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ii Preface The history of science is, to a considerable degree, the history of invention, development and perfecting of imaging methods and devices. Modern science began with the invention and application of optical telescope and microscope in the beginning of 17-th century by Galileo Galilei (1564-1642), Anthony Leeuwenhoek (1632-1723) and Robert Hook (1635-1703). The next decisive stage was invention of photography in the first half of the 19-th century. Photographic plates were the major means in discoveries of X-rays by Wilhelm Conrad Roentgen (1845-1923, The Nobel Prize Laureate, 1901) and radioactivity by Antoine Henri Becquerel (1852-1908, The Nobel Prize Laureate, 1903) at the end of 19-th century. These discoveries, in their turn, almost immediately gave birth to new imaging techniques: to X-ray imaging and radiography. X-rays were discovered by W.C. Roentgen in experiments with cathode rays. Cathode rays were discovered in 1859 by Julius Plücker (1801-1868), who used vacuum tubes invented in 1855 by a German inventor Heinrich Geissler (1815-1879). These tubes, as modified by Sir William Crookes (1832-1919) led eventually to the discovery of the electron and finally brought about the development of electronic television and electron microscopy in 30-th - 40-th of 20-th century. Designer of the first electron microscope E. Ruska (1906-1988) was awarded The Nobel Prize in Physics for 1986. This award was shared with G. Binning and H. Rohrer, who were awarded for their invention of the scanning tunneling microscope. The discovery of diffraction of X-rays by Max von Laue (1889-1960, the Nobel Prize Laureate, 1914) in the beginning of 20-th century marked the advent of a new imaging technique, which we now call computational imaging. Although Von Laue’s motivation was not creating a new imaging technique but rather proving the wave nature of X-rays, lauegrams had very soon become the main imaging tool in crystallography. Lauegrams are not conventional images for visual observation. However, using “lauegrams”, one could numerically reconstruct the spatial structure of atoms in crystals. This shoot gave its crop in less than half a century. One of the most remarkable scientific achievements of 20-th century is based on X-ray crystallography. It is discovery by J. Watson and F. Crick of spiral structure of DNA (the Nobel Prize, 1953). And about at the same time a whole bunch of new computational imaging methods had appeared: holography, synthetic aperture radar, coded aperture imaging, tomography. Two of these inventions were awarded the Nobel Prize: D. Gabor for “his invention and developments of the holographic method” (the Nobel Prize in Physics, 1971) and A. M. Cormack and G. N. Hounsfield for “the development of computer assisted tomography” (the Nobel Prize in Physiology and Medicine, 1979). Denis Gabor invented holography in 1948. This is what D. Gabor wrote in his Nobel Lecture about the development of holography: ” Around 1955 holography went into a long hibernation. The revival came suddenly and explosively in 1963, with the publication of first successful laser hologram by Emmett N. Leith and Juris Upatnieks of the University of Michigan, An Arbor. Their success was due not only to the laser, but to the long theoretical preparation of Emmett Leith (in the field of the “side looking radar”) which started in 1955. Another important development in holography iii [happened] in 1962, just before the “holography explosion”. Russian physicist Yu. N. Denisyuk published an important paper in which he combined holography with the ingenious method of photography in natural colors, for which Gabriel Lippman received the Nobel Prize in 1908”. Denis Gabor received Nobel Prize in 1971. The same year, a paper “Digital holography” was published in Proceedings of IEEE by T. Huang. This paper marked the next step in the development of holography, the use of digital computers for reconstructing, generating and simulating wave fields, and reviewed pioneer accomplishments in this field. These accomplishments prompted a burst of research and publications in early and mid 70-th. At that time, most of the main ideas of digital holography were suggested and tested. Numerous potential applications of digital holography such as fabricating computer-generated diffractive optical elements and spatial filters for optical information processing, 3-D holographic displays and holographic television and computer vision stimulated a great enthusiasm among researchers. However, limited speed and memory capacity of computers available at that time, absence of electronic means and media for sensing and recording optical holograms hampered implementation of these potentials. In 1980-th digital holography went into a sort of hibernation similarly to what happened to holography in 1950-th - 1960-th. With an advent, in the end of 1990-th, of the new generation of high speed microprocessors, high resolution electronic optical sensors and liquid crystal displays, of a technology for fabricating micro lens and mirror arrays digital holography is getting a new wind. Digital holography tasks that required hours and days of computer time in 1970-th can now be solved in almost “real” time for tiny fractions of seconds. Optical holograms can now be directly sensed by high-resolution photo electronic sensors and fed into computers in “real” time with no need for any wet photo-chemical processing. Micro lens and mirror arrays promise a breakthrough in the means for recording computer-generated holograms and creating holographic displays. Recent flow of publications in digital holographic metrology and microscopy indicate revival of digital holography from the hibernation§. The development of optical holography, one of the most remarkable inventions of the XX-th century, was driven by clear understanding of information nature of optics and holography. The information nature of optics and holography is especially distinctly seen in digital holography. Wave field recorded in the form of a hologram in optical, radio frequency or acoustic holography, is represented in digital holography by a digital signal that carries the wave field information deprived of its physical casing. With digital holography and with incorporating digital computers into optical information systems, information optics has reached its maturity. This is not a coincidence that digital holography appeared in the end of 60-th, the same period of time, which digital image processing can be dated back to. In the same way, in a certain sense, as inventing by Ch. Towns, G. Basov and A. Prokhorov lasers in mid 1950-th (the Nobel Prize in Physics, 1964) stimulated development of holography, two events stimulated digital holography and digital image processing: beginning of industrial production of computers in 1960-th and introducing Fast Fourier Transform algorithm made by J. W. Cooley and J. M. Tukey in 1965. iv In this book, we will adhere to meaning of digital holography as of a branch of the imaging science, which deals with numerical reconstruction of digitally recorded holograms and with computer synthesis of holograms and diffractive optical elements. We start, in Ch. 1, with basic principles of physical holography and its mathematical models and introduce diffraction integrals that are used to describe wave propagation from objects to holograms. The fundamental issue of digital holography is discrete representation of optical signals and diffraction integrals. This problem, which has a direct relation to both numerical reconstruction of holograms and to synthesis of computer generated holograms, is addressed in Ch. 2. Then we proceed to methods of digital holography proper. Methods and algorithms for digital recording and numerical reconstruction of holograms, their applicability and appropriate metrological characterization are presented in Ch. 3. Chs. 4 through 6 are devoted to principles of computer-generated holography and mathematical models, to methods of encoding numerical holograms for recording them on spatial light modulators as hologram recording devices and to the analysis of how the results of optical reconstruction of computer-generated holograms depend on the encoding method and on physical parameters of hologram recording devices. Chs. 7 and 8 review applications of computer-generated holograms in optical information processing and for information display and 3D visual communications. In the latter, a especial emphasis is made on using limitations of human 3D vision for reducing the computational complexity of the hologram synthesis and for easing requirements to hologram recording and reconstruction devices. In its methods and applications, digital holography is closely connected with digital image processing. Digital image processing is nowadays a well-established field of information technology covered in many books and educational courses. However some of its aspects that have direct relation to digital holography and its applications deserve discussion and reviewing in the context of the book on digital holography. Therefore, in Ch. 9 we review image processing methods in digital holography: mathematical models of imaging and holographic systems; statistical models of stochastic transformations of optical signals; measuring parameters of random interferences in sensors and imaging and holographic systems; principles of Mean Square Error (MSE)-optimal scalar Wiener filtering for image denoising and deblurring; methods for correcting image gray scale nonlinear distortions; methods for accurate image resampling.