RETOOLING OF COLOR IMAGING IN THE QUATERNION ALGEBRA

Applied Mathematics and Sciences: An International Journal (MathSJ ), Vol. 1, No. 3, December 2014
23
RETOOLING OF COLOR IMAGING IN THE
QUATERNION ALGEBRA
Artyom M. Grigoryan and Sos S. Agaian
Department of Electrical and Computer Engineering,
University of Texas at San Antonio, USA
ABSTRACT
A novel quaternion color representation tool is proposed to the images and videos efficiently. In this work,
we consider a full model for representation and processing color images in the quaternion algebra. Color
images are presented in the threefold complex plane where each color component is described by a
complex image. Our preliminary experimental results show significant performance improvements of the
proposed approach over other well-known color image processing techniques. Moreover, we have shown
how a particular image enhancement of the framework leads to excellent color enhancement (better than
other algorithms tested). In the framework of the proposed model, many other color processing algorithms,
including filtration and restoration, can be expressed.
KEYWORDS
Image Color Analysis, Discrete Fourier Transform, Quaternion Fourier Transformation
1. INTRODUCTION
Color image processing has attracted much interest in the recent years [1],[2] ,[43],[52]. The
reason of these are: a) color features are robust to several image processing procedures (for
example translation and rotation of the regions of interest) b) color features are efficiently used in
many vision tasks, including object recognition, tracking, image segmentation and retrieval,
image registration etc.; c) color is of vital importance in many real life applications such as visual
communications, multimedia systems, fashion and food industries, computer vision,
entertainment, consumer electronics, production printing and proofing, book publishing, digital
photography, digital artwork reproduction, industrial inspection, and biomedical applications
[1],[2],[5],[43],[44]. Over the years, several important contributions were made in the color image
processing systems [2],[52]. Additionally, the traditional color image processing approaches are
based on dealing out each color-channel (red, green, and blue) separately [1],[2],[44]. However,
this methodology fails to capture the inherent correlation between the components and results in
color artefacts [6],[28],[43],[44]. It is natural to ask, how to couple the information contained in
the given color- channels, how to process the three color components as a whole unit without loss
of the spectral relation that is present in them, or how to develop a mathematical color model
that may help to process the color components simultaneously.
Recently, the theory of the quaternion algebra has been used in the application of color science
and color systems which process the three color channels simultaneously [6]-[10]. Quaternions

24
were first discovered by Hamilton in 1843” [3]. A quaternion q is an extension of complex
numbers and has four components; one is a “real” scalar number, and the other three mutually
orthogonal components i,j,k, i.e., q=a+bi+cj+dk, where the coefficients a,b,c, and d are real [6]-
[10],[53]. Currently, quaternions have an awe-inspiring amount of influence on various areas of
mathematics and physics, including group theory, topology, quantum mechanics, computer
graphic, etc [4],[5],[34],[35]. More recently, quaternions have been employed in bioinformatics,
navigation systems [5], and image and video processing [6]-[8],[53]. Quaternion algebra for color
image was first used by Pei and it led to the description of new tools, such as quaternion Fourier
transforms and correlation for image processing by represented the red, green, and blue values at
each pixel in the color image as a single pure quaternion valued pixel [6]. In recent years, there
have been a number of studies on quaternions in color image processing [12]-[13],[32]-[36],[53].
But all these color processing systems are using pure complex quaternions representation but not
the complete quaternions components. Therefore, it is natural to ask, how to use the complete
quaternions representation, or more precisely, how to use the “real” scalar number information in
the color image processing applications, or what the advantage of the use of the complete
representation model over the pure complex quaternions model, particularly in the color image
processing applications.
In this paper, we provide a new view of expressing color images using quaternion-based
representation. We consider a full model for representation and processing color images in the
quaternion algebra. Color images are presented in the threefold complex plane where each color
component is described by a complex image. The key contributions of this work are a) an
extending model for representing and processing color images by describing each color
component as a complex image, b) the practice of the complete quaternions representation models
in color image processing application, c) the advantages of the presented approach by using a
color image enhancement procedure. The rest of this paper is organized as follows. Section II
introduces the background of quaternion algebra and color representation models. Section III
presents a new view of expressing color images using quaternion-based representation. Section
IV gives the experimental results for color image enhancement by using presented new view of
expressing color images using quaternion. Finally, it concludes in Section V that the proposed
new quaternion image model is a powerful tool in color image analysis and processing domain
which may have many other applications.
2. QUATERNION NUMBERS AND COLOR IMAGES
In recent years, the quaternion algebra has been applied more and more in color image
processing. In quaternions the imaginary part of the complex number is extended to three
dimensions, i.e., it has three imaginary parts. The imaginary dimensions are represented as ݅, ݆,
and ݇, which are orthogonal to each other and to real numbers. Any quaternion is represented in
a hyper-complex form as ‫ݍ‬ = ܽ + (ܾ݅ + ݆ܿ + ݀݇)= ܽ + ܾ݅ + ݆ܿ + ݀݇, where the coefficients
ܽ, ܾ, ܿ, and ݀ are real numbers and ݅, ݆, and ݇ are three imaginary units with the following
multiplication laws:
݆݅ = −݆݅ = ݇, ݆݇ = −݆݇ = ݅, ݇݅ = −݅݇ = −݆, ݅ଶ
= ݆ଶ
= ݇ଶ
= ݆݅݇ = −1.
The number ܽ is referred to as the “real” part of q and (ܾ݅ + ݆ܿ + ݀݇) is the “imaginary” part of
‫.ݍ‬ We also will use the following notation for the quaternion number: ‫ݍ‬ = ‫ݍ‬௘ + ݅‫ݍ‬௜ + ݆‫ݍ‬௝ + ݇‫ݍ‬௞.
The quaternion conjugate and modulus of q equal = ܽ − (ܾ݅ + ݆ܿ + ݀݇) and
|‫|ݍ‬ = √ܽଶ + ܾଶ + ܿଶ + ݀ଶ, respectively. The quaternion conjugate is ‫ݍ‬ത = ‫ݍ‬௘ − ݅‫ݍ‬௜ − ݆‫ݍ‬௝ − ݇‫ݍ‬௞.

25
The quaternion can be represented in classic polar form as ‫ݍ‬ = |‫|ݍ‬exp(ߤߴ), where ߤ is a unit
pure quaternion ߤ = ݅ߤ௜ + ݆ߤ௝ + ݇ߤ௞, such that |ߤ| = 1, and ߴ is a real angle in the interval
[0, ߨ]. The exponential number is defined as exp(ߤߴ) = cos(ߴ) + ߤ sin(ߴ). When multiplying
quaternion numbers, it should be noted that commutate property does not hold in quaternion
algebra, i.e., ‫ݍ‬ଵ‫ݍ‬ଶ ≠ ‫ݍ‬ଶ‫ݍ‬ଵ. In matrix form, the product of these numbers is
‫ۏ‬
‫ێ‬
‫ێ‬
‫ۍ‬
(‫ݍ‬ଵ‫ݍ‬ଶ)௘
(‫ݍ‬ଵ‫ݍ‬ଶ)௜
(‫ݍ‬ଵ‫ݍ‬ଶ)௝
(‫ݍ‬ଵ‫ݍ‬ଶ)௞‫ے‬
‫ۑ‬
‫ۑ‬
‫ې‬
=
‫ۏ‬
‫ێ‬
‫ێ‬
‫ێ‬
‫ۍ‬
(‫ݍ‬ଵ)௘ − (‫ݍ‬ଵ)௜ − (‫ݍ‬ଵ)௝ − (‫ݍ‬ଵ)௞
(‫ݍ‬ଵ)௜ + (‫ݍ‬ଵ)௘ − (‫ݍ‬ଵ)௞ + (‫ݍ‬ଵ)௝
(‫ݍ‬ଵ)௝ + (‫ݍ‬ଵ)௞ + (‫ݍ‬ଵ)௘ − (‫ݍ‬ଵ)௜
(‫ݍ‬ଵ)௞ − (‫ݍ‬ଵ)௝ + (‫ݍ‬ଵ)௜ + (‫ݍ‬ଵ)௘‫ے‬
‫ۑ‬
‫ۑ‬
‫ۑ‬
‫ې‬
‫ۏ‬
‫ێ‬
‫ێ‬
‫ۍ‬
(‫ݍ‬ଶ)௘
(‫ݍ‬ଶ)௜
(‫ݍ‬ଶ)௝
(‫ݍ‬ଶ)௞‫ے‬
‫ۑ‬
‫ۑ‬
‫ې‬
.
The quaternion number ‫ݍ‬ = ‫ݍ‬௘ + ݅‫ݍ‬௜ + ݆‫ݍ‬௝ + ݇‫ݍ‬௞ is referred to as a vector ‫ݍ‬ = (‫ݍ‬௘, ‫ݍ‬௜, ‫ݍ‬௝, ‫ݍ‬௞) in
the 4-D real space ܴସ with basic vectors ݁ = (1,0,0,0), i= (0,1,0,0), j= (0,0,1,0), and k=
(0,0,0,1). The dot product of two quaternion numbers ‫ݍ‬ଵ and ‫ݍ‬ଶ is defined as
‫ݍ‬ଵ ∙ ‫ݍ‬ଶ = |‫ݍ‬ଵห|‫ݍ‬ଶ| cos(ߴ൯ = (‫ݍ‬ଵ)௘(‫ݍ‬ଶ)௘ + (‫ݍ‬ଵ)௜(‫ݍ‬ଶ)௜ + (‫ݍ‬ଵ)௝(‫ݍ‬ଶ)௝ + (‫ݍ‬ଵ)௞(‫ݍ‬ଶ)௞.
2.1. Color Image Models
In this section, we consider a few models of colors that are used in color imaging [44].
RGB Model: Three primary color components, R(ed), G(reen), and B(lue) of a pixel are
transferred to three imaginary parts of quaternion numbers with dimensions ݅, ݆, and ݇,
respectively. A discrete color image ݂௡,௠ can therefore be transformed into the imaginary part of
quaternion numbers, by considering the red, green, and blue components of the image as pure
quaternions (with zero real part):
݂௡,௠ = 0 + (‫ݎ‬௡,௠݅ + ݃௡,௠ ݆ + ܾ௡,௠ ݇).
Figure 1 shows the color map of the colors (‫,ݎ‬ ݃, ܾ) into the quaternion space (1, ݅, ݆, ݇).
Figure 1. RBG color cube in the quaternion subspace.

26
The colors in this model are calculated by color components as ‫ܥ‬ = ‫ܴݎ‬ + ݃‫ܩ‬ + ܾ‫.ܤ‬ Practically,
the color is expressed as the triplet (‫,ݎ‬ ݃, ܾ), each component of which can vary from zero to a
defined maximum value. For example, the triplet (‫,ݎ‬ ݃, ܾ) = (255,0,0) is expressed the red color
(1,0,0); the triplet (0,255,0) expresses the green color (0,1,0); the triplet (0,0,255) expresses the
blue color (0,0,1). If the triplet (‫,ݎ‬ ݃, ܾ) is (0,255,255) the result is expressed the magenta color
(‫ܯ‬ = ܴ + ‫ܤ‬ = (1,0, 0) + (0,0,1) = (1,0,1), if all components are at zero, the result is black; if
all components are at maximum, the result is the brightest representable white. The red and green
lights together produce the yellow. Approximately 65% of all cones in the retina are sensitive to
the red light, 33% are sensitive to the green light and about 2% are sensitive to the blue light
(most sensitive). This RGB color model was described by Thomas Young and Herman Helmholtz
in their publication "Theory of trichromatic color vision" (first half of the 19th century) and by
James Maxwell's (color triangle). RGB is a convenient color model for computer graphics and it
is mostly used for recording colors in digital cameras/scanners, including still image and video
cameras. There are various types of models based on commonly used RGB color model, for
example, RGB ProPhoto RGB, scRGB, and CIE RGB and sRGB.
CMYK color model: The mixed colors in this model are the primary colors of pigment, which are
C(yan), M(agenta), and Y(ellow). This model of colors covers a large part of the human color
space. The primary colors from RGB color space are transferred to CMYK space by the
following simple operations:
‫ܥ‬ = 1 − ܴ, ‫ܯ‬ = 1 − ‫,ܩ‬ ܻ = 1 − ‫,ܤ‬
and the additional forth color, black, as ‫ܭ‬ = min(‫,ܥ‬ ‫,ܯ‬ ܻ) with the following change of colors:
‫ܥ‬ = ‫ܥ‬ − ‫,ܭ‬ ‫ܯ‬ = ‫ܯ‬ − ‫,ܭ‬ and ܻ = ܻ − ‫.ܭ‬
HSI color model: The Hue-Saturation-Intensity color model is a non-linear transformation of the
RGB color space. The transformation of colors R, G, and B into the corresponding H, S, and I
values in this model is calculated as follows:
‫ܪ‬ = ൜
ߴ, if ‫ܤ‬ ≤ ‫ܩ‬
360 − ߴ, if ‫ܤ‬ > ‫ܩ‬
‫ܫ‬ =
ܴ + ‫ܩ‬ + ‫ܤ‬
3
ܵ = 1 −
minሼܴ, ‫,ܩ‬ ‫ܤ‬ሽ
‫ܫ‬
.
Here, the angle (in degrees) is calculated by
cos(ߴ) =
1
2
2ܴ − ‫ܩ‬ − ‫ܤ‬
ඥ(ܴ − ‫)ܩ‬ଶ + (ܴ − ‫ܩ()ܤ‬ − ‫)ܤ‬
.
In quaternion space, these three components of the HSI model are defined in the following way
[45]. The value (‫)ܫ‬ component is referred as the norm of the quaternion vector ‫ݍ‬ on the gray axis
(axis of real part of ‫,)ݍ‬ which is (‫ݍ‬ ∙ ߤ)ߤ, where for instance ߤ = (1 + ݆ + ݇)/√3. The saturation
is referred to as the angle between the vectors corresponding to numbers ‫ݍ‬ and ߤ. The hue is

27
defined by a reference vector ‫ݒ‬ which is orthogonal to ߤ, for instance, a vector in red color
direction. These three values of the color model can be calculated as
‫ܪ‬ = atan ቆ
|‫ݍ‬ − ߤ‫|ߤݒݍݒ‬
|‫ݍ‬ − ‫|ݒݍݒ‬
ቇ ,
ܸ = (‫ݍ‬ − ߤ‫,2/)ߤݍ‬ ܵ = |‫ݍ‬ + ߤ‫.2/|ߤݍ‬
CIE XYZ color model: In the XYZ model, a mathematical formula is used to convert the RGB
data to a system of positive integers as values ܺ, ܻ, and ܼ, which are approximately correspond to
red, green, and blue values, respectively. To obtain the ܻܼܺ tristimulus values from the primary
colors R, G, and B, the following formula is used:
൥
ܺ
ܻ
ܼ
൩ =
1
0.17697
൥
0.49 0.31 0.2
0.17697 0.8124 0.01063
0 0.01 0.99
൩൥
ܴ
‫ܩ‬
‫ܤ‬
൩.
The transformation of values ܺ, ܻ, and ܼ into the quaternion space is similar to the RGB color
model, i.e., (ܺ, ܻ, ܼ) → 0 + (݅ܺ + ݆ܻ + ܼ݇).
Since the color information of the image is transformed in quaternions, the discrete color image in
the quaternion algebra is processed as a single matrix. In the traditional approach, the color image
is processed separately by each color component. In other words, the processing of the color
image is reduced to processing of three gray-scale images independently. It was shown in [36],
that the use of quaternions type representation is that a color image is treated as a vector field or
the hyper-complex Fourier transforms can handle color image pixels as vectors and thus offer
scope to process color images holistically; rather than as separated luminance and chrominance,
or separate color space components (example: red, green, blue). The use of the Fourier transform
in color imaging is a new and interesting topic in image processing [24]-[27]. As the
generalization of the traditional Fourier transform, the quaternion Fourier transform was first
defined to process quaternion signals [22]. Later, some practical works related to the quaternion
discrete Fourier transforms (QDFT) and their applications in color image processing were
presented in [23] and [28].
3. MODIFIED COLOR IMAGE REPRESENTATION AND THE 2-D QDFT
In this section, we consider new methods of representation of color image in the quaternion space
and their 2-D QDFTs. Different 2-D quaternion DFTs can be used in image processing, including
the right-side and left-side DQFTs [23],[24],[27]. These two transforms are described similarly.
Therefore, we consider the right-side 2-D DQFT.
The color image ݂௡,௠ is considered to be of size ܰ × ‫.ܯ‬ For the color image in the RGB color
space ݂௡,௠ = (‫ݎ‬௡,௠, ݃௡,௠, ܾ௡,௠) represented in the quaternion algebra as
݂௡,௠ = ݅൫‫ݎ‬௡,௠൯ + ݆൫݃௡,௠൯ + ݇൫ܾ௡,௠൯, (1)

28
the right-side 2-D QDFTs are defined as
‫ܨ‬௣,௦ = ෍ ൭ ෍ ݂௡,௠ܹఓ;ெ
௠௦
ெିଵ
௠ୀ଴
൱ ܹఓ;ே
௡௣
, ‫݌‬ = 0: (ܰ − 1), ‫ݏ‬ = 0: (‫ܯ‬ − 1),
ேିଵ
௡ୀ଴
(2)
where ߤ is an unit pure quaternion ߤ = ݉ଵ ݅ + ݉ଶ݆ + ݉ଷ݇, ߤଶ
= −1. The kernel of the
transform is defined by the periodic exponential functions
ܹఓ;ே
௧
= exp ൬−ߤ
2ߨ‫ݐ‬
ܰ
൰ = cos൬
2ߨ‫ݐ‬
ܰ
൰ − ߤ sin ൬
2ߨ‫ݐ‬
ܰ
൰ , ‫ݐ‬ = 0: (ܰ − 1),
and ܹఓ;ெ
௧
defined similarly. The inverse 2-D QDFT is calculated by
݂௡,௠ =
1
ܰ‫ܯ‬
෍ ቌ෍ ‫ܨ‬௣,௦ܹఓ;ே
ି௡௣
ேିଵ
௣ୀ଴
ቍ ܹఓ;ெ
ି௠௦
, ݊ = 0: (ܰ − 1), ݉ = 0: (‫ܯ‬ − 1).
ெିଵ
௦ୀ଴
(3)
As an example, the color “Lena” image of size 256 × 256 is shown in Figure 2 in part a.
Figure 2. (a) Color image and (b) 2-D QDFT of the quaternion the image.
The 2-D QDFT of the quaternion image ݂௡,௠ in absolute scale and shifted to the center in part b.
3.1. Model with Gray-Scale Average Image
In this section, we consider a few models which are used in our study for color image
enhancement. A quaternion number has four components, and when transforming the color image
݂௡,௠ from the RGB color space into the quaternion algebra, the color image is presented as
݂௡,௠ = (‫ݎ‬௡,௠݅ + ݃௡,௠݆ + ܾ௡,௠݇), i.e., with the real part equal zero. Color imaged can be
represented in different color model for different applications. A color model is an abstract
mathematical model describing a way the colors can be represented as ݊-tuple (ordered list of
elements) of numbers (e.g. (red, green, blue) in the RGB color model and (hue, saturation,
intensity) in HSI model, or four in CMYK (cyan, magenta, yellow and black). Another question
arises here how to handle the 4-tuple (CMYK) cases, and what is a best way to plug the primary
colors into the quaternion representation. Since the 2-D QDFT is defined not only to process

29
color images in the frequency domain, and quaternion images with non zero real parts, we suggest
to fill the real part of the quaternion image by a gray-scale image and use the complete 2-D
QDFT. Figure 3 shows the threefold complex plane ‫ܥ‬଺
or three complex planes intersected
between themselves along one real line ܴଵ in part a. This is a space for all quaternion numbers.
These three complex planes ‫ܥ‬ଶ
of the threefold complex space are colored in the primary colors,
red, green, and blue, since we want to use these planes for the RGB color model. The traditional
representation of color images from the RGB color space into the quaternion subspace of
numbers with zero real parts is shown in part b. In part c, the mapping of quaternions into a subset
of numbers with non zero real parts is given.
Figure 3. Transformations from the 6-D complex space: (a) The threefold complex plane ((‫ܥ‬ଶ
)ଷ
or ‫ܥ‬଺
) of
quaternions, (b) the subset (ܴଷ
) of quaternions for color images in RGB model, and (c) a new subset (ܴସ
) of
quaternions for the model of color images with nonzero gray images.
For model shown in c, the image ܽ௡,௠ = (‫ݎ‬௡,௠ + ݃௡,௠ + ܾ௡,௠)/3 can be considered as such
gray-scale image. Our preliminary results in image enhancement by the quaternion discrete
Fourier transform show, that this real gray-scale component of the quaternion image can be
enhanced together with the color image [28]. This enhancement differs from the gray-scale image
calculated as the average of processed three color components. Therefore, we define the
quaternion-color image by
‫ݍ‬௡,௠ = ܽ௡,௠ + ݂௡,௠ =
‫ݎ‬௡,௠ + ݃௡,௠ + ܾ௡,௠
3
+ ൫‫ݎ‬௡,௠݅ + ݃௡,௠݆ + ܾ௡,௠݇൯. (4)
This quaternion image can be written as a sum of three complex images
‫ݍ‬௡,௠ = ቀ
‫ݎ‬௡,௠
3
+ ‫ݎ‬௡,௠݅ቁ + ቀ
݃௡,௠
3
+ ݃௡,௠݆ቁ + ൬
ܾ௡,௠
3
+ ܾ௡,௠݇൰ (5)
new proposed
traditional
Projections to color space
(a)
(b)
(c)
green
blue
red
k
j
i
0
ܴଵ
ܴଵ
ܴଵܴଵ
‫ܥ‬ଶ
‫ܥ‬ଶ
‫ܥ‬ଶ
k i
j
ܴଵ
ܴଵ
ܴଵ
ܴଵ

30
and
‫ݍ‬௡,௠ = ൬
1
3
+ ݅൰ ‫ݎ‬௡,௠ + ൬
1
3
+ ݆൰ ݃௡,௠ + ൬
1
3
+ ݇൰ ܾ௡,௠ . (6)
The right-side 2-D QDFT over the quaternion image ‫ݍ‬௡,௠ is defined as
ܳ௣,௦ = ෍ ൭ ෍ ‫ݍ‬௡,௠ܹఓ;ெ
௠௦
ெିଵ
௠ୀ଴
൱ܹఓ;ே
௡௣
, ‫݌‬ = 0: (ܰ − 1), ‫ݏ‬ = 0: (‫ܯ‬ − 1).
ேିଵ
௡ୀ଴
(7)
This also can be written as the modified QDFT (mQDFT)
ܳ௣,௦ = ෍ ൭ ෍ ܽ௡,௠ܹఓ;ெ
௠௦
ெିଵ
௠ୀ଴
൱ ܹఓ;ே
௡௣
+ ‫ܨ‬௣,௦ .
ேିଵ
௡ୀ଴
(8)
As an example, Figure 4 shows the gray and color tree images in part a and b, respectively.
Fig. 4. (a) The gray-scale tree image, (b) color three image, (c) 2-D QDFT of the quaternion tree image, and
(d) the difference of 2-D QDFTs of the quaternion and color tree images (in absolute scale).
In this case, the real part ܽ௡,௠ of the quaternion image is the image in a and the imaginary part is
the color image ݂௡,௠ in b. The 2-D QDFT of the quaternion tree image ݃௡,௠ in absolute scale and
shifted to the center is shown in part c, and the difference of 2-D QDFTs of the quaternion and
color tree images in d. The processing of the quaternion image will result in not only a new color
image and a new gray-scale image as well. (An example of processing different gray-scale and
color images in one quaternion image is given in Section IV.)
The number of operations for calculating this 2-D QDFT will increase on the amount required for
calculating ܰ complex ‫-ܯ‬point 1-D QDFTs instead of real ‫-ܯ‬point 1-D QDFTs. Here, we
remind that the complex ‫-ܯ‬point 1-D QDFT can be accomplished by two complex ‫-ܯ‬point
DFTs, and the real ‫-ܯ‬point 1-D QDFT can be accomplished by one complex and one real ‫-ܯ‬
point DFTs, for which fast algorithms can be used [13]-[21].The time difference for calculating

31
the 2-D QDFTs ܳ௣,௦ and ‫ܨ‬௣,௦ is therefore small, as shown in Table 1 for a few cases when ‫ܯ‬ = ܰ
and ܰ is a power of two. The transforms were calculated in MATLAB on a personal computer
with Intel(R) Core(TM) i3 CPU Processor at 3.20GHz speed.
Table 1: Time data for calculating the ܰ × ܰ-point real and complex 2-D QDFTs.
3.2. Model with Gray-Scale Image
Other models of complete quaternion images composed from the color image can also be
considered for the 2-D QDFT. For example, the following quaternion image being a sum of three
complex images can be taken:
‫ݍ‬௡,௠ = ൬
1
3
+ ݅
2
3
൰ ‫ݎ‬௡,௠ + ൬
1
3
+ ݆
2
3
൰ ݃௡,௠ + ൬
1
3
+ ݇
2
3
൰ ܾ௡,௠. (9)
In this model, all three color components of the image are distributed between the real and
imaginary parts in the same way. The color image can be calculated from this quaternion image
as
݂௡,௠ =
3
2
ൣ‫ݍ‬௡,௠ − Real(‫ݍ‬௡,௠) ൧ =
3
2
൤‫ݍ‬௡,௠ −
‫ݎ‬௡,௠ + ݃௡,௠ + ܾ௡,௠
3
൨. (10)
If we denote three imaginary components of the quaternion image ‫ݍ‬௡,௠ by (‫ݍ‬௡,௠)௜, (‫ݍ‬௡,௠)௝, and
(‫ݍ‬௡,௠)௞, the color image can be defined as
‫ݎ‬௡,௠ =
3
2
(‫ݍ‬௡,௠)௜, ܾ௡,௠ =
3
2
(‫ݍ‬௡,௠)௝, ܾ௡,௠ =
3
2
(‫ݍ‬௡,௠)௞.
We now consider a general model of the color image in the quaternion space. Let ܽଵ, ܽଶ, ܽଷ, and
ߚଵ, ߚଶ, and ߚଷ be some numbers from the interval (0,1). The color image ݂௡,௠ can be represented
as the following quaternion image:
‫ݍ‬௡,௠ = (ܽଵ + ݅ߚଵ)‫ݎ‬௡,௠ + (ܽଶ + ݆ߚଶ)݃௡,௠ + (ܽଷ + ݇ߚଷ)ܾ௡,௠ (11)
or
‫ݍ‬௡,௠ = ൫ܽଵ‫ݎ‬௡,௠ + ܽଶ݃௡,௠ + ܽଷܾ௡,௠൯ + ݅ߚଵ‫ݎ‬௡,௠ + ݆ߚଶ݃௡,௠ + ݇ߚଷܾ௡,௠.
To reconstruct the color image, the following calculations can be used when ߚ௡ ≠ 0, ݊ = 1,2,3:

32
‫ݎ‬௡,௠ =
1
ߚଵ
(‫ݍ‬௡,௠)௜, ݃௡,௠ =
1
ߚଶ
(‫ݍ‬௡,௠)௝, ܾ௡,௠ =
1
ߚଷ
(‫ݍ‬௡,௠)௞,
where ݊ = 0: (ܰ − 1) and ݉ = 0: (‫ܯ‬ − 1). Thus, we have a parameterized representation of the
color image in the quaternion space, or threefold complex space ‫ܥ‬଺
. For instance, the coefficients
ܽଵ, ܽଶ, ܽଷ, and ߚଵ, ߚଶ, and ߚଷ can be chosen in such a way that ܽ௡ + ߚ௡ = 1 for ݊ = 1,2,3. When
the coefficients ܽଵ = ܽଷ = ܽଷ = 0, the quaternion image ‫ݍ‬௡,௠ is referred to as the tradition
representation of the color image. The ܽଵ = ܽଷ = ܽଷ = 1 case corresponds to the gray-scale
image ‫ݍ‬௡,௠ = (‫ݎ‬௡,௠ + ݃௡,௠ + ܾ௡,௠)/3.
It should be mentioned, that in the quaternion space, we can consider and process simultaneously
two different images, gray-scale ‫ݒ‬௡,௠ and color ݂௡,௠ images, by combining them into a
quaternion image, for instance, as follows:
‫ݍ‬௡,௠ = (‫ݒ‬௡,௠, ݂௡,௠) = ‫ݒ‬௡,௠ + (݅‫ݎ‬௡,௠ + ݆݃௡,௠ + ܾ݇௡,௠). (12)
Then, after processing this image ‫ݍ‬௡,௠ → ‫ݍ‬ො௡,௠ = (‫ݒ‬ො௡,௠, ݂መ௡,௠) the output gray-scale and color
images are considered to be
‫ݒ‬ො௡,௠ = Real൫‫ݍ‬ො௡,௠൯, ൫݅‫̂ݎ‬௡,௠ + ݆݃ො௡,௠ + ܾ݇෠௡,௠൯ = Imag൫‫ݍ‬ො௡,௠൯,
and color components of the new color image ݂መ௡,௠ are calculated as
‫̂ݎ‬௡,௠ = (‫ݍ‬ො௡,௠)௜, ݃ො௡,௠ = (‫ݍ‬ො௡,௠)௝, ܾ෠௡,௠ = (‫ݍ‬ො௡,௠)௞ .
As an example, Figure 5 shows the gray-scale “Lena” image in part a and color tree image in b.
These two images compose one quaternion image with four components. In parts c and d, the
results of enhancement of the quaternion image are shown. The real component of ‫ݍ‬௡,௠ is shown
in c and the image composed by three color components of the imaginary part in d. “Lena” image
and color tree image were enhanced by a single operator in the quaternion space.

33
Figure 5. (a) The gray-scale image and (b) color image before and (c) gray-scale image and (d) color image
after processing together in the quaternion space.
4. 2-D MQDFT IN IMAGE ENHANCEMENT
In this section, we consider application of the proposed models of color images in the quaternion
space for image enhancement in the frequency domain. The enhancement by the 2-D mQDFT can
be described as shown in Figure 6. The 2-D discrete QDFT of the color image is calculated and
its amplitude only changes by using an operator ‫,ܯ‬ and then, the inverse 2-D QDFT is calculated,
݂௡,௠ → ൛‫ܨ‬௣,௦ = (|‫ܨ‬௣,௦|, ߴ௣,௦)ൟ → ൛‫ܨ‬෠௣,௦ = (‫ܨ|[ܯ‬௣,௦|], ߴ௣,௦)ൟ → ൛݂መ௡,௠ൟ (13)
Here, ߴ௣,௦ is the phase and (|‫ܨ‬௣,௦|, ߴ௣,௦) is a polar representation of ‫ܨ‬௣,௦ .
Figure 6. Block-diagram of the image enhancement.
We consider the well-known method of ߙ-rooting for enhancement of images [20],[29]-[31], [39],
when the magnitude of the quaternion Fourier transform of the image is transformed as
‫ܨ‬௣,௦ → ‫ܨ|[ܯ‬௣,௦|] = |‫ܨ‬௣,௦|ఈ

34
for each frequency-point (‫,݌‬ ‫.)ݏ‬ The value of ߙ is taken from the interval (0,1) and can be
selected by the user, or can be found automatically [19],[20],[28],[37],[41].
To select values of ߙ for image enhancement, we can analyze the color image, by using the
measure CEME introduced in [41]. The discrete color image ݂௡,௠ of size ܰ × ‫ܯ‬ is divided by
݇ଵ݇ଶ blocks of size ‫ܮ‬ଵ × ‫ܮ‬ଶ each, where integers ݇௡ = ‫ܰہ‬௡/‫ܮ‬௡‫,ۂ‬ ݊ = 1,2. Here, ‫ۂ∙ہ‬ denotes the
floor function. The overlapping of these blocks can also be considered [42].The quantitative
measure of enhancement of the color image processed by the 2-D QDFT transform,
݂ = (݂ோ, ݂ீ, ݂஻) → ݂መ = (݂መ௘, ݂መோ, ݂መீ, ݂መ஻),
is defined as follows:
‫ܧ‬௤(ߙ) = ‫ܧܯܧܥ‬ఈ൫݂መ൯ =
1
݇ଵ݇ଶ
෍ ෍ 20
௞మ
௟ୀଵ
௞భ
௞ୀଵ
logଵ଴ ቈ
max௞,௟(݂መோ, ݂መீ, ݂መ஻)
min௞,௟(݂መோ, ݂መீ, ݂መ஻)
቉. (14)
Here, max௞,௟(݂መ) and min௞,௟(݂መ) respectively are the maximum and minimum of colors of the
image ݂መ௡,௠ inside the (݇, ݈)th block, and α is a parameter of the enhancement algorithm.
‫ܧܯܧܥ‬ఈ൫݂መ൯ is called a measure of enhancement, or measure of improvement of the image ݂௡,௠.
The “best” image enhancement parameter α is considered to be the one which maximizes the
value of the ‫,ܧܯܧܥ‬ i.e., ‫ܧܯܧܥ‬ఈ൫݂መ൯ = max ‫݂(ܧܯܧܥ‬መ). When considering the quaternion image
݂መ with non zero real part, the enhancement measure CEME is calculated as
‫ܧ‬௤(ߙ) = ‫ܧܯܧܥ‬ఈ൫݂መ൯ =
1
݇ଵ݇ଶ
෍ ෍ 20
௞మ
௟ୀଵ
௞భ
௞ୀଵ
logଵ଴ ቈ
max௞,௟(݂መ௘, ݂መோ, ݂መீ, ݂መ஻)
min௞,௟(݂መ௘, ݂መோ, ݂መீ, ݂መ஻)
቉. (15)
Now, we consider an example of image enhancement by using the measure ‫.ܧܯܧܥ‬ Figure 7
shows the color image of size 240 × 320 in part a. The image has the measure CEME equal
14.7848 when calculated with blocks of size 7 × 7.
Figure 7. (a) The color image, (b) the 0.8850-rooting by the 2-D DQFT, and (c) the enhancement function
of the image.

35
The curve of the function ‫ܧ‬௤(ߙ) for this image is given in part c, when α runs the interval [0.1,1).
The maximum value of the enhancement is 19.0166 at the point ߙ = 0.8850. The corresponding
0.8850-rooting of the color image is shown in part b.
We also consider an example of enhancement of a quaternion image with non zero real part.
Figure 8 shows the color tree image in part a, and the enhanced image in b, when the 2-D QDFT-
based 0.96-rooting is applied.
Figure 8. (a) Color image and (b) image enhanced by 0.96-rooting.
The enhancement was performed over the image with the real part shown in part a of Figure 9.
The CEME measure of the image has a high value at point ߙ = 0.96. The real part of the inverse
2-D QDFT after ߙ-rooting is shown in b. For comparison, the gray-scale image calculated from
the last three components as their average is shown in c. One can observe that after processing
simultaneously the gray-scale and color tree images, the result of processing of the gray-scale
image is better, than the average of the color components of the quaternion image.
Figure 9. (a) Gray-scale tree image in the quaternion image, (b) enhanced real part of the image, and (c)
average of three imaginary components of the enhanced quaternion image.
Recently, gradient based gray level image enhancement has been introduced [46]-[48]. For color
image in the quaternion space, we apply the following measure of enhancement calculated on the
image gradients:
‫ܧܯܧܥ‬൫݂መ൯ =
1
݇ଵ݇ଶ
෍ ෍ 20
௞మ
௟ୀଵ
௞భ
௞ୀଵ
logଵ଴ ቈ
max௞,௟‫ܩ‬௫ା௬(݂መ)
min௞,௟‫ܩ‬௫ା௬(݂መ)
቉, (16)

36
where the gradient operators ‫ܩ‬௫ା௬ = ‫ܩ‬௫ + ‫ܩ‬௬ or ‫ܩ‬௫ା௬ = (‫ܩ‬௫, ‫ܩ‬௬). Here, the gradients along the
‫-ݔ‬ and ‫-ݕ‬axes are calculated over the components of the quaternion image as
൫‫ܩ‬௫ൣ݂መோ൧, ‫ܩ‬௫ൣ݂መீ൧, ‫ܩ‬௫ൣ݂መ஻൧൯ and ൫‫ܩ‬௬ൣ݂መோ൧, ‫ܩ‬௬ൣ݂መீ൧, ‫ܩ‬௬ൣ݂መ஻൧൯, respectively.
Different gradient operators are used in digital image processing. We apply the following well-
known operations [43],[44]:
Sobel's gradients3 × 3:
‫ܩ‬௫ =
1
4
൥
1 0 −1
2 0 −2
1 0 −1
൩, ‫ܩ‬௬ =
1
4
൥
−1 −2 −1
0 0 0
1 2 −1
൩.
Prewitt's gradients 3 × 3:
‫ܩ‬௫ =
1
3
൥
1 0 −1
1 0 −1
1 0 −1
൩, ‫ܩ‬௬ =
1
3
൥
−1 −1 −1
0 0 0
1 1 −1
൩.
Robert's gradients 3 × 3:
‫ܩ‬௫ = ൥
0 0 −1
0 1 0
0 0 0
൩, ‫ܩ‬௬ =
1
4
൥
−1 0 0
0 1 0
0 0 0
൩.
Frei-Chen's gradients 3 × 3:
‫ܩ‬௫ =
1
1 + √2
൥
1 0 −1
√2 0 −√2
1 0 −1
൩, ‫ܩ‬௬ =
1
1 + √2
቎
−1 −√2 −1
0 0 0
1 √2 −1
቏.
Agaian-Frei-Chen's gradients 5 × 5:
‫ܩ‬௫ =
‫ۏ‬
‫ێ‬
‫ێ‬
‫ێ‬
‫ێ‬
‫ۍ‬ 1 √2 0 −√2 −1
√2 2 0 −2 −√2
2 √8 0 −√8 −2
√2 2 0 −2 −√2
1 √2 0 −√2 −1‫ے‬
‫ۑ‬
‫ۑ‬
‫ۑ‬
‫ۑ‬
‫ې‬
, ‫ܩ‬௬ =
‫ۏ‬
‫ێ‬
‫ێ‬
‫ێ‬
‫ێ‬
‫ۍ‬ 1 √2 2 √2 1
√2 2 √8 2 √2
0 0 0 0 0
−√2 −2 −√8 −2 −√2
−1 −√2 −2 −√2 −1‫ے‬
‫ۑ‬
‫ۑ‬
‫ۑ‬
‫ۑ‬
‫ې‬
.
Many other gradient operators, including the extension of Frei-Chen's gradients of large sizes, can
be found in [49]-[52].
5. CONCLUSION
In this paper, a new view of expressing color images using quaternion-based representation was
provided. In this work, we consider a full model for representation and processing color images in
the quaternion algebra. We have presented a fully quaternion-based color processing framework

37
in which several color analysis problems may be solved. We have shown how a particular image
enhancement in the framework of proposed model leads to an excellent color enhancement (better
than other algorithms tested). Many other color processing algorithms in the framework of the
proposed model can be expressed, including filtration and restoration.
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Authors
Artyom M. Grigoryan is Associate Professor in the Electrical and Computer
Engineering at the University of Texas, San Antonio. He is the author of three books,
three book-chapters, two patents, and many journal papers and specializing in the
theory and application of fast Fourier transforms, image enhancement, computerized
tomography, processing biomedical images, and image cryptography.
Sos S. Agaian is Professor of Electrical and Computer Engineering at the University of
Texas, San Antonio. He has seven books, 500 scientific papers, and holds 14 patents.
He is a Fellow of the International Society for Photo-Optical Instrumentation
Engineers and Fellow Imaging Sciences and Technology (IS&T).

RETOOLING OF COLOR IMAGING IN THE QUATERNION ALGEBRA

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RETOOLING OF COLOR IMAGING IN THE QUATERNION ALGEBRA