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In this paper we introduce and study a two-parameter family of integral operators on the Fock space F 2 (C) . We determine exactly when these operators are bounded and when they are unitary. We show that, under the Bargmann transform, these operators include the classical linear canonical transforms as special cases. As an application, we obtain a new unitary projective representation for the special linear group S L (2 , R) on the Fock space. [ABSTRACT FROM AUTHOR]

We derive Heisenberg uncertainty principles for pairs of Linear Canonical Transforms of a given function, by resorting to the fact that these transforms are just metaplectic operators associated with free symplectic matrices. The results obtained synthesize and generalize previous results found in the literature, because they apply to all signals, in arbitrary dimension and any metaplectic operator (which includes Linear Canonical Transforms as particular cases). Moreover, we also obtain a generalization of the Robertson-Schrödinger uncertainty principle for Linear Canonical Transforms. We also propose a new quadratic phase-space distribution, which represents a signal along two intermediate directions in the time-frequency plane. The marginal distributions are always non-negative and permit a simple interpretation in terms of the Radon transform. We also give a geometric interpretation of this quadratic phase-space representation as a Wigner distribution obtained upon Weyl quantization on a non-standard symplectic vector space. Finally, we derive the multidimensional version of the Hardy uncertainty principle for metaplectic operators and the Paley-Wiener theorem for Linear Canonical Transforms. [ABSTRACT FROM AUTHOR]

Sparse recovery aims to reconstruct signals that are sparse in a linear transform domain from a heavily underdetermined set of measurements. The success of sparse recovery relies critically on the knowledge of transform domains that give compressible representations of the signal of interest. Here we consider two- and three-dimensional images, and investigate various multi-dimensional transforms in terms of the compressibility of the resultant coefficients. Specifically, we compare the fractional Fourier (FRT) and linear canonical transforms (LCT), which are generalized versions of the Fourier transform (FT), as well as Hartley and simplified fractional Hartley transforms, which differ from corresponding Fourier transforms in that they produce real outputs for real inputs. We also examine a cascade approach to improve transform-domain sparsity, where the Haar wavelet transform is applied following an initial Hartley transform. To compare the various methods, images are recovered from a subset of coefficients in the respective transform domains. The number of coefficients that are retained in the subset are varied systematically to examine the level of signal sparsity in each transform domain. Recovery performance is assessed via the structural similarity index (SSIM) and mean squared error (MSE) in reference to original images. Our analyses show that FRT and LCT transform yield the most sparse representations among the tested transforms as dictated by the improved quality of the recovered images. Furthermore, the cascade approach improves transform-domain sparsity among techniques applied on small image patches. [ABSTRACT FROM AUTHOR]

We propose optical image encryption and authentication in the linear canonical transform (LCT) method by using a sparse representation technique. The linearity in the LCT based double random phase encryption (DRPE) system caused the vulnerability of the system. Not only the linearity in the LCT -DRPE system causes the vulnerability of the system but also the LCT order parameter may cause this weakness. Therefore, the encrypted data of double random phase encoding based on LCT is integrated with sparse representation to overcome this weakness. Unlike the traditional DRPE method, only sparse data from the encrypted data is kept for decryption. The numerical simulation proved that the correct authentication results are satisfied despite the use of partially encrypted data. The randomly selective sparse encrypted data reinforces the security of amplitude encoding of DRPE-LCT. Thus, the proposed method added an additional layer to optical security systems. The attacks analysis is implemented in order to show the robustness of the proposed security. In addition, this method has fulfilled the transmission and storage requirements owing to the high reduction of information data. • We propose optical image encryption and authentication method based on LCT. • The sparse strategy was integrated with encryption process to overcome weakness in this method. • The proposed method be able to resist attacks in spite of the use of partially encrypted data. • The method was fulfilled the transmission and storage requirements for reduction of information data. [ABSTRACT FROM AUTHOR]

We describe the structural dynamics of two groups of scientists in relation to the independent simultaneous discovery (i.e., definition and application) of linear canonical transforms. This mathematical construct was built as the transfer kernel of paraxial optical systems by Prof. Stuart A. Collins, working in the ElectroScience Laboratory in Ohio State University. At roughly the same time, it was established as the integral kernel that represents the preservation of uncertainty in quantum mechanics by Prof. Marcos Moshinsky and his postdoctoral associate, Dr. Christiane Quesne, at the Instituto de Física of the Universidad Nacional Autónoma de México. We are interested in the birth and parallel development of the two follower groups that have formed around the two seminal articles, which for more than two decades did not know and acknowledge each other. Each group had different motivations, purposes and applications, and worked in distinct professional environments. As we will show, Moshinsky-Quesne had been highly cited by his associates and students in Mexico and Europe when the importance of his work started to permeate various other mostly theoretical fields; Collins' paper took more time to be referenced, but later originated a vast following notably among Chinese applied optical scientists. Through social network analysis we visualize the structure and development of these two major coauthoring groups, whose community dynamics shows two distinct patterns of communication that illustrate the disparity in the diffusion of theoretical and technological research. [ABSTRACT FROM AUTHOR]

Abstract: In [E. Margolis, Y.C. Eldar, IEEE Trans. Signal Process. 56 (2008) 2728–2745], the authors derived a practical reconstruction formula for bandlimited periodic signals. It was shown that every bandlimited periodic signal can be perfectly reconstructed from finitely many nonuniformly spaced samples taken over a period. In this paper, we generalize this result in a broader sense. We obtain a similar reconstruction formula for a large class of signals, whose linear canonical transforms are bandlimited periodic functions. [Copyright &y& Elsevier]

Abstract: We propose a new method for multiple image encryption using linear canonical transforms and chaotic maps. Three linear canonical transforms and three chaotic maps are used in the proposed technique. The three linear canonical transforms that have been used are the fractional Fourier transform, the extended fractional Fourier transform and the Fresnel transform. The three chaotic maps that have been used are the tent map, the Kaplan–Yorke map and the Ikeda map. These chaotic maps are used to generate the random phase masks and these random phase masks are known as chaotic random phase masks. The mean square error and the signal to noise ratio have been calculated. Robustness of the proposed technique to blind decryption has been evaluated. Optical implementation of the technique has been proposed. Experimental and simulations results are presented to verify the validity of the proposed technique. [Copyright &y& Elsevier]