The Properties and Fast Algorithm of Quaternion Linear Canonical Transform
Abstract: The quaternion linear canonical transform (QLCT) is defined in this paper, with proofs given for its reversibility property, its linear property, its odd-even invariant property and additivity property. Meanwhile, the quaternion convolution (QCV), quaternion correlation (QCR) and product theorem of LCT are deduced. Their physical interpretation is given as classical convolution, correlation and product theorem. Moreover, the fast algorithm of QLCT (FQLCT) is obtained, whose calculation complexity for different signals is similar to FFT. In addition, the paper presents the relationship between the convolution and correlation in LCT domains, and the convolution and correlation can be calculated via product theorem in Fourier transform domain using FFT.

1. Introduction

The linear canonical transform (LCT) is a new tool that comes into being in signal processing - . The LCT is the generalization of the FRFT and so on . Up till now there have been a lot of papers involving the FRFT and the LCT, such as papers - . However, none of them has involved the LCT of quaternion signals (or Hyper-complex signals) even if there has been similar work on FRFT . Quaternion signals can be taken as the generalization of scalar, complex signals and vector, and after the introduction of quaternion signals by Hamilton in 1843 it has become one basic tool for multi-channel and multi-dimensional space. For example, grey image can be taken as scalar, and the analytic signal after Hilbert transformation is complex signal. The color image can be taken as one vector , a quaternion number whose real part is zero. In , the transform, convolution and correlation have been addressed in fractional Fourier transform (FRFT) domain. In this paper we first propose the definition of the QLCT, QCV and QCR in the LCT domain for quaternion signals, which are the generalization of those in . Meanwhile, some properties and the fast algorithm of QLCT are discussed. We also discover the relationship of QCV and QCR in the LCT domain for quaternion signals. We found that QCV and QCR can be implemented via product theorem in the QLCT domain. Thus we not only yield the generalized frame for scalar, complex signal, vector and quaternion signal in the QLCT domain, but also give one new idea and one theoretical base for future engineering use.

In the rest of this paper, we will introduce the definition of QLCT in Section II. We will show the properties in Section III. In Section IV, FRQCV and FRQCR will be addressed. Section V is the fast algorithm. The last section concludes our paper.

2. Definitions of QLCT

For convenience of discussion, we first give some notations used in the following of this paper. $f\left(x,y\right)$denotes 2D signal in time domain; F is classical Fourier transform operator; ${F}^{L\left(a,b,c,d\right)}$( ${F}^{L}$in short) is 1D LCT operator, and ${F}^{L}\left(u\right)$is the 1D LCT of $f\left(x,y\right)$; ${F}^{{L}_{1},{L}_{2}}$is the 2D LCT operator of $f\left(x,y\right)$; ${F}^{Q}$is classical quaternion Fourier transform operator, and ${F}^{Q}\left(u,v\right)$is quaternion Fourier transform of $f\left(x,y\right)$; I is equivalence operator; P is odd-even operator; “*” is classical convolution operator; “ ${}^{-}$” is conjugation operator. “N” is integer set; “R” is real set. Define the product operator of two LCTs’ transform parameter systems:

${L}_{1}{L}_{2}={L}_{1}\left({a}_{1},{b}_{1},{c}_{1},{d}_{1}\right)\cdot {L}_{2}\left({a}_{2},{b}_{2},{c}_{2},{d}_{2}\right)=L\left(a,b,c,d\right)$

where $\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]=\left[\begin{array}{cc}{a}_{1}& {b}_{1}\\ {c}_{1}& {d}_{1}\end{array}\right]\left[\begin{array}{cc}{a}_{2}& {b}_{2}\\ {c}_{2}& {d}_{2}\end{array}\right]$. Quaternion signals are also called Hypercomplex signals, which are the generalization of complex signals. Complex signals have two components: the real part and the imaginary part. However, one quaternion signal has four parts, one real component and three imaginary parts:

$q={q}_{r}+i{q}_{i}+j{q}_{j}+k{q}_{k}$(1)

where ${q}_{r},{q}_{i},{q}_{j},{q}_{k}\in R$, $i,j,k$are three imaginary units, which satisfy the following relations: ${i}^{2}={j}^{2}={k}^{2}=-1$, $ij=-ji=k$, $jk=-kj=i$, $ki=-ik=j$. ${q}_{a}={q}_{r}+i{q}_{i}$, ${q}_{b}={q}_{i}+i{q}_{k}$. If ${q}_{r}=0$, then $q=i{q}_{i}+j{q}_{j}+k{q}_{k}$is called vector, and ${q}_{r}$is called scalar. ${q}_{a}$and ${q}_{b}$are complex signals. Since the sequences of i, j and k will affect the result, the definition of QLCT would take them into account.

Definition 1: For any quaternion signal $f\left(x,y\right)={f}_{r}\left(x,y\right)+i{f}_{i}\left(x,y\right)+j{f}_{j}\left(x,y\right)+k{f}_{k}\left(x,y\right)$( ${f}_{r}\left(x,y\right)$, ${f}_{i}\left(x,y\right)$, ${f}_{j}\left(x,y\right)$, ${f}_{k}\left(x,y\right)$are real ones), the QLCT of $f\left(x,y\right)$is ${F}_{i,j}^{{L}_{1},{L}_{2}}\left(u,v\right)$

$\begin{array}{c}{F}_{i,j}^{{L}_{1},{L}_{2}}\left(u,v\right)={F}_{i,j}^{{L}_{1},{L}_{2}}\left\{f\left(x,y\right)\right\}={F}_{i,j}^{{L}_{1},{L}_{2}}\left\{f\left(x,y\right)\right\}\left(u,v\right)\\ =\underset{-\infty }{\overset{+\infty }{\int }}\underset{-\infty }{\overset{+\infty }{\int }}{K}_{{L}_{1},i}\left(x,u\right)f\left(x,y\right){K}_{{L}_{2},j}\left(y,v\right)\text{d}x\text{d}y\end{array}$(2-1)

where, ${K}_{{L}_{1},i}\left(x,u\right)=\sqrt{\frac{1}{2\text{π}{b}_{1}i}}\mathrm{exp}\left(i\frac{\left({a}_{1}+{d}_{1}\right){u}^{2}}{2{b}_{1}}-i\frac{ux}{{b}_{1}}\right)$, ${K}_{{L}_{2},j}\left(y,v\right)=\sqrt{\frac{1}{2\text{π}{b}_{2}j}}\mathrm{exp}\left(j\frac{\left({a}_{2}+{d}_{2}\right){v}^{2}}{2{b}_{2}}-j\frac{vy}{{b}_{2}}\right)$. Meanwhile, in the following of this paper we assume ${a}_{1}{d}_{1}-{b}_{1}{c}_{1}=1$, ${a}_{2}{d}_{2}-{b}_{2}{c}_{2}=1$and ${b}_{1},{b}_{2}\ne 0$.

The reversibility transform is defined as

$\begin{array}{c}{F}_{i,j}^{-{L}_{1},-{L}_{2}}\left(u,v\right)={F}_{i,j}^{-{L}_{1},-{L}_{2}}\left\{f\left(x,y\right)\right\}={F}_{i,j}^{-{L}_{1},-{L}_{2}}\left\{f\left(x,y\right)\right\}\left(u,v\right)\\ =\underset{-\infty }{\overset{+\infty }{\int }}\underset{-\infty }{\overset{+\infty }{\int }}{K}_{-{L}_{1},i}\left(x,u\right)f\left(x,y\right){K}_{-{L}_{2},j}\left(y,v\right)\text{d}x\text{d}y\end{array}$(2-2)

where, ${K}_{-{L}_{1},i}\left(x,u\right)=\sqrt{\frac{1}{-2\text{π}{b}_{1}i}}\mathrm{exp}\left(-i\frac{\left({a}_{1}+{d}_{1}\right){u}^{2}}{2{b}_{1}}+i\frac{ux}{{b}_{1}}\right)$, ${K}_{-{L}_{2},j}\left(y,v\right)=\sqrt{\frac{1}{-2\text{π}{b}_{2}j}}\mathrm{exp}\left(-j\frac{\left({a}_{2}+{d}_{2}\right){v}^{2}}{2{b}_{2}}+j\frac{vy}{{b}_{2}}\right)$.

If $\left({a}_{1},{b}_{1},{c}_{1},{d}_{1}\right)=\left({a}_{2},{b}_{2},{c}_{2},{d}_{2}\right)=\left(0,-1,1,0\right)$, definition 1 is quaternion Fourier transform; if $\left({a}_{1},{b}_{1},{c}_{1},{d}_{1}\right)=\left(0,-1,1,0\right),\left({a}_{2},{b}_{2},{c}_{2},{d}_{2}\right)=\left(1,0,0,1\right)$, definition 1 is classical 1D Fourier transform of $f\left(x,y\right)$for variable x; if $\left({a}_{1},{b}_{1},{c}_{1},{d}_{1}\right)=\left(1,0,0,1\right)$, $\left({a}_{2},{b}_{2},{c}_{2},{d}_{2}\right)=\left(0,-1,1,0\right)$, definition 1 is classical 1D Fourier transform of $f\left(x,y\right)$for variable y; if $\left({a}_{1},{b}_{1},{c}_{1},{d}_{1}\right)=\left({a}_{2},{b}_{2},{c}_{2},{d}_{2}\right)=\left(1,0,0,1\right)$, definition 1 is equivalence transform of $f\left(x,y\right)$. As shown above, definition 1 is the generalization of the fractional quaternion Fourier transform and the quaternion Fourier transform . The reversibility (or reconstruction) is one important property for one transform, especially for the processing in another domain. The following gives the proof of the reversibility property.

Theorem 1: One quaternion $f\left(x,y\right)$can be reconstructed from ${F}_{i,j}^{{L}_{1},{L}_{2}}\left(u,v\right)$via QLCT.

Proof: The proof is trivial and omitted here.

3. The Properties of QLCT

In the following section we list the properties and present the proof.

Property 1: For any one quaternion signal ${f}_{n}\left(x,y\right)\left(n\in \aleph \right)$, the following relationship is true: ${F}_{i,j}^{{L}_{1},{L}_{2}}\left\{\sum {a}_{n}{f}_{n}\left(x,y\right)\right\}=\sum {a}_{n}\cdot {F}_{i,j}^{{L}_{1},{L}_{2}}\left\{{f}_{n}\left(x,y\right)\right\}$ $\left({a}_{n}\in \Re \right)$.

Proof: Since QLCT is one linear transform, property 1 can be easily obtained from definition 1.

Property 2: ${F}_{i,j}^{{L}_{3},{L}_{4}}{F}_{i,j}^{{L}_{1},{L}_{2}}={F}_{i,j}^{{L}_{1},{L}_{2}}{F}_{i,j}^{{L}_{3},{L}_{4}}={F}_{i,j}^{{L}_{1}{L}_{3},{L}_{2}{L}_{4}}$. Proof: For any one quaternion signal $f\left(x,y\right)$, from definition 1 we can obtain

$\begin{array}{l}{F}_{i,j}^{{L}_{3},{L}_{4}}{F}_{i,j}^{{L}_{1},{L}_{2}}\left\{f\left(x,y\right)\right\}\\ =\underset{-\infty }{\overset{+\infty }{\int }}\underset{-\infty }{\overset{+\infty }{\int }}{K}_{{L}_{3},i}\left(u,s\right)\left\{\underset{-\infty }{\overset{+\infty }{\int }}\underset{-\infty }{\overset{+\infty }{\int }}{K}_{{L}_{1},i}\left(x,u\right)f\left(x,y\right){K}_{{L}_{2},j}\left(y,v\right)\text{d}x\text{d}y\right\}{K}_{{L}_{4},j}\left(v,w\right)\text{d}u\text{d}v\\ =\underset{-\infty }{\overset{+\infty }{\int }}\underset{-\infty }{\overset{+\infty }{\int }}\left\{\underset{-\infty }{\overset{+\infty }{\int }}\underset{-\infty }{\overset{+\infty }{\int }}{K}_{{L}_{3},i}\left(u,s\right){K}_{{L}_{1},i}\left(x,u\right)f\left(x,y\right){K}_{{L}_{2},j}\left(y,v\right){K}_{{L}_{4},j}\left(v,w\right)\text{d}u\text{d}v\right\}\text{d}x\text{d}y\\ =\underset{-\infty }{\overset{+\infty }{\int }}\underset{-\infty }{\overset{+\infty }{\int }}\left\{\left[\underset{-\infty }{\overset{+\infty }{\int }}{K}_{{L}_{3},i}\left(u,s\right){K}_{{L}_{1},i}\left(x,u\right)\text{d}u\right]f\left(x,y\right)\left[\underset{-\infty }{\overset{+\infty }{\int }}{K}_{{L}_{2},j}\left(y,v\right){K}_{{L}_{4},j}\left(v,w\right)\text{d}v\right]\right\}\text{d}x\text{d}y\end{array}$ (3)

For 1D signal the right formula is true :

$\underset{-\infty }{\overset{+\infty }{\int }}{K}_{{L}_{2}}\left(u,{u}^{\prime }\right){K}_{{L}_{1}}\left({u}^{\prime },{u}^{″}\right)\text{d}{u}^{″}={K}_{{L}_{2}{L}_{1}}\left(u,{u}^{″}\right)$(4)

Substitute (4) into (3):

$\begin{array}{l}{F}_{i,j}^{{L}_{3},{L}_{4}}{F}_{i,j}^{{L}_{1},{L}_{2}}\left\{f\left(x,y\right)\right\}\\ =\underset{-\infty }{\overset{+\infty }{\int }}\underset{-\infty }{\overset{+\infty }{\int }}\left\{{K}_{{L}_{3}{L}_{1},i}\left(x,s\right)f\left(x,y\right){K}_{{L}_{4}{L}_{2},j}\left(y,w\right)\right\}\text{d}x\text{d}y={F}_{i,j}^{{L}_{1}{L}_{3},{L}_{2}{L}_{4}}\left\{f\left(x,y\right)\right\}\end{array}$

Therefore,

${F}_{i,j}^{{L}_{3},{L}_{4}}{F}_{i,j}^{{L}_{1},{L}_{2}}={F}_{i,j}^{{L}_{1}{L}_{3},{L}_{2}{L}_{4}}$(5)

The result can be obtained similarly:

${F}_{i,j}^{{L}_{1},{L}_{2}}{F}_{i,j}^{{L}_{3},{L}_{4}}={F}_{i,j}^{{L}_{1}{L}_{3},{L}_{2}{L}_{4}}$(6)

From (5) (6): ${F}_{i,j}^{{L}_{3},{L}_{4}}{F}_{i,j}^{{L}_{1},{L}_{2}}={F}_{i,j}^{{L}_{1},{L}_{2}}{F}_{i,j}^{{L}_{3},{L}_{4}}={F}_{i,j}^{{L}_{1}{L}_{3},{L}_{2}{L}_{4}}$

Property 3: ${F}_{i,j}^{{L}_{3},{L}_{4}}{F}_{i,j}^{{L}_{1},{L}_{2}}={F}_{i,j}^{{L}_{1},{L}_{2}}{F}_{i,j}^{{L}_{3},{L}_{4}}$, ${F}_{i,j}^{{L}_{5},{L}_{6}}\left({F}_{i,j}^{{L}_{3},{L}_{4}}{F}_{i,j}^{{L}_{1},{L}_{2}}\right)=\left({F}_{i,j}^{{L}_{5},{L}_{6}}{F}_{i,j}^{{L}_{3},{L}_{4}}\right){F}_{i,j}^{{L}_{1},{L}_{2}}$.

Proof: This property can be obtained from property 2.

Property 4: If ${F}_{i,j}^{{L}_{1},{L}_{2}}\left\{f\left(x,y\right)\right\}={F}_{i,j}^{{L}_{1},{L}_{2}}\left(u,v\right)$, then ${F}_{i,j}^{{L}_{1},{L}_{2}}\left\{f\left(-x,-y\right)\right\}={F}_{i,j}^{{L}_{1},{L}_{2}}\left(-u,-v\right)$, ${F}_{i,j}^{{L}_{1},{L}_{2}}\left\{f\left(-x,y\right)\right\}={F}_{i,j}^{{L}_{1},{L}_{2}}\left(-u,v\right)$, ${F}_{i,j}^{{L}_{1},{L}_{2}}\left\{f\left(x,-y\right)\right\}={F}_{i,j}^{{L}_{1},{L}_{2}}\left(u,-v\right)$.

Proof: Let ${A}_{1}=\sqrt{2\text{π}{b}_{1}i}$, ${A}_{2}=\sqrt{2\text{π}{b}_{2}j}$, ${C}_{1}=\frac{{d}_{1}}{2{b}_{1}}$, ${C}_{2}=\frac{{d}_{2}}{2{b}_{2}}$, and insert them into (2):

${F}_{i,j}^{{L}_{1},{L}_{2}}\left\{f\left(-x,-y\right)\right\}={A}_{1}{\text{e}}^{i{u}^{2}{C}_{1}}\underset{-\infty }{\overset{+\infty }{\int }}\underset{-\infty }{\overset{+\infty }{\int }}{\text{e}}^{-i\frac{ux}{{b}_{1}}}{\text{e}}^{i{x}^{2}{C}_{1}}f\left(-x,-y\right){\text{e}}^{j{y}^{2}{C}_{1}}{\text{e}}^{-j\frac{vy}{{b}_{2}}}\text{d}x\text{d}y\cdot {A}_{2}{\text{e}}^{j{v}^{2}{C}_{2}}$(7)

Let $s=-x,z=-y$, and substitute them in (7):

$\begin{array}{l}{F}_{i,j}^{{L}_{1},{L}_{2}}\left\{f\left(-x,-y\right)\right\}={F}_{i,j}^{{L}_{1},{L}_{2}}\left\{f\left(s,z\right)\right\}\\ ={A}_{1}{\text{e}}^{i{u}^{2}{C}_{1}}\underset{-\infty }{\overset{+\infty }{\int }}\underset{-\infty }{\overset{+\infty }{\int }}{\text{e}}^{-i\frac{u\cdot \left(-s\right)}{{b}_{1}}}{\text{e}}^{i{s}^{2}{C}_{1}}f\left(s,z\right){\text{e}}^{j{z}^{2}{C}_{2}}{\text{e}}^{-j\frac{v\cdot \left(-z\right)}{{b}_{2}}}\text{d}s\text{d}z\cdot {A}_{\text{2}}{\text{e}}^{j{v}^{2}{C}_{2}}\\ ={A}_{1}{\text{e}}^{i\cdot {\left(-u\right)}^{2}{C}_{1}}\underset{-\infty }{\overset{+\infty }{\int }}\underset{-\infty }{\overset{+\infty }{\int }}{\text{e}}^{-i\frac{\left(-u\right)\cdot s}{{b}_{1}}}{\text{e}}^{i{s}^{2}{C}_{1}}f\left(s,z\right){\text{e}}^{j{z}^{2}{C}_{2}}{\text{e}}^{-j\frac{\left(-v\right)\cdot z}{{b}_{2}}}\text{d}s\text{d}z\cdot {A}_{2}{\text{e}}^{j\cdot {\left(-v\right)}^{2}{C}_{2}}\\ ={F}_{i,j}^{{L}_{1},{L}_{2}}\left(-u,-v\right)\end{array}$

It can be obtained as well: ${F}_{i,j}^{{L}_{1},{L}_{2}}\left\{f\left(-x,y\right)\right\}={F}_{i,j}^{{L}_{1},{L}_{2}}\left(-u,v\right)$and ${F}_{i,j}^{{L}_{1},{L}_{2}}\left\{f\left(x,-y\right)\right\}={F}_{i,j}^{{L}_{1},{L}_{2}}\left(u,-v\right)$.

We can draw the conclusion that transformed signal of the odd is odd, and even is even.

Property 5: If $n\in \aleph$, then ${\left({F}_{i,j}^{{L}_{1},{L}_{2}}\right)}^{n}={F}_{i,j}^{{\left({L}_{1}\right)}^{n},{\left({L}_{2}\right)}^{n}}$.

Proof: From property 2,

$\underset{n}{\underset{︸}{{F}_{i,j}^{{L}_{1},{L}_{2}}{F}_{i,j}^{{L}_{1},{L}_{2}}\cdots {F}_{i,j}^{{L}_{1},{L}_{2}}}}={F}_{i,j}^{\underset{n}{\underset{︸}{{L}_{1}×\cdots ×{L}_{1}}},\underset{n}{\underset{︸}{{L}_{2}×\cdots ×{L}_{2}}}}={F}_{i,j}^{{\left({p}_{1}\right)}^{n},{\left({p}_{2}\right)}^{n}}$

then

${\left({F}_{i,j}^{{L}_{1},{L}_{2}}\right)}^{n}={F}_{i,j}^{{\left({L}_{1}\right)}^{n},{\left({L}_{2}\right)}^{n}}$.

QLCT doesn’t satisfy Parseval’s principle. Meanwhile, it is hard to find one obvious relationship between QLCT and Wigner-Ville time-frequency plane. Some other properties cannot find physical interpretation in QLCT domains.

4. FRQCV and FRQCR

Convolution and correlation play an important role in signal processing, especially for linear system design and filter design, etc. The convolution in time domain is to the product in Fourier transform domain, that is to say, the classical convolution in time domain can be implemented in Fourier transform domain via FFT, which is beneficial for real-time engineering use. In classical time-frequency analysis correlation is special convolution in that the original signals are implemented via conjugation and so on. This is very important for engineering use . The key to this paper is to discover the relationships in fractional quaternion Fourier transform domain between them so that we can find the physical interpretation as that of the classical Fourier transform. Paper yielded fractional convolution and product theorem for 1D signals first, however, it didn’t give the similar physical interpretation as that of the classical theorem. Later papers obtained similar result as the classical theorems. However, they are only for 1D signals. In this section the QCV and QCR of the LCT would be discussed, and can be implemented via FFT.

4.1. Fractional Convolution and Product Theorem

In the following, four theorems are yielded, and theorem 2 and 3 are suitable for scalar and complex signals, and theorem 4 and 5 are suitable for scalar, complex signals, vector and quaternion signals.

Theorem 2: For any real scalar or complex signal $f\left(x,y\right)$and convolution kernel $h\left(x,y\right)$,

$\begin{array}{l}g\left(x,y\right)=f\left(x,y\right)\stackrel{¯}{\ast }h\left(x,y\right)=\left(f\stackrel{¯}{\ast }h\right)\left(x,y\right)\\ \triangleq {A}_{1,2}{\text{e}}^{-i\left({x}^{2}{C}_{1}+{y}^{2}{C}_{2}\right)}\left({\text{e}}^{i\left({x}^{2}{C}_{1}+{y}^{2}{C}_{2}\right)}f\left(x,y\right)\right)\ast \left(h\left(x,y\right){\text{e}}^{i\left({x}^{2}{C}_{1}+{y}^{2}{C}_{2}\right)}\right)\end{array}$

where ${B}_{1}=1/{b}_{1}$, ${B}_{2}=1/{b}_{2}$, ${A}_{1,2}=\frac{1}{2\text{π}i\sqrt{{b}_{1}{b}_{2}}}$, ${C}_{1}=\frac{{a}_{1}}{2{b}_{1}}$, ${C}_{2}=\frac{{a}_{2}}{2{b}_{2}}$, then:

${F}^{{L}_{1},{L}_{2}}\left\{g\left(x,y\right)\right\}={\text{e}}^{-i\left({u}^{2}{C}_{1}+{v}^{2}{C}_{2}\right)}{F}^{{L}_{1},{L}_{2}}\left\{f\left(x,y\right)\right\}\cdot {F}^{{L}_{1},{L}_{2}}\left\{h\left(x,y\right)\right\}$(8)

Proof:

$\begin{array}{c}{F}^{{L}_{1},{L}_{2}}\left\{g\left(x,y\right)\right\}=\underset{-\infty }{\overset{+\infty }{\int }}\underset{-\infty }{\overset{+\infty }{\int }}{K}_{{L}_{1},{L}_{2}}\left(x,y,u,v\right)g\left(x,y\right)\text{d}x\text{d}y\\ ={A}_{1,2}{\text{e}}^{i\left({u}^{2}{C}_{1}+{v}^{2}{C}_{2}\right)}\underset{-\infty }{\overset{+\infty }{\int }}\underset{-\infty }{\overset{+\infty }{\int }}{\text{e}}^{i\left({x}^{2}{C}_{1}+{y}^{2}{C}_{2}\right)}{A}_{1,2}{\text{e}}^{-i\left({x}^{2}{C}_{1}+{y}^{2}{C}_{2}\right)}{\text{e}}^{-i\left(xu{B}_{1}+yv{B}_{2}\right)}\\ \text{\hspace{0.17em}}\text{ }\text{\hspace{0.17em}}\cdot \left({\text{e}}^{i\left({x}^{2}{C}_{1}+{y}^{2}{C}_{2}\right)}f\left(x,y\right)\right)\ast \left(h\left(x,y\right){\text{e}}^{i\left({x}^{2}{C}_{1}+{y}^{2}{C}_{2}\right)}\right)\text{d}x\text{d}y\end{array}$(9)

Substitute (9) with $s=x-\tau ,z=y-\eta$:

$\begin{array}{c}{F}^{{L}_{1},{L}_{2}}\left\{g\left(x,y\right)\right\}={A}_{1,2}^{2}{\text{e}}^{i\left({u}^{2}{C}_{1}+{v}^{2}{C}_{2}\right)}\underset{-\infty }{\overset{+\infty }{\int }}\underset{-\infty }{\overset{+\infty }{\int }}{\text{e}}^{i\left({\tau }^{2}{C}_{1}+{\eta }^{2}{C}_{2}\right)}f\left(\tau ,\eta \right){\text{e}}^{-i\left(\tau u{B}_{1}+\eta v{B}_{2}\right)}\\ \text{\hspace{0.17em}}\text{ }\text{\hspace{0.17em}}\cdot \left\{\underset{-\infty }{\overset{+\infty }{\int }}\underset{-\infty }{\overset{+\infty }{\int }}h\left(s,z\right){\text{e}}^{i\left({s}^{2}{C}_{1}+{z}^{2}{C}_{2}\right)}{\text{e}}^{-i\left(su{B}_{1}+zv{B}_{2}\right)}\text{d}s\text{d}z\right\}\text{d}\tau \text{d}\eta \\ ={A}_{1,2}\underset{-\infty }{\overset{+\infty }{\int }}\underset{-\infty }{\overset{+\infty }{\int }}{\text{e}}^{i\left({\tau }^{2}{C}_{1}+{\eta }^{2}{C}_{2}\right)}f\left(\tau ,\eta \right){\text{e}}^{-i\left(\tau u{B}_{1}+\eta v{B}_{2}\right)}\text{d}\tau \text{d}\eta \cdot {F}^{{L}_{1},{L}_{2}}\left\{h\left(x,y\right)\right\}\\ ={\text{e}}^{-i\left({u}^{2}{C}_{1}+{v}^{2}{C}_{2}\right)}{F}^{{L}_{{}_{1}},{L}_{{}_{2}}}\left\{f\left(x,y\right)\right\}\cdot {F}^{{L}_{{}_{1}},{L}_{{}_{2}}}\left\{h\left(x,y\right)\right\}\end{array}$

From theorem 2 it can be concluded that the convolution of scalar or complex signal is to the product, frequency-modulated by a chirp, of them in linear canonical transform.

Theorem 3: For any real scalar or complex signal $f\left(x,y\right)$and convolution kernel $h\left(x,y\right)$,

$\begin{array}{l}g\left(x,y\right)=\left(f\stackrel{¯}{\stackrel{¯}{\ast }}h\right)\left(x,y\right)\\ \triangleq \left({\text{e}}^{i\left({x}^{2}{C}_{1}+{y}^{2}{C}_{2}\right)}\right)/\text{2π}\cdot \left({\text{e}}^{-i\left({x}^{2}{C}_{1}+{y}^{2}{C}_{2}\right)}f\left(x,y\right)\right)\ast \left(h\left(x,y\right){\text{e}}^{-i\left({x}^{2}{C}_{1}+{y}^{2}{C}_{2}\right)}\right)\end{array}$

where ${B}_{1}=1/{b}_{1}$, ${B}_{2}=1/{b}_{2}$, ${A}_{1,2}=\frac{1}{2\text{π}i\sqrt{{b}_{1}{b}_{2}}}$, ${A}_{1,2}^{-1,-1}=\frac{1}{2\text{π}i\sqrt{\left(-{b}_{1}\right)\left(-{b}_{2}\right)}}$, ${C}_{1}=\frac{{a}_{1}}{2{b}_{1}}$, ${C}_{2}=\frac{{a}_{2}}{2{b}_{2}}$, then

$\begin{array}{l}{F}^{{L}_{1},{L}_{2}}\left\{{\text{e}}^{i\left({u}^{2}{C}_{1}+{v}^{2}{C}_{2}\right)}f\left(x,y\right)g\left(x,y\right)\right\}\\ ={A}_{1,2}^{-1,-1}\left({f}_{{}^{{L}_{1},{L}_{2}}}\left(u,v\right)\stackrel{¯}{\stackrel{¯}{\ast }}{h}_{{}^{{L}_{1},{L}_{2}}}\left(u,v\right)\right)\end{array}$(10)

Proof:

$\begin{array}{l}{F}^{-{L}_{1},-{L}_{2}}\left({A}_{1,2}^{-1,-1}{f}_{{}^{{L}_{1},{L}_{2}}}\left(u,v\right)\stackrel{¯}{\stackrel{¯}{\ast }}{h}_{{}^{{L}_{1},{L}_{2}}}\left(u,v\right)\right)\\ =\frac{{A}_{1,2}^{-1,-1}{\text{e}}^{-i\left({x}^{2}{C}_{1}+{y}^{2}{C}_{2}\right)}}{\text{2π}}\underset{-\infty }{\overset{+\infty }{\int }}\underset{-\infty }{\overset{+\infty }{\int }}{\text{e}}^{-i\left({C}_{1}{u}^{2}+{C}_{2}{v}^{2}\right)}\left\{{A}_{1,2}^{-1,-1}{f}_{{L}_{1},{L}_{2}}\left(u,v\right)\stackrel{¯}{\stackrel{¯}{\ast }}\text{ }\text{ }{h}_{{L}_{1},{L}_{2}}\left(u,v\right)\right\}{\text{e}}^{i\left(xu{B}_{1}+yv{B}_{2}\right)}\text{d}u\text{d}v\end{array}$

Substitute $s=x-\tau ,z=y-\eta$in above equation

$\begin{array}{l}{F}^{-{L}_{1},-{L}_{2}}\left\{{A}_{1,2}^{-1,-1}{f}_{{L}_{1},{L}_{2}}\left(u,v\right)\stackrel{¯}{\stackrel{¯}{\ast }}\text{ }{h}_{{L}_{1},{L}_{2}}\left(u,v\right)\right\}\\ =\frac{{\left({A}_{1,2}^{-1,-1}\right)}^{2}{\text{e}}^{-i\left({x}^{2}{C}_{1}+{y}^{2}{C}_{2}\right)}}{4{\text{π}}^{2}}\underset{-\infty }{\overset{+\infty }{\int }}\underset{-\infty }{\overset{+\infty }{\int }}\left\{\underset{-\infty }{\overset{+\infty }{\int }}\underset{-\infty }{\overset{+\infty }{\int }}{h}_{{L}_{1},{L}_{2}}\left(s,z\right){\text{e}}^{-i\left({C}_{1}{s}^{2}+{C}_{2}{z}^{2}\right)}{\text{e}}^{i\left(xs{B}_{1}+yz{B}_{2}\right)}\text{d}s\text{d}z\right\}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\cdot {\text{e}}^{i\left(x\tau {B}_{1}+y\eta {B}_{2}\right)}{f}_{{L}_{1},{L}_{2}}\left(\tau ,\eta \right){\text{e}}^{-i\left({C}_{1}{\tau }^{2}+{C}_{2}{\eta }^{2}\right)}\text{d}\tau \text{d}\eta \\ =h\left(x,y\right)\frac{{A}_{1,2}^{-1,-1}}{\text{2π}}\underset{-\infty }{\overset{+\infty }{\int }}\underset{-\infty }{\overset{+\infty }{\int }}{\text{e}}^{i\left(x\tau {B}_{1}+y\eta {B}_{2}\right)}{f}_{{L}_{1},{L}_{2}}\left(\tau ,\eta \right){\text{e}}^{-i\left({C}_{1}{\tau }^{2}+{C}_{2}{\eta }^{2}\right)}\text{d}\tau \text{d}\eta \\ =f\left(x,y\right)h\left(x,y\right){\text{e}}^{-i\left({C}_{1}{x}^{2}+{C}_{2}{y}^{2}\right)}\end{array}$

Therefore, ${F}^{{L}_{1},{L}_{2}}\left\{{\text{e}}^{i\left({u}^{2}{C}_{1}+{v}^{2}{C}_{2}\right)}f\left(x,y\right)g\left(x,y\right)\right\}={A}_{1,2}^{-1,-1}\left({f}_{{}^{{L}_{1},{L}_{2}}}\left(u,v\right)\stackrel{¯}{\stackrel{¯}{\ast }}{h}_{{}^{{L}_{1},{L}_{2}}}\left(u,v\right)\right)$.

Theorem 4: For one given quaternion function $f\left(x,y\right)={f}_{a}\left(x,y\right)+{f}_{b}\left(x,y\right)j$and convolution kernel function

$h\left(x,y\right)={h}_{a}\left(x,y\right)+{h}_{b}\left(x,y\right)j$

where ${f}_{a}\left(x,y\right)={f}_{r}\left(x,y\right)+i{f}_{i}\left(x,y\right)$, ${f}_{b}\left(x,y\right)={f}_{j}\left(x,y\right)+i{f}_{k}\left(x,y\right)$, ${h}_{a}\left(x,y\right)={h}_{r}\left(x,y\right)+i{h}_{i}\left(x,y\right)$, ${h}_{b}\left(x,y\right)={h}_{j}\left(x,y\right)+i{h}_{k}\left(x,y\right)$.

Set ${A}_{1,2}=\frac{1}{2\text{π}i\sqrt{{b}_{1}{b}_{2}}}$, ${A}_{1,2}^{-1,-1}=\frac{1}{2\text{π}i\sqrt{\left(-{b}_{1}\right)\left(-{b}_{2}\right)}}$, and define

$\begin{array}{l}g\left(x,y\right)=\left(f\stackrel{¯}{\ast }h\right)\left(x,y\right)\\ \triangleq {A}_{1,2}{\text{e}}^{-i\left({x}^{2}{C}_{1}+{y}^{2}{C}_{2}\right)}\left({\text{e}}^{i\left({x}^{2}{C}_{1}+{y}^{2}{C}_{2}\right)}f\left(x,y\right)\right)\ast \left(h\left(x,y\right){\text{e}}^{i\left({x}^{2}{C}_{1}+{y}^{2}{C}_{2}\right)}\right)\end{array}$

where, ${C}_{1}=\frac{{a}_{1}}{2{b}_{1}}$, ${C}_{2}=\frac{{a}_{2}}{2{b}_{2}}$, ${B}_{1}=1/{b}_{1}$, ${B}_{2}=1/{b}_{2}$, $\alpha =\mathrm{arcsin}{b}_{1}$, $\beta =\mathrm{arcsin}{b}_{2}$, then

$\begin{array}{l}{F}^{{L}_{1},{L}_{2}}\left\{g\left(x,y\right)\right\}={\text{e}}^{-i\left({u}^{2}{C}_{1}+{v}^{2}{C}_{2}\right)}\left\{{F}^{{L}_{1},{L}_{2}}\left[{f}_{a}\left(x,y\right)\right]\cdot {F}^{{L}_{1},{L}_{2}}\left[{h}_{a}\left(x,y\right)\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}-{F}^{{L}_{1},{L}_{2}}\left[{f}_{b}\left(x,y\right)\right]\cdot {F}^{{L}_{1},{L}_{2}}\left[\stackrel{¯}{{h}_{b}}\left(x,y\right)\right]\right\}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }+{\text{e}}^{i\left({u}^{2}{C}_{1}+{v}^{2}{C}_{2}-\alpha -\beta \right)}\left\{{F}^{{L}_{1},{L}_{2}}\left[{f}_{a}\left(x,y\right)\right]\cdot {F}^{-{L}_{1},-{L}_{2}}\left[{h}_{b}\left(-x,-y\right)\right]\\ \text{\hspace{0.17em}}\text{ }\text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+{F}^{{L}_{1},{L}_{2}}\left[{f}_{b}\left(x,y\right)\right]\cdot {F}^{-{L}_{1},-{L}_{2}}\left[\stackrel{¯}{{h}_{a}}\left(-x,-y\right)\right]\right\}\cdot j\end{array}$(11)

Proof:

$\begin{array}{l}\left({\text{e}}^{i\left({x}^{2}{C}_{1}+{y}^{2}{C}_{2}\right)}f\left(x,y\right)\right)\ast \left(h\left(x,y\right){\text{e}}^{i\left({x}^{2}{C}_{1}+{y}^{2}{C}_{2}\right)}\right)\\ =\left({\text{e}}^{i\left({x}^{2}{C}_{1}+{y}^{2}{C}_{2}\right)}\left({f}_{a}\left(x,y\right)+{f}_{b}\left(x,y\right)j\right)\right)\ast \left(\left({h}_{a}\left(x,y\right)+{h}_{b}\left(x,y\right)j\right){\text{e}}^{i\left({x}^{2}{C}_{1}+{y}^{2}{C}_{2}\right)}\right)\\ =\left({\text{e}}^{i\left({x}^{2}{C}_{1}+{y}^{2}{C}_{2}\right)}{f}_{a}\left(x,y\right)\right)\ast \left({h}_{a}\left(x,y\right){\text{e}}^{i\left({x}^{2}{C}_{1}+{y}^{2}{C}_{2}\right)}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }+\left({\text{e}}^{i\left({x}^{2}{C}_{1}+{y}^{2}{C}_{2}\right)}{f}_{a}\left(x,y\right)\right)\ast \left({h}_{b}\left(x,y\right){\text{e}}^{-i\left({x}^{2}{C}_{1}+{y}^{2}{C}_{2}\right)}\right)\cdot j\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }+\left({\text{e}}^{i\left({x}^{2}{C}_{1}+{y}^{2}{C}_{2}\right)}{f}_{b}\left(x,y\right)\right)\ast \left(\stackrel{¯}{{h}_{a}}\left(x,y\right){\text{e}}^{-i\left({x}^{2}{C}_{1}+{y}^{2}{C}_{2}\right)}\right)\cdot j\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }-\left({\text{e}}^{i\left({x}^{2}{C}_{1}+{y}^{2}{C}_{2}\right)}{f}_{b}\left(x,y\right)\right)\ast \left(\stackrel{¯}{{h}_{b}}\left(x,y\right){\text{e}}^{-i\left({x}^{2}{C}_{1}+{y}^{2}{C}_{2}\right)}\right)\end{array}$

From theorem 2 it can be obtained

$\begin{array}{l}{F}^{{L}_{1},{L}_{2}}\left\{{A}_{1,2}{\text{e}}^{-i\left({x}^{2}{C}_{1}+{y}^{2}{C}_{2}\right)}\left({\text{e}}^{i\left({x}^{2}{C}_{1}+{y}^{2}{C}_{2}\right)}{f}_{a}\left(x,y\right)\right)\ast \left({h}_{a}\left(x,y\right){\text{e}}^{i\left({x}^{2}{C}_{1}+{y}^{2}{C}_{2}\right)}\right)\right\}\\ ={\text{e}}^{-i\left({u}^{2}{C}_{1}+{v}^{2}{C}_{2}\right)}\left({F}^{{L}_{1},{L}_{2}}\left\{{f}_{a}\left(x,y\right)\right\}\cdot {F}^{{L}_{1},{L}_{2}}\left\{{h}_{a}\left(x,y\right)\right\}\right)\end{array}$

$\begin{array}{l}{F}^{{L}_{1},{L}_{2}}\left\{{A}_{\alpha ,\beta }{\text{e}}^{-i\left({x}^{2}{C}_{1}+{y}^{2}{C}_{2}\right)}\left({\text{e}}^{i\left({x}^{2}{C}_{1}+{y}^{2}{C}_{2}\right)}{f}_{b}\left(x,y\right)\right)\ast \left(\stackrel{¯}{{h}_{b}}\left(x,y\right){\text{e}}^{-i\left({x}^{2}{C}_{1}+{y}^{2}{C}_{2}\right)}\right)\right\}\\ ={\text{e}}^{-i\left({u}^{2}{C}_{1}+{v}^{2}{C}_{2}\right)}\left({F}^{{L}_{1},{L}_{2}}\left\{{f}_{b}\left(x,y\right)\right\}\cdot {F}^{{L}_{1},{L}_{2}}\left\{\stackrel{¯}{{h}_{b}}\left(x,y\right)\right\}\right)\end{array}$

From the linear property of fractional Fourier transfor

$\begin{array}{l}{F}^{{L}_{1},{L}_{2}}\left\{g\left(x,y\right)\right\}\\ ={\text{e}}^{-i\left({u}^{2}{C}_{1}+{v}^{2}{C}_{2}\right)}\left\{{F}^{{L}_{1},{L}_{2}}\left[{f}_{a}\left(x,y\right)\right]\cdot {F}^{{L}_{1},{L}_{2}}\left[{h}_{a}\left(x,y\right)\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }-{F}^{{L}_{1},{L}_{2}}\left[{f}_{b}\left(x,y\right)\right]\cdot {F}^{{L}_{1},{L}_{2}}\left[\stackrel{¯}{{h}_{b}}\left(x,y\right)\right]\right\}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\text{e}}^{i\left({u}^{2}{C}_{1}+{v}^{2}{C}_{2}-\alpha -\beta \right)}\left\{{F}^{{L}_{1},{L}_{2}}\left[{f}_{b}\left(x,y\right)\right]\cdot {F}^{-{L}_{1},-{L}_{2}}\left[{h}_{b}\left(-x,-y\right)\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{F}^{{L}_{1},{L}_{2}}\left[{f}_{b}\left(x,y\right)\right]\cdot {F}^{-{L}_{1},-{L}_{2}}\left[\stackrel{¯}{{h}_{a}}\left(-x,-y\right)\right]\right\}\cdot j\end{array}$

From theorem 4 we draw the conclusion that the convolution of two quaternion signals is to the summation of product of their components, conjugated or odd-even operated, and the product is frequency modulated by chirps. Meanwhile, it must be noted that the orders of i and j in cannot be disordered.

Theorem 5: For any two quaternion signals

$f\left(x,y\right)={f}_{a}\left(x,y\right)+{f}_{b}\left(x,y\right)j$and $h\left(x,y\right)={h}_{a}\left(x,y\right)+{h}_{b}\left(x,y\right)j$

where ${f}_{a}\left(x,y\right)={f}_{r}\left(x,y\right)+i{f}_{i}\left(x,y\right)$, ${f}_{b}\left(x,y\right)={f}_{j}\left(x,y\right)+i{f}_{k}\left(x,y\right)$, ${h}_{a}\left(x,y\right)={h}_{r}\left(x,y\right)+i{h}_{i}\left(x,y\right)$, ${h}_{b}\left(x,y\right)={h}_{j}\left(x,y\right)+i{h}_{k}\left(x,y\right)$, set

${A}_{1,2}=\frac{1}{2\text{π}i\sqrt{{b}_{1}{b}_{2}}}$and ${A}_{1,2}^{-1,-1}=\frac{1}{2\text{π}i\sqrt{\left(-{b}_{1}\right)\left(-{b}_{2}\right)}}$,

$g\left(x,y\right)=\left(f\stackrel{¯}{\stackrel{¯}{\ast }}h\right)\left(x,y\right)\triangleq \frac{{\text{e}}^{i\left({x}^{2}{C}_{1}+{y}^{2}{C}_{2}\right)}}{2\text{π}}\left({\text{e}}^{-i\left({x}^{2}{C}_{1}+{y}^{2}{C}_{2}\right)}f\left(x,y\right)\right)\ast \left(h\left(x,y\right){\text{e}}^{-i\left({x}^{2}{C}_{1}+{y}^{2}{C}_{2}\right)}\right)$

where ${B}_{1}=1/{b}_{1}$, ${B}_{2}=1/{b}_{2}$, ${C}_{1}=\frac{{a}_{1}}{2{b}_{1}}$, ${C}_{2}=\frac{{a}_{2}}{2{b}_{2}}$, then

$\begin{array}{l}{F}^{{L}_{1},{L}_{2}}\left\{{\text{e}}^{i\left({u}^{2}{C}_{1}+{v}^{2}{C}_{2}\right)}f\left(x,y\right)g\left(x,y\right)\right\}\\ ={A}_{1,2}^{-1,-1}\left\{{\left({f}_{a}\left(x,y\right)\right)}_{{L}_{1},{L}_{2}}\stackrel{¯}{\stackrel{¯}{\ast }}{\left({h}_{a}\left(x,y\right)\right)}_{{L}_{1},{L}_{2}}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-{\left({f}_{b}\left(x,y\right)\right)}_{{L}_{1},{L}_{2}}\stackrel{¯}{\stackrel{¯}{\ast }}{\left(\stackrel{¯}{{h}_{b}}\left(x,y\right)\right)}_{{L}_{1},{L}_{2}}\right\}\left(u,v\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{A}_{1,2}^{-1,-1}\left\{{\left({f}_{a}\left(x,y\right)\right)}_{{L}_{1},{L}_{2}}\stackrel{¯}{\stackrel{¯}{\ast }}{\left({h}_{b}\left(x,y\right)\right)}_{{L}_{1},{L}_{2}}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\left({f}_{b}\left(x,y\right)\right)}_{{L}_{1},{L}_{2}}\stackrel{¯}{\stackrel{¯}{\ast }}{\left(\stackrel{¯}{{h}_{a}}\left(x,y\right)\right)}_{{L}_{1},{L}_{2}}\right\}\left(u,v\right)\cdot j\end{array}$(12)

Proof: Since

$\begin{array}{l}{\text{e}}^{i\left({u}^{2}{C}_{1}+{v}^{2}{C}_{2}\right)}f\left(x,y\right)g\left(x,y\right)={\text{e}}^{i\left({u}^{2}{C}_{1}+{v}^{2}{C}_{2}\right)}\left\{{f}_{a}\left(x,y\right){h}_{a}\left(x,y\right)+{f}_{a}\left(x,y\right){h}_{b}\left(x,y\right)j\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+{f}_{b}\left(x,y\right)\stackrel{¯}{{h}_{a}}\left(x,y\right)j-{f}_{b}\left(x,y\right)\stackrel{¯}{{h}_{b}}\left(x,y\right)\right\}\end{array}$

From theorem 3, it can be obtained:

$\begin{array}{l}{F}^{{L}_{1},{L}_{2}}\left\{{\text{e}}^{i\left({u}^{2}{C}_{1}+{v}^{2}{C}_{2}\right)}{f}_{a}\left(x,y\right){h}_{a}\left(x,y\right)\right\}\\ ={A}_{1,2}^{-1,-1}\left\{{\left({f}_{a}\left(x,y\right)\right)}_{{L}_{1},{L}_{2}}\stackrel{¯}{\stackrel{¯}{\ast }}{\left({h}_{a}\left(x,y\right)\right)}_{{L}_{1},{L}_{2}}\right\}\left(u,v\right)\end{array}$

$\begin{array}{l}{F}^{{L}_{1},{L}_{2}}\left\{{\text{e}}^{i\left({u}^{2}{C}_{1}+{v}^{2}{C}_{2}\right)}{f}_{a}\left(x,y\right){h}_{b}\left(x,y\right)j\right\}\\ ={A}_{1,2}^{-1,-1}\left\{{\left({f}_{a}\left(x,y\right)\right)}_{{L}_{1},{L}_{2}}\stackrel{¯}{\stackrel{¯}{\ast }}{\left({h}_{b}\left(x,y\right)\right)}_{{L}_{1},{L}_{2}}\right\}\left(u,v\right)\cdot j\end{array}$

$\begin{array}{l}{F}^{{L}_{1},{L}_{2}}\left\{{\text{e}}^{i\left({u}^{2}{C}_{1}+{v}^{2}{C}_{2}\right)}{f}_{b}\left(x,y\right)\stackrel{¯}{{h}_{a}}\left(x,y\right)j\right\}\\ ={A}_{1,2}^{-1,-1}\left\{{\left({f}_{b}\left(x,y\right)\right)}_{{L}_{1},{L}_{2}}\stackrel{¯}{\stackrel{¯}{\ast }}{\left(\stackrel{¯}{{h}_{a}}\left(x,y\right)\right)}_{{L}_{1},{L}_{2}}\right\}\left(u,v\right)\cdot j\end{array}$

$\begin{array}{l}{F}^{{L}_{1},{L}_{2}}\left\{{\text{e}}^{i\left({u}^{2}{C}_{1}+{v}^{2}{C}_{2}\right)}{f}_{b}\left(x,y\right){h}_{b}\left(x,y\right)\right\}\\ ={A}_{1,2}^{-1,-1}\left\{{\left({f}_{b}\left(x,y\right)\right)}_{{L}_{1},{L}_{2}}\stackrel{¯}{\stackrel{¯}{\ast }}{\left(\stackrel{¯}{{h}_{b}}\left(x,y\right)\right)}_{{L}_{1},{L}_{2}}\right\}\left(u,v\right)\end{array}$

From the linear property of Fourier transform:

$\begin{array}{l}{F}^{{L}_{1},{L}_{2}}\left\{{\text{e}}^{i\left({u}^{2}{C}_{1}+{v}^{2}{C}_{2}\right)}f\left(x,y\right)g\left(x,y\right)\right\}\\ ={A}_{1,2}^{-1,-1}\left\{{\left({f}_{a}\left(x,y\right)\right)}_{{L}_{1},{L}_{2}}\stackrel{¯}{\stackrel{¯}{\ast }}{\left({h}_{a}\left(x,y\right)\right)}_{{L}_{1},{L}_{2}}-{\left({f}_{b}\left(x,y\right)\right)}_{{L}_{1},{L}_{2}}\stackrel{¯}{\stackrel{¯}{\ast }}{\left(\stackrel{¯}{{h}_{b}}\left(x,y\right)\right)}_{{L}_{1},{L}_{2}}\right\}\left(u,v\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{A}_{1,2}^{-1,-1}\left\{{\left({f}_{a}\left(x,y\right)\right)}_{{L}_{1},{L}_{2}}\stackrel{¯}{\stackrel{¯}{\ast }}{\left({h}_{a}\left(x,y\right)\right)}_{{L}_{1},{L}_{2}}+{\left({f}_{b}\left(x,y\right)\right)}_{{L}_{1},{L}_{2}}\stackrel{¯}{\stackrel{¯}{\ast }}{\left(\stackrel{¯}{{h}_{a}}\left(x,y\right)\right)}_{{L}_{1},{L}_{2}}\right\}\left(u,v\right)\cdot j\end{array}$

From theorem 5 we draw the conclusion that, the product, frequency modulated by a chirp, of two quaternion signals is to the summation, amplitude modulated, of their pseudo convolution.

4.2. FRQCR

Theorem 6 is suitable for scalar and complex signals, and theorem 7 is suitable for scalar, complex signals, vector and quaternion signals.

Theorem 6: For two scalar (or complex) signals $f\left(x,y\right)$and $h\left(x,y\right)$, ${A}_{1,2}=\frac{1}{2\text{π}i\sqrt{{b}_{1}{b}_{2}}}$, ${C}_{1}=\frac{{a}_{1}}{2{b}_{1}}$, ${C}_{2}=\frac{{a}_{2}}{2{b}_{2}}$, $〈f\left(x,y\right),h\left(x,y\right)〉=\underset{-\infty }{\overset{+\infty }{\int }}\underset{-\infty }{\overset{+\infty }{\int }}f\left(\tau ,\eta \right)\stackrel{¯}{h\left(x+\tau ,y+\eta \right)}\text{ }\text{d}\tau \text{d}\eta$, and set $\begin{array}{l}g\left(x,y\right)=f\left(x,y\right)\otimes h\left(x,y\right)\\ ={A}_{1,2}{\text{e}}^{-i\left({C}_{1}{x}^{2}+{C}_{2}{y}^{2}\right)}〈{\text{e}}^{i\left({C}_{1}{x}^{2}+{C}_{2}{y}^{2}\right)}f\left(x,y\right),{\text{e}}^{-i\left({C}_{1}{x}^{2}+{C}_{2}{y}^{2}\right)}h\left(x,y\right)〉,\text{then:}\end{array}$

$f\left(x,y\right)\otimes h\left(x,y\right)=\left(f\left(-x,-y\right)\right)\stackrel{¯}{\ast }\left(\stackrel{¯}{h\left(x,y\right)}\right)$(13)

Proof: the proof is similar with that of FRQCV and is omitted here.

From theorem 6 we draw the conclusion that correlation can be implemented by convolution.

Theorem 7: For any two quaternion signals $f\left(x,y\right)$and $h\left(x,y\right)$, ${A}_{1,2}=\frac{1}{2\text{π}i\sqrt{{b}_{1}{b}_{2}}}$, ${C}_{1}=\frac{{a}_{1}}{2{b}_{1}}$, ${C}_{2}=\frac{{a}_{2}}{2{b}_{2}}$, $〈f\left(x,y\right),h\left(x,y\right)〉=\underset{-\infty }{\overset{+\infty }{\int }}\underset{-\infty }{\overset{+\infty }{\int }}f\left(\tau ,\eta \right)\stackrel{¯}{h\left(x+\tau ,y+\eta \right)}\text{ }\text{d}\tau \text{d}\eta$, and let $\begin{array}{l}g\left(x,y\right)=f\left(x,y\right)\otimes h\left(x,y\right)\\ ={A}_{1,2}{\text{e}}^{-i\left({C}_{1}{x}^{2}+{C}_{2}{y}^{2}\right)}〈{\text{e}}^{i\left({C}_{1}{x}^{2}+{C}_{2}{y}^{2}\right)}f\left(x,y\right),{\text{e}}^{-i\left({C}_{1}{x}^{2}+{C}_{2}{y}^{2}\right)}h\left(x,y\right)〉\end{array}$, “ $\otimes$” is correlation operator, then

$\begin{array}{l}f\left(x,y\right)\otimes h\left(x,y\right)\\ =\left({f}_{a}\left(-x,-y\right)\right)\stackrel{¯}{\ast }\left(\stackrel{¯}{{h}_{a}\left(x,y\right)}\right)+\left({f}_{b}\left(-x,-y\right)\right)\stackrel{¯}{\ast }\left(\stackrel{¯}{{h}_{b}\left(x,y\right)}\right)\\ \text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}-{A}_{1,2}{\text{e}}^{-i\left({x}^{2}{C}_{1}+{y}^{2}{C}_{2}\right)}\left\{\left({\text{e}}^{i\left({x}^{2}{C}_{1}+{y}^{2}{C}_{2}\right)}{f}_{a}\left(-x,-y\right)\right)\ast \left(\stackrel{¯}{{h}_{b}\left(x,y\right)}\text{ }{\text{e}}^{-i\left({x}^{2}{C}_{1}+{y}^{2}{C}_{2}\right)}\right)\right\}\cdot j\\ \text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}+{A}_{1,2}{\text{e}}^{-i\left({x}^{2}{C}_{1}+{y}^{2}{C}_{2}\right)}\left\{\left({\text{e}}^{i\left({x}^{2}{C}_{1}+{y}^{2}{C}_{2}\right)}{f}_{b}\left(-x,-y\right)\right)\ast \left(\stackrel{¯}{{h}_{a}\left(x,y\right)}\text{ }{\text{e}}^{-i\left({x}^{2}{C}_{1}+{y}^{2}{C}_{2}\right)}\right)\text{\hspace{0.17em}}\right\}\cdot j\end{array}$(14)

Proof: The proof is similar with that of FRQCV and is omitted here.

From theorem 7 we draw the conclusion that the correlation of two quaternion signals is to the summation of convolution of their components, conjugated or odd-even operated. It means that correlation can be implemented by convolution via FFT.

5. Fast Algorithm of QLCT

Fast algorithm of QLCT is the key to engineering use. The following discusses the efficient implementation in great detail through the decomposition of quaternion and the definition of the QLCT. For one quaternion function $f\left(x,y\right)$, from definition 1 we have

$\begin{array}{l}{F}_{i,j}^{{L}_{1},{L}_{2}}\left\{f\left(x,y\right)\right\}=\underset{-\infty }{\overset{+\infty }{\int }}\underset{-\infty }{\overset{+\infty }{\int }}{K}_{{L}_{1},i}\left(x,u\right)f\left(x,y\right){K}_{{L}_{2},j}\left(y,v\right)\text{d}x\text{d}y\\ =\sqrt{\frac{1}{2\text{π}{b}_{1}i}}{\text{e}}^{\frac{i{d}_{1}{u}^{2}}{2{b}_{1}}}\underset{-\infty }{\overset{+\infty }{\int }}\underset{-\infty }{\overset{+\infty }{\int }}{\text{e}}^{-i\frac{ux}{{b}_{1}}}{\text{e}}^{i\frac{{x}^{2}{a}_{1}}{2{b}_{1}}}f\left(x,y\right){\text{e}}^{j\frac{{y}^{2}{a}_{2}}{2{b}_{2}}}{\text{e}}^{-j\frac{vy}{{b}_{2}}}\text{d}x\text{d}y\sqrt{\frac{1}{2\text{π}{b}_{2}j}}{\text{e}}^{\frac{j{d}_{2}{v}^{2}}{2{b}_{2}}}\\ ={G}_{i}\left(u\right)\underset{-\infty }{\overset{+\infty }{\int }}\underset{-\infty }{\overset{+\infty }{\int }}{\text{e}}^{-i\frac{ux}{{b}_{1}}}g\left(x,y\right){\text{e}}^{-j\frac{vy}{{b}_{2}}}\text{d}x\text{d}y\cdot {G}_{j}\left(v\right)\end{array}$

where,

${G}_{i}\left(u\right)=\sqrt{\frac{1}{2\text{π}{b}_{1}i}}{\text{e}}^{\frac{i{d}_{1}{u}^{2}}{2{b}_{1}}}$, ${G}_{j}\left(v\right)=\sqrt{\frac{1}{2\text{π}{b}_{2}j}}{\text{e}}^{\frac{j{d}_{2}{v}^{2}}{2{b}_{2}}}$,

$g\left(x,y\right)={\text{e}}^{i\frac{{x}^{2}{a}_{1}}{2{b}_{1}}}f\left(x,y\right){\text{e}}^{j\frac{{y}^{2}{a}_{2}}{2{b}_{2}}}={g}_{r}\left(x,y\right)+i{g}_{i}\left(x,y\right)+j{g}_{j}\left(x,y\right)+k{g}_{k}\left(x,y\right)$

${g}_{r}\left(x,y\right)$, ${g}_{i}\left(x,y\right)$are real signals.

Let

$W\left(u,v\right)=\underset{-\infty }{\overset{+\infty }{\int }}\underset{-\infty }{\overset{+\infty }{\int }}{\text{e}}^{-i\frac{ux}{{b}_{1}}}g\left(x,y\right){\text{e}}^{-j\frac{vy}{{b}_{2}}}\text{d}x\text{d}y$(17)

Then,

$\frac{W\left(u,v\right)+W\left(u,-v\right)}{2}=\underset{-\infty }{\overset{+\infty }{\int }}\underset{-\infty }{\overset{+\infty }{\int }}{\text{e}}^{-i\frac{ux}{{b}_{1}}}g\left(x,y\right)\mathrm{cos}\left(\frac{vy}{{b}_{2}}\right)\text{d}x\text{d}y$(18)

$\frac{W\left(u,v\right)-W\left(u,-v\right)}{2}=\underset{-\infty }{\overset{+\infty }{\int }}\underset{-\infty }{\overset{+\infty }{\int }}{\text{e}}^{-i\frac{ux}{{b}_{1}}}g\left(x,y\right)\mathrm{sin}\left(\frac{vy}{{b}_{2}}\right)\text{d}x\text{d}y\cdot \left(-i\right)$(19)

Therefore,

$\frac{W\left(u,v\right)+W\left(u,-v\right)}{2}+\frac{W\left(u,v\right)-W\left(u,-v\right)}{2}\cdot \left(-k\right)=\underset{-\infty }{\overset{+\infty }{\int }}\underset{-\infty }{\overset{+\infty }{\int }}{\text{e}}^{-i\frac{ux}{{b}_{1}}}g\left(x,y\right){\text{e}}^{-j\frac{vy}{{b}_{2}}}\text{d}x\text{d}y$(20)

Therefore,

${F}_{i,j}^{{L}_{1},{L}_{2}}\left(u,v\right)={G}_{i}\left(u\right)\frac{W\left(u,v\right)\left(1-k\right)+W\left(u,-v\right)\left(1+k\right)}{2}{G}_{j}\left(v\right)$(21)

Then the following task is to implement $W\left(u,v\right)$.

$g\left(x,y\right)$can be expressed as

$g\left(x,y\right)={g}_{r}\left(x,y\right)+i{g}_{i}\left(x,y\right)+j{g}_{j}\left(x,y\right)+k{g}_{k}\left(x,y\right)={g}_{a}\left(x,y\right)+{g}_{b}\left(x,y\right)\cdot j$

where, ${g}_{a}\left(x,y\right)={g}_{r}\left(x,y\right)+i{g}_{i}\left(x,y\right)$, ${g}_{b}\left(x,y\right)={g}_{j}\left(x,y\right)+i{g}_{k}\left(x,y\right)$.

Therefore,

$\begin{array}{c}W\left(u,v\right)=\underset{-\infty }{\overset{+\infty }{\int }}\underset{-\infty }{\overset{+\infty }{\int }}{\text{e}}^{-i\frac{ux}{{b}_{1}}}{\text{e}}^{-i\frac{vy}{{b}_{2}}}{g}_{a}\left(x,y\right)\text{d}x\text{d}y+\left(\underset{-\infty }{\overset{+\infty }{\int }}\underset{-\infty }{\overset{+\infty }{\int }}{\text{e}}^{-i\frac{ux}{{b}_{1}}}{\text{e}}^{-i\frac{vy}{{b}_{2}}}{g}_{b}\left(x,-y\right)\text{d}x\text{d}y\right)\cdot j\\ =F\left\{{g}_{a}\left(x,y\right)\right\}\left(\frac{u}{{b}_{1}},\frac{v}{{b}_{2}}\right)+F\left\{{g}_{b}\left(x,-y\right)\right\}\left(\frac{u}{{b}_{1}},\frac{v}{{b}_{2}}\right)\cdot j\end{array}$(22)

$W\left(u,v\right)$can be Calculated by two 2D FFT and some scaling transform. The steps of calculating QLCT:

1) Calculate $g\left(x,y\right)$from $f\left(x,y\right)$using (16);

2) Calculate $W\left(u,v\right)$from $g\left(x,y\right)$using (22) and (17);

3) Calculate ${G}_{i}\left(u\right)$and ${G}_{j}\left(v\right)$using (16);

4) At last Calculate ${F}_{i,j}^{{L}_{1},{L}_{2}}\left(u,v\right)$using (20) and (21).

For one 2D discrete signal with size M × N, one 2D-DFT needs $MN\cdot {\mathrm{log}}_{2}\left(MN\right)$real number multiplications . To implement $W\left(u,v\right)$, we require $2MN\cdot {\mathrm{log}}_{2}\left(MN\right)$real number multiplications. Therefore, the complexity of quaternion signal $f\left(x,y\right)$is $O\left(2MN\cdot {\mathrm{log}}_{2}\left(MN\right)\right)$. And the complexity of scalar, complex signal and vector is: $O\left(\frac{MN\cdot {\mathrm{log}}_{2}\left(MN\right)}{2}\right)$, $O\left(MN\cdot {\mathrm{log}}_{2}\left(MN\right)\right)$, $O\left(\frac{3MN\cdot {\mathrm{log}}_{2}\left(MN\right)}{2}\right)$.

For any discrete 2D signal $f\left(m,n\right)$( $m\in \left[1,M\right],n\in \left[1,N\right]$), it can be expressed as:

$f\left(m,n\right)={f}_{ee}\left(m,n\right)+{f}_{eo}\left(m,n\right)+{f}_{oe}\left(m,n\right)+{f}_{oo}\left(m,n\right)$

where:

${f}_{ee}\left(m,n\right)=\frac{f\left(m,n\right)+f\left(n,N-n\right)+f\left(M-m,n\right)+f\left(M-m,N-n\right)}{4}$

${f}_{oe}\left(m,n\right)=\frac{f\left(m,n\right)+f\left(n,N-n\right)-f\left(M-m,n\right)-f\left(M-m,N-n\right)}{4}$

${f}_{eo}\left(m,n\right)=\frac{f\left(m,n\right)-f\left(n,N-n\right)+f\left(M-m,n\right)-f\left(M-m,N-n\right)}{4}$

${f}_{oo}\left(m,n\right)=\frac{f\left(m,n\right)-f\left(n,N-n\right)-f\left(M-m,n\right)+f\left(M-m,N-n\right)}{4}$

If in the right side of $f\left(m,n\right)={f}_{ee}\left(m,n\right)+{f}_{eo}\left(m,n\right)+{f}_{oe}\left(m,n\right)+{f}_{oo}\left(m,n\right)$there is only one term, we call $f\left(m,n\right)$symmetric;

If $f\left(M-m,n\right)=±f\left(m,n\right)$, we call $f\left(m,n\right)$symmetric about x;

If $f\left(m,N-n\right)=±f\left(m,n\right)$, we call $f\left(m,n\right)$symmetric about y;

If any above relationship is not true, we call $f\left(m,n\right)$asymmetric.

The symmetry is of great importance to greatly decreasing the calculation complexity of them. Table 1 lists the calculation complexity of different types of signals. It gives the conclusion that the symmetry can decrease the calculation complexity by a few times. Meanwhile, the calculation complexity will increase with the number of components by a few times.

Meanwhile, the calculation complexity of QLCT for different signals is multiplications. Also, the complexity of QCV and QCR for the same type of signals is the same and is much less than calculation in time-domain directly.

Figure 1 shows one intuitive result. The QCR of the quaternion signal $f\left(x,y\right)$and kernel $h\left(x,y\right)$is calculated. We take different signals (scalar, complex, vector and quaternion) as the convolution kernel $h\left(x,y\right)$. The red lines denote the complexity of implementing QCR in time domain directly, and the blue lines denote the complexity of implementing QCR via FFT. For example,

Table 1. The calculation complexity of QLCT for different signals.

Figure 1. The comparison of complexity via FFT and calculation directly.

when the size is 60, there is one nearly-ten-times relationship. Moreover, with the increase of size the gap would become bigger and bigger.

6. Conclusion

One contribution of this paper is that the definition of QLCT is obtained, and its properties are given, and its generalization is proved. The reversibility property disclosed the efficiency of QLCT. The linear property indicated that LCT is linear transform. Another contribution of this paper is that the QCV and QCR of LCT are defined and their relationships and physical interpretation are discovered: the fractional convolution of two quaternion signals is to the summation of product of their components, conjugated or odd-even operated, and the product is frequency modulated by chirps; and the product, frequency modulated by a chirp, of two quaternion signals is to the summation, amplitude modulated, of their pseudo convolution; and the correlation of two quaternion signals is to the summation of convolution of their components, which are conjugated or odd-even operated. The last contribution is that the complexity of QLCT, QCV and QCR are given, and its Fast Algorithm is obtained through implementing them via the product theorem in transformed domain whose complexity is similar to FFT, which is of great importance to engineering use .

Acknowledgements

This work was fully supported by the NSFCs (61471412, 61771020, 61002052, 61250006).

Cite this paper: Zhang, Y. and Xu, G. (2018) The Properties and Fast Algorithm of Quaternion Linear Canonical Transform. Journal of Signal and Information Processing, 9, 202-216. doi: 10.4236/jsip.2018.93012.
References

   Exampleh, V., Ozaktas, M. and Aytur, O. (1995) Fractional Fourier Domains. Signal Process, 46, 119-124.
https://doi.org/10.1016/0165-1684(95)00076-P

   Tao, R., Deng, B. and Wang, Y. (2009) Theory and Application of the Fractional Fourier Transform. Tsinghua University Press, Beijing.

   Moshinsky, M. and Quesne, C. (1971) Linear Canonical Transformations and Their Unitary Representation. Journal of Mathematical Physics, 12, 1772-1783.
https://doi.org/10.1063/1.1665805

   Xu, G.L., Wang, X.T. and Xu, X.G. (2009) The Logarithmic, Heisenberg’s and Windowed Uncertainty Principles in Fractional Fourier Transform Domains. Signal Processing, 89, 339-343.
https://doi.org/10.1016/j.sigpro.2008.09.002

   Xu, G.L., Wang, X.T. and Xu, X.G. (2009) Generalized Hilbert Transform and Its Properties in 2D LCT. Signal Process, 89, 1395-1402.
https://doi.org/10.1016/j.sigpro.2009.01.009

   Mendlovic, D. and Ozaktas, H.M. (1993) Fractional Fourier Transforms and Their Optical Implementation: I. Journal of the Optical Society of America A, 10, 1875-1881.

   Ozaktas, H.M. and Mendlovic, D. (1993) Fractional Fourier Transforms and Their Optical Implementation. II. Journal of the Optical Society of America A, 10, 2522-2531.
https://doi.org/10.1364/JOSAA.10.002522

   Pei, S.C. and Yeh, M.H. (1998) Two Dimensional Discrete Fractional Fourier Transform. Signal Processing, 67, 99-108.
https://doi.org/10.1016/S0165-1684(98)00024-3

   Aytur, O. and Ozaktas, H.M. (1995) Non-Orthogonal Domains in Phase Space of Quantum Optics and Their Relation to Fractional Fourier Transforms. Optics Communications, 120, 166-170.
https://doi.org/10.1016/0030-4018(95)00452-E

   Ozaktas, H.M., Zalevsky, Z. and Kutay, M.A. (2000) The Fractional Fourier Transform with Applications in Optics and Signal Processing. Wiley, New York.

   Hamilton, W.R. (1864) Elements of Quaternions. Longman, London.

   Xu, G.L., Wang, X.T. and Xu, X.G. (2010) On Uncertainty Principle for the Linear Canonical Transform of Complex Signals. IEEE Transactions on Signal Processing, 58, 4916-4918.
https://doi.org/10.1109/TSP.2010.2050201

   Stark, H. (1971) An Extension of the Hilbert Transform Product Theorem. Proceedings of the IEEE, 59, 1359-1360.

   Havlicek, J.P., Havlicek, J.W., Ngao, D., et al. (1998) Skewed 2D Hilbert Transforms and Computed AM-FM Models. Proceedings 1998 International Conference on Image Processing, Chicago, 7 October 1998, 602-606.

   Xu, G.L., Wang, X.T. and Xu, X.G. (2008) Extended Hilbert Transform for Multidimensional Signals. 2008 5th International Conference on Visual Information Engineering, Xi’an, 29 July-1 August 2008, 292-297.

   Hahn, S.L. (1992) Multidimensional Complex Signals with Single-Orthant Spectra. Proceedings of the IEEE, 80, 1287-1300.

   Moxey, C.E., Sangwine, S.J. and Ell, T.A. (2003) Hypercomplex Correlation Techniques for Vector Images. IEEE Transactions on Signal Processing, 51, 1941-1953.
https://doi.org/10.1109/TSP.2003.812734

   Xu, G.L., Wang, X.T. and Xu, X.G. (2007) Neighborhood Limited Empirical Mode Decomposition and Application in Image Processing. 4th International Conference on Image and Graphics, Sichuan, 22-24 August 2007, 149-154.

   Xu, G.L., Wang, X.T. and Xu, X.G. (2008) Fractional Quaternion Fourier Transform, Convolution and Correlation. Signal Processing, 88, 2511-2517.
https://doi.org/10.1016/j.sigpro.2008.04.012

   Sangwine, S.J. and Ell, T.A. (1999) Hypercomplex Auto- and Cross-Correlation of Color Images. Proceedings of the 1999 International Conference on Image Processing, Kobe, 24-28 October 1999, 319-322.

   Ell, T.A. (1993) Quaternion-Fourier Transforms for Analysis of Two-Dimensional Linear Time-Invariant Partial Differential Systems. Proceedings of 32nd IEEE Conference on Decision and Control, San Antonio, 15-17 December 1993, 1830-1841.
https://doi.org/10.1109/CDC.1993.325510

   Xu, G.L., Wang, X.T. and Xu, X.G. (2009) Improved Bi-Dimensional EMD and Hilbert Spectrum for the Analysis of Textures. Pattern Recognition, 42, 718-734.
https://doi.org/10.1016/j.patcog.2008.09.017

   Ell, T.A. (1992) Hypercomplex Spectral Transforms. Ph.D. Dissertation, University of Minnesota, Minneapolis.

   Pei, S.C., Ding, J.J. and Chang, J.H. (2001) Efficient Implementation of Quaternion Fourier Transform, Convolution, and Correlation by 2-D Complex FFT. IEEE Transactions on Signal Processing, 49, 2783-2797.
https://doi.org/10.1109/78.960426

   Duhamel, P. (1986) Implementation of Split-Radix FFT Algorithms for Complex, Real and Real-Symmetric Data. IEEE Transactions on Acoustics, Speech, and Signal Processing, 34, 285-295.
https://doi.org/10.1109/TASSP.1986.1164811

   Almeida, L.B. (1997) Product and Convolution Theorems for the Fractional Fourier Transform. IEEE Signal Processing Letters, 4, 15-17.
https://doi.org/10.1109/97.551689

   Zayed, A.I. (1998) A Convolution and Product Theorem for the Fractional Fourier Transform. IEEE Signal Processing Letters, 5, 101-103.
https://doi.org/10.1109/97.664179

   Akay, O. and Boudreaus, G.F. (1998) Linear Fractionally Invariant Systems: Fractional Filtering and Correlation via Fractional Operators. Conference Record of the 31st Asilomar Conference on Signals, Systems and Computers, Vol. 2, Pacific Grove, 2-5 November 1997, 1494-1498.

   Deng, B., Tao, R. and Wang, Y. (2006) Convolution Theorems for the Linear Canonical Transform and Their Applications. Science in China F, 49, 592-603.
https://doi.org/10.1007/s11432-006-2016-4

   Xu, G., Wang, X. and Xu, X. (2012) On Analysis of Bi-Dimensional Component Decomposition via BEMD. Pattern Recognition, 45, 1617-1625.
https://doi.org/10.1016/j.patcog.2011.11.004

   Xu, G., Zhou, L., Wang, X. and Xu, X. (2017) Assisted Signals Based Mode Decomposition. International Conference on Image, Vision and Computing, Chengdu, 2-4 June 2017, 868-874.

   Xu, G., Wang, X., Zhou, L. and Xu, X. (2018) Image Decomposition and Texture Analysis via Combined Bi-Dimensional Bedrosian’s Principles. IET Image Processing, 12, 262-273.
https://doi.org/10.1049/iet-ipr.2017.0494

Top