A graphics processing platform includes a rasteriser 50 that receives primitives representing an image to be displayed for processing. The rasteriser 50 determines which sets of sampling points of the image include sampling points that are covered by a given primitive, and then generates a fragment for rendering for each set of sampling points found to include a sampling point that is covered by the primitive and passes those fragments to a renderer 51 for rendering. The renderer 51 carries out rendering operations on the fragments that it receives, and stores the rendered fragment data in tile buffers 52. The rendered fragment data is stored in multiple copies in the appropriate sample positions in the tile buffers 52, so as to provide a separate set of fragment data for each individual sample position taken of the image. The data from the tile buffers 52 is input to a downsampling unit 53, and thence output to a frame buffer 54 of a display device 55 for display.