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1 Introduction

An integration method is a mechanism which takes as input a number n of data and combines them to result in a value representative of the information, methods exist which combine information from different sources which can be aggregation operators as arithmetic mean, geometric mean, OWA [1], and so on., In a modular neural network (MNN) is common to use some methods like fuzzy logic Type 1 and Type 2 [2–4], the fuzzy Sugeno integral [5], Interval Type-2 Fuzzy Logic Sugeno Integral [6], a probabilistic sum integrator [7], A Bayesian learning method [8], among others.

The Choquet integral is an aggregation operator which has been successfully used in various applications [9–11], but has not been used as a method of integrating a modular neural network, and in this chapter is proposed for achieving this goal 12.

This chapter is organized as follows: Sect. 2 shows the concepts of Fuzzy Measures and Choquet integral which is the technique that was applied for the combination of the several information sources. Section 3 presents Edge detection based in Sobel and Morphological gradient with type-1 fuzzy system. Section 4 shows the modular neural network proposed and in the Sect. 5 the simulation results are shown. Finally, Sect. 6 shows the conclusions.

2 Fuzzy Measures and Choquet Integral

Sugeno first defined the concept of “fuzzy measure and fuzzy integral” in 1974 [13]. A fuzzy measure is a nonnegative function monotone of values defined in “classical sets”. Currently, when referring to this topic, the term “fuzzy measures” has been replaced by the term “monotonic measures”, “non-additive measures” or “generalized measures” [14–16].When fuzzy measures are defined on fuzzy sets, we speak of fuzzified measures monotonous [16].

2.1 Fuzzy Measures

A fuzzy measure μ with respect to the dataset X, must satisfy the following conditions:

  1. 1.
    $$ \mu \left( X \right) = 1;\;\mu (\emptyset) = 0 $$
  2. 2.
    $$ Si\;A \subset B,\;then\;\mu \left( A \right)\,\le\,\mu \left( B \right) $$

In condition 2 A and B are subsets of X.

A fuzzy measure is a Sugeno measure or λ-fuzzy, if it satisfies the condition (1) of addition for some λ > −1.

$$ \mu \left( {A \cup B} \right) = \mu (A) + \mu (B) + \lambda \mu (A)\;\mu (B) $$
(1)

where λ can be calculated with (2):

$$ f\left( \lambda \right) = \left\{ {\prod\limits_{i = 1}^{n} {\left( {1 + M_{i} \left( {x_{i} } \right)\lambda } \right)} } \right\} - (1 + \lambda ) $$
(2)

The value of the parameter λ is determined by the conditions of the theorem 1.

Theorem 1

Let \( \mu \left( {\left\{ x \right\}} \right) < 1 \) for each x ∈ X and let \( \mu \left( {\left\{ x \right\}} \right) > 0 \) for at least two elements of X. Then (2) determines a unique parameter λ in the following way:

\( If\;\sum\nolimits_{x \in X} {\mu \left( {\{ x\} } \right) < 1} , \;then\;\lambda \;is\;in\;the\;interval\;(0,\infty ). \)

\( If\;\sum\nolimits_{x \in X} {\mu \left( {\{ x\} } \right) = 0} ,\;then\;\lambda = 0; \) That is the unique root of the equation.

\( If\;\sum\nolimits_{x \in X} {\mu \left( {\{ x\} } \right) > 1} ,\;then\;\lambda \;se\;is\;in\;the\;interval\;( - 1,0). \)

The method to calculate Sugeno measures, carries out the calculation in a recursive way, using (3) and (4).

$$ \mu \left( {A_{1} } \right) = \mu \left( {M_{i} } \right) $$
(3)
$$ \mu \left( {A_{i} } \right) = \mu \left( {A_{(i - 1)} } \right) + \mu \left( {M_{i} } \right) + \left( {\lambda \mu \left( {M_{i} } \right)*\mu \left( {A_{(i - 1)} } \right)} \right) $$
(4)

where \( 1 < M_{i} \le \cdots \le {\text{n,}} \) and the values of \( \mu \left( {x_{i} } \right) \) correspond to the fuzzy densities determined by an expert.

To perform this procedure \( \mu \left( {M_{i} } \right) \) should be permuted with respect to the descending order of their respective \( \mu \left( {A_{i} } \right). \)

There are 2 types of Integral, the integral of Sugeno and Choquet Integral.

2.2 Choquet Integral

The Choquet integral can be calculated using (5) or an equivalent expression (6)

$$ \begin{aligned} Choquet =\,& \sum\limits_{i = 1}^{n} {\left\{ {\left[ {A_{i} - A_{(i - 1)} } \right]*D_{i} } \right\}} \\ {\text{With}}\;{\text{A}}_{0} =\,& 0 \\ \end{aligned} $$
(5)

Or also

$$ \begin{aligned} Choquet =&\, \sum\limits_{i = 1}^{n} {A_{i} *\left\{ {\left[ {D_{i} - D_{(i + 1)} } \right]} \right\}} \\ With\;D_{(n + 1)} =&\, 0 \\ \end{aligned} $$
(6)

where \( A_{i} \) represents the fuzzy measurement associated with data \( D_{i} . \)

2.2.1 Pseudocode of Choquet Integral

  • INPUT: Number of information sources n; information sources x1, x2, …, xn; fuzzy densities of information sources M1, M2, …, Mn ∈ (0,1).

  • OUTPUT: Choquet integral (σ(x1), σ(x2), …, σ(xn)).

  • STEP 1: Calculate λ finding the root of the function (2).

  • STEP 2: Fuzzify variable xi.

  • $$ D_{i} = \left\{ {x,\,\mu_{Di} (x) |x \in X} \right\}, \;\mu D_{i} (x) \in \left[ {0,\, 1} \right] $$

  • STEP 3: Reorder Mi with respect to D (xi) in descending order

  • STEP 4: Calculate fuzzy measures for each data with (3), (4).

  • STEP 5: Calculate Choquet integral with (5) or (6).

  • STEP 6: OUTPUT Choquet.

  • STOP

3 Edge Detection

Edge detection can be defined as a method consisting of identifying changes that exist in the light intensity, which can be used to determine certain properties or characteristics of the objects in the image.

We used the database of the ORL [17] to perform the training of the modular neural network, which has images of 40 people with 10 samples of each individual. To each of the images was applied a pre-processing by making use of Sobel edge detector and morphological gradient with type 1 fuzzy logic system [18] in order to highlight features, some of the images can be displayed in Fig. 4a.

3.1 The Morphological Gradient

To perform the method of morphological gradient is calculated every one of the four gradients as commonly done in the traditional method using (7–11), see Fig. 1, however, the sum of the gradients is performed by a fuzzy system type 1 [17] as shown in Fig. 2, and the resulting image can be viewed in Fig. 4b (Fig. 3).

$$ D1 = \sqrt {\left( {z5 - z2} \right)^{2} + \left( {z5 - z8} \right)^{2} } $$
(7)
$$ D2 = \sqrt {\left( {z5 - z4} \right)^{2} + \left( {z5 - z6} \right)^{2} } $$
(8)
$$ D3 = \sqrt {\left( {z5 - z1} \right)^{2} + \left( {z5 - z9} \right)^{2} } $$
(9)
$$ D4 = \sqrt {\left( {z5 - z7} \right)^{2} + \left( {z5 - z3} \right)^{2} } $$
(10)
$$ G = D1 + D2 + D3 + D4 $$
(11)
Fig. 1
figure 1

Calculation of the gradient in the 4 directions

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Fig. 2
figure 2

Variables for the edge detector of morphological gradient type 1

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Fig. 3
figure 3

Variables for the edge detector with the type-1 fuzzy Sobel

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3.2 Sobel

The Sobel operator is applied to a digital image in gray scale, is a pair of 3 × 3 convolution masks, one estimating the gradient in the x-direction (columns) (12) and the other estimating the gradient in the y-direction (rows) (13) [19].

$$ sobel_{x} = \left[ {\begin{array}{*{20}c} { - 1} & 0 & 1 \\ { - 2} & 0 & 2 \\ { - 1} & 0 & 1 \\ \end{array} } \right] $$
(12)
$$ sobel_{y} = \left[ {\begin{array}{*{20}c} 1 & 2 & 1 \\ 0 & 0 & 0 \\ { - 1} & { - 2} & { - 1} \\ \end{array} } \right] $$
(13)

If we have Im,n as a matrix of m rows and r columns where the original image is stored, then gx and gy are matrices having the same dimensions as I, which at each element contain the horizontal and vertical derivative approximations and are calculated by (14) and (15) [19].

$$ g_{x} = \mathop \sum \limits_{i = 1}^{i = 3} \mathop \sum \limits_{j = 1}^{j = 4} Sobel_{x,ij} *I_{r + i - 2,c + j - 2}\quad \begin{array}{*{20}l} {for = 1,2, \ldots ,m} \\ {for = 1,2, \ldots ,n} \\ \end{array} $$
(14)
$$ g_{y} = \mathop \sum \limits_{i = 1}^{i = 3} \mathop \sum \limits_{j = 1}^{j = 4} Sobel_{y,i,j} *I_{r + i - 2,c + j - 2}\quad\,\begin{array}{*{20}l} {for = 1,2, \ldots ,m} \\ {for = 1,2, \ldots ,n} \\ \end{array} $$
(15)

In the Sobel method the gradient magnitude g is calculated by (16).

$$ g = \sqrt {g_{x}^{2} + g_{y}^{2} } $$
(16)

For the type-1 fuzzy inference system, 3 inputs can be used, 2 of them are the gradients with respect to the x-axis and y-axis, calculated with (14) and (15), which we call DH and DV, respectively. The third variable M is the image after the application of a low-pass filter hMF in (17); this filter allows to detect image pixels belonging to regions of the input were the mean gray level is lower. These regions are proportionally affected more by noise, which is supposed to be uniformly distributed over the whole image [19].

$$ hMF = \frac{1}{25}*\left[ {\begin{array}{*{20}c} 1 \hfill & 1 \hfill & 1 \hfill & 1 \hfill & 1 \hfill \\ 1 \hfill & 1 \hfill & 1 \hfill & 1 \hfill & 1 \hfill \\ 1 \hfill & 1 \hfill & 1 \hfill & 1 \hfill & 1 \hfill \\ 1 \hfill & 1 \hfill & 1 \hfill & 1 \hfill & 1 \hfill \\ 1 \hfill & 1 \hfill & 1 \hfill & 1 \hfill & 1 \hfill \\ \end{array} } \right] $$
(17)

After applying the edge detector type 1 with Sobel, the resulting image can be viewed in Fig. 4c.

Fig. 4
figure 4

a Original image, b image with morphological gradient, c image with Sobel

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4 Modular Neural Networks

Were trained a MNN of 3 modules with 80 % of the data of ORL. Each image was divided into 3 sections horizontal and each of which was used as training data in each of the modules, as shown in Fig. 5.

Fig. 5
figure 5

Architecture proposal of the modular neural network

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The integration of the modules of the MNN was made with the Choquet integral.

  • Training parameters were

Training method: gradient descendent with momentum and adaptive learning rate back-propagation (Traingdx).

Each module with two hidden layers [200 200].

Error goal: 0.00001

Epochs: 500

In Table 1 the distribution of the training data is shown.

Table 1 Distribution of the training data
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4.1 The Experiment with a Modular Neural Network Recognition System and Choquet Integral for the Modules Fusion

The experiment consist on applying each evaluated edge detector to obtain a data set of same well know benchmark data base of images like ORL database of faces and then train a neural network to compare the recognition rate using the k-fold cross validation method [20], see Table 2.

Table 2 Procedure performed in the experiment
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5 Simulation Results

In the experiments we performed 27 tests in simulation of the trainings with each edge detectors making variations in fuzzy densities and performing the calculation of the parameter λ with the bisection method. The results obtained with the morphological gradient are shown in Table 3 and with Sobel in Table 4.

Table 3 Results of training with morphological gradient edge detector
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Table 4 Results of training with sobel edge detector
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In Table 5 the percentages of recognition of the Choquet integral with each edge detector are displayed. It can be noted that when using the morphological gradient it was obtained 94 % recognition of the MNN, while with Sobel a 93.125 % was obtained.

Table 5 Results of training with Sobel edge detector
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6 Conclusions

The use of Choquet integral as a integration method answers of a modular neural network applied to face recognition has yielded favorable results when performing the aggregation process of the pre-processed images with the detectors of Sobel edges and morphological gradient, however it is still necessary to use a method that optimizes the value of the Sugeno measure assigned to each source of information because these were designated arbitrarily. Future work could be considering the optimization of the proposed method, as in [7, 21, 22].