A Stochastic Approach for Rendering Multiple Irregular Volumes Naohisa Sakamoto and Koji Koyamada Kyoto University ABSTRACT In this paper, we propose a technique for rendering multiple irregular volumes using a stochastic approach. In this approach, we extend our stochastic projected tetrahedral (SPT) algorithm to treat irregular volumes. The SPT algorithm is a sorting-free volume rendering algorithm that projects a tetrahedral cell onto an image plane and controls the particle rendering using the opacity as the probability in the rasterization process. In the rasterization process, the particle depth is determined as that of a pre-defined location, which may be the front, back or middle point on a ray segment in the SPT algorithm. This determination causes a prominent artifact when treating multiple volumes, although it does not cause any problem when treating a single volume. The artifact occurs because the SPT algorithm controls the particle stochastically in the image plane but not in the depth direction. In the volume rendering process, it is assumed that the number of particles follows a Poisson distribution along the segment of the viewing ray that intersects the tetrahedral cell. The particle spacing follows the exponential distribution. In this case, we construct a cumulative distribution function of the particle spacing and develop a technique for calculating the nearest particle along the interval. We apply this technique to multiple volumes to confirm its effectiveness. Keywords: volume rendering, semi-transparency, irregular volume, multiple volumes. Index Terms: I.3.3 [Computer Graphics]: Picture/Image Generation – display algorithm; I.3.3 [Computer Graphics]: Three-Dimensional Graphics and Realism – color, shading, shadowing, and texture. 1 INTRODUCTION There are many situations in which multiple-volume datasets should be visualized in a single scene. For example, in the numerical simulation research field, scientists often need to visualize multiple datasets generated from several solvers and measuring datasets and confirm the relationships among multiple variables to verify and validate the simulation results. One of the most effective methods for such visualization is a visualization technique that we call fused visualization. In fused visualization, the transparency attributes of the visualization objects must be well-utilized to effectively represent multiple variables in a single pixel. When the transparent objects are rendered, they are processed in the visibility order, which is difficult to determine, especially in the case of irregular grids. Irregular grids are often used to precisely model an object with complex boundaries and are subdivided into multiple regions when the large-scale model is simulated using a high-performance computing environment. In general, to visualize such subdivided regions using a volume rendering technique, these regions are first independently visualized to generate the sub-images. Then, the sub-images are superimposed in the visibility order of the regions. This strategy sometimes fails because the subdivision is not necessarily made by considering the visibility ordering. From the viewpoint of high-performance computing, the subdivision is optimized to minimize the total cost to transfer the information through the boundaries of the subdivided regions. Most previous visualization techniques for multiple regular volumes have employed the data-intermixing technique [1]. Multiple-volume datasets can be multi-valued; i.e., multiple values are defined at positions at which two or more volume datasets intersect. In the data-intermixing technique, a new value is calculated using weighting values. However, there are no specific guidelines for determining these weighting values, which makes multiple-volume rendering complicated. This lack of guidelines is because volume datasets are continuously defined. Previously, interactively render multiple objects has been regarded as a difficult process because a costly visibility sorting is required at each viewing point. Apparently, there are cases for which such a sorting is not possible. In multiple-volume datasets, the volume cells can overlap. Thus, a sorting-free rendering algorithm is absolutely necessary. For this purpose, we proposed a stochastic rendering method that can integrally treat volumes and polygons without visibility sorting. To develop a sorting-free rendering algorithm, we revisited the brightness equation in the volume-rendering algorithm and reconsidered the definition of opacity, which is usually derived from a user-specified transfer function. This led to two approaches, the object-space approach [2] and the image-space approach [3]. In the former approach, we define a density function of emissive opaque particles. According to the density function, we generate particles within a given volume dataset in an object space and project them onto an image plane. Because we use opaque particles, no visibility sorting is required. Although this method exhibits good scalability for treating large-scale irregular volumes [4], the current drawback of this approach is the generation of low-quality images in which particles are visible on the boundary surface polygons upon close inspection. To overcome this problem, in the latter approach, we use a sorting-free method based on projected tetrahedra [5] and the pre-integration technique [6], which simply controls the fragment rendering in the image space by using the evaluated opacity value to calculate the rendered image. In this approach, we regard the brightness equation as the expected value of the luminosity of a ray segment intersected by a grid cell. This method can also visualize volumes and semi-transparent polygons [7]. However, the intersection patterns of grid cells for rendering multiple irregular volumes are not considered in this technique. In general, it is difficult to consider the intersection patterns in the rendering technique based on pre-integration because the number of the patterns might increase as the number of volumes increases. In this paper, we extend the image-space approach [3][7], which we call the SPT algorithm, to multiple-irregular-volume rendering consists of tetrahedral cells and present a novel technique that can determine an exact depth using the pre- integration technique for the stochastic rendering. Our technique e-mail: naohisas@viz.media.kyoto-u.ac.jp e-mail: koyamada.koji.3w@kyoto-u.ac.jp LEAVE 0.5 INCH SPACE AT BOTTOM OF LEFT COLUMN ON FIRST PAGE FOR COPYRIGHT BLOCK 2014 IEEE Pacific Visualization Symposium 978-1-4799-2873-6/14 $31.00 © 2014 IEEE DOI 10.1109/PacificVis.2014.26 272