Image warping refers to the 2-D resampling of a source image onto a target image. In the general case, this requires costly 2-D filtering operations. Simplifications are possible when the warp can be expressed as a cascade of orthogonal 1-D transformations. In these cases, separable transformations have been introduced to realize large performance gains. The central ideas in this area were formulated in the 2-pass algorithm by Catmull and Smith. Although that method applies over an important class of transformations, there are intrinsic problems which limit its usefulness.The goal of this work is to extend the 2-pass approach to handle arbitrary spatial mapping functions. We address the difficulties intrinsic to 2-pass scanline algorithms: bottlenecking, foldovers, and the lack of closed-form inverse solutions. These problems are shown to be resolved in a general, efficient, separable technique, with graceful degradation for transformations of increasing complexity.