Hierarchical Visualization of Large-scale Unstructured Hexahedral Volume Data

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1 Hierarchical Visualization of Large-scale Unstructured Hexahedral Volume Data Jaya Sreevalsan-Nair Lars Linsen Benjamin A. Ahlborn Michael S. Green Bernd Hamann Center for Image Processing and Integrated Computing (CIPIC) Department of Computer Science University of California, Davis, CA Abstract Unstructured hexahedral grids are widely used in numerical simulations, but remained unconsidered by the visualization community due to the hurdles in overcoming the hanging-node problem. In this paper, we present a multi-resolution approach for unstructured hexahedral grid based on energy-minimizing face collapses. We resolve the hanging-node problem in its general form in order to generate visualizations using cutting-planes computation and isosurface extraction. We circumvent discontinuities in cell boundaries in the case of cutting planes by using general local interpolation schemes and in the case of crack-free isosurfaces by using a dual contouring approach. 1 Introduction Scientific volume data sets are typically defined on structured or unstructured 3D grids. Unstructured grids adapt very well to the underlying trivariate function describing some field function (a scalar, vector, or tensor field - or even multiple fields). This adaptivity makes unstructured grids appropriate for most scientific data sets. Though unstructured tetrahedral meshes, for example, have been examined by several scientists, hardly any work has been done on unstructured hexahedral mesh hierarchies, despite the fact that hexahedral grids are commonly used in simulations such as finite element analysis and computational fluid dynamics. Unfortunately, standard data exploration and visualization tools can, in general, not be applied directly to unstructured hexahedral grids. We generate a multiresolution hierarchy of unstructured, rectilinear, hexahedral meshes. The generation of the hierarchy is based on collapsing neighboring hexahedral cells with minimal local variation. This allows us to construct a continuous level-of-detail representation [10]. However, visualization of the resulting data hierarchy is difficult due to the continuity in the levels-of-detail across the grid. Hence, we have developed a cutting plane and an isosurface extraction tool for a general setting, where arbitrary-sized hexahedral building blocks, {sreevals, ahlborn, greenm, hamann}@cs.ucdavis.edu, llinsen@ucdavis.edu. with hanging nodes, are used to discretize a 3D domain. It also helps in preserving the adaptivity of the unstructured grids. Isosurface extraction using conventional cube-based algorithms, like the marching cubes algorithm, would lead to cracks in the resulting isosurface. Fixing such cracks in a general hexahedral setting is a tedious process and may not be free of artifacts. Thus, we chose to use a dual approach (cf. [5] [8]). For cutting plane visualization, trilinear interpolation leads to discontinuities when hanging nodes exist. Our approach uses scattered data interpolation techniques for generating the slices and includes neighboring hanging nodes for interpolation. We compare several techniques, namely Shepard s local method, and Sibson method, for the estimation of function values locally. 2 Related Work Although several multi-resolution approaches exist for unstructured tetrahedral meshes, the efforts for hexahedral meshes were restricted to (adaptive) multiresolution approaches of the structured grids [11, 14, 20, 23, 28]. This unequal treatment to tetrahedral and hexahedral grids is because unstructured hexahedral grids generate hanging nodes during the construction of a multi-resolution hierarchy and hanging nodes are generally intractable during visualization. However this does not deter the use of unstructured hexahedral grids in several numerical simulations such as in finite element applications which exploit the adaptive nature of these grids, as in the case of adaptive hp grids. Hexahedral to tetrahedral mesh transformations for resolving the hanging nodes problem are not free from artifacts, as, there is a conflict between the linear interpolation of values on the diagonal (for splitting a hexahedron to 2 tetrahedra) and the underlying trilinear interpolation model in the interior of hexahedron. Since these transformations do not comply with the underlying volumetric data field, they are futile. Construction of mesh hierarchies is done using either bottom-up or top-down strategies. Our problem requires a bottom-up approach retrieving coarser mesh resolution by simplification of the highest resolution [1]. Renze and Oliver used vertex removal for 3D mesh decimation [22]. The re-triangulation is done using Delaunay triangulation of the vertices adjacent to the re-

2 moved vertex. Irregular tetrahedral mesh simplification has been achieved by Trotts et al. [27] using ordering of approximation errors that would result from a tetrahedral cell collapse (implemented via 3 edge collapses). Staadt et al. [26] extend Hoppes Progressive Mesh Algorithm [7] of collapses to tetrahedra. Hoppe uses an energy function whose minimization drives edge collapses. Meredith et al. [16] use view-dependent error-based resolution selection for generating a hierarchy in order to maintain accuracy at all levels. Energy-based methods are common for surface reconstruction purposes [3]. Pinkall and Polthier [21] use minimum Dirichlet energy to be the exact equivalent of the distortion measure to determine the most optimal triangulation. Desbrun et al. [3] extend this work to obtain intrinsic parametrization by minimization of different distortion measures of the mesh. Levy et al. [9] mention that Dirichlet energy minimizers are harmonic functions and that hierarchical solver will perform well when minimizing a criterion for which the minimum is a harmonic map. These works have inspired us to extend the use of Dirichlet energy minimizers in determining the criterion for a face removal in an unstructured hexahedral 3D mesh. Cutting planes constrain the visualization of 3D data set to an arbitrarily oriented 2D domain. Hence, it is a very useful tool to study changes in the interior of a mesh. For a uniformly resolved mesh, cutting planes can be computed using trilinear interpolation. However, in an adaptive multi-resolution representation, discontinuities along the cell boundaries are inevitable. Hence, we have adopted scattered data interpolation for constructing the slice instead. Isosurfacing is an indispensable tool used in scientific visualization. However hanging nodes pose a problem in the case of adaptively refined grids, as the conventional Marching Cubes (MC) Algorithm approximate the isopoints using linear interpolation on the cell edges. To overcome this problem, we explore the local continuity of the scalar or vector field distribution in the cell interior, and use dual isocontouring techniques. In contrast to the dual contouring methods [5, 8], our algorithm generates vertices lying on the actual isosurface and does not require normal information at vertex locations. We adopt a ray-casting technique to find the points lying in the cell interior. Though use of raycasting techniques is not new in isosurfacing problems, there are differences in the purpose and mechanisms adopted in the previous work and our work. Earlier ray-casting techniques for isosurfacing were used for refining the isosurface generated using the conventional MC Algorithm [2]. Fruhauf [4] pioneered in adopting a ray-casting approach (which usually uses a trilinear interpolation filter) to produce output meshes which were better than ones generated using the MC technique. A number of different interpolation filters have been proposed [6, 15, 17, 18] for more precise computation of the field values and gradients. The advantage of this technique is it allows the adoption of whichever reconstruction filter for interpolation of field values. However its drawbacks are its high time complexity and view-dependence. In our work, we blend ray-casting and dual isocontouring in a view-independent fashion to construct a crack-free isosurface. View-independence is achieved by casting rays along the body diagonals in each cell. 3 Multi-resolution Hierarchy We introduce a novel technique for implementing face removal to build the multi-resolution hierarchy. The underlying fields typically result from physical or chemical simulations, and are approximated using trilinear interpolation in the cell interior. The function is a trivariate harmonic function, as it has continuous second partial derivatives which satisfies the Laplace equation. For such a real-valued harmonic function in a closed domain, one can define the Dirichlet energy to be the closed integral of the square of the modulus of the function gradient over the domain. Relative Dirichlet energies of different cells give an estimate of the relativeness of the gradients of the function. Gradients reflect the behavior of the function in the cells: high gradients indicate significant changes in the scalar field values, and low gradients indicate smooth variation of the function throughout the cell. Our goal while generating coarser resolution in a multi-resolution hierarchy is to preserve as much information as possible. In order to preserve information about where changes occur, we sort the Dirichlet energy contained in each cell to prioritize the order of face removals. The discrete Dirichlet energy in each cell is computed by the gradient of the function in the cell, i.e., by E = 12 i=1 ( ( fi ) 2 + x ( ) 2 fi + y ( ) ) 2 fi z (1) where f i is the function value along the edge i of a cell. The cells in the grid are then sorted in ascending order of Dirichlet energy functional values. A userdefined minimum number of cells with low Dirichlet energy value is considered. From this reduced set of cells, combinations of two cells are selected, which satisfy the following criteria for collapsing: 1. The cells share a common face. 2. The Dirichlet energy of the resulting collapsed cell should be as low as possible. If the first pass fails, the next batch of cells in the initial sorted list is considered. This two-pass sorting process terminates when two appropriate cells, which can be collapsed to one, are found. Intuitively, the combinations occurring in consecutive iterations are in the regions of the grid where the parameter varies smoothly. We chose this two-pass approach over the straightforward one-pass approach in order to reduce the computational overhead.

3 Once the appropriate set of cells is found, the collapse is implemented by removing the common face and appropriately rearranging the connectivity information of the vertices in the resultant cell. Common face removal results in a new hexahedral cell. Shape properties are favorably preserved after the cell collapse. 4 Cutting planes Cutting planes is a projection of the 3D volume data on an arbitrary 2D plane. This helps us to study the distribution of the parameter in the interior of the grid. Trilinear interpolation of the points on the cutting plane is a very simple and hence a tempting technique to be adopted in this scenario of rectilinear grids. However it fails on a coarse grid with hanging nodes and generates artifacts owing to the inherent discontinuities in the cell boundaries. This makes trilinear interpolation a poor choice for cutting plane visualization in certain cases. Thus we turn to other scattered data interpolation techniques like the Sibson interpolation and Shepard s local method for better results. Sibson interpolation method [25] uses information from voxels in the neighborhood of the point on the cutting plane being evaluated. This method thus addresses the drawback of cracks in the cutting plane evaluation in a coarse grid. However the high computational overhead in this case compels us to revisit other scattered data interpolation techniques. Shepard s localized scattered data interpolation method [24], like Sibson interpolation, uses information of the points in the immediate neighborhood of the cutting plane point to be evaluated. Since this technique solely uses the distances of the local neighborhood voxels, there is increased user interactivity as opposed to Sibson interpolation. Though we trade off accuracy for lesser time complexity in this case, the results obtained using Shepard s method are good. 5 Isosurfacing The idea of dual isocontouring is inspired mainly from the work by Ju et al. [8] and Greß et al. [5]. In contrast to the Marching Cubes algorithm [12, 13, 16, 19], dual contouring computes the isopoints computed in the cell interior, rather than on the boundaries. The dual approach is desirable in the case of hanging nodes, owing to the discontinuities in the cell boundaries in the grid. However, in our algorithm we still make use of the case look-up table of the Marching Cubes Algorithm in order to distinguish between the different cases. We shoot rays along the body diagonals and find the intersection points with the trilinearly interpolated surfaces in the cell interior using Cardan s solution for cubic equations. The algorithm is divided into 3 parts, namely, Case Identification, Isopoint Computation and Triangulation. 5.1 Case Identification The preprocessing of our novel isosurfacing algorithm involves the use of the case look-up table of the conventional Marching Cubes Algorithm, in order to identify the specific case in the look-up table to which the cell corresponds. Using the information of the case in the look-up table, we can determine: 1. The rays that are to be shot along the body diagonals: Since we are using the minimum number of rays possible for determining the isopoints, the look up table tells us which rays intersects the isosurface. 2. The number of disjoint surfaces to be expected within each cell: The look-up table tells us how many isosurface components are to expected to be found in the cell and how they are connected to the components detected in the neighboring cells. 5.2 Isopoint Computation This involves finding the intersection of the ray and the trilinear interpolation surface for the given isovalue.the ray is represented as: r = r 1 + p( r 2 r 1 ) (2) where r 1 and r 2 are the position vectors of the endpoints of the ray and p is the parameter, such that p [0, 1]. The trilinear interpolation surface is given as: I = (1 s).(1 t).(1 u).f (1 s).(1 t).u.f (1 s).t.(1 u).f (1 s).t.u.f s.(1 t).(1 u).f s.(1 t).u.f s.t.(1 u).f s.t.u.f 111 (3) where s, t, and u are parameters along x, y, and z axes, respectively and f ijk s are field values at the corresponding vertices of the cell, and I is the isovalue. The parametric representation intersection point in (s, t, u) is substituted in terms of p and substituted in Equation 3. The zeros of the cubic equation in p are found using Cardan s solution. Real roots in the range [0,1] give the required intersection points. There may be more than one desired root, which indicates the existence of disconnected multiple surfaces within a cell. However it is ensured that there will be at least one point per cell containing at least one edge with opposite signs at its ends. 5.3 Triangulation The triangulation of all the interior points is determined by the following rules: 1. Edges are constructed from an interior point to points in at most 3 distinct neighboring cells [8]. 2. In case of boundary cells, Marching Cubes algorithm is used to find points on the cell boundaries for terminating the isosurface.

4 3. In case of only two cells being connected then the isosurface is terminated using the intersection points obtained from the Marching Cubes Algorithm, except on the edges shared by both the cells. 4. Points within a cell are not connected to each other. This is required to maintain the number of disconnected surfaces within each cell. 6 Results For developing our isosurfacing algorithm, small test cases have been used. For finding the minimum number of rays to be cast, a test case of one cube is taken and for constructing the isosurface, a test case of eight neighboring cubes is taken. Some of the isosurfaces that are developed for some test values are as shown in Figures 1 and 2. Figure 1: Cells (a) and (b) corresponding to cases 6 and 7, respectively, of the original MC look-up table. The number of rays (shown as yellow lines) cast is minimal. The white triangles are the MC isosurfaces and the cyan points are the dual isopoints. Figure 2: (a) shows the comparison of MC isosurface (in white) and dual isosurface (in magenta). (b) shows the octahedron (in white) in MC case and its dual, a cube (edges shown in green) generated using our dual method. We applied our novel technique of multi-resolution hierarchy, cutting planes and isosurfacing on a multiblock data set of flow simulations on an oil-water field at different time steps, with data available on parameters like oil pressure, oil concentration, water saturation and oil velocity, for each point in the grid. The data set consists of 876, 148, 576, 152 and 872 cells in 5 different blocks of the multi-block grid, as shown in Figure 3. Each different block is considered as a grid in our present implementation. Our test case is the oil pressure field of the data set, as shown in Figure 3. On developing the multi-resolution hierarchy, as can be seen in Figures 4 and 5, we observe that after 1000 cell collapses, we do not find much change in the color mapping of the oil pressure field. However after 1500 cell collapses, we observe discontinuities in the color mapping (implemented using Gourard shading). Most of the initial cell collapses occurred in the third grid, where the function varies smoothly. The artifacts in the color mapping in the latter case is due to the Gourard shading. Trilinear, Sibson and Shepard s local interpolation schemes are applied on the dataset to get a slice orthogonal to x-z plane, as shown in Figure 6. Sibson interpolation is more time consuming than Shepard s, though the quality of result is more accurate in former than in other two, where discontinuities can be very obviously seen (Figure 6). However, we weigh the lower time cost in this trade-off between time complexity and accuracy. Hence Shepard s method is more favored than the Sibson. Our isosurfacing algorithm is applied to the oil pressure field. We observe that before performing the cell collapses, our algorithm as well as the Marching Cubes algorithm give similar results (Figure 7). Except in the latter, there is a discontinuity at the boundary between two different blocks. This can be addressed to by treating the boundary cases specifically. 7 Conclusions & Future Work As can be observed from our test study, the three different operations of scientific visualization work as expected. However certain issues that need to be addressed to are : 1. One cell collapse per run of the algorithm is not very economical. The algorithm should improved in order to allow more than one collapse per iteration so that more than one pair of cell with similar energy functional values can be collapsed together. 2. Optimization of the isosurfacing algorithm is needed so that it can be efficient on large-scale data sets too. 3. Isosurfacing needs to be improved upon for cases including more complex isosurfaces, like the tunnels [12]. 4. A formal look-up table for the isosurfacing needs to be made. Specific boundary cases need to be visited again. This is essential for applying the new algorithm on the cases of hanging nodes. Acknowledgments This work was supported by the National Science Foundation under contracts ACI (CAREER Award) and ACI , through the Large Scientific and Software Data Set Visualization (LSSDSV) program under contract ACI , and through the National Partnership for Advanced Computational Infrastructure (NPACI); the National Institute of Men-

5 tal Health and the National Science Foundation under contract NIMH 2 P20 MH A2; and the Lawrence Livermore National Laboratory under ASCI ASAP Level-2 Memorandum Agreement B and under Memorandum Agreement B We also acknowledge the support of ALSTOM Schilling Robotics and SGI. We thank the members of the Visualization and Graphics Research Group at the Center for Image Processing and Integrated Computing (CIPIC) at the University of California, Davis, and the members of the Data Science Group at the Center for Applied Scientific Computing (CASC) at the Lawrence Livermore National Laboratory, Livermore, California. References [1] P. Cignoni, D. Costanza, C. Montani, C. Rocchini, and R. Scopigno, Simplification of tetrahedral meshes with accurate error evaluation, in Proceedings of Visualization 2000, Los Alamitos, CA, 2000, IEEE Computer Society Press, pp [2] P. Cignoni, F. Ganovelli, C. Montani, and R. Scopigno, Reconstruction of topologically correct and adaptive trilinear isosurfaces, Computer & Graphics, 24(3) (2000), pp [3] M. Desbrun, M. Meyer, and P. Alliez, Intrinsic parameterization of surface meshes, in Proceedings of Eurographics 2002, 2002, pp [4] T. Fruhauf, Raycasting opaque isosurfaces in nonregularly gridded cfd data, in Visualization in Scientific Computing (1995 Eurographics WS Proc.), Springer KG, Wien, Chia, 3-5 May 1995, pp [5] A. Greß and R. Klein, Efficient representation and extraction of 2-manifold isosurfaces using kd-trees, in Proceedings of Pacific Graphics 2003, IEEE CS Press, October [6] B. Hamann, E. LaMar, and K. I. Joy, Highquality rendering of smooth isosurfaces, The Journal of Visualization and Computer Animation, 10 (1999), pp [7] H. Hoppe, Progressive meshes, in Proceedings of Siggraph 1996, ACM Press, New York, NY, 1996, pp [8] T. Ju, F. Losasso, S. Schaefer, and J. Warren, Dual contouring of hermite data, in Proceedings of Siggraph 2002, ACM Press, New York, NY, 2002, pp [9] B. Levy, S. Petitjean, N. Ray, and J. Mailot, Least squares conformal maps for automatic texture atlas generation, in Proceedings of SIGGRAPH 2002, ACM Press, New York, NY, 2002, pp [10] P. Lindstrom, D. Koller, W. Ribarsky, L. F. Hodges, N. Faust, and G. A.Turner, Real-time, continuous level of detail rendering of height fields, in Proceedings of the SIGGRAPH 1996, ACM Press, 1996, pp [11] L. Lippert, M. H. Gross, and C. Kurmann, Compression domain volume rendering for distributed environments, in Proceedings of the Eurographics 97, vol. 14, COMPUTER GRAPHICS Forum, 1997, pp [12] A. Lopes and K. Brodlie, Improving the robustness and accuracy of the marching cubes algorithm for isosurfacing, IEEE Transactions on Visualization and Computer Graphics, 9(1) (2003), pp [13] W. E. Lorensen and H.E.Cline, Marching cubes: A high-resolution 3d surface construction algorithm, in Proceedings of SIGGRAPH 1987, vol. 21(4), Computer Graphics, 1987, pp [14] D. Maegher, Geometric modeling using octree encoding, Computer Graphics and Image Processing, 19 (1982), pp [15] S. R. Marschner and R. J. Lobb, An evaluation of reconstruction filters for volume rendering, in Proceedings of IEEE Visualization 1994, Tyson Corner, VA, 1994, R. D. Bergeron and A. E. Kaufman, pp [16] J. Meredith and K.-L. Ma, Multiresolution view-dependent splat-based volume rendering of large irregular data, in Proceedings of IEEE Symposium on Parallel and Large Data Visualization and Graphics, 2001, pp [17] T. Möller, R. Machiraju, K. Müeller, and R. Yagel, Classification and local error estimation of interpolation and derivative filters for volume rendering, in Proceedings of IEEE Symposium on Volume Visualization 1996, 1996, pp [18] T. Möller, R. Machiraju, K. Müeller, and R. Yagel, A comparison of normal estimation schemes, in IEEE Visualization, 1997, pp [19] G. M. Nielson, On marching cubes, IEEE Transactions on Visualization and Computer Graphics, 9(3) (2003), pp [20] M. Ohlberger and M. Rumpf, Hierarchical and adaptive visualization on nested grids, Computing, 59 (1997), pp [21] U. Pinkall and K. Polthier, Computing discrete minimal surfaces, Experimental Mathematics, 2(1) (1993), pp [22] K. J. Renze and J.H.Oliver, Generalized unstructured decimation, IEEE Computer Graphics & Applications, 16(6) (1996), pp

6 [23] R. Shekhar, E. Fayyad, R. Yagel, and J. F. Cornhill, Octree-based decimation of marching cubes surfaces, in Proceedings of IEEE Conference on Visualization 1997, R. Yagel and G. M. Nielson, eds., IEEE, IEEE Computer Society Press, 1996, pp [24] D. Shepard., A two-dimensional interpolation function for irregularly spaced data, in Proceedings of the 23rd National Conference, ACM, August 1968, pp [25] R. Sibson, A brief description of natural neighbor interpolation, in Interpreting Multivariate Data, V.Barnett, ed., John Wiley & Sons, New York, 1981, pp Figure 5: (a) and (b) show the mesh after 1500 collapses. Note that most of the collapses have occurred in the grid with smooth varying function. [26] O. G. Staadt and M.H.Gross, Progressive tetrahedralization, in Proceedings of IEEE Visualization 1998, IEEE Computer Society, Washington DC, 1998, pp [27] I. J. Trotts, B. Hamann, K. I. Joy, and D. F. Wiley, Simplification of tetrahedral meshes, in Proceedings of IEEE Visualization 1998, IEEE Computer Society, Washington DC, 1998, pp [28] R. Westermann, L. Kobbelt, and T. Ertl, Real-time exploration of regular volume data by adaptive reconstruction of isosurfaces, The Visual Computer, (1999), pp Figure 6: Comparison of (a) Trilinear, (b) Sibson, and (c) Shepard s local interpolation schemes for the cutting plane visualization on grid after 500 collapses. Figure 3: (a) shows the mesh of the oil field, with different colors identifying different blocks. (b) shows the color mapping of the Oil Pressure field. Figure 4: (a) and (b) show the mesh after 1000 collapses. Figure 7: (a) shows the isosurface obtained using MC algorithm and (b) shows the one obtained using our algorithm. The discontinuity in the surface in (b) is due to boundary between 2 different blocks, and not cell boundary discontinuity due to cell collapse.