Evolutionary Image Segmentation
Based on Multiobjective Clustering
Shinichi Shirakawa and Tomoharu Nagao

Abstract— In the fields of image processing and recognition,
image segmentation is an important basic technique in which
an image is partitioned into multiple regions (sets of pixels).
In this paper, we propose a method for evolutionary image
segmentation based on multiobjective clustering. In this method,
two objectives, overall deviation and edge value, are optimized
simultaneously using a multiobjective evolutionary algorithm.
These objectives are important factors for image segmentation.
The proposed method finds various solutions (image segmentation results) by the use of an evolutionary process. We apply the
proposed method to several image segmentation problems andconfirm that various solutions are obtained. In addition, we use
a simple heuristic method to select one solution from the original
Pareto solutions and show that a good image segmentation
result is selected.

I. I NTRODUCTION
Image processing and recognition technologies are becoming increasingly important. In the fields of image processing
and recognition, image segmentation is a basic and important
technique in which an image is partitioned into multiple
regions (sets of pixels). Each of the pixels in a regionis similar to the other with respect to some characteristic
or computed property, such as color, intensity, or texture.
Recently, many image segmentation methods have been
proposed and their effectiveness have been demonstrated [1],[2]. Typical examples include clustering methods, histogrambased methods, region growing methods [3], and graphpartitioning methods [4], [5]. In clustering methods, the kmeans algorithm is the most popular technique, and is used
to partition an image into k clusters. The quality of thesolution using k-means depends on the initial set of clusters
and the value of k. Region growing methods examine theneighboring pixels of initial seed points and determine
whether the pixel should be added to the seed point.
The minimum spanning tree (MST) technique is also usedin image segmentation [6], [7]. Each pixel is a node in agraph. The weight of an edge is a measure of the similarity
between pixels. The image is partitioned into segments byremoving the edges whose weight is more than the threshold.
Evolutionary computation (EC) techniques are also applied
to image segmentation problems. Several algorithms using
genetic algorithms (GAs) have been proposed [8], [9]. Poli
applied genetic programming (GP) [10] to construct pixelclassification-based segmentation algorithms [11].
S. Shirakawa and T. Nagao are with the Department ofInformation
Media
and
Environment,
Graduate
SchoolofEnvironment
andInformation
Sciences,
Yokohama
National
University, 79-7 Tokiwadai, Hodogaya-ku, Yokohama, Kanagawa
240-8501, Japan shirakawa-shinichi-bg@ynu.ac.jp,

nagao@ynu.ac.jp

The multiobjective clustering algorithm, multi-objective
clustering with automatic k-determination (MOCK), wasdeveloped by Handl and Knowles [12], [13], and various
improvements and applications have been investigated [14]–[18]. This algorithm optimizes two complementary objectives
based on cluster compactness and connectedness using a
multiobjective evolutionary algorithm (MOEA) [19], [20].
MOCK has been applied mainly to artificial datasets. MOCK
is reported to find partitions better than other conventional
clustering algorithms, such as the k-means method.
In this paper, we propose a method for evolutionary
image segmentation based on multiobjective clustering. In
the proposed method, two objectives, overall deviation andedge value, are optimized simultaneously using an MOEA.
These objectives are important factors in image segmentation. The proposed method finds various solutions (image
segmentation results) through an evolutionary process. We
apply the proposed method to several image segmentation
problems and confirm that various solutions are obtained.
Since there is no general solution to image segmentation
problems, image segmentation techniques often have to be
combined with domain knowledge to solve such problems
effectively for a problem domain. Therefore, we believe that
it is significant to obtain various types of image segmentation
results simultaneously. In addition, we use a simple heuristic
method to select one solution from the original Pareto
solutions and show that a good image segmentation result
is selected.
The next section of this paper presents an overview of
MOCK. In Section 3, we describe our proposed method
for evolutionary image segmentation based on multiobjective clustering. Next, in Section 4, we apply the proposed
method to image segmentation problems and show several
experimental results. Finally, in Section 5, we describe ourconclusions and future work.
II. M ULTIOBJECTIVE C LUSTERING WITH AUTOMATIC
K- DETERMINATION (MOCK) [12], [13]MOCK is a recent clustering algorithms based on an
MOEA. MOCK optimizes two clustering objectives, overall
deviation and connectivity, which reflect two fundamentallydifferent aspects of a good clustering solution. Therefore,
MOCK returns not just one solution, but an entire set ofsolutions. The first objective, overall deviation, is defined inequation (1). This equation described the overall summed
distances between data items and their corresponding cluster
center.

Dev(C) =∑ ∑δ(i, µk )(1)

Ck ∈C i∈Ck

where C is the set of all clusters, µk is the centroid ofcluster Ck , and δ() is the distance function. Overall deviation
should be minimized. Minimizing overall deviation increases
the number of clusters.
The second objective, connectivity, is defined in equation(2). This objective evaluates the degree to which neighboringdata points have been placed in the same cluster.

Conn(C) ={where xr,s =

N ∑
L
∑

xi,nni (j) ,i=1 j=1
1j0

if ∄Ck : r, s ∈ Ck
otherwize(2)

where N is the number of data points, nni (j) is the jthnearest neighbor of datum i, and L is a parameter determining the number of neighbors that contribute to the connectivity measure. Connectivity is also minimized. By minimizingthese two objectives, a variety of different solutions are
generated in MOCK.
Each individual in MOCK can be represented as a graph.
Each datum is represented as a node, and an edge between
two nodes indicates that these data are in the same cluster.
Each individual has an N -length gene (N is the number ofdata points). The range of the gene value is [1, N ]. Whenthe ith gene value is j, data item i links to data item j. The
decoding of this representation requires the identification of
all subgraphs. Thus, the number of clusters is determined
automatically in the decoding step.
In the MOCK algorithm, MST is used for the initializationof individuals. MST is the shortest tree that contains all nodes
in the graph and has no loops. The shortest tree means thatthe total cost of all edges is the minimum. MOCK uses
Prim’s algorithm to create the MST. In the initialization of
the ith individual in the population, the (i − 1) long links are
removed from the MST individual. In this manner, MOCK
creates high-quality initial solutions.
In addition, MOCK has a scheme for determining thenumber of clusters automatically (automatically selecting thebest solution). This scheme is based on the Gap statistic[21]. In this scheme, random data are created based on theprincipal component of original data, and these data are
divided into clusters by MOCK. The solutions of the random
and original data are different. The appropriate number ofclusters is determined using these solutions of random andoriginal data.
III. E VOLUTIONARY I MAGE S EGMENTATION BASED ON
M ULTIOBJECTIVE C LUSTERINGIn this section, we describe the proposed method forevolutionary image segmentation based on multiobjective
clustering.

Fig. 1. Example of the phenotype (graph representation) and genotype(string of integers) of an individual in the proposed method. The graphcorresponds to an image plane.

A. Representation of individuals
Figure 1 illustrates the phenotype and genotype of anindividual in the proposed method. The representation ofindividuals is a graph structure. The phenotype corresponds
to an image plane. Each pixel in the image is represented
as a node, and an edge between two nodes indicates thatthese pixels are in the same region (cluster). Each node canconnect only with either itself or the four neighboring pixels. This graph representation indicates image segmentation.
Each individual has an N -length gene (N is the number ofpixels). When the ith gene value is j, pixels i and j arein the same region. For instance, nodes (pixels) numbered
1, 2, 3, 5, 6, and 7 in Figure 1 are in the same region.
The decoding of this representation requires identificationof all subgraphs. Thus, the number of regions is determined
automatically in the decoding step. Because of the constraint
on connections, regions far from each other on the image
plane do not appear as the same region. A very large search
space can be limited using this constraint, and it also reduces
the computational time required to create the MST. We usean MOEA to optimize image segmentation represented by
the graph structure.
In the proposed algorithm, MST under the constraint on
node connections is used for the initialization of individuals.
We use Prim’s algorithm, as used in MOCK, to create the
MST. We assume to the initial individuals, MST is used
as it is (do not remove the links from MST). Therefore,
initial individuals have the potential for creating high-quality
solutions.

B. Multiobjective evolutionary algorithm
We adopt an evolutionary method to obtain the optimum
structure. The genotype of the proposed method is a linear
string, as in MOCK. Therefore, the method can use atypical simple genetic operator. In this study, we use uniform
crossover and mutation as the genetic operators. In uniform
crossover, randomly selected genes are swapped between twoparents, which generate offsprings. In mutation, randomly selected genes with a mutation rate Pm are randomly changed
under the constraints on connections.
We use Strength Pareto Evolutionary Algorithm 2(SPEA2) [22] as the MOEA technique. SPEA2 assigns
fitness values to individuals according to dominance anddensity criteria. The selection of individuals is also basedon these criteria. It maintains two populations, a current
population and an archive population. In each generation,
the nondominated solutions in the current population and thearchive are copied into the archive of the next generation.
Selection occurs from both the population and archive. The
procedure of the SPEA2 algorithm is as follows:
1) Initialization: Generate an initial population P0 andcreate the empty archive population P̄0 = ∅. Set thegeneration counter t = 0.
2) Fitness assignment: Calculate fitness values of individuals in Pt and P̄t .
3) Environmental selection: Copy all nondominated individuals in Pt and P̄t to P̄t+1 . If the size of P̄t+1
exceeds N̄ (archive size), reduce P̄t+1 by means of
the truncation operator; otherwise, if the size of P̄t+1
is less than N̄ , fill P̄t+1 with dominated individuals in
Pt and P̄t .
4) Termination: If a certain specified condition is satisfied, return nondominated individuals in P̄t+1 as theoutputs of the system.
5) Mating selection: Select parents from P̄t+1 based on
binary tournament selection.
6) Variation: Apply crossover and mutation operators to
the parents and set Pt+1 to the resulting population.
Increment generation counter (t = t + 1) and go tostep 2.
In the fitness assignment step, a strength value S(i) anda fitness F (i) are calculated for each individual i in the
population Pt and archive P̄t . S(i) shows the number ofdominated solutions by the ith individual. F (i) is the sumof the strength values of its dominators. In the environmental
selection step, the nondominated individuals whose fitness is
0 are placed in P̄t+1 . If the number of such individuals isgreater than the archive size, the archive is truncated using a
special operator. In this operator, the individual that has theminimum distance to another individual is chosen at eachstage. If the nondominated individuals cannot fill the archive
exactly, the best dominated individuals are copied to P̄t+1 .
C. Objective functions
In the proposed method, two objectives, overall deviationand edge value, are optimized simultaneously using an

MOEA. These objectives are important factors of image
segmentation.
Overall deviation is defined in equation (3). This equation
gives the overall summed distances between the pixels andthe center value of the corresponding region (cluster) they
belong to. Overall deviation is a measure of the similarity of
pixels in the same region.
∑ ∑Dev(R) =δ(i, µk )(3)
Rk ∈R i∈Rkwhere R is the set of all regions, µk is the centroid of the
pixels in the region Rk , and δ() is the distance function.
Overall deviation should be minimized. Minimizing overall
deviation roughly increases the number of regions (clusters).
We calculate the distance function δ() defined as a Euclidean
distance, and use either RGB or CIE L*a*b* [23] as the color
space. The distance function using the RGB color space isdefined in equation (4).
√
δRGB = ∆R2 + ∆G2 + ∆B 2
(4)
The distance function using the CIE L*a*b* color space isdefined in equation (5). The CIE L*a*b* color space has auniform chromaticity scale.
√(5)δL∗a∗b∗ = ∆L ∗2 +∆a ∗2 +∆b∗2
The second objective, the edge value, is defined in equation (6). This objective evaluates the average distance onboundaries between the regions. This value is a measure of
the difference in the boundary between the regions.
1 ∑∑xi,j ,B i=1
N

Edge(R) = −

j∈F

i
{
δ(r, s) if ∄Rk : r, s ∈ Rk
where xr,s =0otherwize(6)

where N is the number of pixels, Fi indicates the fourneighboring pixels of pixel i, B is the number of boundary
pixels, and δ() is the distance function. This edge value
is also minimized, which roughly decreases the number ofregions (clusters). By minimizing these two objectives, avariety of different image segmentations are generated. In
other words, the proposed method returns a range of solutions
that have different numbers of regions.
D. Solution selection method
The main objective of this study is to obtain a varietyof image segmentation results. However, it is necessary toobtain one solution suitable for a circumstances. In MOCK,
the number of clusters is determined based on the Gap
statistic. However, its determination scheme requires random
data clustering in several trials. Therefore, the computationaltime for determining the number of clusters is high [17].
In the proposed approach, we use a simple heuristic
method to select one solution and find the “knee” of theoriginal Pareto solutions. The most interesting solutions of

(a) pepper

Fig. 2.

(b) sailboat

Example of the selection scheme in the proposed method.

TABLE IS ETTINGS OF PARAMETERS USED IN THE EXPERIMENTS .
Parameters
Gene length
Color space
MOEA model
Number of generations
Population size
Archive size
Crossover
Crossover rate Pc
Mutation rate Pm
Constraints

Valuesimage size
(96 × 96[pixels] or 128 × 96[pixels])
RGB or CIE L*a*b*
SPEA2
300
50
50
Uniform crossover
0.7
0.0001
1 < region num < 50region size > 100 [pixels](c) terra(d) paprika
the Pareto-optimal front are those where a small improvement in one objective would lead to a large deterioration inat least one other objective. These solutions are sometimes
also called knees [24].
Our selection scheme is as follows:
1) Normalize the fitness value of all the solutions to therange of [0.0, 1.0] using the original Pareto solutions.
2) Calculate the sum of “normalized Dev(R)” and “normalized Edge(R)”, and select the solution with theminimal value as the best solution.
In other words, the solution that has the minimal distancefrom the line y = −x is selected as the adequate solution inthis selection scheme. An example of our selection scheme
is shown in Figure 2.
IV. E XPERIMENTS AND R ESULTS
In this section, we apply the proposed method to image
segmentation problems, and confirm that a variety of image
segmentation results are obtained. In addition, we select one
solution from the original Pareto solutions.
A. Settings of the experiments
The parameters used by the proposed method are shownin Table I. We use either RGB or CIE L*a*b* as the color
space. We constrain the number of pixels in one cluster to be

Fig. 3.

Original color images used in the experiments.

less than 100 to prevent the existence of very small regions(clusters). The original color images used in the experiments
are shown in Figure 3. In these experiments, we use four
images; (a) pepper, (b) sailboat, (c) terra, and (d) paprika.
B. Results and discussion
The obtained Pareto solutions by the proposed method
are shown in Figures 4 and 5. The values of the axes arenormalized. Figure 4 shows the obtained Pareto solutions
using the RGB color space, and Figure 5 shows those using
CIE L*a*b* color space. The obtained solutions of each
image have different Pareto front. The solutions obtainedwith the RGB and CIE L*a*b* color spaces are also different. Although the shapes of the Pareto front are different,a wide range of solutions is found in all cases. Therefore, we
conclude that the evolutionary algorithm is efficient. We use
SPEA2 as the MOEA model in the experiments. It would
be interesting to analyze the behaviour of NAGA-II [25] andother MOEA models. Nebro et al. show that the convergence
speed of SPEA2 is slowest for test problems [26].
Another objective of this study is to verify whether various

(a) pepper(b) sailboat(c) terra(d) paprika

Fig. 4. Obtained Pareto solutions by the proposed method using the RGB color space. The horizontal axis represents normalized edge values, and thevertical axis represents normalized overall deviation.

(a) pepper(b) sailboat(c) terra(d) paprika

Fig. 5. Obtained Pareto solutions by the proposed method using the CIE L*a*b* color space. The horizontal axis represents normalized edge values, andthe vertical axis represents normalized overall deviation.

image segmentation results are obtained. Figure 6 shows
examples of the image segmentation results in the case of
the “pepper” image using the RGB color space. The blacklines in the images indicate the boundaries of the regions.
The results show that various image segmentations (various
numbers and shapes of regions) are obtained. Also, the image
segmentations are considered to be relatively good.
Next, we select one solution from the Pareto front using
the selection scheme. Figure 7 shows the selected solutions.
In general, these image segmentations are considered goodresults. In the case of “paprika” the three peppers are
partitioned into a single region when the CIE L*a*b* color
space is used. Thus, the best solution is not selected in thiscase.
The results of image segmentation using the k-means
method, which is one of the clustering methods, are shownin Figure 8. In this case, we apply the k-means method
to the “paprika” image using the RGB color space. In thismethod, the user has to choose an appropriate number ofclusters. Image segmentation is overdone when there aremany clusters. Also, the k-means method assigns regions
separated on the image plane to the same region. It is
confirmed that the proposed method obtains good solutions
compared with the k-means method.

In the experiment, we only compared the proposed methodwith k-means method. We might have to compare the proposed method with other methods such as graph partitioning
based methods [4], [5] and ensemble based methods [27].

V. C ONCLUSIONS AND F UTURE W ORK
In this paper, we proposed evolutionary image segmentation based on multiobjective clustering. In the proposed
method, two objectives, overall deviation and edge value, areoptimized simultaneously using an MOEA. These objectives
are important factors in image segmentation. The proposed
method finds various solutions (image segmentation results)
by an evolutionary process. We applied the proposed method
to several image segmentation problems and confirmed thatvarious solutions are obtained. In addition, one solution was
selected from the original Pareto solutions by a simple heuristic method. We also show that a good image segmentation
result is selected.
In future work, we will apply the proposed method toother types of images, such as medical images. Moreover,
we plan to examine more appropriate objectives for image
segmentation.

Overall deviation: 0.948
Edge value: 0.174
Region num: 3

Overall deviation: 0.860
Edge value: 0.289
Region num: 5Overall deviation: 0.845
Edge value: 0.316
Region num: 5Overall deviation: 0.698
Edge value: 0.362
Region num: 5Overall deviation: 0.832
Edge value: 0.334
Region num: 6

Overall deviation: 0.627
Edge value: 0.381
Region num: 6

Overall deviation: 0.438
Edge value: 0.509
Region num: 6

Overall deviation: 0.590
Edge value: 0.438
Region num: 7Overall deviation: 0.410
Edge value: 0.512
Region num: 7Overall deviation: 0.562
Edge value: 0.462
Region num: 8Overall deviation: 0.249
Edge value: 0.625
Region num: 10

Overall deviation: 0.149
Edge value: 0.738
Region num: 11

Overall deviation: 0.178
Edge value: 0.724
Region num: 13

Overall deviation: 0.062
Edge value: 0.863
Region num:14

Overall deviation: 0.063
Edge value: 0.844
Region num: 15

Overall deviation: 0.000
Edge value: 1.000
Region num: 17

Fig. 6. Examples of the image segmentation results in the case of the “pepper” image using the RGB color space. The black lines indicate the boundariesof the regions. Each result also shows the normalized overall deviation, normalized edge value, and number of regions.

pepper
Color space: RGB
Overall deviation: 0.302
Edge value: 0.567
Region num: 9

pepperColor space: CIE L*a*b*
Overall deviation: 0.290
Edge value: 0.422
Region num: 9

sailboat
Color space: RGB
Overall deviation: 0.114
Edge value: 0.710
Region num: 7

sailboat
Color space: CIE L*a*b*
Overall deviation: 0.056
Edge value: 0.651
Region num: 10

terra
Color space: RGB
Overall deviation: 0.240
Edge value: 0.666
Region num: 9

terra
Color space: CIE L*a*b*
Overall deviation: 0.366
Edge value: 0.195
Region num: 7

paprika
Color space: RGB
Overall deviation: 0.042
Edge value: 0.883
Region num: 9

paprika
Color space: CIE L*a*b*
Overall deviation: 0.526
Edge value: 0.076
Region num: 4

Fig. 7. Selected image segmentation results obtained by the proposed method. The black lines indicate the boundaries of the regions. The normalized
overall deviation, normalized edge value, and number of regions are also given for each selected solution.

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C ODEThe python implementation is available at https://github.com/shirakawas/mock-segmentation.
E RRATA
Equation (6)•In the original paper:
The second objective, the edge value, is defined in equation (6). This objective evaluates the overall summed
distances on boundaries between the regions. This value
is a measure of the difference in the boundary between
the regions.
Edge(R) = −N ∑∑xi,j ,

i=1 j∈Fi{
δ(r, s) if ∄Rk : r, s ∈ Rk
where xr,s =0otherwize
where N is the number of pixels, Fi indicates the fourneighboring pixels of pixel i, and δ() is the distancefunction.
• In this revised paper (corrected version):
The second objective, the edge value, is defined in equation (6). This objective evaluates the average distance onboundaries between the regions. This value is a measureof the difference in the boundary between the regions.
1 ∑∑xi,j ,B i=1
N

Edge(R) = −

j∈F

i
{
δ(r, s) if ∄Rk : r, s ∈ Rk
where xr,s =0otherwize

where N is the number of pixels, Fi indicates thefour neighboring pixels of pixel i, B is the number ofboundary pixels, and δ() is the distance function.