Fractional Transforms in Optical Information Processing
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Fractional Transforms in Optical Information Processing Tatiana Alieva Facultad de Ciencias F´ısicas, Universidad Complutense de Madrid, Ciudad Universitaria, 28040 Madrid, Spain Email: [email protected]
Martin J. Bastiaans Faculteit Elektrotechniek, Technische Universiteit Eindhoven, Postbus 513, 5600 MB Eindhoven, The Netherlands Email: [email protected]
Maria Luisa Calvo Facultad de Ciencias F´ısicas, Universidad Complutense de Madrid, Ciudad Universitaria, 28040 Madrid, Spain Email: [email protected] Received 31 March 2004 We review the progress achieved in optical information processing during the last decade by applying fractional linear integral transforms. The fractional Fourier transform and its applications for phase retrieval, beam characterization, space-variant pattern recognition, adaptive filter design, encryption, watermarking, and so forth is discussed in detail. A general algorithm for the fractionalization of linear cyclic integral transforms is introduced and it is shown that they can be fractionalized in an infinite number of ways. Basic properties of fractional cyclic transforms are considered. The implementation of some fractional transforms in optics, such as fractional Hankel, sine, cosine, Hartley, and Hilbert transforms, is discussed. New horizons of the application of fractional transforms for optical information processing are underlined. Keywords and phrases: fractional Fourier transform, fractional convolution, fractional cyclic transforms, fractional optics.
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INTRODUCTION
During the last decades, optics is playing an increasingly important role in computing technology: data storage (CDROM) and data communication (optical fibres). In the area of information processing, optics also has certain advantages with respect to electronic computing, thanks to its massive parallelism, operating with continuous data, and so forth [1, 2, 3]. Moreover, the modern trend from binary logic to fuzzy logic, which is now used in several areas of science and technology such as control and security systems, robotic vision, industrial inspection, and so forth, opens up new perspectives for optical information processing. Indeed, typical optical phenomena such as diffraction and interference inherit fuzziness and therefore permit an optical implementation of fuzzy logic [4]. The first and highly successful configuration for optical data processing—the optical correlator—was introduced by Van der Lugt more than 30 years ago [5]. It is based on This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
the ability of a thin lens to produce the two-dimensional Fourier transform (FT) of an image in its back focal plane. This invention led to further creation of a great variety of optical and optoelectronic processors such as joint correlators, adaptive filters, optical differentiators, and so forth [6]. More sophisticated tools such as wavelet trans
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