Coherence-sensitive solid fitting

Abstract

In the previous reports (in: G.M. Nielson, D. Silver (Eds.), Proceedings of the IEEE Visualization ’95, IEEE Computer Society Press, Los Alamitos, CA, ’95, pp. 151–8; IEEE Transactions on Visualization and Computer Graphics 2 (2) (1996) 144–55; in: H. Hagen, G.M. Nielson, F. Post (Eds.), Proceedings of the Dagstuhl ’97 Scientific Visualization, IEEE Computer Society Press, Los Alamitos, CA, 2000, pp. 65–78), solid fitting has been presented as a generalized indirect volume visualization method, which relies primarily on a simple, but powerful geometric volume model, termed interval volume, to allow one to represent a 3D sub-volume for which the associated scalar values lie within a specified closed interval. The field interval-based specification of volumetric regions of interest has various advantages to play a complementary role with isosurfacing and direct volume rendering.

This paper presents a new concept, termed coherence-sensitive solid fitting, which uses a global measure of volumetric coherence to estimate the spatial/temporal complexities of interval volume to be extracted, and adaptively controls the shape and retinal properties of interval volume to realize an interactive and effective volume exploration environment.

Experiments with many well-known testbed datasets and an attractive simulation dataset from CFD research are performed to prove the feasibility of the present concept empirically.

Introduction

Solid fitting is an effective indirect volume visualization method which generalizes the traditional surface fitting[1], [2], [3]. The method relies primarily on a simple, but powerful geometric volume model, termed interval volume, to allow one to represent a 3D sub-volume IV(α,β) for which the associated scalar field values f(x,y,z) lie within a specified closed interval [α,β]. Interval volume IV(α,β) has different meanings according to how we specify its interval [α,β]. First, IV(f_min,f_max) becomes the entire volume, where f_min and f_max are the minimum and maximum scalar field values, respectively. Next, IV(α,α) reduces to IS(α), which is an isosurface extracted with the scalar field value α. And finally, $\text{IV(α−ε,α+ε),}\text{(ε⪡f}{}_{\max }\text{−f}{}_{\min }\text{)}$ can represent a practical counterpart of isosurface IS(α) taking into account some allowance 2ε about α. In these respects, interval volume can serve as a generalized isosurface.

Introducing the field interval-based specification of volumetric regions of interest (ROIs) allows one:

• to produce more intuitive and informative images efficiently through boundary surface transparency and pseudo-color coding of cross-sections;

• to compute morphological quantities such as the surface area, total volume, and field average over the ROIs; and

• to control the location and length of the interval to investigate the properties such as the magnitude of errors/noises due to structural ambiguity inherent to natural objects, measurement, mathematical modeling, or numerical computation, and to search for the target isosurface in a stepwise manner (field focusing).

For example, consider an analytic trivariate function defined by

$$\text{f(x,y,z)=}\text{(2}{}^8\text{−1)}\text{92}\text{x}{}^3\text{+y}{}^3\text{+}\text{3}\text{2}\text{z}{}^2\text{(x+y+1)−3xy}\text{+}\text{2}{}^8\text{−1}\text{2}\text{(|x,y,z|⩽2).}$$

Indeed, this function is a 3D version of Folium of Descartes, whose implicit surface defined by the equation f(x,y,z)=127.5=(2⁸−1)/2 has the origin of the volume as its singular point [3]. Fig. 1(a) shows five isosurfaces extracted from the volume. Notice that due to some digitization error, the third isosurface whose field value is 127.5 splits into two parts, and does not pass the center (origin) of the volume. This may give a misleading interpretation about the topological feature of the volumetric function. On the other hand, Fig. 1(b) shows that an interval volume having a very thin interval around 127.5 holds the correct topology of the target isosurface, from which the existence of the singular point can be readily anticipated.

Geometric modeling of field interval-based volumetric ROIs has recently begun to draw special attention from visualization researchers. For instance, Udupa proposes a cuberille model, called shell, for the compact representation and fast manipulation of regular volumetric datasets containing fuzzy boundaries between adjacent materials [4]. Guo presents interval sets with the aim of unifying both surface fitting and direct volume rendering (DVR) [5]. Crawfis exploits the concept of data space slicing in the context of realtime interaction with volumetric objects in a virtual environment [6].

From the viewpoint of producing a truly semi-transparent image of the entire volumetric dataset, solid fitting is superseded by DVR algorithms, which allows one to peer inside the dataset without the aid of intermediate geometrical representations. However, DVR algorithms are inherently computationally expensive, in spite of many efforts for reasonable rendering timings by algorithmic optimization, usage of parallel computation, and development of special hardware architectures [7]. Thus, the advance use of solid fitting can be thought of as a significant step towards the settled usage of DVR algorithms. For example, field focusing makes it possible to decide proper color/opacity transfer functions and viewing-related parameter values for DVR algorithms to produce final high-quality images so as to convey the most significant aspects of the target objects. Moreover, interval volume could be used as a matting (reference) data structure for polygon-assisted acceleration of DVR algorithms.

In order for solid fitting to serve as a key tool for preliminary volumetric exploration, the solid representation of volumetric ROIs must be more spatially efficient than the original lattice structure, and hence promising economical storage/transmission and extraction/manipulation. In general, it is known that the spatial efficiency of geometric structures extracted from a volumetric dataset is closely related to its spatial coherence, which is the fact that its associated field values do not frequently change in adjacent voxels [2]. In other words, the more coherent the field value distribution becomes, the fewer high-frequency (noise) components it has. Therefore, estimating the coherence of a given dataset prior to the geometric structure extraction plays a key role in realizing a time-critical environment for indirect volume visualization, where responsive interactivity keeps one to immerse one's self in exploratory visualization.

The main aim of this paper is three-fold:

• developing a measure of volumetric coherence on the basis of a well-known, second-order, gray-level statistics [8] for characterizing 2D textures (Section 2);

• investigating the relationship between the coherence measure and various statistics related to geometric structures of interval volume, and exploiting a type of coherence-sensitive solid fitting, which strives to resolve accuracy versus efficiency tradeoffs in terms of topological disambiguation (Section 3); and

• exploiting yet another type of coherence-sensitive solid fitting, which is designated for adaptive multivariate visualization (Section 4).

Section 5 concludes the paper with a few remarks on future research issues.