Elsevier

Computers & Graphics

Volume 37, Issue 7, November 2013, Pages 873-884
Computers & Graphics

Technical Section
Point-based rendering of implicit surfaces in R4

https://doi.org/10.1016/j.cag.2013.06.005Get rights and content

Highlights

  • A robust method for point sampling an implicit surface in R4.

  • A new illumination model for two-dimensional objects in a four-dimensional space.

  • A GPU implementation of our algorithm is also discussed.

Abstract

We present a point-based algorithm for rendering implicit surfaces in R4. Our algorithm combines a new method for approximating an implicit surface with points which uses interval arithmetic for topological robustness with a new 4D illumination model that together with a color transfer function enhances the visualization of a two-dimensional surface in four-dimensional space.

Introduction

An implicit object is the set of all solutions of an equation F(p)=0, where F:ΩRnRm [1]. Of special interest to computer graphics are implicit curves in R2(n=2,m=1) and implicit surfaces in R3(n=3,m=1), but several problems in visual computing can be formulated as high-dimensional implicit problems [2], [3], [4]. In this paper, we study rendering schemes for the visualization of implicit surfaces in R4(n=4,m=2), including graphs of complex functions, which are important for mathematical visualization [5], [6].

Previous work. Interest in studying the visual aspect of mathematical objects in four-dimensional space dates from the beginning of the 20th century [7], [8]. The advent of computers made it possible to try to visualize higher-dimensional objects. In 1986, Banchoff [9] introduced a method for analyzing point clouds around two-dimensional surfaces by using interactive computer graphics and also gave applications to the graphing of complex functions. Hoffmann and Zhou [4] in 1991 presented a pipeline for visualizing implicit surfaces in R4 that used a “polygonalization before projection” strategy to achieve better interaction rates. They also pointed out some applications of their four-dimensional visualization method in offset curve geometry and in collision detection. Another interesting application of visualization in four-dimensional space is in the study of complex-valued contours [10], whose purpose is to analyze the solution set of G1(0), where G:C2C is a complex function of two complex variables (C is the complex plane). Note that solving G1(0) is equivalent to finding the inverse image of 0 of a real function F:R4R2. Weigle and Banks [10] in 1996 presented a meshing algorithm to approximate G1(0). Nieser, Poelke, and Polthier [11] in 2010 proposed an algorithm for meshing Riemann surfaces in R3 from explicitly given branch points with corresponding branch indices. Their meshing approach and their proposed coloring scheme for the complex plane provide a straightforward visualization of the topological structure of Riemann surfaces, which are important mathematical objects.

It is well known that computing a polygonal approximation of an implicit object is a challenging problem [12] for two main reasons: it is difficult to find points on the object [13] and it is difficult to connect isolated points into a mesh [14]. Moreover, these two difficulties grow exponentially with the dimension of the ambient space, especially when one seeks adaptive meshes [15].

One direction for avoiding having to compute a polygonal approximation just for rendering is to use ray tracing (see [16], [17] and the references therein). Another direction is point-based rendering, pioneered by Witkin and Heckbert [18], who rendered implicit surfaces by placing small opaque disks at points sampled on an implicit surface using physically based methods [14]. Balsys and Suffern [19] described an interval method for point-based rendering of hypersurfaces in R4(n=4,m=1). Our focus here is on the case m=2.

Contributions. Our method for point-based rendering of implicit surfaces in R4 has two components: a robust method for point sampling an implicit surface and a new illumination model for two-dimensional objects in four-dimensional space. Section 2 presents an overview of our method.

As has been done for implicit curves in R2 [20], for implicit surfaces in R3 [21], and for non-manifold objects in R2 [22], our method uses interval arithmetic to locate the surface and guide the point sampling, thus improving the topological robustness of the approximation (Section 3). Our illumination model is combined with a color transfer function to enhance the visualization (Section 4). We also briefly discuss a GPU implementation of our algorithm. Section 5 shows some results and Section 6 contains our conclusions and suggestions for future work.

Section snippets

Overview of our rendering algorithm

An implicit surface S in R4 is the inverse image of (0,0) of a function F:ΩR4R2. More precisely S=F1(0,0)={(x,y,z,w)ΩR4:F(x,y,z,w)=(0,0)}We assume that F is smooth (continuously differentiable) and that (0,0) is a regular value of F, which means that the derivative of F does not vanish at any point of S. By the implicit function theorem, this implies that S is an embedded two-dimensional manifold in R4, that is, a non-singular surface.

Our algorithm first computes a topologically robust

Point sampling

Our method for sampling points on an implicit surface in R4 has three steps: domain exploration using spatial adaptive subdivision (Section 3.1), generation of seed points (Section 3.2), and refinement (Section 3.3).

Our method extends to R4 the shape and tone depiction algorithm proposed by Brazil et al. [23] for non-photorealistic rendering of implicit surfaces in R3. It is also an extension of the work of Balsys et al. [22] that proposed a point-based technique to rendering non-manifold

Rendering pipeline

We now propose a rendering pipeline for surfaces in R4. We start by describing how to orient four-dimensional space using Euler angles and the projection scheme (Section 4.1). We then describe the illumination model (Section 4.2) and how to enhance the visualization with transfer functions (Section 4.3). Finally, we describe how to implement this pipeline on the GPU (Section 4.4).

Results

We now report experiments with our method using the functions listed in Table 1 with the domain Ω taken as the 4-box [h]=[2,2]4. We ran all the experiments on an Intel Core i7 2.6 GHz with 4 Gbytes of memory and Intel HD Graphics 3000 on board.

Conclusions and future work

In this paper, we presented a new point-based rendering scheme for surfaces in four-dimensional space. We introduced a simple and robust point-based approximation for implicit surfaces and a suitable illumination model for a two-dimensional surface in R4. We also proposed an animation scheme, called the kaleidoscope of surfaces, that interpolates between two different observer positions to facilitate the surface visualization.

One limitation of our method is that it requires that the function

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