Automatic adjustment of the pulse-coupled neural network hyperparameters based on differential evolution and cluster validity index for image segmentation

Applied Soft Computing Journal xxx (xxxx) xxx

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Automatic adjustment of the pulse-coupled neural network hyperparameters based on differential evolution and cluster validity index for image segmentation ∗

Wilfrido Gómez-Flores , Juanita Hernández-López Center for Research and Advanced Studies of the National Polytechnic Institute, ZIP 87130, Ciudad Victoria, Tamaulipas, Mexico

highlights

graphical

abstract

• Automatic image segmentation is

•

•

•

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performed by the pulse-coupled neural network (PCNN). The hyperparameters of the PCNN are tuned by the differential evolution algorithm. A cluster validity index (CVI) is used as fitness function to guide the search procedure. The conventional entropy principle is outperformed by the CVI named Silhouette index. The proposed approach is applied to natural images as well as medical images.

article

info

Article history: Received 23 May 2017 Received in revised form 28 May 2019 Accepted 31 May 2019 Available online xxxx Keywords: Image segmentation Pulse-coupled neural network Differential evolution Cluster validity index

a b s t r a c t The pulse-coupled neural network (PCNN) is a cortical model that can be used in image segmentation applications. The performance of the PCNN depends on adjusting its hyperparameters, where population-based metaheuristics, such as evolutionary algorithms, have been used to perform this task by optimizing a fitness function. In this regard, the entropy criterion is a common fitness function used to evaluate the quality of potential PCNN solutions. However, maximizing the entropy is related to maximize the inter-group separation, but the intra-group cohesion is unconsidered. In this regard, a cluster validity index (CVI) can be used as a fitness function, which defines a relationship between inter-group separation and intra-group cohesion. Therefore, we propose using a CVI to quantify the segmentation quality generated by the PCNN given a set of hyperparameters adjusted by the differential evolution algorithm. The proposed approach is tested on a dataset of natural images, where every image has three reference segmentations; thus, the Jaccard index is used to measure the segmentation performance. The experimental results reveal that a simplified PCNN, when used jointly with the Silhouette index, obtains the best performance with a mean Jaccard value of 0.77, whereas the entropy criterion attains 0.41. Additionally, the proposed approach is tested on two modalities of medical images to show its applicability in other kinds of images. The results suggest that using a CVI instead of the entropy criterion can improve the segmentation performance of the PCNN. © 2019 Elsevier B.V. All rights reserved.

1. Introduction ∗ Correspondence to: CINVESTAV, Ciudad Victoria, Tamaulipas, Mexico. E-mail address: [email protected] (W. Gómez-Flores).

Artificial neural networks (ANN) are mathematical models inspired by the biological neurons in the nervous system of the

https://doi.org/10.1016/j.asoc.2019.105547 1568-4946/© 2019 Elsevier B.V. All rights reserved.

Please cite this article as: W. Gómez-Flores and J. Hernández-López, Automatic adjustment of the pulse-coupled neural network hyperparameters based on differential evolution and cluster validity index for image segmentation, Applied Soft Computing Journal (2019) 105547, https://doi.org/10.1016/j.asoc.2019.105547.

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animals, which can be described as mapping an input space to an output space [1]. In this context, the pulse-coupled neural network (PCNN) is a bio-inspired approach that is based on Eckhorn’s model, which simulates the oscillating behavior of a cat’s visual cortex to create binary images [2]. The PCNN has been used in different image processing applications, such as image segmentation, image fusion, noise filtering, object and edge detection, and so on [3,4]. Moreover, several model simplifications have been proposed to reduce the complexity of the standard PCNN and to increase the computing speed [5,6]. In this sense, the intersecting cortex model (ICM) [7] and the simplified PCNN model (SPCNN) [8] are typical examples of modifications to the standard PCNN. PCNN addresses the segmentation problem in the unsupervised learning domain, that is, clustering pixels based on intensity similarity. In addition, other kinds of ANN models have been used for unsupervised segmentation such as self-organizing maps (SOM) [9], hybrid approaches that combine SOM and PCNN [10], and fully convolutional networks (FCN) [11,12]. On the other hand, supervised learning is widely used to produce pixel-wise classification (known as semantic segmentation), where ANN models such as convolutional neural networks (CNN) [13,14], recurrent neural networks (RNN) [15], and lattice neural networks (LNN) [16] have made image segmentation. Semantic segmentation approaches require large training datasets in which objects of different classes are manually labeled by experts, which can be labor-intensive and costly. Hence, if it is quite challenging to obtain a large-enough dataset for training, then unsupervised segmentation is an alternative to extract the objects of interest by clustering similar local features such as color, intensities, and textures [17]. Independent of the neural network model selected for image segmentation, the adjustment of hyperparameters is an important task that should be addressed in order to obtain an adequate performance [8]. Evolutionary algorithms (EAs) have been widely used to design the architecture of ANN, in which the ‘optimum’ hyperparameters should be found such as the numbers of layers and neurons, the type of activation functions, and the synaptic weights [18,19]. Discovering ANN architectures by using EAs is convenient because the hyperparameters are stored into a coding scheme to represent potential solutions [20]. Also, EAs are useful in finding solutions close to the global optimum in reasonable computation time [21], and they are robust in achieving global optimization in comparison to gradient-based methods [22,23]. The adjustment of PCNN hyperparameters can be viewed as an optimization problem, where EAs such as genetic algorithm (GA) [24,25] and differential evolution (DE) [26,27] have been used to perform this task. Nevertheless other kinds of metaheuristics have also been used to adjust the PCNN hyperparameters including particle swarm optimization (PSO) [28], artificial immune system (AIS) [29], spiral optimization algorithm (SOA) [30], bacterial foraging optimization (BFO) [31], fly fruit optimization algorithm (FOA) [32], and ant colony optimization (ACO) [33]. Several of the metaheuristics mentioned above maximize the entropy principle to find the PCNN hyperparameters that produce a binary image that contains two groups of pixels (or clusters) that are related to the objects of interest and their background. However, the entropy criterion only provides a measure of the overlap between the intensity probability distributions that are related to the objects and their background [34], that is, how different are the objects relative to the background intensities, and it does not indicate how similar the intensity levels of the objects are. The intensity of similarities and dissimilarities can be measured by a cluster validity index (CVI), which defines a relationship between cluster cohesion and cluster separation to estimate

the quality of a clustering solution [35]. A CVI is an adequate way to quantify the segmentation quality of a PCNN, and it can be used as a fitness function to guide the search procedure of a metaheuristic algorithm. In our previous work in [26], we evaluated the standard PCNN and four CVIs to segment 30 natural images, where the DE algorithm is the underlying metaheuristic. This evolutionary algorithm is also used herein due to the simplicity of its implementation, its fast convergence, it codifies real-valued problems, and it has been demonstrated to outperform other metaheuristics such as particle swarm optimization and genetic algorithm in several global optimization problems [36]. Additionally, DE does not require the calculation of analytical expressions such as gradients and it is useful for problems with fitness functions that are nondifferentiable, non-continuous, non-linear, multi-dimensional or have many local minima [23]. We found in [26] that using the Xie–Beni index (a prototype-based CVI) is a feasible way to improve the entropy criterion. However, our proposed approach can be improved by using simplified PCNN models as well as other CVIs that measure the cohesion and separation between all pixels in the image, such as the Silhouette index. Therefore, the purpose of this study is to extend our previous work by including two simplified PCNN models, jointly with six different CVIs (including the entropy principle). In addition to the segmentation of natural images, to test the potential applications of the proposed approach, medical images (ultrasound and mammography) are included within the experiments to segment breast lesions. The remainder of this paper is organized as follows. In Section 2, the PCNN tuning defined as an optimization problem is presented. In Section 3 the PCNN network models, the cluster validity indices, and the differential evolution algorithm are described. In Section 4 the proposed segmentation approach is presented. In Sections 5 and 6, the experimental setup and the results are given. Lastly, Section 7 and 8 present a discussion and the conclusions of this study. 2. Problem definition The problem of image segmentation can be addressed as a clustering problem (i.e., unsupervised segmentation), where an input image S is partitioned into two groups, c1 and c2 , containing the pixels that belong to the objects of interest and their background, respectively, to form a grouping denoted by C = {c1 , c2 } that should satisfy the following three conditions [37]: 1. ck ̸ = ∅ for k = 1, 2; 2. c1 ∪ c2 = S; 3. c1 ∩ c2 = ∅. On the other hand, let x = [x1 , . . . , xD ] be the vector containing the D hyperparameters of a PCNN that generates a segmentation (or clustering) C for the input image S. Let X = {x1 , . . . , xN } be the set of vectors of PCNN hyperparameters that generates N feasible partitions of S. Then, the problem of finding the best clustering can be formulated as an optimization problem, where Ω = {Cx1 , . . . , CxN } is the set of candidate groupings of the pixels in S given the set X, so the optimal segmentation C∗ ∈ Ω should satisfy

∀C ∈ Ω : f (C∗ ) < f (C)

(1)

where f (·) is a fitness function given in terms of a CVI, which measures the relationship between cluster cohesion and cluster separation. Note that f (·) is minimized without loss of generality.

Please cite this article as: W. Gómez-Flores and J. Hernández-López, Automatic adjustment of the pulse-coupled neural network hyperparameters based on differential evolution and cluster validity index for image segmentation, Applied Soft Computing Journal (2019) 105547, https://doi.org/10.1016/j.asoc.2019.105547.

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Fig. 1. Typical model of a standard PCN. The hyperparameters that should be tuned are shown in gray.

3

Fig. 2. The ICM model. The hyperparameters that should be tuned are shown in gray.

3. Basic concepts 3.1. Network models 3.1.1. Standard PCNN model Eckhorn et al. proposed the linking field network (LFN), which is a bio-inspired neural model based on the visual cortex of cats. Their experiments revealed that a cat’s midbrain has an oscillatory behavior that creates binary images from visual impressions [2]. From a computational perspective, if a digital image represents the input data of the LFN, then the image pixels are grouped based on spatial proximity and intensity similarity. By doing so, Johnson and Ritter proposed the PCNN, which is a modification of the LFN that is used to adapt its functioning for image processing algorithms [38]. The PCNN is a bidimensional single layer that is a laterally connected network of integrate-and-fire neurons, with a 1-to-1 correspondence between the image pixels and network neurons. The output images at different iterations typically represent some segment or edge information of the input image. The structure of a standard pulse-coupled neuron (PCN) has three sections, as shown in Fig. 1: (1) input field, which receives stimulus signals, that is, the pixel intensity Sij and the surrounding PCN outputs linked by a synaptic weights matrix Wr with radius r; (2) modulation field, which integrates both linking Lij and feeding Fij signals to create the internal activity Uij ; and (3) pulse generator, here the neuron generates a trigger event Yij when Uij is larger than an adaptive threshold θij . The behavior of a single PCN is described iteratively as [5] Fij [t ] = e−αF Fij [t − 1] + VF (Wr ∗ Y [t − 1])ij + Sij ,

(2)

Lij [t ] = e−αL Lij [t − 1] + VL (Wr ∗ Y [t − 1])ij ,

(3)

Uij [t ] = Fij [t ](1 + β Lij [t ]),

(4)

1

if Uij [t ] > θij [t − 1]

0

otherwise,

(5)

θij [t ] = e−αθ θij [t − 1] + Vθ Yij [t − 1],

(6)

{ Yij [t ] =

wpq =

0

if p = q = m

1/r

otherwise,

where r is the distance from the central element to entry wpq and m is half of the linear dimension of Wr . The standard PCNN has seven hyperparameters to be adjusted: αF , αL , and αθ are attenuation time constants; β is the linking coefficient; and VF , VL , and Vθ denote voltage potentials. In addition, the radius r of the synaptic matrix and the maximum number of iterations tmax should be defined. 3.1.2. Simplified PCNN models The standard PCNN has seven hyperparameters to tune simultaneously that increase its computational complexity. Therefore, different simplified PCNN models have been proposed to reduce the number of hyperparameters and to increase the computing speed of the standard PCNN model. The ICM model is a special case of PCNN where there are no linking neurons and the linking coefficient is set to zero, as illustrated in Fig. 2. The ICM has the same three sections as the standard PCNN: input field, modulation field, and pulse generation, whose behavior is depicted by the following expressions [7]: Fij [t ] = fFij [t − 1] + Sij + (Wr ∗ Y [t − 1])ij ,

{

where ij is the position of the PCN in the network, with 0 ≤ i ≤ M − 1, 0 ≤ j ≤ N − 1; M and N represent the height and width of the PCNN (i.e., the same size as the input image), respectively; ‘∗’ denotes the convolution operator; and, the entries of the synaptic weight matrix are computed as [4]

{

Fig. 3. The SPCNN model. The hyperparameters that should be tuned are shown in gray.

(7)

Yij [t ] =

1

if Fij [t ] > θij [t − 1]

0

otherwise,

θij [t ] = g θij [t − 1] + hYij [t ],

(8) (9) (10)

where r, f , h, and g are hyperparameters that should be adjusted. On the other hand, the SPCNN eliminates all of the leaky integrators in the feeding and linking fields, as illustrated in Fig. 3. The feeding input only receives the external stimulus of the input

Please cite this article as: W. Gómez-Flores and J. Hernández-López, Automatic adjustment of the pulse-coupled neural network hyperparameters based on differential evolution and cluster validity index for image segmentation, Applied Soft Computing Journal (2019) 105547, https://doi.org/10.1016/j.asoc.2019.105547.

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image, whereas the feeding field receives the surrounding neuron outputs linked by a synaptic weights matrix. The behavior of the SPCNN is determined by the following equations [8]: Fij [t ] = Sij ,

(11)

Lij [t ] = (Wr ∗ Y [t − 1])ij ,

(12)

Uij [t ] = Fij [t ](1 + β Lij [t ]),

(13)

where 1 ∑ d(gi , c¯ k ). d¯ k = nk

(19)

gi ∈ck

• PBM index (PBM↑ ) [44]. This index is defined as the product

{ Yij [t ] =

θij [t ] = e

1

0 −αθ

if Uij [t ] > θij [t − 1]

(14)

otherwise,

θij [t − 1] + Vθ Yij [t − 1],

3.2. Cluster validity indices A CVI evaluates the goodness of a particular clustering structure (i.e., a segmented image) by using only information inherent in the data (i.e., intensity levels) [39,40]. Commonly, a CVI defines a relationship between cluster cohesion (within-group scatter) and cluster separation (between-group scatter) to estimate the quality of a clustering solution [41]. These measures are defined in terms of a distance metric, where the Euclidean distance is probably the most common. Herein, both the cohesion and the separation are measured from the intensity levels of the image; thus, the Euclidean distance becomes the absolute difference between two intensity values gi and gj as

⏐ ⏐ d(gi , gj ) = ⏐gi − gj ⏐ .

• A point g is a single pixel value in the input image S. • An input image can be seen as an array of n intensity values denoted by S = [g1 , g2 , . . . , gn ]. • A segmentation (or clustering), denoted by C = {ck | k = 1, 2}, refers to a set of mutually disjoint clusters that partition S into two groups related to the objects in the image and their background. • The number of points in a cluster ck is denoted by n∑ k = |ck |. • The centroid of a cluster is expressed as c¯ k = n1 gi ∈ck gi , k whereas the centroid of S (i.e., total mean intensity) is ∑ denoted as g¯ = 1n g ∈S gi . i

The five CVIs considered in this study are described next. An acronym is defined to identify each CVI, followed by an up (↑) or down (↓) arrow to indicate whether the index is maximized or minimized:

• Calinski–Harabasz index (CH↑ ) [42]. This is a ratio-type index in which the separation is measured by the sum of the distances from the centroids to global mean and the cohesion is estimated by the sum of the distances of the points in a cluster to their respective centroids:

∑

ck ∈C

∑

¯ 2 nk d(c¯ k , g)

∑

d(gi , C¯ k )

ck ∈C

gi ∈ck

2

.

(17)

• Davies–Bouldin index (DB↓ ) [43]. The cohesion is estimated by the mean distance of the objects to their respective centroids and the separation quantifies the distance between centroids: DB(C) =

d¯ 1 + d¯ 2 d(c¯ 1 , c¯ 2 )

PBM(C) =

,

d(c¯ 1 , c¯ 2 ) ·

∑

ck ∈C

(18)

∑

gi ∈S

∑

gi ∈ck

¯ d(gi , g)

d(gi , c¯ k )

)2 .

(20)

• Silhouette index (SI↑ ) [45]. This index is a normalized summation-type index in which the cohesion is measured by the sum of the distances between all of the points in the same cluster and the separation is based on the nearest neighbor distance between points in different groups: SI(C) =

b (gi , ck ) − a (gi , ck )

1∑ ∑ n

max {b (gi , ck ) , a (gi , ck )}

ck ∈C gi ∈ck

,

(21)

where a(gi , ck ) = b(gi , ck ) =

1

∑

nk − 1 1 nr

d(gi , gj ),

(22)

gj ∈ck

∑

d(gi , gj ).

(23)

gj ∈cr ̸ =ck

• Xie–Beni index (XB↓ ) [40]. This is the ratio of the total variation to the minimum separation of the clusters:

(16)

Additionally, some definitions are required before defining the CVIs that are considered herein:

CH(C) =

(

(15)

where r, β , αθ , and Vθ are hyperparameters that should be adjusted. Notice that the maximum number of iterations tmax should be defined for ICM and SPCNN models.

(n − 2) ·

of cohesion and separation measures. The former is the sum of all point distances in a cluster to their respective centroids, whereas the latter is estimated by the distance between the centroids:

∑ XB(C) =

ck ∈C

∑

gi ∈ck

d(gi , c¯ k )2

n · d(c¯ 1 , c¯ 2 )2

.

(24)

For the sake of comparison, the entropy criterion is also included, which is the most common fitness function used by metaheuristics to adjust the PCNN hyperparameters. The entropy criterion (ENT↑ ) measures the inter-class entropy of the whole image by summing the objects entropy and the background entropy as [25,27–31]: ENT(C) = −

∑

pk log2 pk ,

(25)

ck ∈C

where the probability of the kth group is pk =

1 n

∑

gi ∈ck

gi .

3.3. Differential evolution DE is one of the most popular nature-inspired metaheuristics that is used to solve real-parameter optimization problems. This algorithm is inspired by the natural evolution of individuals within a population; that is, the survival of the fittest. DE maintains a population of potential solutions that mutate and recombine to produce new individuals, which are further evaluated and selected based on their fitness. The DE process involves the following basic steps [21,46]: 1. Initialization: the population with N individuals is deg g noted by the set Xg = {x1 , . . . , xN }, where g denotes a generation counter. For the ith individual, a d-dimensional g g g vector is defined by xi = [xi,1 , . . . , xi,d ]. At g = 0, the N individuals are randomly initialized within the search space range [xLj , xUj ], with j = 1, . . . , d, representing the lower (L) and upper bounds (U) of each variable: x0ij = xLj + U (0, 1) · (xUj − xLj ),

(26)

where U (0, 1) is a uniformly distributed random number between 0 and 1.

Please cite this article as: W. Gómez-Flores and J. Hernández-López, Automatic adjustment of the pulse-coupled neural network hyperparameters based on differential evolution and cluster validity index for image segmentation, Applied Soft Computing Journal (2019) 105547, https://doi.org/10.1016/j.asoc.2019.105547.

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g

2. Mutation: for the ith target vector xi , a mutant vector vi g is created by combining the current best individual, xbest , and two individuals randomly chosen from the current g g population, xr1 and xr2 , such that r1 ̸ = r2 ̸ = i. The mutant vector is generated by using the current-to-best strategy as [47] g

g

g

g

g

g

vi = xi + F · (xbest − xi ) + F · (xr1 − xr2 ),

(27)

where F ∈ (0, 1) is the scaling factor that controls the amplification of the vector differences. g 3. Crossover: a trial vector ui is created by exchanging the g g elements of the target vector xi and the mutant vector vi , which is performed by the binomial crossover as

{ g

ui,j =

vig,j if U (0, 1) < CR ∨ j = jr g xi,j otherwise,

(28)

where CR ∈ (0, 1) is the crossover rate that controls the amount of information that is copied from the mutant to the trial vector and jr is a random integer number generated between [1, d]. 4. Penalty: to prevent the solution falling outside of the search space limits, the bounce-back strategy is used to reset out-of-bound trial variables by selecting a new value that lies between the target variable value and the bound being violated [48]:

{ g ui,j

=

g

g

xi,j + U (0, 1) · (xLj − xi,j ) g

g

xi,j + U (0, 1) · (xUj − xi,j )

g

if ui,j < xLj g

if ui,j > xUj .

(29)

Notice that this reinitialization strategy takes progress toward the optimum into account and generates vectors that are located even closer to the bounds. g 5. Selection: if the fitness of the trial vector f (ui ) is better g g than the fitness of the target vector f (xi ), then ui replaces g xi in the next generation, which is expressed by

{ g +1 xi

=

g

if f (ui ) ≤ f (xi )

g

otherwise,

ui

xi

g

g

(30)

where f (·) is the fitness function minimized without loss of generality. To improve the convergence properties of DE, Das et al. [49] adjusted the mutation factor (F ) and the crossover rate (CR) parameters as follows. At each generation, the scale factor is randomly generated in the range (0.5, 1) by using the relation F = 0.5 · (1 + U (0, 1)).

(31)

This scheme allows stochastic variations of the difference vector and thus helps retain population diversity as the search progresses. On the other hand, CR is linearly decreased along the generations from 1 to 0.5 as CR = 1 − 0.5 ·

g gmax

,

(32)

where g is the current generation and gmax is the maximum number of generations. If CR = 1.0, all the elements of the target vector are replaced by the components of the mutant vector, whereas if CR decreases, more elements of the target vector are then inherited by the trial vector. Such a variation of CR helps to explore the search space at the beginning and finely adjust the movements of trial solutions during the later search stages so that the interior of a relatively small space is explored, in which the suspected global optimum lies [49].

5

4. Proposed approach The pseudo-code of the proposed segmentation method based on PCNN and DE is shown in Algorithm 1. Note that the standard PCNN model is considered to exemplify the segmentation procedure, although the other PCNN variants that are defined in Section 3.1.2 can be used. In addition, the XB index is used as the fitness function f (·) that is minimized, although many of the CVIs described in Section 3.2 can be employed. It is worth mentioning that the input image S is normalized so that the largest pixel value is the unity [5]. Algorithm 1: Proposed segmentation algorithm based on a PCNN whose hyperparameters are adjusted by DE. Require: grayscale image (S), population size (N), maximum number of generations (gmax ), lower and upper limits of each variable ([xLj , xUj ]) Ensure: segmented image (C∗ ), best PCNN parameters (x∗ ) Initialize population using [xLj , xUj ]: X0 = {x01 , . . . , x0N } (Eq. (26)) 0

0

Segment S with PCNN using X0 : Ω 0 = {Cx1 , . . . , CxN } (Eq. (14)) Evaluate initial solutions: f (Ω 0 ) (Eq. (24)) for g = 1 to gmax do for i = 1 to N do g Apply mutation strategy: vi (Eq. (27)) g Apply binomial crossover: ui (Eq. (28)) g Apply penalty strategy to ui (Eq. (29)) g

g

Segment S with PCNN using ui : Cui (Eq. (14)) g ui

g xi

if f (C ) < f (C ) then g g Replace target vector with trial vector: xi ← ui (Eqs. (24) and (30)) else Keep the target vector in the population end if end for end for Get best individual x∗ and its associated segmentation C∗ (Eq. (1)) On the other hand, each individual in the population codifies the hyperparameters of the PCNN models, as summarized in Table 1, which are randomly initialized in agreement to their respective lower (xL ) and upper (xU ) limit values as defined in Eq. (26) [26]. In addition, the parameters of the DE algorithm are set as: number of individuals N = 50 and maximum number of generations gmax = 50. 5. Experimental setup The proposed approach is applied on natural and medical images to evaluate the segmentation performance. The dataset of natural images was obtained from the ‘Segmentation Evaluation Database’ [50],1 which considers 49 gray-scale photos of distinct scenarios, including animals, persons, vegetation, buildings, and so on. The size of all of the images is 255 × 300 pixels. Every image in the dataset includes three reference segmentations of the objects of interest, which are defined manually by three different human subjects. The medical images dataset considers two imaging modalities: breast ultrasound (BUS) and mammography (X-ray). The BUS dataset contains 836 ultrasonographies of breast tumors, with 1 www.wisdom.weizmann.ac.il/~vision/Seg_Evaluation_DB/index.html.

Please cite this article as: W. Gómez-Flores and J. Hernández-López, Automatic adjustment of the pulse-coupled neural network hyperparameters based on differential evolution and cluster validity index for image segmentation, Applied Soft Computing Journal (2019) 105547, https://doi.org/10.1016/j.asoc.2019.105547.

6

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Table 1 Hyperparameters of the PCNN models and their lower (xL ) and upper limit (xU ) values. Symbol ‘✓’ indicates that the variable is considered by the model, otherwise symbol ‘✗’ is used. Variable

Description

αL αF αθ

Attenuation constants

VL VF Vθ

Voltage potentials

β

r tmax f g h

Linking coefficient Synaptic matrix radius Number of iterations ICM scalars

L

U

x

x

PCNN

SPCNN

ICM

0.01 0.01 0.01 0.01 0.01 1.00 0.01 1 2 0.01 0.01 1.00

2.50 2.50 2.50 2.50 2.50 25 2.50 9 20 2.50 2.50 25

✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✗ ✗ ✗

✗ ✗ ✓ ✗ ✗ ✓ ✓ ✓ ✓ ✗ ✗ ✗

✗ ✗ ✗ ✗ ✗ ✗ ✗ ✓ ✓ ✓ ✓ ✓

514 benign and 322 malignant lesions, which were manually outlined by a radiologist. This dataset was acquired by our research group during breast diagnostic procedures at the National Cancer Institute (INCa) of Rio de Janeiro, Brazil. The mammography dataset was obtained from the ‘Breast Cancer Digital Repository’ (BCDR).2 This dataset considers 574 mammographies, with 322 benign and 252 malignant lesions. Every image has a breast tumor that has been manually outlined by a radiologist. Regarding the medical images, a region of interest (ROI) containing a breast tumor is cropped from the manual delineation. Next, the ROI is preprocessed to enhance the tumor region by using a constraint Gaussian function, as proposed in [51,52]. The ROIs mean size of the BUS dataset is 135 × 206 pixels, whereas for the mammography dataset it is 261 × 260 pixels. In addition, for the sake of comparison, both medical image datasets were segmented by an algorithm based on the watershed transformation, which was developed specifically for segmenting breast lesions and whose implementation details can be found in [53]. To evaluate the segmentation performance of the proposed approach, the output of the segmentation algorithm (C∗ ) is compared with a reference segmentation (R) by using the Jaccard index that is defined by [54] J(C∗ , R) =

|C∗ ∩ R| , |C∗ ∪ R|

(33)

where |·| denotes number of logical true pixels. This index returns a value in the range [0,1], where ‘1’ indicates perfect similarity between both segmentations and ‘0’ indicates total disagreement. Because three network models and six CVIs are considered, 18 combinations are evaluated. For instance, the nomenclature ICM-SI indicates that the ICM model and the Silhouette index are used in the Algorithm 1. To statistically determine the segmentation performance of the proposed approach, 31 runs are considered for each algorithm combination. Then, the one-way analysis of variance test (ANOVA, α = 0.05), with the Tukey–Kramer correction, is used to statistically compare the mean segmentation performance of every algorithm combination. In addition, the wall clock time to segment and evaluate the individual fitness is measured. The testing platform employed a computer with 4 cores at 3.5 GHz (Intel i7 4770k) and 32 GB of RAM. All of the algorithms were developed in MATLAB 2014a (The MathWorks, Natick, MA, United States). The source code of the proposed segmentation method is available from the authors on request.

2 bcdr.inegi.up.pt.

6. Results 6.1. Segmentation performance in natural images The segmentation results in terms of the Jaccard index are shown in Fig. 4(a). The height of the bars corresponds to the Jaccard mean value of the 31 experiments considering all of the images in the dataset and the three reference segmentations. It is noticeable that the Silhouette index obtained the highest Jaccard values for all of the network models, whereas the lowest performance was obtained by the entropy criterion. By comparing the segmentation performance of all the CVIs for every single network model, the ANOVA test with Tukey–Kramer correction reveals that the Silhouette and Xie–Beni indices perform statistically similarly (p > 0.150) for all network models, whereas the Silhouette index is statistically superior (p < 0.001) from the remaining CVIs. In addition, the entropy criterion is statistically inferior (p < 0.001) to the evaluated CVIs for all of the network models. By comparing the variants PCNN-SI, ICM-SI, and SPCNN-SI, the ANOVA test indicates that there are no statistical differences between variants (p = 0.419), although the SPCNN-SI performed slightly superior (0.77 ± 0.11) to its counterparts. Fig. 4(b) illustrates the computation time required by an individual of the population to segment a single image and to measure its fitness. In general, the SPCNN model required less computation time than the other network models. It is also notable that the entropy criterion aggregates more computation cost to measure the individual fitness. For the SPCNN-SI variant, which obtained the best segmentation performance, the computation time is statistically different from PCNN-SI (p < 0.001) and it is not statistically different from ICM-SI (p = 0.212). These results confirm that the computation time diminishes significantly by reducing the number of hyperparameters without degrading the segmentation performance. Hence, considering the segmentation performance and the computation time, both the SPCNN-SI and the ICM-SI variants are adequate choices for the proposed algorithm. Fig. 5 shows ten examples of natural images segmented by the SPCNN model considering all of the tested CVIs. A reference segmentation is also shown. Note that SPCNN-SI and SPCNN-XB variants generate segmentations that are quite similar to the reference image, where the objects of interest are well segmented. In addition, the SPCNN-PBM variant is able to adequately segment images where the objects are relatively large in relation to the entire image size. Although for images where the objects are small (such as third and fourth rows), the segmentations are not satisfactory. Finally, SPCNN-CH, SPCNN-DB, and SPCNN-ENT failed to properly segment almost all of the images. SPCNN-DB tends to undersegment the input image, whereas SPCNN-CH and SPCNN-ENT tend to oversegment the objects. 6.2. Segmentation performance in medical images Based on the segmentation results of Section 6.1, the SPCNNSI variant is applied to BUS and mammography datasets for segmenting breast tumors. The watershed-based method [53] is also applied to both datasets. The boxplots of the segmentation results are shown in Fig. 6. In addition, the statistical results of the Jaccard index for both datasets are summarized in Table 2. In the boxplots of Fig. 6, it is notable that the SPCNN-SI variant tends to have less variability than the watershed-based method regarding both image datasets; the watershed-based method presents several outliers close to zero, that is, some images are mis-segmented. For the BUS dataset, the ANOVA test revealed that there are no statistical differences (p = 0.320) between SPCNN-SI

Please cite this article as: W. Gómez-Flores and J. Hernández-López, Automatic adjustment of the pulse-coupled neural network hyperparameters based on differential evolution and cluster validity index for image segmentation, Applied Soft Computing Journal (2019) 105547, https://doi.org/10.1016/j.asoc.2019.105547.

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Fig. 4. Results for the natural images dataset: (a) segmentation performance and (b) computation time. In the superior part, the corresponding mean±standard deviation is shown for each combination.

Fig. 5. Examples of natural images segmented by the proposed approach using the SPCNN model and considering all of the tested CVIs. The number on the right-hand side of each segmented image corresponds to its Jaccard value.

Fig. 6. Boxplots of the Jaccard index results for the medical images datasets: (a) breast ultrasound and (b) mammography. The black points indicate outliers.

Please cite this article as: W. Gómez-Flores and J. Hernández-López, Automatic adjustment of the pulse-coupled neural network hyperparameters based on differential evolution and cluster validity index for image segmentation, Applied Soft Computing Journal (2019) 105547, https://doi.org/10.1016/j.asoc.2019.105547.

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Table 2 Statistical results (mean±standard deviation) for the medical images datasets. Method

All

Benign

Malignant

0.77 ± 0.06 0.78 ± 0.10

0.77 ± 0.05 0.76 ± 0.11

0.79 ± 0.07 0.75 ± 0.17

0.75 ± 0.06 0.67 ± 0.18

Breast ultrasound SPCNN-SI Watershed-based

0.77 ± 0.06 0.77 ± 0.10 Mammography

SPCNN-SI Watershed-based

0.77 ± 0.07 0.71 ± 0.18

and watershed-based methods. Although for the mammography dataset, both methods are statistically different (p < 0.001). In addition, segmenting malignant tumors is often more difficult than segmenting benign ones because malignant lesions tend to have irregular and ill-defined shapes, whereas benign lesions tend to have more regular shapes. Fig. 7 illustrates the segmentation results of the SPCNN-SI variant and the watershed-based method in ten medical images. These images could be considered challenging because the tumors present irregular shapes, ill-defined borders, and heterogeneous intensities. Notice that the SPCNN-SI variant is able to generate segmentations close to the radiologist’s delineations, whereas the watershed-based method tends to undersegment the lesions. 7. Discussion Image segmentation is a challenging task because the objects of interest should be extracted automatically with acceptable accuracy. In this sense, the cortical models are attractive approaches because they attempt to simulate the behavior of the visual cortex of animals such as cats or guinea pigs [5]. The PCNN is based on the cortical model proposed by Eckhorn et al. [2], which has a coupling relationship between neighboring neurons that fire simultaneously when the neurons meet certain conditions of pixel similarity; hence, it has been widely used for image segmentation applications. However, the performance of the PCNN strongly depends on selecting adequate hyperparameters, which has been addressed as an optimization problem, where population-based metaheuristics are commonly used for this purpose. Herein, the DE was used as the underlying global optimization method. The goal is to find a set of hyperparameters to segment the objects of interest in an unsupervised way accurately. Because the intensity distributions of the objects of interest and their background are assumed to be well-separated, maximizing the entropy criterion is a common choice in PCNN tuning by metaheuristics [25,27–31]. Hence, only the inter-group separation is measured; that is, how different are the intensities between the objects and their background. However, the entropy criterion does not consider the intra-group cohesion; that is, how similar are the intensity levels of the objects. Therefore, maximizing the inter-group separability does not guarantee that the objects of interest present similar intensity values. To overcome the limitations of the entropy criterion, the CVIs can be used to measure the segmentation quality of the PCNN given a set of hyperparameters. The CVIs have been proposed and used in cluster analysis to estimate the quality of a clustering solution [41]. The experimental results in natural images demonstrated that the CVIs significantly outperformed the entropy criterion, where the Silhouette index (SI) presented the best performance. This satisfactory performance happens because both the cohesion and the separation are computed by considering the distances between all of the pixels in the image, whereas the reaming evaluated CVIs (i.e., CH, DB, PBM, and XB) simplify the computation by considering group centroids (or prototypes).

Three networks models were evaluated: the standard PCNN and two simplified models, namely ICM and SPCNN. The segmentation results revealed that these three network models performed statistically similarly by considering the same CVI, although the computing time significantly diminished in the simplified models. Therefore, the SPCNN-SI variant is an adequate option for image segmentation because it obtained the best segmentation performance in reduced computing time. The SPCNN-SI variant was applied to the specific case of breast lesion segmentation in ultrasonography and mammography. This application is particularly relevant in the context of computeraided diagnosis (CAD). The main goal of CAD is to increase the efficiency and effectiveness of breast cancer screening by using a computer system as a second reader to assist the radiologist in clinical recommendations [55]. Commonly, the segmented lesions are further classified into benign and malignant classes; thus, lesion segmentation is an important task to describe the lesion shape accurately. The SPCNN-SI outperformed a segmentation method based on the watershed transform, which was designed for breast lesion segmentation [53]. The improvement in the segmentation performance is due to the objective function that is used to evaluate the quality of a potential lesion segmentation. The proposed approach optimizes a CVI that is computed directly from the intensities of the image, whereas the watershed-based method maximizes a gradient-based function called average radial derivative (ARD) function [52]. The ARD is particularly useful when the lesions present well-defined contours (i.e., strong intensity gradients) and the internal intensities are relatively homogeneous. However, if the lesion presents blurry edges (i.e., weak intensity gradients) and different intensities, then the ARD tends to undersegment the lesion, as illustrated in the examples of Fig. 7. Hence, the proposed approach could be used in scenarios where the image presents weak intensity changes between the lesion and the adjacent tissue. In this study, the framework of the DE algorithm allowed to easily incorporate a CVI as an objective function, although other metaheuristics can optimize a CVI for adjusting the PCNN hyperparameters and potentially obtaining similar results. Therefore, we deduce that the satisfactory performance is mainly because of a CVI defines a relationship between cluster cohesion and cluster separation to measure the quality of a potential segmentation. The feedback from CVI guides the optimization process to find the PCNN hyperparameters that contribute to partition the input image into two well-separated groups of pixels, that is, the objects of interest and their background. 8. Conclusions An evolutionary segmentation approach based on the PCNN was presented. The PCNN requires its hyperparameters to be adjusted to perform adequately in image segmentation tasks, where population-based metaheuristics are commonly used. The maximization of the entropy criterion is a common fitness function that is employed to measure the segmentation quality of a potential solution. However, because image segmentation is basically a clustering of intensity levels, the entropy criterion only measures the inter-cluster separability while the intra-cluster cohesion is unconsidered. A CVI defines a relationship between cluster cohesion and cluster separation to estimate the quality of a clustering solution. Hence, using a CVI as a fitness function within the DE algorithm was proposed herein. The results demonstrate that the proposed approach significantly outperforms the entropy criterion under the same experimental conditions. Therefore, optimizing a CVI instead of the entropy criterion is a feasible approach to improve the segmentation performance of the PCNN. In addition, the SPCNN jointly with

Please cite this article as: W. Gómez-Flores and J. Hernández-López, Automatic adjustment of the pulse-coupled neural network hyperparameters based on differential evolution and cluster validity index for image segmentation, Applied Soft Computing Journal (2019) 105547, https://doi.org/10.1016/j.asoc.2019.105547.

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Fig. 7. Medical images segmented by radiologist, SPCNN-SI, and watershed-based methods (black and white contours): (a) breast ultrasound and (b) mammographies. The number on the right-hand side of each segmented image corresponds to its Jaccard value.

the Silhouette index (SI) accurately segmented images in reducing computation time. The SPCNN-SI variant was tested in natural images, to segment objects of interest, as well as in medical images, to segment breast tumors. It is worth mentioning that the effectiveness of the proposed approach is limited to images where the objects of interest are relatively well-contrasted with their background to obtain binary segmentation. In addition, real-time applications are not considered herein; nevertheless, parallelization strategies and hardware like field programmable gate arrays (FPGAs) may accelerate the algorithm. Under this scenario, the SPCNN-SI method can be potentially extended to other kinds of medical images such as vessel extraction in retinal images, brain tumor segmentation in magnetic resonance images, blood cell segmentation and extraction of cell structures in microscopy images, and so on. Also, applications such as defect detection in materials, robot navigation, and target tracking, could be addressed by the proposed method. Acknowledgments This research was supported by a Fondo SEP-Cinvestav 2018, Mexico grant (No. 145) and a National Council of Science and Technology (CONACyT, Mexico) research scholar grant (No. 463795). Declaration of competing interest

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No author associated with this paper has disclosed any potential or pertinent conflicts which may be perceived to have impending conflict with this work. For full disclosure statements refer to https://doi.org/10.1016/j.asoc.2019.105547.

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