Dynamic Tubular Grid: An Efficient Data Structure and Algorithms for High Resolution Level Sets

Abstract

Level set methods [Osher and Sethian. Fronts propagating with curvaturedependent speed: algorithms based on Hamilton-Jacobi formulations. J. Comput. Phys. 79 (1988) 12] have proved very successful for interface tracking in many different areas of computational science. However, current level set methods are limited by a poor balance between computational efficiency and storage requirements. Tree-based methods have relatively slow access times, whereas narrow band schemes lead to very large memory footprints for high resolution interfaces. In this paper we present a level set scheme for which both computational complexity and storage requirements scale with the size of the interface. Our novel level set data structure and algorithms are fast, cache efficient and allow for a very low memory footprint when representing high resolution level sets. We use a time-dependent and interface adapting grid dubbed the "Dynamic Tubular Grid" or DT-Grid. Additionally, it has been optimized for advanced finite difference schemes currently employed in accurate level set computations. As a key feature of the DT-Grid, the associated interface propagations are not limited to any computational box and can expand freely. We present several numerical evaluations, including a level set simulation on a grid with an effective resolution of 1024 3 .

High Resolution Sharp Computational Methods for Elliptic and Parabolic Problems in Complex Geometries

“…We also mention that other local level-set methods have been proposed, see e.g [105,99,72] and the references therein. Finally, we note that true local-level-set methods, i.e.…”

Fast Sparse Level Sets on Graphics Hardware

“…Therefore, unlike other state-of-theart CPU methods (e.g. [11], [25]), our method is limited in the maximum resolution that can be achieved, by the (relatively small) amount of memory available on mainstream GPUs. However, in terms of GPU level sets, resolutions of 4096 3 voxels have not previously been achieved.…”

“…Recently, Nielsen and Museth introduced the "Dynamic Tubular Grid" (DT-Grid) [25], a recursive, compressed levelset representation inspired by the compressed-row-storage technique used to represent sparse matrices. They showed that the memory requirement of DT-Grid is optimal, i.e., it is proportional to the size of the interface.…”

“…The active neighbouring tiles of a given tile can be found in constant time, so that updating the level-set values of all tiles is linear in the number of active tiles. The STL method was found to be faster than the recent approaches mentioned above [4], [11], [25] and more importantly, the algorithm can greatly benefit from both fine-and coarsegrain parallelization by leveraging SIMD and/or multi-core configurations.…”

“…By contrast, the DT-Grid [25] requires potentially different handling of every voxel, and furthermore it relies on more complex iterator structures specific for every neighbouring voxel, which would result in more registers being used in a CUDA implementation. Additionally, to implement in CUDA the 'push' operation used to insert grid points to the DT-Grid data structure, one has to compare the last inserted coordinate to the current one and execute potentially different code consisting of write operations in a random-access fashion.…”