PAPER A Half-Skewed Octree for Volume Ray Casting

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1 IEICE TRANS. INF. & SYST., VOL.E90 D, NO.7 JULY PAPER A Half-Skewed Octree for Volume Ray Casting Sukhyun LIM a), Nonmember and Byeong-Seok SHIN b), Member SUMMARY Ahierarchicalrepresentationformedbyanoctreefora volume ray casting is a well-known data structure to skip over transparent regions requiring little preprocessing and storage. However, it accompanies unnecessary comparison and level shift between octants. We propose a new data structure named half-skewed octree, which is an auxiliary octree to support the conventional octree. In preprocessing step, a half-skewed octree selects eight different child octants in each generation step compared with the conventional octree. During rendering, after comparing an octant of the conventional octree with corresponding octant of the half-skewed octree simultaneously at the same level, a ray chooses one of two octants to jump over transparent regions farther away. By this method, we can reduce unnecessary comparison and level shift between octants. Another problem of a conventional octree structure is that it is difficult to determine a distance from the boundary of a transparent octant to opposite boundary. Although we exploit the previously proposed distance template, we cannot expect the acceleration when a ray direction is almost parallel to the octant s boundary. However, our method can solve it without additional operations because a ray selects one octant to leap farther away. As a result, our approach is much faster than the method using conventional octree while preserving image quality and requiring minimal storage. key words: volume visualization, volume ray casting, space-leaping, hierarchical representation, octree 1. Introduction Volume visualization is a research area dealing with various techniques to extract meaningful and visual information from volume data [1], [2]. One of the most frequently used techniques is direct volume rendering, producing highquality images directly from volume data without intermediate representation. Volume ray casting is a well-known direct volume rendering method [1]. In general, it is composed of two steps [3]: after a ray advances through transparent regions, it integrates colors and opacities as it penetrates an object boundary. Although this produces highquality images, the rendering time is too long. In order to solve this problem, a number of researchers have concentrated on skipping transparent regions that do not contribute to the final image [3] [5]. The distancemap-based approach is a typical spaceleaping method, which computes the distance from each voxel to a nontransparent voxel and stores the value into the structure [6], [7]. Since the distance value for each voxel is the minimum distance to the object boundary, a ray leaps Manuscript received July 18, Manuscript revised January 31, The authors are with the Dept. of Computer Science and Information Engineering, Inha University, Korea. a) s.lim@inha.ac.kr b) bsshin@inha.ac.kr DOI: /ietisy/e90 d over transparent regions quickly without entering into nontransparent objects. Yagel and Shi introduced a method to accelerate the process of volume rendering in a sequence of images based on a distancemap [8]. It exploits the temporal coherence between consecutive images. Wan et al. placed a sphere at every empty voxel position where the sphere radius indicates the distance to the closest nontransparent voxel [9]. Sramek and Kaufman proposed a view-sensitive extension of the distancemap-based approach [10]. Levoy classified volume into a binary representation according to the opacity of voxels [4]. Using this classification, a pyramid is constructed for space leaping and adaptive termination of ray casting. Wilhelms and Gelder introduced a method to enhance the polygonization speed of isosurfacing [11]. It uses a pyramid containing the minimum and maximum voxel values of subvolumes at one level lower. This method was later surveyed as a multi-resolution acceleration technique, called homogeneity acceleration, for volume ray tracing [12]. Stander and Hart attempted to reduce the rendering time by storing the Lipschitz range in each octree node [13]. Grimm et al. presented an efficient hybrid removal and skipping technique using a granular resolution octree [14]. A lot of methods used a hierarchical octree structure for space leaping [15] [20]. To determine the transparency of octants, a lot of methods are proposed [11] [14]. However, an octree still has some problems. The first is to accompany unnecessary comparison and level shift between octants. The second is that it is difficult to determine a distance from an octant s boundary to opposite boundary if the octant is regarded as transparent (we call it as transparent octant). Although the distance template is proposed to solve the second problem [21], we cannot expect the acceleration if a ray direction is almost parallel to the octant s boundary. To solve the above two problems simultaneously, we propose a new data structure, half-skewed octree. It is an auxiliary octree to support the conventional octree. In octree generation step, the half-skewed octree selects different eight child octants where the position criterion to choose them is half skewed at the same level. During the rendering, a ray chooses one octant after comparing an octant of the conventional octree with corresponding octant of the half-skewed octree simultaneously to jump over transparent regions farther away from the current sample point. Our method improves the rendering speed while solving the problems of conventional octree. In Sect. 2, we explain our data structure in detail. Ex- Copyright c 2007 The Institute of Electronics, Information and Communication Engineers

2 IEICE TRANS. INF. & SYST., VOL.E90 D, NO.7 JULY perimental results are shown in Sect. 3. Finally, we discuss and conclude our work. 2. A Half-Skewed Octree We explain our data structure and how to reduce unnecessary comparison and level shift between octants. Also, the problem of the distance template and its solution are presented. In the conventional octree, eight child octants create a single parent octant (see Eq. (1)). In the half-skewed octree, eight child octants are also exploited. Compared with the conventional octree, however, the criterion to select an octant is half skewed at the same level (see Eq. (2)). That is, a parent octant of the half-skewed octree is generated by choosing one child octant per neighboring eight parent octants of the conventional octree. Therefore, we call it as a half-skewed octree. A 2D example of the conventional octree and half-skewed octree is illustrated in Fig Problem of Conventional Octree and Its Solution In an octree generation step, eight octants create a single parent octant until it reaches the root node recursively. In general, each octant holds standard deviation [12], minimum and maximum range [11], [14] [16], [18] [20], or Lipschitz range [13] in order to determine its transparency. The acceleration methods using an octree can skip transparent regions because a ray leaps over transparent octants. As the octree s hierarchy increases, however, unnecessary comparison and level shift to find nontransparent octants are required. Since a ray does not recognize whether transparent octants exist or not in current level before referring values to determine transparency, it has to examine the entire octants for all levels. Therefore, it increases rendering time. Figure 1 shows an example of a ray-traversal example. If an octant is transparent, according to the spatial coherence, the values of neighboring octants can be confined within a transparent range. We assume that the dimension of a volume dataset is N 3, and denote an octant in l-level as Oli, j,k (where, i, j, and k is an octant index for an x-, y-, and z-axis, and 2l N). The Ci,l j,k and Hi,l j,k means a parent octant of the conventional octree and half-skewed octree in l-level, respectively. Fig. 1 A procedure of ray traversal using a conventional octree: unnecessary comparison (shaded boxes) and level shift (thick arrows) between octants are required. For simplicity, we assume the second level hierarchy. Ci,l j,k = 1 l 1 O2i+a,2 j+b,2k+c (1) l 1 O2i+a+1,2 j+b+1,2k+c+1 (2) a,b,c=0 Hi,l j,k = 1 a,b,c=0 A ray-traversal procedure of our method is as follows: after a ray is fired from each pixel on an image plane, the stored values to determine transparency in two octants of the conventional octree and half-skewed octree are referred. In selecting one octant, we have to consider four cases (see Table 1). The first case occurs when the conventional octree s octant is transparent and half-skewed octree s octant is nontransparent. In this case, a ray jumps to the boundary of the conventional octree s octant since the half-skewed octree s octant is nontransparent. In the second case, a ray jumps to the boundary of the half-skewed octree s octant when the Fig. 2 Structures of the conventional octree and half-skewed octree. For simplicity, we use a quadtree structure of which the size is 82, and assume that the level of child octant is l-1. The half-skewed quadtree chooses four neighboring child quadrants where the position to select them is halfskewed.

3 LIM and SHIN: A HALF-SKEWED OCTREE FOR VOLUME RAY CASTING 1087 Table 1 Four cases for space leaping using the conventional octree and the half-skewed octree. mode conventional octant half-skewed octant action case 1 transparent nontransparent jumps to the conventional octant s boundary case 2 nontransparent transparent jumps to the half-skewed octant s boundary case 3 transparent transparent selects an octant between conventional octant and half-skewed one case 4 nontransparent nontransparent examines the child octants Fig. 3 Pseudocode for ray traversal of our method. conventional octree s octant is nontransparent and the halfskewed octree s octant is transparent. If the two octants are all transparent, a ray selects one octant to jump over transparent regions farther away from the current sample point. In order to determine the criterion for selecting one octant, we exploit the distance template [21]. It is a pre-defined distancemap to reduce the cost for determining the distance between a boundary and opposite boundary (we explain it in Sect. 2.2, in detail). By comparing two distance values in each template of the conventional octree and half-skewed octree, we select one octant containing large distance value (that is, the ray leaps farther away by selected octant). The fourth case occurs when both octants are nontransparent. Only in this case, a ray shifts down to the lower level and refers to their child octants. These procedures continue until the ray meets a nontransparent octant. If an octant of the leaf node is confined within a nontransparent range, a resampling filter is performed to compute an accurate scalar value. At the leaf level of the conventional octree and halfskewed octree, their sizes are all two. Therefore, there is no gain in leaf level, so we do not generate the leaf node in case of the half-skewed octree. During ray traversal, if rays reach the leaf level, our method exploits only the conventional octree. In this case, the half-skewed octree requires only 12.5% additional data storage and preprocessing time compared with the conventional octree. Figure 3 shows the pseudocode for ray traversal. Figure 4 shows an example applying the same viewing conditions as in Fig. 1. Since a ray chooses the half-skewed octree s octant, unnecessary comparison and level shift between octants are reduced. That is, five steps (see Fig. 1) are decreased as two steps (in this case, our method includes the comparison time between an octant of the conventional octree and that of the half-skewed octree), and it causes increasing the rendering speed. As a result, closely combining two octrees supports efficient space-leaping.

4 1088 IEICE TRANS. INF. & SYST., VOL.E90 D, NO.7 JULY 2007 Fig. 4 An example of ray traversal when applying the same viewing conditions of Fig. 1. Dotted circles represent disregarded sample points. A ray selects the half-skewed octree s octant after comparing the octant of the conventional octree with corresponding octant of the half-skewed octree. Fig. 5 An example of the distance template based on Euclidian distance transform of which the size is 8 3. A ray refers a distance values when it lies on arbitrary position in the distance template. 2.2 Problem of Distance Template and Its Solution The methods using an octree structure can skip transparent regions quickly because rays jump over transparent octants without further considerations. In order to skip over a transparent octant, however, a distance value to the octant s boundary should be computed by using mathematical computations. That is, we calculate the distance between a ray and an AABB (Axis-Aligned Bounding Box). However, due to expensive computation cost, rendering performance is degraded. Moreover, in case of jumping over several octants to reach nontransparent (interesting) objects, this computation is a burden to fast rendering. We proposed the distance template in our previous work to solve the problem [21]. A distance template is a pre-defined distancemap to reduce the cost for determining the distance between the entry point and the exit point of a ray with respect to an octree. This structure always holds the nearest distance value to reach an octant s boundary when a sample point lies on arbitrary position in an octree. Figure 5 shows the structure of the distance template. We use the Euclidian distance transform approximated integers as 10, 14 and 17 according to directions [10]. Since the size of an octant is always fixed as 2 l (once again, l is the current octree level), the template is not dependent on the viewing conditions, the volume datasets, and OTF (Opacity Transfer Function). Therefore, if it is generated at once we can reuse it continuously. Moreover, because the half-skewed octree is an octree structure (that is, its size is same with that of the conventional octree at the same level), we can exploit it without additional operations. Figure 6 illustrates an example of the ray traversal using the distance template. A ray has to jump over the transparent octant based on the distance template. While the conventional method computes it by mathematical computations, our method uses different distances according to the distance template. Fig. 6 A procedure of ray traversal using a distance template. For simplicity, we suppose that the current octant s size is 8 3. A ray jumps over the transparent octants by referencing the distance value stored in the template. Although the distance template leaps to the boundary of transparent octants quickly and reliably because it can directly access the distance value to reach their boundaries, it cannot achieve acceleration if a ray advances along with the boundary of each octant (that is, a ray direction is almost parallel to the boundary). Since distance values of the template is the minimum distances to reach the octant s boundary, the values diminish moving toward the boundary (See Fig. 7). However, when we exploit our half-skewed octree, a ray selects larger distance values after comparing an octant of the conventional octree with corresponding octant of the half-skewed octree simultaneously. Therefore, a ray quickly skips over transparent regions without additional operations even if a sample point is located in the boundary regions of each octant as shown in Fig. 8. So, our method reduces unnecessary comparison and level shift between octants, and solves the problem of the distance template, simultaneously.

5 LIM and SHIN: A HALF-SKEWED OCTREE FOR VOLUME RAY CASTING 1089 Fig. 7 Another ray traversal example using the distance template. If a ray traverses transparent octant along with the boundary of each octant, we cannot expect acceleration. Fig. 8 Even when a sample point locates boundary regions, a ray leaps over transparent octants efficiently. 3. Experimental Results All the methods were implemented on a PC equipped with AMD Athlon64 2 TM CPU and 2 GB main memory. We exploited six datasets for experiment. The first dataset was an engine block with a two cylinder scanned at a resolution of 512 3, the second was a CT scan of a bonsai with a resolution of 512 3, and the third dataset was an aneurysm of a human brain vessel with resolution. The fourth dataset was the Boston teapot at a resolution of , and the fifth was a piggy bank with a CT scan that had a resolution of The last was the Chapel Hill CT head with a resolution of Table 1 shows the preprocessing time and required memory for the datasets. We set the maximum level of the conventional octree as six (that is, that of the halfskewed octree is five), and the minimum and maximum range at each octree block are stored for determining transparency [11], [14] [16], [18] [20]. Table 2 indicates that the half-skewed octree requires approximately 12.5% memory and preprocessing time compared with the conventional octree. All the results represented the average computation time. In addition, to utilize the dual core CPU for the volume ray casting, we applied the TLP (thread-level parallelism) scheme [22]. To show that our method is still efficient, we experimented on the Euclidean distancemap-based method [6] [8], [10]. Compared with it, the preprocessing time of our method is faster about 70 times. The rendering speed of our method is about 31% faster than that of the method using the conventional octree (see Table 3). When we compare the rendering performance with the distancemap approach, our method is slow approximately 12%. Figure 9 shows the image quality. To obtain comparisons before/after applying our method for each dataset, we compare the pixel values of those images. We conclude that there are no differences between the images. This verifies that our method renders volumetric scenes without loss of image quality. 4. Discussion The rendering speed of our method was approximately 30% faster than the method using only conventional octree under fixed viewing conditions. When we compare the performance with the distancemap approach, the rendering speed of our method was reduced as 10%. However, the distancemap-based methods require more storage (in general, the size is the same as that of the volume dataset) and long preprocessing time. In addition, when the OTF is changed, we have to rebuild it. However, our method does not have the problems of the distancemap-based methods while requiring less preprocessing time and memory. Although our method reduces the rendering time, the efficiency is degraded when the distance from the viewing position to the first-hit position of the nontransparent voxel is short because the possibility to refer only the conventional octree is increased. In addition, since our method is one of the empty space-leaping methods to skip over transparent regions, it has all disadvantages of the space-leaping approaches. For example, if a volume dataset does not have the nontransparent regions, or the transparent and nontransparent voxels are laid repeatedly by voxel-by-voxel, we cannot expect the acceleration. 5. Conclusion We propose a half-skewed octree to reduce the rendering time for a CPU-based volume ray casting without loss of image quality. Our method is used in most cases for volume visualization to skip over transparent regions because it is based on the conventional octree structure that is widely used for space leaping. It means that it can be used for all applications using an octree structure. Compared with the conventional octree, the half-skewed octree selects eight different child octants in each generation step. Using the halfskewed octree, the conventional octree, and their distance

6 1090 IEICE TRANS. INF. & SYST., VOL.E90 D, NO.7 JULY 2007 Table 2 Preprocessing time and memory requirement for experimental datasets. In case of the half-skewed octree, its generation time and memory requirement are included. preprocessing time (sec) required memory (MB) dataset Euclidian conventional half-skewed Euclidian conventional half-skewed distancemap octree octree distancemap octree octree engine bonsai aneurysm teapot piggy head Table 3 Comparison of rendering time. All results are rendered to a perspective projection with pixels. The sign of (+) means that the speed of our method is faster than that of a comparison method, and vice versa. rendering time (sec) efficiency (%) dataset Euclidian conventional half-skewed (A):(C) (B):(C) distancemap (A) octree (B) octree (C) engine bonsai aneurysm teapot piggy head average Fig. 9 The images in the top row is obtained with parallel projection and rendered translucently to show inner structures, and the images in the bottom row is projected to perspective view and the OTFs are set as the isosurface-like shape. templates, rays can efficiently skip over transparent regions by selecting one octant possessing the larger distance value. Since the half-skewed octree shares the leaf nodes with the conventional octree, additional requirement for preprocessing time and memory storage is minimal. Another advantage is that the conventional octree, half-skewed octree, and their distance templates are all independent on the viewing conditions and OTF. In addition, because the distance template is not oriented to the volume datasets, we can reuse it after it is generated once. Experimental results show that our method produces high-quality images as in conventional volume ray casting, and requires less rendering time compared with the conventional acceleration methods. Recently, an octree is widely used for GPU-based methods for interactive volume rendering. Future work should be focused on applying our data structure to GPU-based approaches for empty space-leaping.

7 LIM and SHIN: A HALF-SKEWED OCTREE FOR VOLUME RAY CASTING 1091 Acknowledgment This work was supported by INHA UNIVERSITY Research Grant. References [1] M. Levoy, Display of surface from volume data, IEEE Comput. Graph. Appl., vol.8, no.3, pp.29 37, [2] A. Kaufman, ed., Volume visualization, IEEE Computer Society Press, [3] B. Krishnamurthy and C. Bajaj, ed., Data visualization techniques, John Wiley & Sons, [4] M. Levoy, Efficient ray tracing of volume data, ACM Transactions on Graphics, vol.9, pp , [5] R. Yagel, D. Cohen, and A. Kaufman, Discrete ray tracing, IEEE Comput. Graph. Appl., vol.12, no.5, pp.19 28, [6] K. Zuiderveld, A. Koning, and M. Viergever, Acceleration of ray-casting using 3D distance transforms, Proc. Visualization in Biomedical Computing 1992, pp , [7] D. Cohen and Z. Sheffer, Proximity clouds - An acceleration technique for 3D grid traversal, The Visual Computer, vol.10, pp.27 38, [8] R. Yagel and Z. Shi, Accelerating volume animation by spaceleaping, Proc. IEEE Visualization 1993, pp.62 69, [9] M. Wan, Q. Tang, A. Kaufman, Z. Liang, and M. Wax, Volume rendering based interactive navigation within the human colon, Proc. IEEE Visualization 1999, pp , [10] M. Sramek and A. Kaufman, Fast ray-tracing of rectilinear volume data using distance transforms, IEEE Trans. Vis. Comput. Graphics, vol.6, no.3, pp , [11] J. Wilhelms and A.V. Gelder, Octree for faster isosurface generation, ACM Trans. Graphics, vol.11, no.3, pp , [12] J. Danskin and P. Hanrahan, Fast algorithms for volume ray tracing, Proc. IEEE Visualization 1992, pp.91 98, [13] B. Stander and J. Hart, A Lipschitz method for accelerated volume rendering, Proc. IEEE Visualization 1994, pp , [14] S. Grimm, S. Bruckner, A. Kanitsar, and E. Gröller, Memory efficient acceleration structures and techniques for CPU-based volume raycasting of large data, Proc. IEEE Volume Visualization 2004, pp.1 8, [15] S. Parker, P. Shirley, Y. Livnat, C. Hansen, and P. Sloan, Interactive ray tracing for isosurface rendering, Proc. IEEE Visualization 1998, pp , [16] G. Knittel, The UltraVis system, Proc. IEEE Volume Visualization 2000, pp.71 79, [17] S. Guthe and W. Straßer, Real-time decompression and visualization of animated volume data, Proc. IEEE Volume Visualization 2001, pp , [18] B. Mora, J.P. Jessel, and R. Caubet, A new object order ray-casting algorithm, Proc. IEEE Volume Visualization 2002, pp , [19] M. Hadwiger, C. Sigg, H. Scharsach, K. Bühler, and M. Gross, Real-time ray-casting and advanced shading of discrete isosurfaces, Comput. Graph. Forum, vol.24, pp , [20] S. Lim and B.S. Shin, Efficient space-leaping using optimal block sets, IEICE Trans. Inf. & Syst., vol.e88-d, no.12, pp , Dec [21] S. Lim and B.S. Shin, Reliable space leaping using distance template, Lecture Notes in Computer Science, vol.3337, pp.60 66, [22] S. Grimm, S. Bruckner, A. Kanitsar, and E. Gröller, A refined data addressing and processing scheme to accelerate volume raycasting, Comput. Graph., vol.28, pp , Sukhyun Lim is a research professor in Inha University, Korea. He simultaneously received the BS degree in Computer Science & Engineering and Physics from Inha University, Korea, in And he received MS and Ph.D degrees in Computer Science & Engineering from Inha University in 2001 and 2006, respectively. His research interests include volume visualization, terrain rendering, and hardware-based rendering. Byeong-Seok Shin is an associate professor in the school of computer engineering, Inha University, Korea. Current research interests include volume rendering, terrain rendering, realtime graphics, stereoscopic, virtual reality, augmented reality, and medical imaging such as MIP, virtual endoscopy and virtual surgery. He received BS, MS, and PhD in computer engineering from the Seoul National University in Korea.