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 twoand three-dimensional domains and on surfaces in an intuitively understandable way. Here, a new approach based on anisotropicnonlinear diffusion is introduced. It enables an easy perception of vector field data and serves as an appropriate scale space methodfor the visualization of complicated flow pattern. The approach is closely related to nonlinear diffusion methods in image analysis whereimages 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 aparabolic PDE problem. It is discretized only in the final implementational step. Therefore, many important qualitative aspects canalready be discussed on a continuous level. Applications are shown for flow fields in 2D and 3D, as well as for principal directions ofcurvature on general triangulated surfaces. Furthermore, the provisions for flow segmentation are outlined. Index TermsÐFlow visualization, multiscale, nonlinear diffusion, segmentation.æ 1 INTRODUCTIONTHE visualization of field data, especially of velocityfields from CFD computations, is one of the funda-mental tasks in scientific visualization. A variety ofdifferent approaches has been presented. The simplestmethod of drawing vector plots at nodes of some overlaidregular grid in general produces visual clutter because ofthe typically different local scaling of the field in thespatial domain, which leads to disturbing multiple over-laps in certain regions, whereas, in other areas, smallstructures such as eddies cannot be resolved adequately.This gets even worse if tangential fields on highly curvedsurfaces are considered.The central goal is to come up with intuitively betterreceptible methods which give an overall, as well as adetailed, view on the flow patterns. Single particle linesonly partially enlighten features of a complex flow field.Thus, we want to define a texture which represents the fieldglobally on a 2D or 3D domain and on surfaces, respec-tively. Here, we confine ourselves to stationary fields. In theEuclidean case, we suppose v : ! IR for some domainIR, whereas, in the case of a manifoldM embedded inIR, we consider a tangential vector field v. We ask for amethod generating stretched streamline type patternswhich are aligned to the vector field v…x†. Furthermore,the possibility of successively coarsening this pattern isobviously a desirable property. Methods which are basedon such a scale of spaces and enhance certain structures ofimages are well-known in image processing analysis.Actually, nonlinear diffusion allows the smoothing of grayor color images while retaining and enhancing edges [18].Now, we set up a diffusion problem, with strong smoothingalong integral lines and edge enhancement in the orthogo-nal directions. Applying this to some initial random noiseimage intensity, we generate a scale of successively coarserpatterns which represent the vector field. Finite elements inspace and a semi-implicit time stepping are applied to solvethis diffusion problem numerically. Furthermore, a suitablemodification of the approach allows the identification oftopological regions.Before we explain in detail the method, let us discussrelated work on vector field visualization and imageprocessing. Later on we will identify some of the well-known methods as equivalent to special cases or asymptoticlimits of the presented new method, respectively. 2 RELATED WORKThe spot noise method proposed by van Wijk [25]introduces spot-like texture splats which are aligned bydeformation to the velocity field in 2D or on surfaces in 3D.These splats are plotted in the fluid domain, showing strongalignment patterns in the flow direction. The original firstorder approximation to the flow was improved by de Leeuwand van Wijk in [6] by using higher order polynomialdeformations of the spots in areas of significant vorticity. Inan animated sequence, these spots can be moved alongstreamlines of the flow. Furthermore, in 3D, van Wijk [26]applies the integration to clouds of oriented particles andanimates them by drawing similar moving transparent andilluminated splats.The Line Integral Convolution (LIC) approach of Cabraland Leedom [4] integrates the fundamental ODE describingstreamlines forward and backward in time at everyIEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS, VOL. 6, NO. 2, APRIL-JUNE 2000139 . The authors are with the Institute for Applied Mathematics, University ofBonn, 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:[email protected], and reference IEEECS Log Number 111480. 1077-2626/00/$10.00 ß 2000 IEEE pixelized point in the domain, convolves a white noisealong these particle paths with some Gaussian type filterkernel, and takes the resulting value as an intensity valuefor the corresponding pixel. According to the strongcorrelation of this intensity along the streamlines and thelack of any correlation in the orthogonal direction, theresulting texturing of the domain shows dense streamlinefilaments of varying intensity. Stalling and Hege [21]increased the performance of this method, especially byreusing portions of the convolution integral alreadycomputed on points along the streamline. Forssell [10]proposed a similar method on surfaces and Max et al. [17]discussed flow visualization by texturing on contoursurfaces. Max and Becker [16] presented a method forvisualizing 2D and 3D flows by animating textures.Shen and Kao [20] applied an LIC type method tounsteady flow fields. Recently, a method [2] has beenpresented which generates streakline type patterns bynumerical calculation of the transport of inlet coordinatesand inlet position. Interrante and Grosch [12] generalizedline integral convolution to 3D in terms of volumerendering of line filaments.In [24], Turk and Banks discuss an approach whichselects a certain number of streamlines. They are auto-matically equally distributed all over the computationaldomain to characterize, in a sketch-type representation, thesignificant aspects of the flow. An energy minimizingprocess is used to generate the actual distribution ofstreamlines.Especially for 3D velocity fields, particle tracing is a verypopular tool. But, a few particle integrations released by theuser can hardly scope with the complexity of 3D vectorfields. Stalling et al. [22] use pseudorandomly distributed,illuminated, and transparent streamlines to give a denserand more receptible representation, which shows theoverall structure and enhances important details.Van Wijk [27] proposed the implicit stream surfacemethod. For a stationary flow field, the transport equationsv r ˆ 0 are solved for given v and certain inflow andoutflow boundary conditions in a precomputing step. Then,isosurfaces of the resulting function are streamsurfacesand 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 recomputa-tion. For instance, if we ask for a finer or coarser scale of theline integral convolution pattern, the computation has to berestarted with a coarser initial image intensity. In the case ofspot noise, larger spots have to be selected and theirstretching along the field has to be increased. The approachto be presented here will incorporate a successive coarsen-ing as time proceeds in the underlying diffusion problem.As already mentioned in the introduction, our method ofanisotropic nonlinear diffusion to visualize vector fields isderived from well-known image processing methodology.Discrete diffusion type methods have been known for along time. Perona and Malik [18] introduced a continuousdiffusion model which allows the denoising of imagestogether with the enhancing of edges. Alvarez et al. [1]established a rigorous axiomatic theory of diffusive scalespace methods. Kawohl and Kutev [14] investigate aqualitative analysis of the Perona and Malik model. Therecovering of lower dimensional structures in images isanalyzed by Weickert [28], who introduced an anisotropicnonlinear diffusion method, where the diffusion matrixdepends on the so-called structure tensor of the image. Afinite element discretization and its convergence propertieshave been studied by Kacur and Mikula [13].Concerning the application of diffusion type methods onsurfaces, a general introduction to differential calculus onmanifolds can be found for instance in the book bydo Carmo [7]. Dziuk [8] presented an algorithm for thesolution of partial differential equations on surfaces and, in[9], he discussed a numerical method for geometricdiffusion applied to the surface itself which coincides withthe mean curvature motion. 3 THE NONLINEAR DIFFUSION PROBLEMLet us now derive our method based on a suitable PDEproblem. At first, we confine ourselves to the case of planardomains in 2D and 3D. Here, nonlinear anisotropicdiffusion applied to some initial random noisy image willenable an intuitive and scalable visualization of compli-cated vector fields. Therefore, we pick up the idea of lineintegral convolution, where a strong correlation in theimage intensity along integral lines is achieved by convolu-tion of an initial white noise along these lines. As proposedalready by Cabral and Leedom [4], a suitable choice for theconvolution kernel is a Gaussian kernel. On the other hand,an appropriately scaled Gaussian kernel is known to be thefundamental solution of the heat equation. Thus, lineintegral convolution is nothing else than solving the heatequation in 1D on an integral line parameterized withrespect to arc length. On pixels which are located ondifferent integral lines, the resulting image intensities arenot correlated. Hence, the thickness of the resulting imagepatterns in line integral convolution is of the size of therandom initial patterns, in general, a single pixel. Increasingthis size, as has been proposed by Kiu and Banks [15], leadsto broader stripes and, unfortunately, less sharp transitionsacross streamline patterns. As described so far, line integralconvolution is a discrete pixel-based method. If we ask for awell-posed continuous diffusion problem with similarproperties, we are led to some anisotropic diffusion, nowcontrolled by a suitable diffusion matrix.To begin with, let us at first introduce a general nonlineardiffusion method from image processing and then discussthe selection of the appropriate diffusion tensor and therighthand side. Here, we consider first the case of an imagein Euclidean space either in 2D or 3D. In Section 6, we thengeneralize this with respect to textures on surfaces. Weconsider a function :IR0 ! IR which solves theparabolic problem @@t ÿ div A…r †r…† ˆ f… † in IR‡ ;…0; † ˆ 0 on ;@@ ˆ 0 on IR‡ @ :for given initial density 0 : ! ‰0; 1Š. Here, ˆ is amollification of the current density, which will later on turn140IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS, VOL. 6, NO. 2, APRIL-JUNE 2000