The automatic formal verification of multiplier designs has been pursued since the introduction of BDDs. We present a new rewriter-based method for efficient and automatic verification of signed and unsigned integer multiplier designs. We have proved the soundness of this method using the ACL2 theorem prover, and we can verify integer multiplier designs with various architectures automatically, including Wallace, Dadda, and 4-to-2 compressor trees, designed with Booth encoding and various types of final stage adders. Our experiments have shown that our approach scales well in terms of time and memory. With our method, we can confirm the correctness of $$1024\times 1024$$ -bit multiplier designs within minutes.

Recent work demonstrates that interactive ray tracing is possible on desktop systems, but there is still much debate as to how to most efficiently support advanced visual effects such as soft shadows, smooth freeform surfaces, complex shading, and animated scenes. With these challenges in mind, we reconsider the options for designing a rendering system and present Razor , a new software rendering architecture for distribution ray tracing designed to produce high-quality images with high performance on future single-chip many-core hardware. Razor includes two noteworthy capabilities: a set of techniques for quickly building the kd-tree acceleration structure on demand every frame from a scene graph and a system design that allows for crack-free multiresolution geometry with each ray independently choosing its geometry resolution. Razor's per-frame kd-tree build is designed to robustly handle arbitrarily scene animation, while its per-ray multiresolution geometry provides continuous level of detail driven by ray and path differentials. Razor also decouples shading from visibility computations using a two-phase shading scheme inspired by the REYES system, and caches tessellated representations of freeform surfaces at multiple levels of detail. We present experimental results gathered from a prototype system implemented on eight CPU cores, and discuss which aspects of the system are most successful and which would benefit from further investigation.

A key problem in applications such as soft shadows and defocus blur is to identify points or primitives which are inside a volume of space. For example, the soft shadow computation involves finding surfaces which pass in front of an area light as viewed from a point p in the scene. The desired surfaces are those which are inside a frustum defined by the light and p , and can be found by intersecting the frustum with an acceleration structure over geometry. However, accurately computing this intersection is computationally intensive. In this article, we introduce a homogeneous transform which reduces the computation required to determine the set of points or primitives which are inside a tetrahedral volume. The transform converts tetrahedra into axis-aligned boxes, substantially reducing the cost of intersection with an axis-aligned acceleration structure over points or primitives. We describe the application of this transform to soft shadows and defocus blur, and briefly consider potential uses of the underlying mathematical approach in higher-dimensional problems.

We introduce a straightforward, robust, and efficient algorithm for rendering high-quality soft shadows in dynamic scenes. Each frame, points in the scene visible from the eye are inserted into a spatial acceleration structure. Shadow umbrae are computed by sampling the scene from the light at the image plane coordinates given by the stored points. Penumbrae are computed at the same set of points, per silhouette edge, in two steps. First, the set of points affected by a given edge is estimated from the expected light-view screen-space bounds of the corresponding penumbra. Second, the actual overlap between these points and the penumbra is computed analytically directly from the occluding geometry. The umbral and penumbral sources of occlusion are then combined to determine the degree of shadow at the eye-view pixel corresponding to each sample point. An implementation of this algorithm for the Larrabee architecture yields from 27 to 33 frames per second in simulation for scenes from a modern game, and produces significantly higher image quality than other recent methods in the real-time domain.

Abstract Ray tracing has long been a method of choice for off‐line rendering, but traditionally was too slow for interactive use. With faster hardware and algorithmic improvements this has recently changed, and real‐time ray tracing is finally within reach. However, real‐time capability also opens up new problems that do not exist in an off‐line environment. In particular real‐time ray tracing offers the opportunity to interactively ray trace moving/animated scene content. This presents a challenge to the data structures that have been developed for ray tracing over the past few decades. Spatial data structures crucial for fast ray tracing must be rebuilt or updated as the scene changes, and this can become a bottleneck for the speed of ray tracing. This bottleneck has recently received much attention by researchers and that has resulted in a multitude of different algorithms, data structures and strategies for handling animated scenes. The effectiveness of techniques for ray tracing dynamic scenes vary dramatically depending on details such as scene complexity, model structure, type of motion and the coherency of the rays. Consequently, there is so far no approach that is best in all cases, and determining the best technique for a particular problem can be a challenge. In this State of the Art Report (STAR), we aim to survey the different approaches to ray tracing animated scenes, discussing their strengths and weaknesses, and their relationship to other approaches. The overall goal is to help the reader choose the best approach depending on the situation, and to expose promising areas where there is potential for algorithmic improvements.

Topology has been an important tool for analyzing scalar data and flow fields in visualization. In this work, we analyze the topology of multivariate image and volume data sets with discontinuities in order to create an efficient, raster-based representation we call IStar. Specifically, the topology information is used to create a dual structure that contains nodes and connectivity information for every segmentable region in the original data set. This graph structure, along with a sampled representation of the segmented data set, is embedded into a standard raster image which can then be substantially downsampled and compressed. During rendering, the raster image is upsampled and the dual graph is used to reconstruct the original function. Unlike traditional raster approaches, our representation can preserve sharp discontinuities at any level of magnification, much like scalable vector graphics. However, because our representation is raster-based, it is well suited to the real-time rendering pipeline. We demonstrate this by reconstructing our data sets on graphics hardware at real-time rates.