We propose a method and present applications of this method that converts a diffraction pattern into an elemental image set in order to display them on an integral imaging based display setup. We generate elemental images based on diffraction calculations as an alternative to commonly used ray tracing methods. Ray tracing methods do not accommodate the interference and diffraction phenomena. Our proposed method enables us to obtain elemental images from a holographic recording of a 3D object/scene. The diffraction pattern can be either numerically generated data or digitally acquired optical data. The method shows the connection between a hologram (diffraction pattern) and an elemental image set of the same 3D object. We showed three examples, one of which is the digitally captured optical diffraction tomography data of an epithelium cell. We obtained optical reconstructions with our integral imaging display setup where we used a digital lenslet array. We also obtained numerical reconstructions, again by using the diffraction calculations, for comparison. The digital and optical reconstruction results are in good agreement.