Fourier-based approaches to calculate the Fresnel diffraction of light provide one of the most efficient algorithms for holographic computations because this permits the use of the fast Fourier transform (FFT). This research overcomes the limitations on sampling imposed by Fourier-based algorithms by the development of a fast shifted Fresnel transform. This fast shifted Fresnel transform is used to develop a tiling approach to hologram construction and reconstruction, which computes the Fresnel propagation of light between parallel planes having different resolutions.