Fast volume rendering using a shear-warp factorization of the viewing transformation

Abstract

Several existing volume rendering algorithms operate by factoring the viewing transformation into a 3D shear parallel to the data slices, a projection to form an intermediate but distorted image, and a 2D warp to form an undistorted final image. We extend this class of algorithms in three ways. First, we describe a new object-order rendering algorithm based on the factorization that is significantly faster than published algorithms with minimal loss of image quality. Shear-warp factorizations have the property that rows of voxels in the volume are aligned with rows of pixels in the intermediate image. We use this fact to construct a scanline-based algorithm that traverses the volume and the intermediate image in synchrony, taking advantage of the spatial coherence present in both. We use spatial data structures based on run-length encoding for both the volume and the intermediate image. Our implementation running on an SGI Indigo workstation renders a 2563 voxel medical data set in one second. Our second extension is a shearwarp factorization for perspective viewing transformations, and we show how our rendering algorithm can support this extension. Third, we introduce a data structure for encoding spatial coherence in unclassified volumes (i.e. scalar fields with no precomputed opacity). When combined with our shear-warp rendering algorithm this data structure allows us to classify and render a 2563 voxel volume in three seconds. The method extends to support mixed volumes and geometry and is parallelizable.

Fast re-rendering of volume and surface graphics by depth, color, and opacity buffering

“…Our method bears some similarity to that sketched by Lacroute (Lacroute, 1995). (12) where N max is the total number of merged ray sets.…”

“…Condition 1 is the scheme adopted by the Lacroute shear/warp factorization to reduce the storage overhead for the classified volume by run-length encoding the data (RLE) and speed up the subsequent compositing for each scanline -basically the renderer only does work when there is a nontransparent voxel run (Lacroute and Levoy, 1994). Condition 2 is an approximation control factor which deals with nonuniformity in the opacity over a voxel run.…”

Fast maximum intensity projection algorithm using shear warp factorization and reduced resampling

“…2. The shear warp factorization method operates by factorizing the viewing transformation matrix into a 3D shear parallel to the data slices to form an intermediate but distorted projection image and then applying a 2D warp to form an undistorted final image (11). For affine viewing transformation matrix (M view ) concerned in this study, the shear warp factorization includes a permutation (represented by matrix P), a 3D shear (represented by matrix M shear ), and a 2D warp (represented by matrix M warp ): M view ϭ M warp M shear P. The permutation matrix P is associated with the choice of the principal viewing axis.…”

“…The details for the calculation of M shear and M warp from a given M view can be found in Ref. 11 and are summarized in the Appendix.…”

“…We provide here a summary of the basic formulation for the shear warp algorithm (11) for the convenience of readers interested in implementation. Let (X,Y,Z) be the object coordinate and (x,y,z) be the image coordinate.…”