Steven E. Molnar

Logarithmic perspective shadow maps

We present a novel shadow map parameterization to reduce perspective aliasing artifacts for both point and directional light sources. We derive the aliasing error equations for both types of light sources in general position. Using these equations we compute tight bounds on the aliasing error. From these bounds we derive our shadow map parameterization, which is a simple combination of a perspective projection with a logarithmic transformation. We formulate several types of logarithmic perspective shadow maps (LogPSMs) by replacing the parameterization of existing algorithms with our own. We perform an extensive error analysis for both LogPSMs and existing algorithms. This analysis is a major contribution of this paper and is useful for gaining insight into existing techniques. We show that compared with competing algorithms, LogPSMs can produce significantly less aliasing error. Equivalently, for the same error as competing algorithms, LogPSMs can produce significant savings in both storage and bandwidth. We demonstrate the benefit of LogPSMs for several models of varying complexity.

Practical logarithmic rasterization for low-error shadow maps

Logarithmic shadow maps can deliver the same quality as competing shadow map algorithms with substantially less storage and bandwidth. We show how current GPU architectures can be modified incrementally to support rendering of logarithmic shadow maps at current GPU fill rates. Specifically, we modify the rasterizer to support rendering to a nonuniform grid with the same watertight rasterization properties as current rasterizers. We also describe a depth compression scheme to handle the nonlinear primitives produced by logarithmic rasterization. Our proposed architecture enhancements align with current trends of decreasing cost for on-chip computation relative to off-chip bandwidth and storage. For a modest increase in computation, logarithmic rasterization can greatly reduce shadow map bandwidth and storage costs.

Practical logarithmic shadow maps

Computer games and simulators are among the most demanding real-time computer graphics applications. In these applications, shadows greatly contribute to the visual realism and provide important depth cues. However, computing high-quality shadows at interactive rates in a complex and dynamic environment remains a challenging problem. In this paper, we restrict ourselves to generating hard shadows from point light sources. One of the popular techniques for interactive shadow generation is shadow mapping [Williams 1978]. A major disadvantage of shadow maps is that they are prone to aliasing artifacts if they have inadequate resolution. Aliasing error can be classified as perspective or projection aliasing [Stamminger and Drettakis 2002]. Current practical shadow mapping solutions reduce perspective aliasing by reparameterizing the shadow map to allocate more resolution to the undersampled regions of a scene [Stamminger and Drettakis 2002; Wimmer et al. 2004]. These algorithms use the 4× 4 projective transformation matrices available on current GPUs to obtain a non-uniform parameterization. The optimal shadow mapping parameterization, however, involves logarithmic transformations. Since perspective and logarithmic functions can diverge significantly, approximations with projective transformations can require much higher shadow map resolution to avoid perspective aliasing.