Two-dimensional nonseparable discrete linear canonical transform based on CM-CC-CM-CC decomposition

05/26/2017
by   Soo-Chang Pei, et al.
0

As a generalization of the two-dimensional Fourier transform (2D FT) and 2D fractional Fourier transform, the 2D nonseparable linear canonical transform (2D NsLCT) is useful in optics, signal and image processing. To reduce the digital implementation complexity of the 2D NsLCT, some previous works decomposed the 2D NsLCT into several low-complexity operations, including 2D FT, 2D chirp multiplication (2D CM) and 2D affine transformations. However, 2D affine transformations will introduce interpolation error. In this paper, we propose a new decomposition called CM-CC-CM-CC decomposition, which decomposes the 2D NsLCT into two 2D CMs and two 2D chirp convolutions (2D CCs). No 2D affine transforms are involved. Simulation results show that the proposed methods have higher accuracy, lower computational complexity and smaller error in the additivity property compared with the previous works. Plus, the proposed methods have perfect reversibility property that one can reconstruct the input signal/image losslessly from the output.

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1 Introduction

Linear canonical transform (LCT), first introduced in [1, 2]

, is a generalization of the fractional Fourier transform (FRFT). It unifies a variety of transforms such as Fourier transform (FT), FRFT and Fresnel transform. It has four parameters with three degrees of freedom, and thus more important and useful in optics

[3, 4, 5]

and many signal processing applications including filter design, radar system analysis, signal synthesis, phase reconstruction, time-frequency analysis, pattern recognition, encryption and modulation

[6, 7, 8, 9, 10, 11]. To extend the 1D LCT to two dimensions , an easy and straightforward approach is performing two independent 1D LCTs in the two transverse directions and , respectively. Since the two-dimensional (2D) kernel can be separated, this 2D transform is called 2D separable LCT (2D SLCT) [12]. The 2D SLCT can produce affine transformations in the and planes, where and are the spatial-frequency coordinates with respect to and . Since the 1D LCT has three degrees of freedom, the 2D SLCT has six degrees of freedom.

A further generalization of the 2D SLCT is the 2D nonseparable LCT (2D NsLCT) [13, 14, 15], named after its nonseparable 2D kernel. The 2D NsLCT provides four more (i.e. ten) degrees of freedom to represent all transformations not only in and planes but also in , , and planes. The 1D/2D FT, FRFT and Fresnel transform, as well as the 1D LCT, 2D SLCT and gyrator transform [16] are all its special cases. All the applications of these special cases in optics, signal processing and digital image processing can be extended and become more flexible by the 2D NsLCT. For example, in [14], the authors show that the noise with nonseparable term cannot be removed clearly by the 2D SLCT filter but can be by the 2D NsLCT filter. In optical system analysis, the 2D NsLCT is more effective when analyzing systems containing quadratic phase components misaligned in both and axes [17, 18].

Consider an affine transformation in the space-spatial-frequency plane:

(1)

where and are the spatial-frequency coordinates with respect to and , respectively. The transformation matrix in (1) is called ABCD matrix and also denoted by in this paper. Assume and . The 2D NsLCT that can result in the space-spatial-frequency transformation in (1) is given by [13, 14, 15]

(2)

where and are the 2D input and output signals, respectively. The ABCD matrix for a valid 2D NsLCT should satisfy

(3)

or equivalently

(4)

Either (3) or (4) leads to six linear equations, i.e. six constraints. Although there are a total of parameters in , , and , the number of degrees of freedom is 10 due to the six constraints. It is obvious that the definition in (1) is valid only when , i.e. is invertible. When , the 2D NsLCT reduces into a 2D affine transformation multiplied by a 2D chirp function:

(5)

When but , the definition of 2D NsLCT is subdivided into several different cases. One can refer to [14] for a detailed details. The digital implementation of the 1D LCT has been widely studied in many papers such as [19, 20, 21, 22, 23, 24, 25, 26]. The digital implementation of the 2D SLCT can be easily realized by performing any of these 1D implementation techniques twice, one in the direction and one in the direction. However, there are less works regarding the digital implementation of the 2D NsLCT [27, 28, 29, 30, 31]. Zhao et al.’s works [29, 30, 31] mainly focus on the sampling of the 2D NsLCT to ensure unitary property. Koç et al.’s work [27] and Ding et al.’s work [28] focus on the development of digital implementation algorithms to improve complexity and accuracy.

To develop the 2D NsDLCT, the simplest method is to discretize the 2D NsLCT in (5) by sampling and summation:

(6)

where and are the input and output sampling points, respectively. This 2D NsDLCT is named direct method in this paper. The direct method is very inefficient. In order to reduce computational complexity, the 2D NsLCT is decomposed into several simpler 2D operations, and it follows that one can develop a low-complexity 2D NsDLCT by connecting several low-complexity 2D discrete operations. Decomposition methods have been widely used in the digital implementation of 1D/2D FRFT, 1D LCT, gyrator transform, etc. In [27], the authors decomposed the 2D NsLCT into one 2D chirp multiplication (2D CM), one 2D FRFT and two 2D affine transformations. However, 2D affine transformations will introduce interpolation error in the discrete case. According, in [28], another decomposition that involves two 2D CMs, one 2D FT and only one 2D affine transformation is proposed.

In this paper, the accuracy is further improved by decomposing the 2D NsLCT into two 2D CMs and two 2D chirp convolutions (2D CCs), called CM-CC-CM-CC decomposition. More precisely, only 2D CMs, 2D FTs and 2D inverse FTs (2D IFTs) are used. No 2D affine transformations are involved. Based on this decomposition, two types of 2D NsDLCT are proposed. Compared to each other, one has higher accuracy while the other one has lower complexity. All the proposed 2D NsDLCTs have lower complexity and higher accuracy than the previous works [27, 28]. Plus, the proposed methods have lower error in additivity property, which is a useful property in applications such as filtering and encryption/decryption. Besides, another decomposition called CM-CC-CM-CC decomposition is introduced such that the proposed methods have perfect reversibility property. That is, the input discrete signal/image can be perfectly recovered from the output of the proposed 2D NsDLCTs.

2 Basic 2D Discrete Operations

The 2D nonseparable discrete LCTs (NsDLCTs) proposed in this paper will be compared with Koç’s method [27] and Ding’s method [28]. In this section, some basic 2D discrete operations used in these 2D NsDLCTs are introduced. The computational complexity of these operations is also analyzed. Denote as the output of a 2D continuous operation with being the input. Given sampled input signal , the corresponding 2D discrete operation is designed to generate output that can approximate .

2.1 2D Discrete Chirp Multiplication (Discrete CM)

Consider that the ABCD matrix is given by

(7)

is symmetric, i.e. , because of the constraints in (3) or (4). Thus, this ABCD matrix has only three degrees of freedom. Since , the definition in (5) is used, and the 2D NsLCT becomes the multiplication with a 2D chirp function, called 2D chirp multiplication (CM) for short:

(8)

Sampling (8) with sampling intervals and , the 2D discrete CM, denoted by , is given by

(9)

Supposing the exponential kernel function in (9) can be precomputed and stored in memory, only one pointwise product that involves complex multiplications is required.

2.2 2D Discrete Fourier Transform (DFT) and inverse DFT (IDFT)

In (1), if ABCD matrix is given by

(10)

the 2D NsLCT reduces to the 2D Fourier transform (FT) or the 2D inverse FT (IFT) multiplied by constant phase :

(11)

If , the discrete version of (11), denoted by or , are simply the 2D DFT or 2D IDFT multiplied by some constant:

(12)
(13)

The 2D DFT and IDFT can be implemented by 2D FFT with

complex multiplications. Zero-padding the input signal to size

, where , can reduce the output sampling intervals to and , but the cost is higher computational complexity.

2.3 2D Discrete Chirp Convolution (Discrete CC)

Suppose the ABCD matrix is of the following form

(14)

is symmetric because of the constraints in (3) or (4) and thus has only three degrees of freedom. In (1), leads to

(15)

which is a 2D convolution with a 2D chirp function and called 2D chirp convolution (CC) for short. In the discrete case, directly calculating the sampled version of (2.3) by summation leads to computational complexity up to . Fortunately, the ABCD matrix in (14) can be decomposed as

(16)

Taking the benefit of the additivity property of 2D NsLCT, the above equality implies that the 2D CC can be alternatively implemented by one 2D IFT, one 2D CM and one 2D FT, i.e.

(17)

Therefore, the 2D discrete CC with chirp matrix , denoted by , is defined as a cascade of one 2D IDFT, one 2D discrete CM with chirp matrix , and one 2D DFT:

(18)

Two 2D FFTs and one pointwise product are used and totally require complex multiplications.

2.4 2D Discrete Fractional Fourier Transform (DFRFT)

The 2D FRFT with two parameters and is a special case of the 2D NsLCT. If , , and are given by

(19)

the 2D NsLCT reduces to the 2D FRFT [32] with some constant phase difference:

(20)

where and are 1D FRFT kernels [33, 34] with fractioanl angles and , respectively:

(21)

Obviously, the 2D FRFT is separable and can be implemented by two 1D FRFTs in two transverse directions, and .

There are a variety of implementation algorithms for 1D FRFT, and a review of some of them is given in [19, 35]. Here, we introduce the algorithm used in Koç’s method [27]. If , the sampled version of the kernel in (21) is given by

(22)

For the minus-plus sign in the above kernel, minus is used when while plus is used when . (22) shows that the 1D DFRFT can be implemented by two discrete CMs and one DFT/IDFT.

Once the 1D DFRFT is developed, the 2D DFRFT can be commutated by two separate 1D DFRFTs in two transverse directions, and :

(23)

According to (22), if , , and , the 2D DFRFT can be implemented by two 2D discrete CMs and one 2D DFT:

(24)

where chirp matrix is given by

(25)

One 2D FFT and two pointwise products are used in and thus totally involve complex multiplications.

2.5 2D Discrete Affine Transformation

When and are both , one has since :

(26)

From (5), the above ABCD matrix leads to

(27)

which is a 2D affine transformation. Sampling (2.5) with and yields

(28)

However, is often not available when there are a limited number of input samples. Accordingly, 2D interpolation is necessary.

Here, we introduce the bilinear interpolation method that is used in Ding’s method [28]. With the discrete input , the 2D discrete affine transformation, denoted by , based on bilinear interpolation is given by

(29)

where

(30)

The symbol denotes floor function. In (2.5), if , , and are pre-computed, each output sample requires real multiplications, or equivalently complex multiplications. Therefore, the total number of required complex multiplications is .

Smaller and can decrease the approximation error between and , but the additional upsampling preprocess requires more computation time.

3 Review of Previous Works

Most digital implementation methods of the 1D FRFT can be extended to the 1D LCT. However, the 2D NsLCT is much more complicated. Most digital implementation methods of the 2D FRFT is not suitable for the 2D NsLCT or need extensive modification. In this section, the implementation algorithms of Koç’s 2D NsDLCT [27] and Ding’s 2D NsDLCT [28] are introduced . These two methods are based on the additivity property of 2D NsLCT and decompositions of ABCD matrix. Again, assume the discrete input and output are both of size .

3.1 Koç’s 2D NsDLCT [27]

In [27], Koç, Ozaktas and Hesselink proposed a fast algorithm to compute the 2D NsLCT. It is based on the Iwasawa decomposition [36, 15] that decomposes the ABCD matrix into

(31)

where

(32)

In (31), the first and second matrices correspond to 2D CM and 2D affine transformation, respectively. The last matrix can be further decomposed as follows:

(33)

where the first and third matrices correspond to 2D affine transformations (more precisely, geometric rotations) while the second matrix corresponds to 2D FRFT:

(34)

The fractional angles and can be obtained from

(35)

And then the values of rotation angles and can be determined by solving the following two equations:

(36)

Substituting (33) into (31) leads to

(37)

Based on this decomposition, Koç et al.’s developed a 2D NsDLCT, denoted by , consisting of one 2D discrete CMs, two 2D discrete affine transformation and one 2D DFRFT:

(38)

These basic discrete operations used above have been defined in (9), (12), (13), (24) and (2.5), and is given by

(39)

If the input of the 2D DFT is zero-padded to where , the output of the 2D DFT will have smaller sampling intervals. Then, as mentioned in Sec. 2.2.5, the interpolation in the 2D discrete affine transformation will have higher accuracy. But the cost is higher computational complexity. If two-times upsampling () is employed, the number of complex multiplications used in Koç’s method becomes

(40)
(41)

The six terms in (40) (from left to right) are the numbers of complex multiplications used in the six discrete operations in (3.1) (from left to right), respectively.

3.2 Ding’s 2D NsDLCT [28]

Since 2D discrete affine transformations will introduce interpolation error, Ding, Pei and Liu proposed a more accurate 2D NsDLCT in which only one 2D discrete affine transformation is used. In their work, the ABCD matrix is decomposed into another form:

(42)

Based on (42), Ding et al.’s developed another type of 2D NsDLCT, denoted by , which connects a 2D discrete CM, a 2D discrete affine transformation, a 2D DFT and another 2D discrete CM in series:

(43)

These basic discrete operations are defined in (9), (12) and (2.5).

Again, suppose two-times upsampling is performed in for higher accuracy in the 2D discrete affine transformation . Assume two-times upsampling () is employed. The number of complex multiplications required by Ding’s method is given by

(44)

Comparing (41) and (44), Ding’s method has lower computational complexity than Koç’s method.

4 Proposed 2D CM-CC-CM-CC NsDLCTs

Since the 2D discrete affine transformation will introduce interpolation error, in this paper, some 2D NsDLCTs are developed with no 2D discrete affine transformation involved. In [23], Koç et al. have introduced a variety of decompositions for 1D LCT, where the parameter matrix is decomposed into three, four or five matrices. The CM-CC-CM decomposition [23, 26] is the only one without using scaling operation. Therefore, to avoid affine transformations in two dimensions, a possible method is decomposing the ABCD matrix into CM matrices and CC matrices . Accordingly, all the proposed methods are composed of only the 2D discrete CMs and 2D discrete CCs. More precisely, only the 2D discrete CMs, 2D DFTs and 2D IDFTs are used. These basic discrete operations have been defined in (9), (12), (13) and (18).

4.1 2D NsDLCT Based on CM-CC-CM Decomposition When and

First, suppose the 2D NsLCT can also be decomposed into CM-CC-CM. Then, it implies that the ABCD matrix can be expressed as the following form:

(45)

Obviously, must be invertible. And since all the matrix in (45) should satisfy the constraints in (3) or (4), the necessary and sufficient condition is that is symmetric. Therefore, if

(46)

the 2D NsDLCTs based on the CM-CC-CM decomposition is given by

(47)

In this method, two 2D FFTs and three pointwise products are utilized and totally require complex multiplications. The 2D FRFT and gyrator transform [16, 37] are the special cases of the 2D NsLCT with and , and thus can be digitally implemented by (4.1).

4.2 2D NsDLCT Based on CM-CC-CM-CC Decomposition When or

The CM-CC-CM decomposition in (45) is valid only when and . Consider the more general case that ABCD matrix is arbitrary (but the constraints in (3) or (4) should be satisfied) and at most has ten degrees of freedom. The CM matrix and CC matrix have only three degrees of freedom. Four CM/CC matrices are required to describe the ABCD matrix. Accordingly, the CM-CC-CM-CC decomposition that has twelve degrees of freedom is considered. First, decompose the ABCD matrix into

(48)

where , and . If is symmetric and invertible, according to (45), the ABCD matrix can be further decomposed as follows:

(49)

This decomposition is valid even if or . The 2D NsDLCT based on the above CM-CC-CM-CC decomposition is given by

(50)

Since the CM-CC-CM-CC decomposition has two more degrees of freedom, there are infinite number of possible decomposition results (i.e. infinite choices of , and ).

There are two approaches to determine , and . Firstly, if is invertible, once is determined, can be obtained from and then from . A valid for the CM-CC-CM-CC decomposition should satisfy the following three conditions:

(51)

that is,

(52)
(53)

Since and need to satisfy the constraints in (3), it is required that and . Thus, one has , and it follows that :

(54)

Secondly, if is non-invertible, one has to determine first, and then obtain and from and , respectively. A valid should lead to

(55)

doesn’t need to be considered because it is true when is true:

(56)

Since there are infinite number of solutions of to (55) (or to (51)), the problem is what the best choice of (or ) is. In the following, two types of 2D NsDLCT are proposed based on two types of . One is for high accuracy while the other one is for low complexity.

4.2.1 2D High-Accuracy NsDLCT (HA-NsDLCT)

It has been shown in (1) that the 2D NsLCT will produce an affine transformation in the space-spatial-frequency plane. The 2D CM with chirp matrix will produce shearing in spatial-frequency domain:

(57)

Consider the simple case that the input signal occupies in space domain and in spatial-frequency domain, i.e. has space-spatial-bandwidth product . After the shearing in (4.2.1), the space-spatial-bandwidth product becomes , where the ratio function is defined as

(58)

From (1), the 2D CC with chirp matrix will lead to shearing in space domain: