FOA-Kmeans-Color-Image-Segmentation

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FOA-Kmeans-Color-Image-Segmentation

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Clustering analysis using an evolutionary optimization algorithm based on nature, Forest Optimization Algorithm

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# Color-Based Image Segmentation using hybrid Forest Optimization Algorithm and K-means

**Alberto Villagómez Vargas, Hermes Espínola González, Humberto Atondo
Martín del Campo**

## Abstract

In this article, we do clustering analysis using an evolutionary
optimization algorithm based on nature, Forest Optimization Algorithm
(FOA), perfect for continuous nonlinear optimization problems, similar
to Particle Swarm Optimization Algorithm (PSO), for cluster analysis.
Cluster analysis is a frequently used applied statistical technique that
helps reveal hidden structures and clusters in large data sets, in other
words it is a collection of objects. Cluster analysis has a wide range
of applications in a vast number of fields such as in medicine,
telecommunications, image processing, data mining, voice mining, web
mining and the list just keeps going. Forest Optimization Algorithm is
an algorithm inspired by how trees in a forest can survive for several
decades while others will die within a few years. These algorithm
simulations are based in seeding procedure of the trees, where some
seeds will fall right under the tree while others will be distributed in
a wide range of area by natural procedures and by the animals that fed
on this seeds. The use of this type of optimization algorithms in
cluster analysis is applied to find the most optimal cluster with the
help of different objective functions.

## 1. Introduction

According to Rajabioun, optimization is the process of improving
something and making it better. In this case optimization is the process
of adjusting the inputs, which consist of variables, to find the global
minimum or maximum result of a function, which is called cost or
fitness. Many optimization algorithms are inspired by nature, some
examples are the Particle Swarm Optimization Algorithm (PSO) (*Kennedy &
Eberhart, 1995*) and the Genetic Algorithm (GA) (*Sivanandam & Deepa,
2007*). Another example of algorithms based on nature is the Forest
Optimization Algorithm (FAO) (*Ghaemi, M & Feizi-Derakhshi, 2014*) which
is an algorithm inspired in the formation of forests and is the
algorithm that we will use to make cluster analysis.

Trees have been governing planet earth for millions of years in the form
of forests, they have had presence in several earth’s eras and have
adapted by using different ways to survive and have manage to preserve
their existence. Because trees are living organisms they will age and
eventually die, that’s why trees need to replace the old trees with new
ones and rarely some of this trees get to live for several decades. The
distinguished trees that are able to survive for long periods of time,
are often the ones that are in suitable geographical habitats and have
the best growing conditions. To do this trees disperse their seeds
immediately to place them in safe sites and ensure their survival. What
the Forest Optimization Algorithm (FOA) does is try to find this
distinguished trees which are the most optimal solution in the forest
with help of different procedure such as seed dispersal.

There are many different natural procedures that help the distribution
of the seeds in the forest, this procedures are known as seed dispersal.
Seed dispersal can be affected in many ways and by different factors.
Seed dispersal is categorize into two sub-divisions depending on how far
they are from their mother tree. This sub-divisions are the local seed
dispersal and the long-distance seed dispersal.

When seeding process begins, some seeds fell right into the ground, just
below the tree they were born from or near it. This procedure is named
local seed dispersal and we will refer to this process as “Local
seeding”. There is also something we know as long-distance dispersal and
is when seeds travel long distances to grow in different parts of the
forest. Long-distance seed dispersal takes place because there are many
animals who feed on seeds and move them far away, seeds can also be
moved by water and wind or trees can give them “wings” so that they can
travel long distances. We will refer to long-distance seed dispersal as
“Global seeding”.

Long-distance seed dispersal have been proved to be of extreme
importance in comparison to local seed dispersal for some trees
survival. This is proved by two hypothesis, the “Escape Hypothesis” and
the “Colonization Hypothesis”. The Escape Hypothesis implies that it
exists a disproportionate success for seeds that travel far away from
their parent, comparing it to the seeds that have fallen nearby or right
under the tree. Colonization Hypothesis says that parents can use the
advantage of uncompetitive environment thanks to long-distance seed
dispersal.

Seeds which have manage to survive long enough to become trees will
begin to sprout as soon as possible. But not every seed will grow to
become a tree, there are many reasons for this. Some trees have strange
behavior, such as creating many empty and dead seeds, trees do this to
discourage animals that feed on seeds to eat theirs. Another reason
because why not all seeds make it to become trees is that some trees
need very specific conditions and requirements to grow.

There exist 3 main factors that affect the death of trees: biotic,
mechanical and suppression. Biotic is the evidence that some trees are
killed by pathogens and insects. Mechanical is the evidence of crushing,
snapping, or uprooting of the trees. At last we have suppression and it
is the evidence of natural competition among species to survive and it
plays a very significant role in trees survival.

When seeds fall right under the parent tree, a hard competition among
neighbors take place for sonlight and other resources at a very young
age. Thanks to this observations we can notice that as local density of
trees increases, mortality does as well and the competition for
essential resources will remove nearby weaker or younger neighbors. We
have observed that because of this reason the primary source of
mortality is actually competition. So often, the winners of this
competition are the seedling that began to grow first, having this time
advantage helps them reach sunlight and other essential resources but,
blockin that resources from their nearby neighbors. When one of this
seedling succeeds to be the winner, no other tree will be able to grow
in below that tree for a some years.

Trees can make use of long-distance seed dispersal to colonize empty or
less competitive spaces so that more seeds can grow into trees.
Long-distance seed dispersal is of crucial importance because as we have
said before, many of the seeds that fall right under their parent tree
will die. Seeds that travel long distances to become trees have a better
chance of succeeding at it.

In this paper we will use the Forest Optimization Algorithm (FOA), an
algorithm based on the survival and competition of trees in the forest.
We will use this procedures to solve a problem of image clustering
analysis. We will do this by simulating the local seed dispersal and
long-distance seed dispersal of the trees and we will reference to them
as local seedling and global seedling respectively. Local seedling will
help us to explore while global seedling will help us to exploit.

## 2. Background

### *2.1 FOA*

The Forest Optimization Algorithm is an evolutionary algorithm inspired
by few trees in the forests which can survive for decades, while other
trees could live for a limited period. Proposed by Manizheh Ghaemi and
Mohammad-Reza Feizi-Derakhshi from the University of Tabriz in 2014. The
FOA algorithm is as follows:

#### *2.1.1 Initialize trees*

In FOA, a tree is a potential solution of each problem. Each tree has
the values of the variables and an age. The age of each newly generated
tree is 0, and the value of the variables are initialized randomly in
the domain of the objective function.

#### *2.1.2 Local seeding of the trees*

Each tree with age 0 drops “Local seeding changes” seeds near the parent
tree, and the age of the parent tree is increased by 1.

#### *2.1.3 Population limiting*

All trees that which age is greater than “life time” is removed from the
forest and added to the candidate population list, also, sort the forest
and remove the worst trees to keep the forest with at most “Area limit”
trees.

#### *2.1.4 Global seeding of the trees*

Pick “Transfer rate” percentage of the candidate population to perform
global seeding, this is, copy its values and modify “Global Seeding
Changes” values randomly and add the new tree to the forest.

#### *2.1.5 Updating the best so far tree*

The age of the tree with the highest fitness is set to 0, so it is able
to optimize locally.

#### *2.1.6 Stop condition*

Stop the main loop when the maximum number of iterations is reached or
when the best fitness value so far has not changed for several
iterations.

### *2.2 K-means*

K-means is a clustering method proposed by MacQueen in 1967 that is
popular in cluster analysis in data mining. K-means aims to partition n
observations into k clusters in which each observation belongs to the
cluster with the nearest mean. Let X be a dataset containing n data
points located in a d-dimensional space. The definition of clustering is
to divide the n data points into m clusters. The clustering problem is
converted into optimization problem in such a way that minimizing the
summation of distance between the entire data points with its nearest
cluster.

![](https://wikimedia.org/api/rest_v1/media/math/render/svg/5337aaa52e8956d39a6d4c0d9b022d26540b4a56)

## 3. Proposed method

### Clustering using hybrid FOA-KM

In this paper we propose a novel algorithm for image segmentation based
on FOA and KM. The image segmentation is done using K-means clustering,
which is optimized by the Forest Optimization Algorithm. The flowchart
of the hybrid FOA-KM is shown in figure1. The following are the steps
used in hybrid FOA-KM algorithm.

![Hybrid FOA-KM flowchart](https://raw.githubusercontent.com/hermesespinola/FOA-Kmeans-Color-Image-Segmentation/master/FOAClustering/FOA-KM.png)

#### Step 1. *Initialization*

The potential solution of each problem is considered as a tree, A tree
is a bidimensional matrix of clusters, the age and fitness of the trees.
The age of each tree is set to ‘0’ for each each newly generated tree -
as result of local seeding or global seeding. After each local seeding
stage the age of each trees, except for the newly generated trees, is
increased by ‘1’. This is used as later as a controlling mechanism in
the number of trees in the forest.

#### *Step 2. Local seeding*

Local seeding of the trees attempts to simulate this procedure of the
nature. This operator is performed on the trees with age 0 and adds some
neighbors of each tree to the forest. The number of seeds that fall on
the land near the trees is a parameter named “Local Seeding Changes” or
“LSC”. A variable of the tree is selected randomly and is added a random
number between \[-Δx, Δx\], where Δx is a small constant. This way the
search procedure is done in a limited interval and local search can be
simulated. This procedure is repeated LSC times for each tree with age
0.

In the case of adding values, it may be situations where the values of
the variables become less or more than the related variables lower and
upper limits. In order to avoid these problems, values less than the
lower limit and values bigger than the upper limit are truncated to the
limits.

Local seeding operator adds many trees to the forest, so there must be a
limitation on the number of trees in the forest.

Additionally trees with age greater than “Life time” should be removed
from the forest and added to the candidate populated, which will be used
to perform global seeding in order to avoid bad local searching and also
avoid local optima points.

#### *Step 3. Population limiting*

The number of trees in the forest must be limited in order to prevent
infinite expansion of the forest. The first limitation is “Life time”,
the second limitation is “Area limit”. Area limit is a parameter that
constrains the size of the forest. In this step, the objective function,
k-means euclidean distance, is evaluated as the fitness of each tree,
then the forest is sorted according to this fitness value.
consecutively, if the number of trees is bigger than the limitation of
the forest, extra trees are removed from the end of the forest, that is,
the trees with the worst fitness, and they are added to the candidate
population.

#### *Step 4. Global seeding*

This step attempts to simulate the natural process where natural factors
in which seeds are distributed in the entire forest and as a result, the
habitat of the trees become wider.

Global seeding is performed on a predefined percentage of the candidate
population from the previous stage. This percentage is a predefined
parameter named as “Transfer rate”.

“Transfer rate” trees are selected from the candidate population. Then
some of the variables of each tree each tree is selected randomly. This
time, the value of each selected variable is exchanged with another
randomly generated value in the related variable’s range. The number of
variables that will be changed is another parameter and is named “Global
Seeding Changes” or “GSC”.

#### *Step 5. Updating the best so far tree*

The tree with the highest fitness value is selected as the best tree.
Then the age of best tree is set to ‘0’ in order to avoid the aging of
the best tree as the result of local seeding stage. In this way, the
best tree will be able to locally optimize its location by local seeding
operator, because local seeding is only performed on trees with age ‘0’.

#### *Step 6. Stop condition*

Three stop conditions can be considered: 1. predefined number of
iterations, 2. observance of no change in the best fitness value and 3.
Reaching the specified level of accuracy.

## 4. Experiments with FOA to minimize mathematical functions

First we are going to run the Forest Optimization Algorithm with some
mathematical functions to get their global minimum in order to test the
completeness of the algorithm.

The values of the limits are adjusted according to the function.

The next parameters of the algorithm are fixed as follows:

-   Initial number of trees =100

-   Maximum number of trees = 100

-   Lifetime of a tree = 6

-   LSC = 3

-   Transfer rate = 0.1

-   GSC = 2

**Sinusoidal function**

![](https://raw.githubusercontent.com/hermesespinola/FOA-Kmeans-Color-Image-Segmentation/images/sinFun.png)

0&lt;x, y&lt;10

Global minimums f(x) = -18.5547at different positions (9.039, -8.668),
(9.039, 8.668), (-9.039, 8.668)

| Number of iterations   | Number of variables to optimize  |  Positions       | Last function evaluation|
| :------------- | :----------- | :--------------- | :------------- |
| 10             | 2            | 8.9534, -8.6080  | -17.9640       |
| 50             | 2            | -9.0389, 8.6681  | -18.5547       |
| 100            | 2            | 9.0390, 8.6682   | -18.5547       |

**Sum of different powers function**

![](http://www.sfu.ca/~ssurjano/sumpow2.png)

-1≤ xi ≤1

Global minimum f(x) = 0 at xi = 0 for i=0, 1, 2… n

| Number of iterations | Number of variables to optimize  | Positions                              | Last function evaluation  |
  | :------------------- | :------------------------------- | :------------------------------------- | :------------------------ |
  | 10                   | 5                                | -0.0177 -0.1534 0.0829 0.1443 -0.1472  | 0.0040                    |
  | 50                   | 5                                | -0.0003 0.0012 0.0258 -0.0337 -0.0888  | 1.0680e-06                |
  | 100                  | 5                                | 0.0000 0.0001 -0.0002 -0.0001 -0.0017  | 5.9970e-11                |

**Eggholder function**

![](http://www.sfu.ca/~ssurjano/egg2.png)

-512&lt;xi&lt;512

Global minimums f(x) = -959.6407 at (512, 404.2319)

| Number of iterations  | Number of variables to optimize | Positions          | Last function evaluation |
  | :-------------------- |:------------------------------- | :----------------- | :----------------------- |
  | 10                    | 2                               | -467.8004 381.2903 |  -886.1135               |
  | 50                    | 2                               | 511.2238 401.3568  |  -951.5988               |
  | 100                   | 2                               | 510.0009 401.2261  |  -950.5605               |

**4. Rastrigin function**

![](http://www.sfu.ca/~ssurjano/rastr2.png)

-5.12&lt;xi&lt;5.12

Global minimums f(x) = 0 at xi = 0 for i=0, 1, 2… n

| Number of iterations  | Number of variables to optimize  | Positions                                         | Last function evaluation |
  | :-------------------- | :------------------------------- | :------------------------------------------------ | :----------------------- |
  | 10                    | 5                                |  1.0387 -0.1816 1.0455 0.1107 -0.0802             |  12.3188                 |
  | 50                    | 5                                |  0.9958 -0.0004 -0.9954 0.9943 -0.9959            |  3.9803                  |
  | 100                   | 5                                |  0.9950 -0.9944 0.0000 -0.0000 0.0002             |  1.9900                  |
  | 1000                  | 5                                |  1.0e-04 \*(-0.0511 0.1245 -0.3778 0.0225 0.5014) |  8.1888e-07              |

**5. Griewank function**

![](http://www.sfu.ca/~ssurjano/griewank2.png)

-600&lt;xi&lt;600

Global minimums f(x) = 0 at xi = 0 for i=0, 1, 2… n

| Number of iterations   | Number of variables to optimize   | Positions                                    | Last function evaluation |
  | :-------------------- | :------------------------------- | :------------------------------------------ | :---------------------- |
  |10                     |5                                 |38.3575 -74.0117 -73.7770 -15.8375 -1.4636   |4.1577                   |
  |50                     |5                                 |-15.6588 9.0056 26.9700 18.9150 -35.2602     |0.6801                   |
  |100                    |5                                 |15.6967 4.4584 -16.3073 -43.9101 14.0469     |0.6652                   |
  |1000                   |5                                 |9.4132 13.3206 -5.4336 -0.0022 -21.0215      |0.1847                   |
  |10000                  |5                                 |-0.0004 -4.4383 5.4333 -6.2707 7.0076        |0.0345                   |

**5. Experiments with FOA to minimize k-means algorithm for image
clustering**

For this experiments we use different sizes of images in pixels:

-   Icon: 10x10

-   Small: 128x128

-   Medium: 400x400

-   Large: 680x680

The experiments were made with the image’s original composition, its RGB
values and without resizing the original photo.

The next parameters of the algorithm are fixed as follows:

-   Number of the iterations = 100

-   Number of clusters = 3

|Image size  |  RGB values per cluster |   First k-means evaluation |   Last k-means evaluation
|------------| ------------------------| -------------------------- | -------------------------
|  Icon      | \[0, 0, 0\], \[251, 116, 49\], [46, 176, 131\]        | 7,866.06                   | 28.00
|  Small     | \[0, 1, 0\], \[228, 230, 226\], \[121, 183, 37\]       | 1,586,918.28               | 1,025,600.00
|  Medium    | \[116, 239, 255\], \[37, 240, 221\], \[255, 221, 0\]  | 8,744,951.20             |  4,007,400.00
|  Large     | \[0, 0, 0\], \[248, 231, 156\], \[15, 66, 109\]       |  37,104,666.71             | 6,109,100.00

## 6. Conclusion

In this article we discuss an algorithm based on a nature process of
trees survival, this algorithm is the Forest Optimization Algorithm
(FOA). This algorithm is inspired by some procedures in the forest,
specifically, the seed dispersal procedure. This procedure consist
basically that in the forest, the trees with with most sunlight and
better growth conditions will live longer that other trees and they use
different methods to ensure future generations for many years. When
seedling begins, some seeds fall right under their parent tree while
others tend to travel long distances in search of better growing
conditions and less competition. When they fall below their parent tree
it is called local seed dispersal and when seeds travel long distances
it is called long-distance seed dispersal. Seeds that travel long
distance have better chances to survive and to become a tree because
most of the seeds will fall under their parent tree, limiting resources
and having a heavy competition for sunlight. In this article we used
this algorithm to optimize the k-means objective function and then do
color-based image segmentation with the results of the optimization of
the objective function.

## 7. References

Ghaemi, M., & Feizi-Derakhshi, M. R. Forest Optimization Algorithm.
*Expert System with Applications* (2014),
[*http://dx.doi.org/10.1016/j.eswa.2014.05.009*](http://dx.doi.org/10.1016/j.eswa.2014.05.009)

Binu, D. Cluster analysis using optimization algorithms with newly
designed objective functions. Expert Systems with Applications (2015),
[*http://dx.doi.org/10.1016/j.eswa.2015.03.031*](http://dx.doi.org/10.1016/j.eswa.2015.03.031)

Cuevas, E., Oliva Navarro, D., Díaz Cortés, M. and Osuna Enciso, J.
(2016). OPTIMIZACIÓN Algoritmos Programados con MATLAB. Alfaomega.

Surjanovic, S. & Bingham, D. (2013). Virtual Library of Simulation
Experiments: Test Functions and Datasets. Retrieved from
http://www.sfu.ca/\~ssurjano.

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