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Rendering volumes represented as a 3D grid of voxels requires an overwhelming amount of processing power. In this paper we investigate efficient techniques for rendering semi-transparent volumes on vector and parallel processors. Parallelism inherent in a regular grid is obtained by decomposing the volume into geometric primitives called beams, slices and slabs of voxels. By using the adjacent properties of voxels in beams and slices, efficient incremental transformation schemes are developed. The slab decomposition of the volume allows the implementation of an efficient parallel feed-forward renderer which includes the splatting technique for image reconstruction and a back-to-front method for creating images. The authors report the implementation of this feed-forward volume renderer on a hierarchical shared memory machine with individual pipelined processors.