In this chapter, we outline physical principles of holography and introduce mathematical models of recording and reconstruction of hologram and diffraction integrals used in these models.

This chapter is devoted to discrete representation of diffraction integrals. It expounds principles of the discretization and their applications to derivation of different modifications of Discrete Fourier Transforms, Discrete Fresnel Transforms and Kirchhoff-Rayleigh-Sommerfeld transforms. The chapter is supplemented with appendices that contain relevant mathematical details, description of the new algorithm for signal convolution via fast cosine transform and a summary of discrete diffraction transforms.

Digital recording and numerical reconstruction of holograms is one of the main tasks of digital holography with numerous applications. This chapter exposes off-axis and on-axis methods of hologram digital recording, introduces the notion of the point-spread function of numerical reconstruction of hologram, which is the main metrological characteristics of the process, presents results of evaluation of point-spread functions of numerical reconstruction of Fourier and Fresnel holograms and discusses zones of applicability of different reconstruction algorithms

This chapter introduces basic principles and mathematical models of computer-generated holography and reviews spatial light modulators for recording of computer-generated holograms.

The purpose of hologram encoding is converting complex valued samples of mathematical holograms obtained in the result of synthesis of computer-generated holograms into data that can be used to directly control physical parameters of recording media. This chapter overviews different hologram encoding methods oriented on different recording media that can be used for this purpose.

Computer-generated holograms are used as optical elements in optical setups. This chapter analyses how methods of generating, encoding and recording computer-generated hologram affect the results of their optical reconstruction.

A remarkable property of lenses and parabolic mirrors is their ability of performing, in parallel and at the speed of light, Fourier transform of input wave fronts and to act as “chirp”- spatial light modulator. This basic property enables creating optical information processing systems for implementation of optical Fourier analysis, image convolution and correlation. In this chapter, basic properties of lenses and parabolic mirrors relevant to optical information processing are briefly explained, elements of the theory of optical correlators for reliable target location in images are an introduced and their different implementations of optical correlators are described.

This chapter is devoted to synthesis and application of computer-generated display holograms for 3D visualization and communication. It introduces a computer generated hologram based 3-D communication paradigm, discusses properties and limitations of human visual system relevant to 3D visualization and describes several specific methods for synthesis of computer-generated hologram capable of reproducing 3D sensation and possible approaches to designing 3D holographic displays

Work in digital holography always involves image processing, such as image denoising, deblurring, resampling, geometrical transformation and alike. In this chapter, basic approaches and methods of image processing most relevant to applications of digital holography are reviewed.

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