Path Jacobians: Theory and Applications

In accordance with Fermat's Variation Principle, a ray path connecting two arbitrary points in a scene via multiple reflectors is given by a non-linear system. If we fix one of the two points and let the other change, the system can be considered as a function relating the reflection points along the path to the varying point. In this paper, we present the concept of path Jacobians for such a ray path, which are the derivatives of the reflection points along the path with respect to its varying endpoint. Path Jacobians provide a first-order approximation to the path perturbation as a result of the changes of the endpoint and set a mathematical foundation for exploiting path coherence in incremental rendering. We also derive the analytic expression for path Jacobians from the Implicit Function Theorem. To illustrate its use, three related applications are described: image warping, stereoscopic ray tracing and caustic contour generation.