A computer-implemented method of transforming an image file by an image file transformation apparatus, the method including 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.