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The Fractional Fourier Transform: with Applications in Optics and Signal Processing

TL;DR: The fractional Fourier transform (FFT) as discussed by the authors has been used in a variety of applications, such as matching filtering, detection, and pattern recognition, as well as signal recovery.
Abstract: Preface. Acknowledgments. Introduction. Signals, Systems, and Transformations. Wigner Distributions and Linear Canonical Transforms. The Fractional Fourier Transform. Time-Order and Space-Order Representations. The Discrete Fractional Fourier Transform. Optical Signals and Systems. Phase-Space Optics. The Fractional Fourier Transform in Optics. Applications of the Fractional Fourier Transform to Filtering, Estimation, and Signal Recovery. Applications of the Fractional Fourier Transform to Matched Filtering, Detection, and Pattern Recognition. Bibliography on the Fractional Fourier Transform. Other Cited Works. Credits. Index.
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Proceedings ArticleDOI
01 Sep 2001
TL;DR: An overview of applications which have so far received interest are given and some potential application areas remaining to be explored are noted.
Abstract: A brief introduction to the fractional Fourier transform and its properties is given. Its relation to phase-space representations (time- or space-frequency representations) and the concept of fractional Fourier domains are discussed. An overview of applications which have so far received interest are given and some potential application areas remaining to be explored are noted.

481 citations


Cites background from "The Fractional Fourier Transform: w..."

  • ...Those interested in learning more are referred to a recent book on the subject [1] or the chapter-length treatment [2]....

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  • ...Such two-dimensional functions 3 5 with variables and 5 are known as rectangular time-order or space-order representations of the function 3 5 , depending on whether the variable 5 is interpreted as time or space (or something else) [1]....

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  • ...A more comprehensive treatment and an extensive list of references may once again be found in [1] and [2]....

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  • ...The fractional Fourier transform has spurred interest in many other fractional transforms; see [1] for further references....

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  • ...It is also possible to write convolution and multiplication properties for the fractional Fourier transform, though these are not of great simplicity [1]....

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Journal ArticleDOI
TL;DR: Experiments show that, for rich-grained digital image, the capability of nonlinearly enhancing complex texture details in smooth area by fractional differential-based approach appears obvious better than by traditional integral-based algorithms.
Abstract: In this paper, we intend to implement a class of fractional differential masks with high-precision. Thanks to two commonly used definitions of fractional differential for what are known as Grumwald-Letnikov and Riemann-Liouville, we propose six fractional differential masks and present the structures and parameters of each mask respectively on the direction of negative x-coordinate, positive x-coordinate, negative y-coordinate, positive y-coordinate, left downward diagonal, left upward diagonal, right downward diagonal, and right upward diagonal. Moreover, by theoretical and experimental analyzing, we demonstrate the second is the best performance fractional differential mask of the proposed six ones. Finally, we discuss further the capability of multiscale fractional differential masks for texture enhancement. Experiments show that, for rich-grained digital image, the capability of nonlinearly enhancing complex texture details in smooth area by fractional differential-based approach appears obvious better than by traditional integral-based algorithms.

403 citations

Journal ArticleDOI
TL;DR: This paper is geared toward signal processing practitioners by emphasizing the practical digital realizations and applications of the FRFT, which is closely related to other mathematical transforms, such as time-frequency and linear canonical transforms.

335 citations


Cites background or methods from "The Fractional Fourier Transform: w..."

  • ...It should be mentioned that the LCT is also called the affine Fourier transform, the generalized Fresnel transform, the Collins formula, the ABCD transform, or the almost Fourier and almost Fresnel transformation [1,45]....

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  • ...The LCT is defined by [1,45,46]...

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  • ...Further examples can be found in [1,7,126]....

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  • ...The above equation can be considered as the spectral decomposition of the kernel function of FRFT [1]....

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  • ...Introductory and review contributions [1,4–15]...

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Journal ArticleDOI
TL;DR: There are hundreds of papers published in this field, some with hundreds of citations as discussed by the authors, and some possible technical applications and connections to other research areas, such as complementarity relations, Bell's inequality violation and entanglement.

318 citations

Journal ArticleDOI
TL;DR: An optical multiple random phase mask encoding system is applied, and one of the phase-only masks is selected and laterally translated along a preset direction during the encryption process.
Abstract: In this Letter, we propose a method for optical image encryption based on diffractive imaging. An optical multiple random phase mask encoding system is applied, and one of the phase-only masks is selected and laterally translated along a preset direction during the encryption process. For image decryption, a phase retrieval algorithm is proposed to extract a high-quality plaintext. The feasibility and effectiveness of the proposed method are demonstrated by numerical results. The proposed method can provide a new strategy instead of conventional interference methods, and it may open up a new research perspective for optical image encryption.

314 citations