Fast Non-Linear Projections Using Graphics Hardware

Linear perspective projections are used extensively in graphics. They provide a non-distorted view, with simple computations that map easily to hardware. Non-linear projections, such as the view given by a fish-eye lens are also used, either for artistic reasons or in order to provide a larger field of view, e.g. to approximate environment reflections or omnidirectional shadow maps. As the computations related to non-linear projections are more involved, they are harder to implement, especially in hardware, and have found little use so far in practical applications. In this paper, we apply existing methods for non-linear projections [Lloyd et al. 2006; Hou et al. 2006; Fournier 2005] to a specific class: non-linear projections with a single center of projection, radial symmetry and convexity. This class includes, but is not limited to, paraboloid projections, hemispherical projections and fish-eye lenses. We show that, for this class, the projection of a 3D triangle is a single curved triangle, and we give a mathematical analysis of the curved edges of the triangle; this analysis allows us to reduce the computations involved, and to provide a faster implementation. The overhead for non-linearity is bearable and can be balanced with the fact that a single nonlinear projection can replaces as many as five linear projections (in a hemicube), with less discontinuities and a smaller memory cost, thus making non-linear projections a practical alternative.