Color Image Segmentation Based On Principal Component Analysis With Application of Firefly Algorithm And Gaussian Mixture Model

Giuliani Donatella
International Journal of Image Processing (IJIP), Volume (12) : Issue (4) : 2018 101
Color Image Segmentation Based On Principal Component
Analysis With Application of Firefly Algorithm And
Gaussian Mixture Model
Giuliani Donatella giulianidonatella@libero.it
University of Bologna
Bologna, Italy
Abstract
In this paper we propose a segmentation method for multi-spectral images in the HSV space,
based on the Principal Component Analysis to generate grayscale images. Then the Firefly
Algorithm has been applied on the gray-level images in a histogram-based research of cluster
centroids. The FA is a metaheuristic optimization algorithm, centered on the flashing behaviour of
fireflies. The Firefly Algorithm is performed to determine automatically the number of clusters and
to select the gray levels for partitioning pixels into homogeneous regions. Successively, these
gray values are employed during the initialization step of a Gaussian Mixture Model for estimation
of parameters, evaluated through the Expectation-Maximization technique. The coefficients of the
linear super-position of Gaussians can be regarded as the prior probabilities of each component.
Applying the Bayes rule, the posterior probabilities have been estimated and their maxima are
used to assign each pixel to the clusters, according to their gray values.
Keywords: Color Image Segmentation, Image Clustering, Firefly Algorithm, Gaussian Mixture
Model.
1. INTRODUCTION
Image segmentation is the process of separating an image into disjoint and homogeneous
components. Because of the enormous variety and complexity of color images, robust and
efficient segmentation approaches are far from satisfying, especially when applied on images of
the real world. Color image segmentation attracts more and more interest, mainly because it
provides a greater amount of information respect to grayscale images. Grayscale image
segmentation methods are categorized recurring to the two properties utilized for partitioning:
discontinuity and similarity, the most common are the region-based and edge-based methods. In
more details, region-based methods are founded on similarity, they segment an image recurring
to low-level features of it, such as intensity, texture or color, assuming that pixels belong to the
same region if they have similar characteristics. Hence they can work for colored images as well.
On the contrary, color image edge-detection techniques are extension of monochromatic
methods, they operate on the three color channels independently combining their final results.
More explicitly, the gradient operators proposed for grayscale segmentation approaches, are
extended to color images by taking the vector sum of gradients for each component [1]. The
paradigms of gray level image segmentation, pixel-based, region-based or edge-based can be
also used for color images. Besides, we must include the so called physics-based methods which
take into account information about the exposure to light during image acquisition processes as
well. In summary, color image segmentation methods may be classified into three diﬀerent
groups depending on the technique employed: 1) feature-space based techniques; 2) image
domain based techniques; and 3) physics based techniques. The feature-space methods work in
a certain color space, assuming that color is a constant property of the surface of each object
within an image. Thus these methods map pixels into regions having homogenous characteristics
or create some ad hoc histograms upon color features, such as hue. Generally, objects will
appear as clusters or in correspondence of peaks within histograms, in the latter case. The

Giuliani Donatella
image-domain based techniques, such as split and merge or region growing methods, perform a
spatial grouping of pixels according to similarity or dissimilarity criteria to guarantee spatial
compactness. Lastly, in the physics-based approach, all algorithms examine the objects
portrayed in color images taking into account light exposure, highlights, and shadows. These
phenomena frequently cause appearance of color variations of uniformly colored surfaces. To
avoid over segmented regions, these algorithms analyze how light interacts with colored
materials [2].
2. PRELIMINARIES
A color is perceived by human beings as a combination of the tri-stimuli captured by three
different kinds of photoreceptors in the retina, called cones, whose spectral responses are
centered on the wavelengths of red, green, blue. Therefore, a digital color image is represented
by a two dimensional array of three dimensional vectors which represent the pixel's red, green
and blue values, the three primary colors which form the color space. Although RGB image
representation is the most often used, it is not adequate for performing color recognition in image
analysis since this space has three independent channels, so any color can be created by
combination of these three color bases. RGB color space is suitable for color display but it is not
good for color scene segmentation and analysis because of the high correlation among the R, G,
B components and the great sensitivity to variations in illumination [3]. For this reason, we have
adopted HSV color space (Hue, Saturation, Value) given that it structures any color in the same
way in which it is perceived by human eyes [4]. More precisely, hue is the color tone, i.e. red or
yellow, saturation defines the amount of each color, i.e. dark green or pale green, finally the
value, also called intensity or lightness, allows to distinguish between a dark or a light color. An
accurate comparison study between different color spaces revealed that the HSV space provides
good results in image segmentation [4].
3. METHODOLOGY
In this paper we propose a segmentation method based on pixel intensity without recurring to
thresholding techniques. Firstly, the proposed segmentation algorithm performs the Principal
Component Analysis on color images. PCA is a standard tool of modern data analysis that
transforms a number of possible correlated variables into a smaller number of uncorrelated ones,
preserving the most relevant information of the original data set. Then we proceeded to a
histogram-based research of cluster centroids applying a meta-heuristic algorithm, the Firefly
Algorithm (FA), belonging to a novel class of non-deterministic optimization methods.
Over the last few decades, researchers started to use metaheuristic algorithms for clustering to
overcome the weakness of the existing conventional methodologies, generally deterministic [5].
Metaheuristic algorithms are a class of approximate methods that allow to discover possible
solutions exploring the search space in order to find near-optimal solutions. Shortly, metaheuristic
algorithms are iteratively processes developed to search a solution that is good enough in a time
that is small enough [6]. These algorithms are frequently nature-inspired, they have the
advantages of finding global optima due to the action of multiple search agents, randomly
generated [7].
Nature inspired metaheuristic algorithms are applied for multilevel thresholding image
segmentation in a comparative study between Particle Swarm Optimization (PSO), Artificial Bee
Colony Optimization (ABC), Ant Colony Optimization (ACO) and Cuckoo Search (CS) algorithms,
that are investigated in partitioning gray-scale images [8]. The well-known K-means clustering
algorithm is applied by an optimization technique based on FA algorithm, with the squared error
as objective function. The performance of the FA is compared with other two nature-inspired
methods, ACO and PSO, and other nine traditional approaches [9]. A bio-inspired segmentation
algorithm is proposed for solving crop type classiﬁcation problems using a multispectral satellite
image [10]. A colour image partitioning procedure is implemented applying a multi-level
thresholding approach on the three R, G and B histograms, analyzed independently [11]. The
predefined number of optimal thresholds for each colour component are attained by maximizing

Giuliani Donatella
Otsu’s between-class variance function, recurring to the Firefly Algorithm. The maximum entropy
thresholding method is widely used in literature, this approach has been performed in partitioning
multi-channels images employing Firefly Algorithm [12], in this context entropy operators yield
high values in regions characterized by drastic chromaticity variations. For an exhaustive review
of the more conventional segmentation methods refer to [13], [14], [15].
In the present work we started with a pre-processing step, during which the original image has
been de-correlated in order to improve its appearance and to enhance the contrast between pixel
colors. Then applying PCA, the image dimensionality has been reduced from the three-
dimensional HSV color space to the one-dimensional gray representation [16]. To this aim, given
a color image
with components , let be the vector mean, expressed by:
the covariance matrix C is evaluated as:
So, if we denote with 1  2  3 and , respectively, the eigenvalues and eigenvectors
of C, the components of de-correlated image may be computed as:
Through PCA, the de-correlated image is projected along the direction of the principal
component in which it has been registered the highest variability, therefore the intensity image
I is generated by:
Once the gray image has been obtained, we proceed in evaluating automatically the number of
clusters and the associated representative levels through a histogram-based segmentation
approach by means the Firefly Algorithm.
The Firefly Algorithm, developed by Yang X.S. [17], employs fireflies as search agents making
use of their idealized flashing characteristics. In describing the Firefly Algorithm, three
fundamental rules may be delineated [18],[19],[20],[21]:
(1) All fireflies are unisex so that one firefly will be attracted to other fireflies regardless of
their sex;
(2) Attractiveness is proportional to their brightness, thus for any two flashing fireflies, the
less bright one will move towards the brighter one. The attractiveness is proportional to
brightness, decreasing as their distance increases. If there is no brighter one, a generic
firefly will move randomly in the search space;
(3) The brightness of a firefly is determined by an objective function. For a maximization
problem, the brightness can simply be proportional to the objective function.
The attractiveness (r) of a firefly is proportional to the brightness and decreases with distance r,
because the light is absorbed in the media, as a consequence luminosity becomes weaker with
the distance from source, so we have:

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where is the attractiveness at and γ is the light absorption coefficient. As a firefly i is
attracted to another brighter j, its movement is described at time t by:
(1)
where represents the distance between the two fireflies. The second term is due to the
attraction, while the third term defines the random component through the casual values ,
drawn from a Gaussian distribution at a given iteration. Thus, describes a simple random
walk. The value of expresses the variation of attractiveness,  = 0 corresponds to constant
attractiveness and, conversely, as  →∞ the attractiveness tends to zero, which is equivalent to a
totally random search. Given the relevant level of variability of gray intensity histograms for real
images, in this unsupervised clustering method, we proposed the Firefly Algorithm because it
fulfils the aim of searching local and global maxima simultaneously, due to the decreasing of
attractiveness (r) with distance. In order to capture the most of significant histogram peaks, the
objective function is defined as the relative change of pixel frequency, therefore a firefly is
attracted towards the others close by it but having meaningfully greater values of pixel frequency.
In the following applications, the fireflies used as search agents are 30, the average scale L is set
equal to 256 which is the number of gray levels, and the light absorption coefficient  is set equal
to . In order to avoid an excessive erratic motion in the search space, the random
component is reduced during iterations, as with and . After
evaluating automatically the number K of clusters through the global maxima derived by the FA
algorithm, the corresponding gray-levels intensities have been assigned as initial means
to the Gaussian Mixture Model [22], [23]. The initial standard deviations are
approximately computed through the differ--rences respect to the nearest centroids.
A Gaussian Mixture Model (GMM) is a parametric probability density function represented as a
weighted sum of Gaussian components. In this work GMM parameters are estimated from data
using the iterative Expectation-Maximization (EM) algorithm. Let us recall that a univariate
Gaussian density distribution is expressed by:
where represents a Gaussian distribution of mean and standard deviation A
Gaussian mixture model is a weighted sum of K components of Gaussian densities, analytically
given by:
(2)
Equation (2) represents a linear superposition of probability densities in which the mixing
coefficients indicate the weight of each distribution. According to the first and second axiom of
probability theory, these coefficients must satisfy the following relations:
Thus, a Gaussian mixture model is parametrized by means , variances and the mixture
weights of all components. We can think of the mixing coefficients as prior probabilities
corresponding to each component:

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where is the number of image points and are the numbers of data belonging to
the k-th cluster. The GMM parameters have been calculated using the Expectation Maximization
(EM) algorithm. This algorithm is an iterative optimization technique that starts from some initial
estimation of parameters and then proceeds to iteratively update them until convergence is
detected. In short it consists of four basic steps:
1) The Initialization Step: define initial guesses for parameters, in this context they have been
derived by the FA algorithm
2) The Expectation Step: compute the responsibilities Eq. (3), namely the posterior
probabilities of a given gray intensity x to belong to the k-th cluster, according to the Bayes
rule we have:
(3)
3) The Maximization Step: re-estimate the parameters using Eq. (4) and the current
responsabilities, given by Eq. (3):
(4)
4) Compute convergence by the value of log likelihood after each iteration, halting if it does
not change in a significant manner from one iteration to the next:
Iterate Expectation Step (2) and Maximization Step (3) until convergence.
After determining the parameter values with EM technique, we have proceeded to the assignment
of pixels of a given gray level to the cluster by means the evaluation of the maximum value of
responsibilities:
all the pixels with gray level are attributed to the -th cluster, given that, by definition,
represents the probability of the -th GMM’s component to have generated the value . The
proposed segmentation method performs classification of pixels in a straightforward and effective
way reducing greatly the computational costs [24].
4. EXPERIMENTAL RESULTS
In this section we present some experimental results obtained basically with segmentation of test
images extracted from Berkeley Segmentation Dataset BSDS300 [25]. Firstly, we applied our
method on Iris, the test image shown in Fig.1. A de-correlated image has been produced on
HSV space, as we can see in Fig.1 (on the left) the outcome gives prominence to pixel colors,
increasing the level of contrast.

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FIGURE 1: Original and De-correlated Images.
Afterwards, the de-correlated image has been projected on the principal component , the
derived image is represented in Fig.2. Given the relevant level of variability of gray intensity
histograms for real images (see the enlarged histogram in Fig.3), in this unsupervised clustering
method we propose the use of a meta-heuristic algorithm, more specifically the Firefly Algorithm
(FA). The FA fulfils the aim of searching simultaneously local and global maxima, due to the
decreasing of attractiveness (r) with distance. In order to capture the most of significant
histogram peaks, the objective function is defined as the relative change of pixel frequency,
therefore a firefly is attracted towards the others close by it but having meaningfully greater
values of pixel frequency. In the present application, the fireflies used as search agents are 30,
the average scale L is set equal to 256 which is the number of gray levels, the light absorption
coefficient  is defined as . In order to avoid an excessive erratic motion in the search
space, the random component may be reduced during iterations, so decreases as
with and .
-
FIGURE 2: Iris Grayscale Image with PCA.
After evaluating automatically the number K of clusters through the maxima derived by FA (Fig.3
on then left), the corresponding gray-levels intensities have been assigned as initial means
to the Gaussian Mixture Model [26], [27]. The initial standard deviations are
approximately computed through the differences respect to the nearest centroids, on the contrary
the initial centroids of clusters are issued by FA algorithm.

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FIGURE 3: Grayscale Image Histogram and Maxima Computed with FA.
FIGURE 4: Grayscale Histogram and Gaussian Mixture Model.
To evaluate the consistency of this proposed clustering algorithm, the quality of final outcomes is
estimated by using several validation measures, more precisely: the Root Mean Square Error
(RMSE), the Normalized Correlation Coefficient (NK) and finally the Davies-Bouldin (DB) index
[28], [29]. The RMSE is the simplest validity index, a large value of RMSE means that the image
is of poor quality, given an image and the segmented image
, it is defined as:
The normalized correlation coefficient (NK) quantifies in percentage the level of correlation
between the two images and it is expressed as follows:
Finally, for testing the consistency of the method, we have calculated the Davies-Bouldin index,
that is based on similarity measures between clusters, obtained through the dispersion of
each cluster and the dissimilarity measures . Usually, given a partition of an initial dataset I in
disjoint clusters of cardinality , is defined in the following way:

Giuliani Donatella
where represents the intra-cluster standard deviation and
measures the distance between two different clusters and , through the
distance of their centre points. The Davies-Bouldin index estimates the average of similarity
between each cluster as:
even with this validation index a lower value implies a better clustering segmentation.
FIGURE 5: Gray segmented image of Iris.
The segmented image with four clusters is displayed in Fig.5, Table 1 reports the GMM
parameters and the estimated validation indices, in the last row. We point out that the final value
of standard deviations of some clusters are rather different respect to the initial ones. To this end
we recall that, during the starting step, variances were evaluated approximately by minimizing the
deviations from the initial cluster centroids.
TABLE 1: Cluster Parameters Evaluated with GMM.
Finally, using the original data set as reference image, we have performed a colorization process
on the segmented grayscale image, recurring to the means of the three colour channels
computed on pixels belonging to the same cluster, the final result is shown in Fig.6.
K = 4 Data Clusters
(N. iterations EM =10)
N.pixels:
162 x 149 = 24138
k initial 39.00 76.29 183.23 198.05
k final 42.18 71.65 187.63 193.22
k
initial
5.66 2.86 4.57 5.32
k final 8.18 4.19 10.43 19.47
k  13255 1804 4338 4741
kfinal 0.574 0.049 0.196 0.179
 RMSE
0.069
NK
0.981
DB
2.814

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FIGURE 6: Color Segmented Image of Iris.
As a second test image let us consider the butterfly displayed in the training image 138 of Dataset
BSDS300:
FIGURE 7: Butterfly original image and grayscale image with PCA.
Through the FA algorithm the number of derived maxima is K = 5 (Fig. 8) In Table 2 are explicitly
specified the initial and final values of centroids. After ten iterations of the Expectation
Maximization algorithm, the convergence to final values is reached within a predefined precision
level. These values do not present significant differences respect to the initial ones. The
validation indices confirm the reliability of this feature-space based method for image
segmentation.
TABLE 2: Cluster Parameters Evaluated with GMM.
K = 5 Data Clusters
(N. iterations EM =10)
N.pixels:
321x481 = 154401
k initial 86.00 115.73 166.30 208.94 251.68
k final 84.66 115.71 167.39 210.90 246.50
k
initial
9.15 4.93 4.77 4.86 2.04
k final 17.91 13.74 12,45 12.81 4.18
k  63476 26001 29369 33012 2543
kfinal 0.408 0.170 0.192 0.213 0.015
 RMSE
0.023
NK
0.988
DB
0.198

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FIGURE 8: Grayscale Image Histogram and Maxima Computed with FA.
FIGURE 9: Grayscale Histogram and Gaussian Mixture Model.
FIGURE 10: Gray and Color Segmented Images of Butterfly.
The colorization process performed on the segmented grayscale image of Fig.10 (on right) puts in
evidence the five predominant colour features appearing in the original image.
5. CONCLUSIONS
This research has addressed on colour image segmentation, one of the fundamental task of
image processing. To this aim, it has been developed a feature-space based approach recurring
to PCA, the nature-inspired Firefly Algorithm and the Gaussian Mixture Model. This method,
applied on several images extracted from Berkeley Segmentation Dataset BSDS300, is able to
find out automatically the number of groups in which an image may be partitioned and their

Giuliani Donatella
centroids. Their initial estimates, computed by a metaheuristic algorithm, are used as initial
values for evaluating the GMM parameters. The results appear fairly solid and reliable, as can be
inferred by the validation analysis performed on the test image analysed in this paper. A
noteworthy advantage of the proposed methodology derives from the use of the maxima of
responsibilities for pixel assignment, that implies a consistent reduction of computational costs.
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