High-Quality, Semi-Analytical Volume Rendering for AMR Data St´ ephane Marchesin and Guillaume Colin de Verdi` ere Fig. 1. Close-up of an AMR dataset showing a meteorite falling into the sea rendered using our system. Abstract—This paper presents a pipeline for high quality volume rendering of adaptive mesh refinement (AMR) datasets. We intro- duce a new method allowing high quality visualization of hexahedral cells in this context; this method avoids artifacts like discontinuities in the isosurfaces. To achieve this, we choose the number and placement of sampling points over the cast rays according to the an- alytical properties of the reconstructed signal inside each cell. We extend our method to handle volume shading of such cells. We propose an interpolation scheme that guarantees continuity between adjacent cells of different AMR levels. We introduce an efficient hybrid CPU-GPU mesh traversal technique. We present an implementation of our AMR visualization method on current graphics hardware, and show results demonstrating both the quality and performance of our method. Index Terms—Volume rendering, AMR data, Volume shading. 1 I NTRODUCTION AND MOTIVATION Adaptive mesh refinement (AMR) is a mesh refinement strategy aimed at reducing the cost of numerical simulations while maintaining high accuracy results. This design allows both simple programming thanks to identical cell shapes and implicit connectivity, and an efficient use of processing resources thanks to adaptive local refinement. Due to its simplicity and computational efficiency, this scheme is widely used in the numerical simulation field. Figure 2 shows an example of such a mesh in two dimensions. As an AMR mesh can be viewed (locally) as a tree, we define the AMR cell level as the level of cell size in the AMR tree, 0 being the biggest cells at the top of the tree, and higher levels being the smaller cells. We define an AMR patch as a cuboid of homogeneous cells, i.e. where all AMR cells have the same level. In this paper, we focus on the specific case of AMR meshes carrying vertex-centered data (as opposed to cell-centered data). In order to analyze the data resulting from AMR simulations, visualization tools are wanted. In particular, volume rendering is a powerful exploration method for 3D data. As of today, volume rendering of AMR data presents three major challenges: 1. A view-order cell traversal technique is needed. On the CPU, this problem can be solved easily and efficiently with existing data • St´ ephane Marchesin and Guillaume Colin de Verdi` ere, CEA, DAM, DIF, F-91297, Arpajon, France. E-mail: marchesi@ocre.cea.fr, guillaume.colin-de-verdiere@cea.fr. Manuscript received 31 March 2009; accepted 27 July 2009; posted online 11 October 2009; mailed on 5 October 2009. For information on obtaining reprints of this article, please send email to: tvcg@computer.org . Fig. 2. AMR mesh example with three different cell levels. structures. However, direct adaptation of these structures to the GPU is difficult, and can result in a waste of memory or subopti- mal performance. As we are interested in interactive visualization, such data structures should allow an efficient implementation. In particular, using additional cells to achieve simpler mesh traversal is not always desirable since it decreases the performance of the system. 2. An interpolation method is required inside the cells to reconstruct a continuous scalar function from the discrete AMR data. This implies defining an interpolant function inside the cells, not only for simple hexahedral cells, but also for cells lying at the border between patches of different levels. In the case of hexahedral cells, the most widespread choice for an interpolation function inside a given cell is the trilinear reconstruction. However, one still needs to restore the continuity at the borders between cells of different levels. This is commonly achieved by splitting cells into tetrahedra or pyramids, though this leads to a more complex mesh, and also needs an additional interpolation scheme per new cell type. This is further complicated by the requirement that different interpolation schemes must be coherent across shared cell faces. 3. High accuracy cell rendering techniques are highly sought after. 1611 1077-2626/09/$25.00 © 2009 IEEE Published by the IEEE Computer Society IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS, VOL. 15, NO. 6, NOVEMBER/DECEMBER 2009