An Interactive System for Heterogeneous 3D Volumetric Data Visualization

Transcription

1 The 5th International Conference on Computer Science & Education Hefei, China. August 24 27, 2010 ThP13.10 An Interactive System for Heterogeneous 3D Volumetric Data Visualization Huanhuan Xu 1,2, Wuyi Yu 3,LiWei 3, Ning Zhang 3, and Xin Li 1,2,* 1 Electrical & Computer Engineering,Louisiana State University,Baton Rouge, LA 70803, USA 2 Center for Computation & Technology,Louisiana State University,Baton Rouge, LA 70803, USA 3 Department of Automation, Xiamen University, Xiamen, Fujian ,China * xinli@lsu.edu Abstract With the development of 3D scanning and imaging technologies, large amount of 3D surface and volumetric geometric data have been routinely collected these days. Compared with surface data, volumetric data are usually equipped with rich material, texture, intensity, and other heterogeneous structural information in their interior regions. Effectively visualizing such information can be very important since it facilitates many data analysis tasks. This paper studies the effective visualization of heterogeneous volumetric data which can have not only geometry but also other equipped scalar fields. We have developed an interactive system that can efficiently visualize the geometry and scalar field simultaneously via different cross-sections. A simple data structure used for handling large-scale 3D data is presented. Vector fields on the volumetric data such as 3D mapping, B-Spline fitting and fitting error field can also be effectively visualized in our framework. Effectiveness of our system is demonstrated and compared with existing volumetric data rendering system. Index Terms Volumetric Model Visualization, Volumetric Mapping Visualization, B-Spline Fitting Visualization, Tetrahedral Mesh Cutting. I. INTRODUCTION The inside compositions of the object play an important role for people to understand the object. Technology, at its most basic level, seeks to understand how the world behaves. Over the past three decades, computer graphics methods have fueled a growing understanding of physical phenomena, which in true have helped scientists and engineers substantially improve the quality of life. Since some objects have too complicated structures and it is difficult for people to analyze the behaviors of these objects, the visualization techniques can help human better observe and understand them. The rapid advancement of 3D scanning techniques makes it easier to acquire massive 3D data nowadays. 3-D scientific visualization has made a major impact on the display of properties or behaviors of these data. It is a reference work for professionals who use computing techniques to simulate/analyze data, and for the academic and software communities who support them. Many engineering design and analysis applications share common components of modeling, analysis, and the visualization of results. These terms describe a process which is as shown in Figure 1. In such an environment, the analysis itself may govern geometric changes, and the results can be visualized as the analysis proceeds. Within this Fig. 1. The current trend towards an integrated design, analysis and visualization environment framework, one can still describe the individual components of this analysis cycle as modeling, analysis or visualization activities [6]. To produce things better, faster, and less expensively than before, visualization tools have evolved to help scientists and engineers understand more about how things behave. In the process we have gained an insight into a whole new realm of information which has heretofore been hidden from us. In this paper, we develop an effective way to visualize heterogeneous 3D volumetric models and their application, such as mapping and fitting. Our algorithm works for general polygonal meshes, although in the remaining part of paper we only illustrate the algorithm on tetrahedral and hexahedral meshes. The main contributions of this work is that (1) We build an efficient mesh representation data structure for interactive visualization of large-scale 3D heterogenous data. (2) Upon our data structure, we develop a real time rendering system, where given a reference point and a normal direction, the corresponding cross-section is visualized to illustrate interior structure of a volumetric model. (3) The cut regions are triangulated and interpolated smoothly and efficiently, so that scalar fields and vector fields can be nicely illustrated by color-encoding or texture mapping. II. ALGORITHM DESIGN In this section, we will give the detail information for our algorithm according to the data structure we used and the processing steps of the algorithm /10/$ IEEE 1745

2 A. Data structure A well know data structure for representing tetrahedral meshes is the half-face structure, where vertices, half-edges, edges, half-faces, faces, and tetrahedra elements are linked together for efficient topological and geometric local traversals. However, half-face data structure stores about 3 to 4 times larger storage space to trade for efficiency in geometric processing. For large-scale models, it costs a lot of space as well as time to visualize different cross-sections using this structure. Meanwhile, main functional of the viewer is just to efficiently retrieve different regions of the model, without the necessity of frequent updating. A much simpler data structure is adequate for our application. We are only saving vertices position and their relationship with their tetrahedra. Using KDtree to organize all the tetrahedrons of the object improves the efficiency of tetrahedral traversal. B. Get a Cutting Plane We design a user-friendly GUI for the cross-section visualization. On this GUI, users can chose the coordinates of the original point and the normal which can determine a cutting plane uniquely. When we get the original point, expressed as p 0 =(x 0,y 0,z 0 ) and the normal direction n =(a, b, c) of a cutting plane, the cutting plane can be formulated as a(x x 0 )+b(y y 0 )+c(z z 0 )=0, (1) C. Tetrahedron Classification A cutting plane passing through a volumetric model represented by a tetrahedral mesh, all tetrahedra can be classified into three categories: 1) Below the cutting plane; 2) above the cutting plane); and 3) intersected by the cutting plane. For tetrahedra type 1), they form several connected components and we will directly render their boundary triangles. For tetrahedra type 3), we will render part of them after the intersection computation. Simply, we can put the coordinates of vertexes form each tetrahedron into the left side of the equation (1). By checking which sides all tetrahedral vertices locate, with respect to the cutting plane, we can straightforwardly get the type of each tetrahedron. However, in common situation, there are enormous tetrahedra in a tetrahedral mesh representation of a big volumetric model. If whenever a cutting pane is given, or is adjusted, we need to iterate through all the tetrahedra to check the type then determine the rendering strategy, the system will be very slow due to such a large computational burden. To improve the efficiency, we design a preprocessing procedure for all tetrahedra so that the type of tetrahedrons can be determined faster. The system speed can significantly improve. First, we compute the bounding sphere BS i for each tetrahedron T i. The bounding sphere of a tetrahedron is a hyper-sphere that completely encompasses the tetrahedron. The main motivation for the usage of bounding sphere is due to the following observations: If a tetrahedron T i intersects with a plane, its bounding sphere BS i must intersect with the plane as well. The tetrahedron is type 3 in this case. If a bounding sphere BS i intersects with a plane, the intersection of tetrahedron it encompasses may not. So when we find all intersected spheres, we need to further determine the type each tetrahedron. If a bounding sphere BS i doesn t intersect with a plane, the tetrahedron it encompasses will not intersect with the plane either. The tetrahedron is either type 1 or 2. In this paper, we simply use the mean of the coordinates of four vertices in each tetrahedron as the center C i of bounding sphere. And the radius R i of bounding sphere is the max distance from vertices to center. Fig. 2 shows an example of bounding sphere. Fig. 2. This figure shows a tetrahedron with its bounding sphere. The red point is the center of bounding sphere, and the green line is the radius of the bounding sphere Second, we compute the item D i of each bounding sphere BS i. and D i =(ax i + by i + cz i + d)/ a 2 + b 2 + c 2, (2) d = (ax 0 + by 0 + cz 0 ), (3) where x i,y i,z i is the coordinates of sphere center. As it shows in the equation, D i means the distance from the sphere center of BS i to the cutting plane. And we could get the side of sphere center by D i : { inside D i 0 (x i,y i,z i )is (4) outside D i > 0 Only using D i is not enough to know the side of whole bounding sphere. We define two other items K1 i and K2 i for each bounding sphere: and K1 i = D i R i, (5) K2 i = D i + R i, (6) When K1 i > 0, the bounding sphere is outside the cutting panel; if K2 i < 0, the bounding sphere is inside the cutting panel. Here we need two KD trees for getting the tetrahedrons of type 1 and 2 robustly. We insert all the tetrahedrons into KD tree T 1 and T 2 according to their value of K1 i and K2 i, By traveling the nodes of right sub-tree in T 1 whose K1 i value is close to 0, we can easily get all nodes outside the cutting panel. We save these tetrahedrons into a set S out.in 1746

3 the same way, we can get the inside tetrahedrons from T 2 and put them into set S in. We define another set S all including all the tetrahedrons in the model. The set S inter contains the tetrahedrons had the possibility of intersection and it can be described as: shows, when we find edge AD, we can find the next face ACD using the same method. We mark the intersection B,C,D along this path. Thus we can get a section B C D. S inter = S all S in S out, (7) The further calculation of intersecting points for S inter will be described in following step. Our method will have a good performance in real time operations with the user only translating the cutting plane later. When user move the panel along normal direction, and the changed cutting panel P is: a(x x 0 )+b(y y 0 )+c(z z 0 )=Δd, (8) and D i of each bounding sphere is: D i = D i Δd/ a 2 + b 2 + c 2, (9) changed K1 i, K2 i should be: K1 i = D i R i Δd/ a 2 + b 2 + c 2, (10) K2 i = D i + R i Δd/ a 2 + b 2 + c 2, (11) In the classifying step, we do not need to rebuild KD trees according to the new value. For getting S out, the condition K1 i > 0 equals K1 i > Δd/ a 2 + b 2 + c 2, and we can get the outside tetrahedrons by traveling the right sub-tree of node whose value is closest to Δd/ a 2 + b 2 + c 2 in T 1 ; for S in, the condition K2 i < 0 equals K2 i < Δd/ a 2 + b 2 + c 2, and we can get the outside tetrahedrons by traveling the left sub-tree of node whose value is close to Δd/ a 2 + b 2 + c 2 in T 2.So we can easily get the classification of tetrahedra by traveling original KD trees, rather than rebuild the trees; it becomes more efficient. D. Computing Intersection In the classification step, we get a classification of all tetrahedra in the model. And we get a set of tetrahedra which is probably intersected. We further check their intersection situations by putting the coordinates of the vertexes it contains into the left side of formula 1. If the values are all larger than 0, the tetrahedron is outside the cutting plane. Meanwhile, if the values are all less than 0, the tetrahedron is inside the cutting plane. In the case of the tetrahedrons whose vertexes values are not all larger or less than 0, then tetrahedron is intersected by the cutting plane. We need a intersection computation process for these tetrahedra. Suppose a tetrahedron is called ABCD. Any three of the four vertices build a face of the tetrahedron. We get one face arbitrarily, for example, face ABC. We estimate its location relationship with the cutting plane. When we find the first edge that gets intersected by the cutting plane, edge AB for example, we call the intersection point B. Then turn to the next face ABD by rotate around the axis of edge AB. As the Figure 3 Fig. 3. The intersection points for one tetrahedron. (a)the original tetrahedron (b) three intersection points for the tetrahedron (c) four intersection points for the tetrahedron In this step, we must compute the points of intersection, like B,C,D. It is computing intersection of a line and a plane. From the two vertices of an edge, we can get the equation of the line. Compute the equation of line and the equation of plane together, we can get the intersecting point. Once the section B,C,D is computed, we need to make the triangle coherently oriented. The expression is used: n(b D C ) n(cuttingplane) > 0. If the normal of the section satisfy the expression, it keeps unchanged. Otherwise, we flip it over. There are two cases on intersections when the bounding sphere is intersected by the cutting plane: three intersection points and four intersection points. We discussed the threepoint-intersected case above, while the four-point-intersected case can be handled similarly, only one more edge needs to be checked. Tetrahedra are checked by searching the KD-tree. And we get vertices of the intersected planar triangle mesh, and they together compose the tessellation of the entire crosssection, for the subsequent visualization. III. IMPLEMENTATION AND DISCUSSION We implemented our algorithm under Microsoft Visual Studio platform using FLTK for GUI design, and using OPENGL for graphics [1]. Time complexity analysis. When given a cutting plane, we arrange all tetrahedral in the KD-tree which takes an O(n log n) offline preprocessing. When translating the cutting plane, we find all intersected tetrahedral/hexahedral cells in the KD-tree which takes O(log n) time. For the intersected tetrahedron or hexahedron, our algorithm takes O(1) operations to get the polygon on the intersection plane. Therefore, extracting the entire polygonal mesh for rendering on the intersection plane needs O(k) time, where k is the number of intersected cells. In whole, our complexity of intersection computation is reduced to O(logn + k). IV. EXPERIMENTAL RESULTS For input tetrahedron data, we use color coding scheme to visualize these scalar(vector) field. One way we get the scalar(vector) field is computer distance fields. Another way is to generate noise model for tetrahedron data. In the following, 1747

4 we applied our algorithm into tetrahedron mapping and tetrahedron fitting. When we handle tetrahedron mapping, we use distance field as our scalar field. When we handle tetrahedron fitting, we use noise generating model to visualize. For models represented by hexahedral meshes, they can be cut using a similar scheme. Distance field. Let S be a surface in R 3. Then the distance field of a surface S is a scalar function D S : R 3 R such that for all p R 3, D s (p) =sign(p).min{d(p, q) q S}, where { 1 if p inside, sign(p) = +1 if p outside. In other words, D s tells for any point p the distance to the closet point on the surface S [2]. A. Heterogenous 3D Volumetrical Data In this section, we applied our algorithm to visualize heterogenous 3D volumetrical data directly, and using distance filed as our scalar field. Figure 4(a,b) shows using our algorithm to visualize Omontondo model with different cutting-plane. Figure 4(d,e) shows using our algorithm to visualize Sphere model with different cutting-plane. We also compare our algorithm with the previous algorithm which is based on volume grid, Figure 4(c,f) shows using previous algorithm to visualize the Omontondo and sphere objects. Through this comparison, it can be seen that our algorithm has following advantages: 1. In the previous algorithm we can only cut the objects along with X-axis, Y-axis and Z-axis. While in our algorithm, the users can define random cutting plane describe to a normal and an original point. This improvement can help us to get more information from 3D objects. 2. The biggest disadvantage of method based on volume grid is the disconnect along edge. While in our new algorithm, it avoid this problem very successful. B. Mapping 3D Volumetrical Data In this section, we applied our algorithm to visualize 3D mapping effects. For the volumetric mapping method, we adopt the algorithm from [5], [4], [3] which use MFS to do volumetric mapping and get accurate and efficient results. Figure 5 shows the original 3D model, and we are going to map a 3D cube to a 3D head. The number of vertices of these two data is 500K. Figure 6 shows the mapping effects using our algorithm. The Figure 6(a) shows the original field of the cube, the red means the larger value, the blues means the smallest value. The color bar on the top of the cube shows the range of value.the Figure 6(b) shows the distances of the head model. Since we map the cube to the head, then in the cube we could see a head distance field. Figure 6(c) shows using our algorithm to visualize the mapping effect, which express the mapping effect very clearly and pretty. Fig. 5. The original cube model and head model (a)the cube model (b) The head model C. Fitting 3D Volumetrical Data In this section, we apply our algorithm to the visualize B- Spline fitting effect which requires highly precise. We generate noise vector(scalar)field for head model (500K). Figure 7(a)(d) shows the original field for noise head model with different cutting-plane. Figure 7(b)(e) shows the fitting results using B- spline method to fit original noise head model. Figure 7(c)(f) shows the error field of this fitting. Through our visualization method, it is easy for us to analyze which part has larger fitting error (the red means bigger value, the blue means smaller value). For example, in here, we could see the ear of the head has larger error value than other part. Then we could refine the ear part to improve the whole fitting effect. V. CONCLUSION We have presented an effective system to visualize heterogenous 3D volumetric data. 3D volumetric data and their scalar/vector fields are visualized via sequential cross-sections decided by users interactively. Our simple data structure and clipping/interpolating algorithm guaranties the efficiency of this system. The developed system can be a useful tool for many 3D heterogeneous data processing and modeling tasks. ACKNOWLEDGMENT Volumetric data are from Stanford Shape Repository and Aim@Shape Repository. This work is supported by Louisiana Board of Regents Research Competitive Subprogram (RCS) LEQSF( )-RD-A-06 and PFund: NSF(2009)-PFUND Huanhuan Xu is supported in part by the Mark and Carolyn Guidry doctoral fellowhsip from Electrical and Computer Engineering Department of LSU. REFERENCES [1] D.Shreiner, M. Woo, J.Neider, and T.Davis. OpenGL(R) Programming Guide: The Official Guide to Learning OpenGL(R). Addison-Wesley Professional; 6 edition, [2] E. Langetepe and G. Zachmann. Geometric Data Structures for Computer Graphics. A K Peters, Ltd. Wellesley, Massachusetts, [3] X. Li, X. Guo, H. Wang, Y. He, X. Gu, and H. Qin. Harmonic volumetric mapping for solid modeling applications. In Proc. ACM symp. on Solid and physical modeling, pages , [4] X. Li, X. Guo, H. Wang, Y. He, X. Gu, and H. Qin. Meshless harmonic volumetric mapping using fundamental solution methods. IEEE Trans. on Automation Science and Engineering, 6, [5] X. Li, H.H. Xu, S.H. Wan, Z. Yin, and W.Y. Yu. Feature-aligned harmonic volumetric mapping using mfs. Computer Graphics, 34, [6] R.S.Gallagher. Computer Visualization. CRC Press,

5 Fig. 4. Visualizing Omotondo model using the current system (a,b,d,e) and previous system (c,f). Fig. 6. Cube Mapping of the Max-Planck Model. (a) color-encoded distance field of polycube; (b) distance field of the Max-Planck; (c) Visualizing transferred distance field (induced by the volumetric mapping) upon the cube domain. Fig. 7. The noise fitting field for head model (a,d)the original field for head model with different cutting plane (b,e) the fitting results for head model using B-Spline (c) the fitting error field. 1749