Real-time approximate sorting for self shadowing and transparency in hair rendering

When rendering materials represented by high frequency geometry such as hair, smoke or clouds, standard shadow mapping or shadow volume algorithms fail to produce good self shadowing results due to aliasing. Moreover, in all of the aforementioned examples, properly approximating self shadowing is crucial to getting realistic results. To cope with this problem, opacity shadow maps have been used. I.e., an opacity function is rendered into a set of slices parallel to the light-plane. The original Opacity Shadow Map technique [Kim and Neumann 2001] requires the geometry to be rendered once for each slice, making it impossible to render complex geometry into a large set of slices in real time. In this paper we present a method for sorting n line primitives into s number of sub-sets, where the primitives of one set occupy a single slice, in O(nlog(s)), making it possible to render hair into opacity maps in linear time. It is also shown how the same method can be used to roughly sort the geometry in back-to-front order for alpha blending, to allow for transparency. Finally, we present a way of rendering self shadowed geometry using a single 2D opacity map, thereby reducing the memory usage significantly.