#### Abstract

General convolution theorems for two-dimensional quaternion Fourier transforms (QFTs) are presented. It is shown that these theorems are valid not only for real-valued functions but also for quaternion-valued functions. We describe some useful properties of generalized convolutions and compare them with the convolution theorems of the classical Fourier transform. We finally apply the obtained results to study hypoellipticity and to solve the heat equation in quaternion algebra framework.

#### 1. Introduction

Convolution is a mathematical operation with several applications in pure and applied mathematics such as numerical analysis, numerical linear algebra, and the design and implementation of finite impulse response filters in signal processing. In [1–3], the authors introduced the Clifford convolution. It is found that some properties of convolution, when generalized to the Clifford Fourier transform (CFT), are very similar to the classical ones.

On the other hand, the quaternion Fourier transform (QFT) is a nontrivial generalization of the classical Fourier transform (FT) using quaternion algebra. The QFT has been shown to be related to the other quaternion signal analysis tools such as quaternion wavelet transform, fractional quaternion Fourier transform, quaternionic windowed Fourier transform, and quaternion Wigner transform [4–9]. A number of already known and useful properties of this extended transform are generalizations of the corresponding properties of the FT with some modifications, but the generalization of convolution theorems of the QFT is still an open problem. In the recent past, several authors [10–13] tried to formulate convolution theorems for the QFT. But they only treated them for real-valued functions which is quite similar to the classical case. In [14], the authors briefly introduced, without proof, the QFT of the convolution of two-dimensional quaternion signals.

In this paper, we establish general convolutions for QFT. Because quaternion multiplication is not commutative, we find new properties of the QFT of convolution of two quaternion-valued functions. These properties describe closely the relationship between the quaternion convolution and its QFT. The generalization of the convolution theorems of the QFT is mainly motivated by the Clifford convolution of general geometric Fourier transform, which has been recently studied in [15, 16]. We further establish the inverse QFT of the product of the QFT, which is very useful in solving partial differential equations in quaternion algebra framework.

This paper consists of the following sections. Section 2 deals with some results on the real quaternion algebra and the definition of the QFT and its basic properties. We also review some basic properties of QFT, which will be necessary in the next section. Section 3 establishes convolution theorems of QFT and some of their consequences. Section 4 presents an application of QFT to study hypoellipticity and to solve the heat equation in quaternion algebra. Some conclusions are drawn in Section 5.

#### 2. Quaternion Algebra

For convenience, we specify the notation used in this paper. The quaternion algebra over , denoted by , is an associative noncommutative four-dimensional algebra,
which obeys the following multiplication rules:
For a quaternion , is called the *scalar* part of denoted by and is called the *vector* (or *pure*) part of . The vector part of is conventionally denoted by . Let , and let , be their vector parts, respectively. It is common to write for short
Then, (2) yields the quaternionic multiplication as
The quaternion conjugate of , given by
is an anti-involution; that is,
From (5) we obtain the norm or modulus of defined as
It is not difficult to see that
Using the conjugate (5) and the modulus of , we can define the inverse of as
which shows that is a normed division algebra. As in the algebra of complex numbers, we can define three nontrivial quaternion involutions [10]:

Hereinafter, besides the quaternion units , , and and the vector part of a quaternion , we will use the real vector notation: and so on when there is no confusion. This gives the following definition.

*Definition 1 (see [10]). *A function is called quaternionic Hermitian if, for the involutions and ,
for each .

For any unit quaternion and for any vector the action of the operator on is equivalent to a rotation of the vector through an angle about as the axis of rotation.

It is convenient to introduce an inner product for two functions as follows: In particular, for , we obtain the scalar product of the above inner product (15) given by

##### 2.1. Multiindices and Derivatives

A couple of nonnegative integers is called a *multiindex*. We denote
and for ,
Derivatives are conveniently expressed by multiindices:
Denote by the standard basis of . The *vector differential* along the direction is defined by
where .

##### 2.2. QFT and Its Properties

*Definition 2. *The QFT of is the transform given by the integral
where is called the quaternion Fourier transform operator or the quaternion Fourier transformation.

Using the Euler formula for the quaternion Fourier kernel , we can rewrite (21) in the following form:

*Definition 3. *The inverse QFT of is the transform given by the integral

Some important properties of the QFT are stated in the following lemmas proved in [12, 17].

Lemma 4. *Let . If , then
*

In particular, if , then And if , then

Lemma 5 (scalar QFT Parseval). *The scalar product of and its QFT are related by
**
And in particular, with , the Plancherel theorem indicates that
*

This shows that the total signal energy computed in the spatial domain is equal to the total signal energy computed in the quaternion domain.

#### 3. Convolution of QFT

In this section, we establish the quaternion convolution of the QFT which extends the classical convolution to quaternion fields. Let us first define the convolution of two quaternion-valued functions.

*Definition 6. *The convolution of and , denoted by , is defined by

*Example 7. *To illustrate the general noncommutativity , let us compute the convolution of and . Although , we can still define the convolution of and , because decays rapidly at infinity. A simple calculation gives
On the other hand, we have

In the following, we summarize the elementary properties of quaternion convolution as shown in Table 1 (compared to Folland [18]).

Lemma 8 (linearity). *For quaternion functions , and and quaternion constants and one gets
**
One also gets for real constants and (due to the noncommutativity of the quaternion multiplication, (33) does not hold for quaternion constants and )
*

Lemma 9 (shifting). *Given a quaternion function , let denote the shifted (translated) function defined by , where . Then one gets
*

*Proof. *For (34), a direct calculation gives
which finishes the proof.

*Remark 10. *From (34) and (35), it is not difficult to see that and .

Lemma 11 (conjugation). *For all quaternion functions one has
*

*Proof. *A straightforward computation gives
This finishes the proof.

Lemma 12 (see [2, 3]; vector differential). *For all quaternion functions one has
*

Ell and Sangwine [19] distinguish between right and left discrete quaternion convolution due to the non-commutative property of the quaternion multiplication. Here, we only consider one kind of quaternion convolutions. We come now to the main theorem (generalization of the QFT of the quaternion convolution in general geometric Fourier transform is investigated in [15]. It can easily be seen that the result is closely related to equation of [15]) of this paper. This theorem describes the relationship between the convolution of two quaternion functions and its QFT.

Theorem 13. *Let and be two quaternion-valued functions, then the QFT of the convolution of and is given by
*

*Proof. *In this proof we will use the decomposition of quaternion functions and their QFTs. Let and denote the QFT of and , respectively. Expanding the QFT of the left-hand side of (40), we immediately get

By the change of variables , the above transform can be written as
where the assumption for is used in the fourth line. This gives the desired result.

The following lemmas are special cases of Theorem 13.

Lemma 14. *Let , where
**
If , then (40) takes the form
**
On the other hand, if , then
*

*Proof. *We only prove expression (44) of Lemma 14, with the other being similar. Following the steps of (41) we immediately get
which was to be proven.

Lemma 15. *Let , where
**
If , then
**
which is of the same form as a convolution of the classical Fourier transform [20]. *

*Remark 16. *It is important to notice that, if , where
then Lemma 15 reduces to
where .

Table 2 compares convolution theorems of the QFT and classical FT for .

The following theorem is useful for solving the heat equation in quaternion algebra.

Theorem 17. *If and , then
*

*Proof. *By the QFT inversion, we get, after some simplification,
where, in the second line, we have used the assumption . This completes the proof of (51).

As an immediate consequence of Theorem 17, we get the following corollaries.

Corollary 18. *If and , where
**
then (51) reduces to
*

Corollary 19. *Let
**
And consider the quaternionic Gabor filter
*

Then,

*Proof. *The QFT of is given by
and the QFT of is given by
Therefore, using Corollary 18, we obtain (57).

#### 4. Applications of QFT

In [19], the authors proposed to use quaternions in order to define a Fourier transform applicable to color images. Their framework makes it possible to compute a single, holistic, Fourier transform which treats a color image as a vector field. In image processing, taking a given image as the initial value, the forward solution to the heat equation or a diffusion equation in general, produces blurred images and the backward solution produces sharpen images for example, see [21, pages 342–350].

In this section, we present two applications of QFT to partial differential equations in quaternion algebra.

##### 4.1. Hypoellipticity

In this paper, since we only deal with QFT in the framework, we will discuss the hypoellipticity in this framework; that is, we will only deal with solutions for linear partial differential operators with constant quaternion coefficients: The noncommutativity of quaternion gives different aspects of with constant complex coefficients .

*Example 20. *Let , , and . (i)Since , we have when . But, when , as in general, we cannot have in general.(ii)We have when . But, by the same reason as (i), when , we cannot have in general.

Let us start with the definition of our version of hypoellipticity (compared to [22, page 110]).

*Definition 21. *The linear partial differential operator in is said to be -hypoelliptic if, given any subset of and any solution in such that is a function in , then all its components are a function in .

*Definition 22. *Given a linear partial differential operator of (60) with the quaternion constant coefficients. One says that a solution of , where is the delta function, is called a fundamental solution of .

Let and be subsets of . Define the sum by .

Theorem 23. *Assume that there is one fundamental solution of which is a function in , and the identities
**
are satisfied for arbitrary sufficiently smooth quaternion-valued functions and such that is a compactly supported quaternion function with of being quaternion constant coefficients and of being real constant coefficients. Then, the linear partial differential operator is -hypoelliptic in . *

*Proof. *Firstly, let be an arbitrary open subset of and a solution in with values in such that is a function in . Let be an arbitrary point in . It will suffice to show that is a function in some open neighborhood of . Take an open disc such that . There exists a function such that and in . Then, we have
where every term of contains a derivative of of nonzero order; therefore , where the derivatives of vanish, especially in and outside of . For the fundamental solution , the hypothesis (61) implies
Hence
But is a compactly supported function and the convolution of any function with any compactly supported function is a function. Therefore, it suffices to show that is a function in an open neighborhood of , because is also a function in an open neighborhood of and in .

Finally, we will show that is a function in an open neighborhood of . Let us select such that . Then, the open disc is a neighborhood of . Let , another cutoff function, be equal to one for and to zero for . We have
The hypothesis implies that , and therefore . Since
is contained in the -neighborhood of . We have already seen that . Hence, vanishes in , and, therefore, is a function in .

##### 4.2. Parabolic Initial Value Problem

Let us consider the parabolic initial value problem with where is the quaternion Schwartz space. Applying the QFT, we easily obtain The general solution of (69) is given by where is a quaternion constant. We impose the initial condition to obtain Notice that the QFT of a Gaussian quaternion function is also a Gaussian quaternion function (compared to Bahri et al. [13]). Hence Applying the inverse QFT, we have Since then we can apply the convolution theorem of (51) to get where , and , . By Definition 6 of the convolution, we finally obtain In an actual application, one often takes the quaternionic Gabor filter (see [6, 10]) as Therefore, the above identity will reduce to

#### 5. Conclusion

Due to the non-commutative property of quaternion multiplication, there are three different types of two-dimensional QFTs. These three QFTs are the so-called left-sided QFT, right-sided QFT, and a double-sided QFT, respectively. In this work, we have established convolution theorem of the double-sided QFT applied to real fields and quaternion fields . Some important properties of the QFT convolution are investigated. We have shown that the QFT convolution is useful to study hypoellipticity and to solve the heat equation in quaternion algebra framework. It can easily be seen that the solution of generalized heat equation is extension of solution of the classical heat equation.

The future work will establish the convolution theorems of the right-sided QFT. We compare some properties of the convolution theorems of the two types of QFTs. We will apply the properties to find the solution of partial differential equations in quaternion algebra framework. The solutions of generalized partial differential equations using the properties of the three types of two-dimensional QFTs will be compared too.

#### Acknowledgments

The authors would like to thank the reviewers whose deep and extensive comments greatly contributed to improve this paper. The first author is partially supported by Hibah Penelitian Kompetisi Internal Tahun 2013 (no. 110/UN4-.42/LK.26/SP-UH/2013) from the Hasanuddin University, Indonesia. The second author is partially supported by JSPS.KAKENHI (C)22540130 and (C)25400202 of Japan and the third author is partially supported by NSERC of Canada.