Image warping refers to the 2D resampling of a source image onto a target image. Despite the variety of techniques proposed, a large class of image warping problems remains inadequately solved: mapping between two images which are delimited by arbitrary, closed, planar curves, e.g., handdrawn curves. This paper describes a novel algorithm to perform image warping among arbitrary planar shapes whose boundary correspondences are known. A generalized polar coordinate parameterization is introduced to facilitate an efficient mapping procedure. Images are treated as collections of interior layers, extracted via a thinning process. Mapping these layers between the source and target images generates the 2D resampling grid that defines the warping. The thinning operation extends the standard polar coordinate representation to deal with arbitrary shapes.