7.2 Commuting Operator of Offset Linear Canonical Transform
7.3.3 Discrete OLCT
The eigenfunctions of the OLCT can form an orthonormal set. Therefore, the eigenfunc-tions can be used to implement the OLCT. For example, when|a + d| < 2, express the input signal x(t) in terms of hβ,γM,m(t) as follows:
x(t) =
∑∞
m=0
amhβ,γM,m(t), where am =
∫∞
−∞
x(t)hβ,γM,m(t)dt. (7.30)
Since hβ,γM,m(t) are the eigenfunctions of the OLCT with eigenvalues λm given in (7.27) when|a + d| < 2, the OLCT of x(t) can be obtained from
XMβ,γ(u) =
∑∞
m=0
amLβ,γM
{
hβ,γM,m(t)}=
∑∞
m=0
λmamhβ,γM,m(u). (7.31)
However, for the discrete case, we cannot digitally implement the OLCT by sampling (7.30) and (7.31) with some sampling period ∆ directly. The sampled eigenfunctions, hβ,γM,m(n∆), are no longer orthogonal to each other. Therefore, we need to find an orthog-onal set of discrete functions that can approximate hβ,γM,m(n∆). A simple solution is to develop the discrete version ofCβ,γM, denoted by Cβ,γM, and then find an orthonormal set of discrete eigenfunctions (i.e. eigenvectors) from Cβ,γM.
To obtain Cβ,γM , a method similar to that in [100] is used. Assume the sampling period is ∆ =
√
2π/N , and the discrete input is given by x[n] = x ((n− (N − 1)/2)∆) where 0≤ n ≤ N − 1. In (7.20), Cβ,γM is composed of two kinds of operators, t− β and Dt− jγ.
It is straightforward to define the discrete version of t− β as an N × N diagonal matrix T− βI, where I is the identity matrix and
[T]kn=
(
n−N2−1) √2πN, n = k
0, n ̸= k , (7.32)
where 0 ≤ n, k ≤ N − 1. Let F denote the Fourier transform. It is well known that FDt= jtF. Therefore, Dt− jγ = jF−1tF − jγ and its discrete version can be designed
(a)
1st, 6th and 11th eigenvectors of Cβ,γM
−10 0 10
Figure 7.2: Time-frequency distributions: (a) 1st, (b) 6th and (c) 11th eigenvectors of Cβ,γM used to approximate the sampled OLCT eigenfunctions hβ,γM,m(n∆) with m = 0, 5, 10, where M = [0.53, 0.63;−0.67, 1.09], β = −2, γ = 3, N = 127 and ∆ =√2π/N .
as jFHTF− jγI, where F is an N × N centered DFT matrix:
[F]kn= e−j2πN(k−N2−1)(n−N−12 ), 0 ≤ n, k ≤ N − 1. (7.33)
Accordingly, the discrete version ofCβ,γM is given by
Cβ,γM = b(jFHTF− jγI)2 + ja− d 2
[
(T− βI)(jFHTF− jγI)
+(jFHTF− jγI)(T− βI)]+ c(T− βI)2. (7.34)
Performing eigendecomposition on Cβ,γM, the N orthonormal eigenvectors, denoted by hβ,γM,m[n], can approximate hβ,γM,m(n∆) with some constant difference. For example, assume M = [0.53, 0.63;−0.67, 1.09], β = −2, γ = 3 and N = 127. The sampled OLCT eigenfunctions hβ,γM,m(n∆) for m = 0, 5, 10 are approximated by the 1st, 6th and 11th eigenvectors of Cβ,γM, the time-frequency distributions of which are shown in Fig. 7.2(a), (b) and (c).
Therefore, a discrete OLCT can be developed as follows
XMβ,γ[k] =
This discrete OLCT features perfect reconstruction because hβ,γM,m[n]’s are orthonormal.
In the following, a simulation is given, using the same parameters as in Fig. 7.2. The
−10 −5 0 5 10
Figure 7.3: (a) Discrete input x[n], (b) time-frequency distribution of x[n], (c) output XMβ,γ[k] of the discrete OLCT and (d) time-frequency distribution of XMβ,γ[k], where M = [0.53, 0.63;−0.67, 1.09], β = −2, γ = 3 and N = 127.
shown in Fig. 7.3(a). The time-frequency distribution of x[n] is depicted in Fig. 7.3(b).
The output XMβ,γ[k] of the discrete OLCT is shown in Fig. 7.3(c), and its time-frequency distribution is depicted in Fig. 7.3(d).
7.4 Conclusion
In this chapter, a different definition of OLCT is proposed. The proposed OLCT has a more concise form of inverse transform than the conventional definition of OLCT. A lin-ear operator that commutes with the proposed OLCT is derived. The eigenfunctions and eigenvalues of the commuting operator are analyzed. We prove that the commuting op-erator and the proposed OLCT share common eigenfunctions with different eigenvalues.
We also derive the discrete version of the commuting operator, the eigenvectors of which can be used to develop a discrete OLCT. The proposed discrete OLCT has an important property: perfect reversibility.
Chapter 8
Discrete Gyrator Transforms:
Computational Algorithms and Applications
As an extension of the 2D fractional Fourier transform (FRFT) and a special case of the 2D linear canonical transform (LCT), the gyrator transform was introduced to produce rotations in twisted space/spatial-frequency planes. It is a useful tool in optics, signal processing and image processing. In this chapter, we develop discrete gyrator transforms (DGTs) based on the 2D LCT. Taking the advantage of the additivity property of the 2D LCT, we propose three kinds of DGTs, each of which is a cascade of low-complexity operators. These DGTs have different constraints, characteristics and properties, and are realized by different computational algorithms. Besides, we propose a kind of DGT based on the eigenfunctions of the gyrator transform. This DGT is an orthonormal transform, and thus its comprehensive properties, especially the additivity property, make it more useful in many applications. We also develop an efficient computational algorithm to significantly reduce the complexity of this DGT. At the end, a brief review of some im-portant applications of the DGTs is presented, including mode conversion, sampling and reconstruction, watermarking and image encryption.
8.1 Introduction
Fractional Fourier transform (FRFT) [60–62, 137–140], as a generalization of the Fourier transform, is very useful in many applications such as optical system analysis, phase re-trieval, filter design and pattern recognition. The FRFT is a linear canonical integral trans-form that produces a rotation in the time/frequency plane (x, ωx). To extend the FRFT to two dimensions (x, y), an easy and straightforward approach is performing two sepa-rate 1D FRFTs on two transverse directions, x and y, respectively [141]. Accordingly, this 2D separable FRFT generates rotations in the space/spatial-frequency planes, (x, ωx) and (y, ωy). In [142], another kind of 2D linear canonical integral transform, called gy-rator transform, was proposed to produce rotations in the twisted space/spatial-frequency planes, i.e. (x, ωy) and (y, ωx) planes. Given a 2D signal g(x, y), the gyrator transform with rotation angle α is
G(u, v) = GTα{g(x, y)} = |csc α|
2π
∫∞
−∞
∫∞
−∞
exp
[j (uv + xy)
tan α − j(uy + vx) sin α
]
g(x, y)dxdy.
(8.1)
It is obvious that the above definition is singular at α = kπ. When α = 2kπ, the gyrator transform is defined as G(u, v) = g(u, v); and when α = (2k+1)π, G(u, v) = g(−u, −v).
If G1(ωx, ωy) is defined as the 2D Fourier transform of g(x, y), the gyrator transform with α = π/2 reduces to the reflection of G1(ωx, ωy), i.e. G(u, v) = G1(v, u). The gyrator transform cannot be separated into two 1D transforms, and thus it is sometimes classified as a kind of 2D nonseparable FRFT.
In [142], the optical implementation of the gyrator transform has been discussed. And several properties of the gyrator transform have been derived in [142, 143]. The focus of this chapter is on the digital implementations of the gyrator transform, called discrete gyrator transforms (DGTs) for short. Suppose the sampling intervals in space domain and spatial-frequency domain are (∆x, ∆y) and (∆u, ∆v), respectively:
g[m, n]= g (m∆∆ x, n∆y) , G[p, q]= G (p∆∆ u, q∆v) . (8.2)
The simplest way to derive the DGT is sampling the continuous gyrator transform and computing it directly by summation:
G[p, q] = DGTα{g[m, n]}
= |csc α|
2π
∑
m
∑
n
exp
[j (pq∆u∆v+ mn∆x∆y)
tan α −j (pn∆u∆y + qm∆v∆x) sin α
]
g[m, n]∆x∆y. (8.3)
The advantage of this discrete transform is that there are no constraints on (∆x, ∆y) and (∆u, ∆v), but it has very high computational complexity and is thus time-consuming.
In [143,144], some low-complexity DGTs implemented by discrete Fourier transform (DFT) or convolution were proposed. These DGTs are derived directly from (8.1) and (8.3). In this chapter, we develop DGTs from the point of view of 2D linear canonical transform (LCT). The gyrator transform is a special case of the 2D LCT. Using the addi-tivity property of the 2D LCT, the gyrator transform can be factorized into a sequence of low-complexity transforms. With a different decomposition method, a different DGT can be developed. In this chapter, three kinds of DGTs are proposed based on the 2D LCT. The first one is realized by 2D linear convolution, the second uses the 2D DFT, and the last one is implemented by 2D circular convolution. Since different computational algorithms are utilized, they have different constraints on the sampling intervals, different characteristics and properties, and different computational complexity. The DGTs in [143, 144] are the special cases of the proposed DGTs. The first two proposed DGTs are singular at α = kπ, while the third one is singular at α = (2k + 1)π. When α is close to kπ or (2k + 1)π, these DGTs suffer from low-accuracy and overlapping (aliasing) problems. Accordingly, a method is proposed to help the DGTs avoid these problems.
The DGTs mentioned above have unitary and reversibility properties. However, they don’t satisfy the additivity property, which is useful in many signal/image processing ap-plications. Accordingly, we develop the 4th kind of DGT, which is based on the eigenfunc-tions of the gyrator transform. It has been shown in [143] that rotated Hermite Gaussian functions (RHGFs) are the eigenfunctions of the continuous gyrator transform. For the
dis-crete case, we generate disdis-crete orthonormal RHGFs from 1D disdis-crete Hermite Gaussian functions (HGFs) given by [145]. The DGT based on the discrete HGFs is an orthonor-mal transform, and therefore it satisfies many properties including unitary, reversibility and additivity. To reduce the complexity of this DGT, we also develop an efficient com-putational algorithm. In the end of this chapter, to emphasize the importance of the pro-posed DGTs, some applications are introduced, including mode conversion, sampling and reconstruction, watermarking and image encryption.