Abstract: Whilst the Deep Shadow Maps algorithm for accelerating the rendering of volumetric shadows is a fitting solution for offline applications, it can consume an unbounded amount of memory and is difficult to map well to current graphics hardware. For these reasons alternative methods have been proposed for interactive applications based on the idea of accumulating opacity from volume primitives in depth-based buckets or slices. However, these slice-based methods can introduce objectionable and unstable slice-like artefacts due to undersampling of the depth range, and often many slices are required to overcome these artefacts. Further refinements have been proposed to address this, but they constrain the generality of the algorithm, for example by mandating an assumption of uniform opacity or by dropping support for pre-filtering. We introduce a novel algorithm called Fourier Opacity Mapping (FOM) and we show that it is a good choice for rendering artefact-free pre-filtered volumetric shadows in cases where spatial opacity variations are smooth (e.g. smoke, gas and low-opacity hair). We also show how the algorithm can be generalised to other orthonormal bases.