This paper develops a family of multi-pass image resampling algorithms that use one-dimensional filtering stages to achieve high-quality results at low computational cost. Our key insight is to perform a frequency-domain analysis to ensure that very little aliasing occurs at each stage in the multi-pass transform and to insert additional stages where necessary to ensure this. Using one-dimensional resampling enables the use of small resampling kernels, thus producing highly efficient algorithms. We compare our results with other state of the art software and hardware resampling algorithms.