曲面造型技术

给出了一个新的基于任意多面体网格骨架的构造性自由曲面造型算法.算法首先由每个给定骨架构造出一个距离场,然后利用隐函数光滑过渡技术和CSG(constructive solid geometry)表示技术将所构造的隐式曲面自由地两两粘合成一张光滑曲面.隐式曲面的多边形化算法则用来生成最终曲面网格.以任意骨架作为基本体素,突破了传统隐式曲面以点为基本骨架的限制.而且,距离曲面很好地逼近了原骨架形状,使用户可直观地对复杂曲面进行交互设计.而形变函数的引入,则极大地丰富了此方法的造型能力.实验结果表明,基于该算法的原型系统能够方便、直观地构造复杂的自由曲面.     

Warp-based helical implicit primitives

Implicit modeling with skeleton-based primitives has been limited up to now to planar skeleton elements, since no closed-form solution was found for convolution along more complex curves. We show that warping techniques can be adapted to efficiently generate convolution-like implicit primitives of varying radius along helices, a useful 3D skeleton found in a number of natural shapes. Depending on a single parameter of the helix, we warp it onto an arc of circle or onto a line segment. For those latter skeletons closed-form convolutions are known for entire families of kernels. The new warps introduced preserve the circular shape of the normal cross section to the primitive.     

A new implicit blending technique for volumetric modelling

Current implicit blending techniques are mostly designed for use in surface modelling, where only boundaries of the object defined by the implicit primitives are important. In contrast, in volumetric implicit modelling the interior of the object is also significant, which requires different and more suitable techniques for combining implicit primitives. In this paper, we first discuss irregularities that occur using the current techniques. Then, a new technique for blending implicit primitives, especially appropriate in volumetric modelling (e.g., cloud modelling), is introduced. It overcomes these abnormalities and gives us better results than current techniques.     

Interactive Visualization of Implicit Surfaces with Singularities

This paper presents work on two methods for interactive visualization of implicit surfaces: physically-based sampling using particle systems and polygonization followed by physically-based mesh improvement which explicitly makes use of the surface-defining equation. While most previous work applied to bounded manifolds without singularities and without boundary (topological spheres) we broaden the scope of the methods to include surfaces with such features, in particular cusp points and surface self-intersections. These aspects are not (yet) essential for computer graphics modelling with implicit surfaces but they naturally occur in simulations of interest in mathematical visualization. In this paper we use the Kummer family of algebraic surfaces as an example.     

Radial Supershapes for Solid Modeling

In the previous work, an efficient method has been proposed to represent solid objects as multiple combinations of globally deformed supershapes. In this paper, this framework is applied with a new supershape implicit function that is based on the notion of radial distance and results are presented on realistic models composed of hundreds of hierarchically globally deformed supershapes. An implicit equation with guaranteed differential properties is obtained by simple combinations of the primitives' implicit representations using R-function theory. The surface corresponding to the zero-set of the implicit equation is efficiently and directly polygonized using the primitives' parametric forms. Moreover, hierarchical global deformations are considered to increase the range of shapes that can be modeled. The potential of the approach is illustrated by representing complex models composed of several hundreds of primitives inspired from CAD models of mechanical parts.     

Robust creation of implicit surfaces from polygonal meshes

Implicit surfaces are used for a number of tasks in computer graphics, including modeling soft or organic objects, morphing, collision detection, and constructive solid geometry. Although operating on implicit surfaces is usually straightforward, creating them is not. We introduce a practical method for creating implicit surfaces from polygonal models that produces high-quality results for complex surfaces. Whereas much previous work in implicit surfaces has been done with primitives such as "blobbies," we use implicit surfaces based on a variational interpolation technique (the three-dimensional generalization of thin-plate interpolation). Given a polygonal mesh, we convert the data to a volumetric representation to use as a guide for creating the implicit surface iteratively. We begin by seeding the surface with a number of constraint points through which the surface must pass. Iteratively, additional constraints are added; the resulting surfaces are evaluated, and the errors guide the placement of subsequent constraints. We have applied our method successfully to a variety of polygonal meshes and consider it to be robust     

Segment Tracing Using Local Lipschitz Bounds

We introduce Segment Tracing, a new algorithm that accelerates the classical Sphere Tracing method for computing the intersection between a ray and an implicit surface. Our approach consists in computing the Lipschitz bound locally over a segment to improve the marching step computation and accelerate the overall process. We describe the computation of the Lipschitz bound for different operators and primitives. We demonstrate that our algorithm significantly reduces the number of field function queries compared to previous methods, without the need for additional accelerating data-structures. Our method can be applied to a vast variety of implicit models ranging from hierarchical procedural objects built from complex primitives, to simulation-generated implicit surfaces created from many particles.     

Extending the CSG Tree. Warping, Blending and Boolean Operations in an Implicit Surface Modeling System

Automatic blending has characterized the major advantage of implicit surface modeling systems. Recently, the introduction of deformations based on space warping and Boolean operations between primitives has increased the usefulness of such systems. We propose a further enhancement which will extend the range of models that can be easily and intuitively defined with a skeletal implicit surface system. We describe a hierarchical method which allows arbitrary compositions of models that make use of blending, warping and Boolean operations. We call this structure the BlobTree . Blending and space warping are treated in the same way as union, difference and intersection, i.e. as nodes in the BlobTree . The traversal of the BlobTree is described along with two rendering algorithms; a polygonizer and a ray tracer. We present some examples of interesting models which can be made easily using our approach that would be very difficult to represent with conventional systems.     

Controlled Metamorphosis Between Skeleton‐Driven Animated Polyhedral Meshes of Arbitrary Topologies

Enabling animators to smoothly transform between animated meshes of differing topologies is a long‐standing problem in geometric modelling and computer animation. In this paper, we propose a new hybrid approach built upon the advantages of scalar field‐based models (often called implicit surfaces) which can easily change their topology by changing their defining scalar field. Given two meshes, animated by their rigging‐skeletons, we associate each mesh with its own approximating implicit surface. This implicit surface moves synchronously with the mesh. The shape‐metamorphosis process is performed in several steps: first, we collapse the two meshes to their corresponding approximating implicit surfaces, then we transform between the two implicit surfaces and finally we inverse transition from the resulting metamorphosed implicit surface to the target mesh. The examples presented in this paper demonstrating the results of the proposed technique were implemented using an in‐house plug‐in for Maya?.     

A survey of partial differential equations in geometric design

Computer-aided geometric design is an area where the improvement of surface generation techniques is an everlasting demand, since faster and more accurate geometric models are required. Traditional methods for generating surfaces were initially mainly based upon interpolation algorithms. Recently, partial differential equations (PDE) were introduced as a valuable tool for geometric modelling, since they offer a number of features from which these areas can benefit. This work summarizes the uses given to PDE surfaces as a surface generation technique together with some other applications to computer graphics.     

Isotopic meshing of implicit surfaces

Implicit surfaces are given as the zero set of a function F:ℝ³→ℝ. Although several algorithms exist for generating piecewise linear approximations, most of these are based on a user-defined stepsize or bounds to indicate the precision, and therefore cannot guarantee topological correctness. Interval arithmetic provides a mechanism to determine global properties of the implicit function. In this paper we present an algorithm that uses these properties to generate a piecewise linear approximation of implicit curves and surfaces, that is isotopic to the curve or surface itself. The algorithm is simple and fast, and is among the first to guarantee isotopy for implicit surface meshing.     

Embedded Implicit Stand-Ins for Animated Meshes: A Case of Hybrid Modelling

In this paper, we address shape modelling problems, encountered in computer animation and computer games development that are difficult to solve just using polygonal meshes. Our approach is based on a hybrid-modelling concept that combines polygonal meshes with implicit surfaces. A hybrid model consists of an animated polygonal mesh and an approximation of this mesh by a convolution surface stand-in that is embedded within it or is attached to it. The motions of both objects are synchronised using a rigging skeleton. We model the interaction between an animated mesh object and a viscoelastic substance, which is normally represented in an implicit form. Our approach is aimed at achieving verisimilitude rather than physically based simulation. The adhesive behaviour of the viscous object is modelled using geometric blending operations on the corresponding implicit surfaces. Another application of this approach is the creation of metamorphosing implicit surface parts that are attached to an animated mesh. A prototype implementation of the proposed approach and several examples of modelling and animation with near real-time preview times are presented.     

Neural Representation of Open Surfaces

Neural implicit surfaces have emerged as an effective, learnable representation for shapes of arbitrary topology. However, representing open surfaces remains a challenge. Different methods, such as unsigned distance fields (UDF), have been proposed to tackle this issue, but a general solution remains elusive. The generalized winding number (GWN), which is often used to distinguish interior points from exterior points of 3D shapes, is arguably the most promising approach. The GWN changes smoothly in regions where there is a hole in the surface, but it is discontinuous at points on the surface. Effectively, this means that it can be used in lieu of an implicit surface representation while providing information about holes, but, unfortunately, it does not provide information about the distance to the surface necessary for e.g. ray tracing, and special care must be taken when implementing surface reconstruction. Therefore, we introduce the semi-signed distance field (SSDF) representation which comprises both the GWN and the surface distance. We compare the GWN and SSDF representations for the applications of surface reconstruction, interpolation, reconstruction from partial data, and latent vector analysis using two very different data sets. We find that both the GWN and SSDF are well suited for neural representation of open surfaces.     

Representation of arbitrary shapes using implicit quadrics

This paper is concerned with free-form surface constructions using implicit quadrics. More specifically, we are interested in the following problem: given a polyhedron with triangular facets and tangent planes prescribed at its vertices, fit a smooth (tangent-plane continuous), implicit, and piecewise quadric surface through the vertices of the polyhedron so that the surface is tangent to the prescribed tangent plane at each vertex. We show that in order to solve this problem without splitting the facets of the polyhedron, the prescribed tangent planes must satisfy a condition, and under this condition we give a local scheme for constructing the smooth piecewise quadric surface. Using this scheme, we can represent arbitrary shapes by quadric primitives. The implementation results are reported.