Abstract: Extremely dense spatial sampling is often needed to prevent aliasing when rendering objects with high frequency variations in geometry and reflectance. To accelerate the rendering process, we introduce characteristic point maps (CPMs), a hierarchy of view-independent points, which are chosen to preserve the appearance of the original model across different scales. In preprocessing, randomized matrix column sampling is used to reduce an initial dense sampling to a minimum number of characteristic points with associated weights. In rendering, the reflected radiance is computed using a weighted average of reflectances from characteristic points. Unlike existing techniques, our approach requires no restrictions on the original geometry or reflectance functions.