A first-order analysis of lighting, shading, and shadows

The shading in a scene depends on a combination of many factors---how the lighting varies spatially across a surface, how it varies along different directions, the geometric curvature and reflectance properties of objects, and the locations of soft shadows. In this article, we conduct a complete first-order or gradient analysis of lighting, shading, and shadows, showing how each factor separately contributes to scene appearance, and when it is important. Gradients are well-suited to analyzing the intricate combination of appearance effects, since each gradient term corresponds directly to variation in a specific factor. First, we show how the spatial and directional gradients of the light field change as light interacts with curved objects. This extends the recent frequency analysis of Durand et al. [2005] to gradients, and has many advantages for operations, like bump mapping, that are difficult to analyze in the Fourier domain. Second, we consider the individual terms responsible for shading gradients, such as lighting variation, convolution with the surface BRDF, and the object's curvature. This analysis indicates the relative importance of various terms, and shows precisely how they combine in shading. Third, we understand the effects of soft shadows, computing accurate visibility gradients, and generalizing previous work to arbitrary curved occluders. As one practical application, our visibility gradients can be directly used with conventional ray-tracing methods in practical gradient interpolation methods for efficient rendering. Moreover, our theoretical framework can be used to adaptively sample images in high-gradient regions for efficient rendering.