Level Set and PDE Methods for Visualization

ÐVector field visualization is an important topic in scientific visualization. Its aim is to graphically represent field data on two and three-dimensional domains and on surfaces in an intuitively understandable way. Here, a new approach based on anisotropic nonlinear diffusion is introduced. It enables an easy perception of vector field data and serves as an appropriate scale space method for the visualization of complicated flow pattern. The approach is closely related to nonlinear diffusion methods in image analysis where images are smoothed while still retaining and enhancing edges. Here, an initial noisy image intensity is smoothed along integral lines, whereas the image is sharpened in the orthogonal direction. The method is based on a continuous model and requires the solution of a parabolic PDE problem. It is discretized only in the final implementational step. Therefore, many important qualitative aspects can already be discussed on a continuous level. Applications are shown for flow fields in 2D and 3D, as well as for principal directions of curvature on general triangulated surfaces. Furthermore, the provisions for flow segmentation are outlined. Index TermsÐFlow visualization, multiscale, nonlinear diffusion, segmentation. æ 1 INTRODUCTION THE visualization of field data, especially of velocity fields from CFD computations, is one of the fundamental tasks in scientific visualization. A variety of different approaches has been presented. The simplest method of drawing vector plots at nodes of some overlaid regular grid in general produces visual clutter because of the typically different local scaling of the field in the spatial domain, which leads to disturbing multiple overlaps in certain regions, whereas, in other areas, small structures such as eddies cannot be resolved adequately. This gets even worse if tangential fields on highly curved surfaces are considered. The central goal is to come up with intuitively better receptible methods which give an overall, as well as a detailed, view on the flow patterns. Single particle lines only partially enlighten features of a complex flow field. Thus, we want to define a texture which represents the field globally on a 2D or 3D domain and on surfaces, respectively. Here, we confine ourselves to stationary fields. In the Euclidean case, we suppose v : ! IR for some domain IR, whereas, in the case of a manifoldM embedded in IR, we consider a tangential vector field v. We ask for a method generating stretched streamline type patterns which are aligned to the vector field v…x†. Furthermore, the possibility of successively coarsening this pattern is obviously a desirable property. Methods which are based on such a scale of spaces and enhance certain structures of images are well-known in image processing analysis. Actually, nonlinear diffusion allows the smoothing of gray or color images while retaining and enhancing edges [18]. Now, we set up a diffusion problem, with strong smoothing along integral lines and edge enhancement in the orthogonal directions. Applying this to some initial random noise image intensity, we generate a scale of successively coarser patterns which represent the vector field. Finite elements in space and a semi-implicit time stepping are applied to solve this diffusion problem numerically. Furthermore, a suitable modification of the approach allows the identification of topological regions. Before we explain in detail the method, let us discuss related work on vector field visualization and image processing. Later on we will identify some of the wellknown methods as equivalent to special cases or asymptotic limits of the presented new method, respectively. 2 RELATED WORK The spot noise method proposed by van Wijk [25] introduces spot-like texture splats which are aligned by deformation to the velocity field in 2D or on surfaces in 3D. These splats are plotted in the fluid domain, showing strong alignment patterns in the flow direction. The original first order approximation to the flow was improved by de Leeuw and van Wijk in [6] by using higher order polynomial deformations of the spots in areas of significant vorticity. In an animated sequence, these spots can be moved along streamlines of the flow. Furthermore, in 3D, van Wijk [26] applies the integration to clouds of oriented particles and animates them by drawing similar moving transparent and illuminated splats. The Line Integral Convolution (LIC) approach of Cabral and Leedom [4] integrates the fundamental ODE describing streamlines forward and backward in time at every IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS, VOL. 6, NO. 2, APRIL-JUNE 2000 139 . The authors are with the Institute for Applied Mathematics, University of Bonn, Wegelstraûe 6, 53115 Bonn, Germany. E-mail: {diewald, tpreuss, rumpf}@iam.uni-bonn.de. Manuscript received 15 Mar. 2000; accepted 3 Apr. 2000. For information on obtaining reprints of this article, please send e-mail to: tvcg@computer.org, and reference IEEECS Log Number 111480. 1077-2626/00/$10.00 ß 2000 IEEE pixelized point in the domain, convolves a white noise along these particle paths with some Gaussian type filter kernel, and takes the resulting value as an intensity value for the corresponding pixel. According to the strong correlation of this intensity along the streamlines and the lack of any correlation in the orthogonal direction, the resulting texturing of the domain shows dense streamline filaments of varying intensity. Stalling and Hege [21] increased the performance of this method, especially by reusing portions of the convolution integral already computed on points along the streamline. Forssell [10] proposed a similar method on surfaces and Max et al. [17] discussed flow visualization by texturing on contour surfaces. Max and Becker [16] presented a method for visualizing 2D and 3D flows by animating textures. Shen and Kao [20] applied an LIC type method to unsteady flow fields. Recently, a method [2] has been presented which generates streakline type patterns by numerical calculation of the transport of inlet coordinates and inlet position. Interrante and Grosch [12] generalized line integral convolution to 3D in terms of volume rendering of line filaments. In [24], Turk and Banks discuss an approach which selects a certain number of streamlines. They are automatically equally distributed all over the computational domain to characterize, in a sketch-type representation, the significant aspects of the flow. An energy minimizing process is used to generate the actual distribution of streamlines. Especially for 3D velocity fields, particle tracing is a very popular tool. But, a few particle integrations released by the user can hardly scope with the complexity of 3D vector fields. Stalling et al. [22] use pseudorandomly distributed, illuminated, and transparent streamlines to give a denser and more receptible representation, which shows the overall structure and enhances important details. Van Wijk [27] proposed the implicit stream surface method. For a stationary flow field, the transport equations v r ˆ 0 are solved for given v and certain inflow and outflow boundary conditions in a precomputing step. Then, isosurfaces of the resulting function are streamsurfaces and can be efficiently extracted with interactive frame rates, even for larger data sets. Most of the methods presented so far have in common, that the generation of a coarser scale requires a recomputation. For instance, if we ask for a finer or coarser scale of the line integral convolution pattern, the computation has to be restarted with a coarser initial image intensity. In the case of spot noise, larger spots have to be selected and their stretching along the field has to be increased. The approach to be presented here will incorporate a successive coarsening as time proceeds in the underlying diffusion problem. As already mentioned in the introduction, our method of anisotropic nonlinear diffusion to visualize vector fields is derived from well-known image processing methodology. Discrete diffusion type methods have been known for a long time. Perona and Malik [18] introduced a continuous diffusion model which allows the denoising of images together with the enhancing of edges. Alvarez et al. [1] established a rigorous axiomatic theory of diffusive scale space methods. Kawohl and Kutev [14] investigate a qualitative analysis of the Perona and Malik model. The recovering of lower dimensional structures in images is analyzed by Weickert [28], who introduced an anisotropic nonlinear diffusion method, where the diffusion matrix depends on the so-called structure tensor of the image. A finite element discretization and its convergence properties have been studied by Kacur and Mikula [13]. Concerning the application of diffusion type methods on surfaces, a general introduction to differential calculus on manifolds can be found for instance in the book by do Carmo [7]. Dziuk [8] presented an algorithm for the solution of partial differential equations on surfaces and, in [9], he discussed a numerical method for geometric diffusion applied to the surface itself which coincides with the mean curvature motion. 3 THE NONLINEAR DIFFUSION PROBLEM Let us now derive our method based on a suitable PDE problem. At first, we confine ourselves to the case of planar domains in 2D and 3D. Here, nonlinear anisotropic diffusion applied to some initial random noisy image will enable an intuitive and scalable visualization of complicated vector fields. Therefore, we pick up the idea of line integral convolution, where a strong correlation in the image intensity along integral lines is achieved by convolution of an initial white noise along these lines. As proposed already by Cabral and Leedom [4], a suitable choice for the convolution kernel is a Gaussian kernel. On the other hand, an appropriately scaled Gaussian kernel is known to be the fundamental solution of the heat equati