Fast and accurate computation of two-dimensional non-separable quadratic-phase integrals

Abstract

We report a fast and accurate algorithm for numerical computation of two-dimensional non-separable linear canonical transforms (2D-NS-LCTs). Also known as quadratic-phase integrals, this class of integral transforms represents a broad class of optical systems including Fresnel propagation in free space, propagation in gradedindex media, passage through thin lenses, and arbitrary concatenations of any number of these, including anamorphic/astigmatic/non-orthogonal cases. The general two-dimensional non-separable case poses several challenges which do not exist in the one-dimensional case and the separable two-dimensional case. The algorithm takes ϳÑ log Ñ time, where Ñ is the two-dimensional space-bandwidth product of the signal. Our method properly tracks and controls the space-bandwidth products in two dimensions, in order to achieve information theoretically sufficient, but not wastefully redundant, sampling required for the reconstruction of the underlying continuous functions at any stage of the algorithm. Additionally, we provide an alternative definition of general 2D-NS-LCTs that shows its kernel explicitly in terms of its ten parameters, and relate these parameters bidirectionally to conventional ABCD matrix parameters.

2D non-separable linear canonical transform (2D-NS-LCT) based cryptography

“…The 2D-NS-LCT can represent a wide variety of non-orthogonal, non-axially symmetric and anamorphic systems [21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40] . Among its special cases are the FT, FRT, and FST, gyrator transform, chirp transform, homogeous coordinate/affine transform (including e.g.…”

“…rotation transform, and shearing (interferometer) transform). The continuous 2D-NS-LCT of a signal is defined as 22 , ,…”

“…The number of independent parameters in the matrix M is ten 22 . One of the most important properties of the continuous 2D-NS-LCT is the unitary property, i.e., they are always invertible as follows 40 , .…”

“…It however is widely accepted that the security of any encryption system depends on the keys used and therefore the larger the key-space the more the security will be 5 . In addition to these approaches, in this paper, for the first time, a novel 2D-NS-LCT [21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39] based optical DRPE system is proposed. This proposed system offers encrypting information in multiple (10) degrees of freedom (see Table 1).…”

Constraints to solve parallelogram grid problems in 2D non separable linear canonical transform

“…The continuous two dimensional non-separable linear canonical transform (2D-NS-LCT) of an input signal can be given by [1,2], where M is the transformation matrix of the LCT system, where are 2 2 sub-matrices, e.g. .…”