Discrete transforms, fast algorithms, and point spread functions of numerical reconstruction of digitally recorded holograms

In digital reconstruction of holograms, holograms are sampled first by CCD or CMOS photosensitive cameras and the array of numbers obtained is then subjected in computer to digital transformations that imitate wave back propagation from the camera to the object plane. As a result, samples of the object wave front amplitude are obtained. Accuracy and computational complexity of numerical implementation of the wave propagation transforms are of primary importance for digital holography. The chapter addresses these problems. Specifically, (i) different versions are introduced of discrete representations of the integral Fourier, Fresnel and Kirchhoff-Reileigh-Zommerfeld transforms that correspond to different geometries of hologram and reconstructed image sampling, including canonical discrete Fourier transform (DFT), scaled shifted DFT, rotated DFT, affine DFT, canonical discrete Fresnel transform (DFrT), scaled shifted DFrT, partial DFrT, convolutional DFrT; (ii) fast computational algorithms for these transforms are outlined, and (iii) point spread functions of different hologram reconstruction algorithms are derived that show how reconstruction results depend on the holographic setup and photographic camera physical parameters such as objectto-camera distance, radiation wavelength, camera size, pitch, fill factor, and the like.