Ravi Ramamoorthi  Columbia University 
Dhruv Mahajan  
Peter Belhumeur 
Abstract
The shading in a scene depends on a combination of many factorshow
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
paper, 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 for 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. 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. As one practical application,
our theoretical framework can be used to adaptively sample images in
highgradient regions for efficient rendering. Third, we understand
the effects of soft shadows, computing accurate visibility gradients.
We generalize previous work to arbitrary curved occluders, and develop
a local framework that is easy to integrate with conventional
raytracing methods. Our visibility gradients can be directly used in
practical gradient interpolation methods for efficient rendering.
BackgroundThis paper starts a new direction in reflection analysis. So far, mathematical analysis has focused on frequency domain and spherical harmonic techniques. Waveletbased approaches have also gained prominence, but more for practical computation rather than analytic formulae. While there has been some previous work on perturbationbased methods, those approaches have not so far considered a complete light field analysis. We believe this paper unifies and extends much of the previous theory in rendering, developing a complete first order or gradient analysis of reflection. ResultsLight reflection from Curved SurfacesWe are able to study the full process of light reflection from curved surfaces, showing how the spatioangular variation in the light field transforms as a result of the basic shading steps. We are also able to extend the analysis to normal or bump maps. Analysis of First Order TermsWe combine the analysis of the different gradient terms in a unified formula which controls the relative importance of spatial variation, angular variation and surface curvature. We analyze the effects of these terms in a variety of situations, and also consider an extension to secondorder Hessians. A practical application is efficient image sampling based on gradient magnitude. Analysis of Visibility GradientsWe derive new analytic expressions for soft shadow gradients, that for the first time consider the effects of general curved or polygonal blockers. Our formulation is completely local, based only on the angular visibility discontinuities at a single spatial location. It can be used directly for gradientbased interpolation. PaperPaper can be downloaded. This paper will appear in the January 2007 issue of the ACM Transactions on Graphics 

 
 
