1. A computer-implemented method of transforming an image file by an image file transformation apparatus, the method comprising: providing an image file in a pixel-based format having a plurality of pixels; dividing the pixels into a plurality of patches; sampling the pixels to generate boundary conditions relating to each of the patches; deriving Fourier coefficients of a solution to a partial differential equation according to the boundary conditions; and outputting the Fourier coefficients for each of the patches as a transformed image file.