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We will contact you if necessary. To learn more about Copies Direct watch this short online video. Need help? A similarity measure, that is, a function that indicates how much two fuzzy sets are equal, is at the core of a GEB-FAM model. In this paper, we present a detailed study on the use of cardinality-based similarity measures in the definition of a GEB-FAM.
Sobretudo, avaliamos o desempenho das GEB-FAMs usando tais medidas de similaridade em um problema de reconhecimento de faces. The Hopfield neural network, proposed by J. Hopfield in , is a widely known neural network model able to implement an autoassociative memory for the storage of binary or bipolar vectors 11 , 13 , Despite its various applications 14 , 26 , 25 , the Hopfield neural network suffers from a low absolute storage capacity Such limitation motivated many researchers to develop improved versions of the Hopfield neural network 12 , 10 , 9.
In particular, Chiueh and Goodman introduced the exponential correlation associative memory ECAM , a high-capacity autoassociative model designed for the storage of bipolar patterns 3 or the heteroassociative case, Jeng et al. Many applications of associative memories, however, require storage of vectors with real components or fuzzy sets 1 , 2 , 7 , 6 , 19 , 29 , 32 rent exponential fuzzy associative memories 28 , 29 GRE-FAMs are designed for the storage and recall of a finite family of fuzzy sets.
In general terms, a similarity measure is a function that indicates how much two fuzzy sets are equal. In previous works, we considered a normalized version of the similarity measure proposed by Xuecheng In this work, however, we adopt a more general definition proposed by De Baets and De Meyer 4 and focus on cardinality-based similarity measures 5 , 4.
Furthermore, we performed extensive computational experiments in order to evaluate the performance of GEB-FAMs based on these similarity measures in a face recognition problem. This work is organized as follows. Computational experiments, performed to evaluate the performance of GEB-FAMs in a face recognition problem, are described and analyzed in Section 4. We finish the paper with the concluding remarks in Section 5. We begin this section by recalling the definition of fuzzy sets Subsequently, we present some cardinality-based fuzzy similarity measures proposed by De Baets and De Meyer in 4.
A fuzzy similarity measure is a function that associates to a given pair of fuzzy sets a real number on the interval [0, 1] that indicates the degree of equality of these fuzzy sets. The definition of a fuzzy similarity measure may vary according to the context. In our previous works 28 , 29 , we adopted a normalized version of the axiomatic definition proposed by Xuecheng Finally, given a t-norm T , we say that S is T -transitive if the inequality.
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In 4 , De Beats and De Meyer introduced the following class of rational similarity measures based on the cardinality of fuzzy sets on a finite universe of discourse:. Table 1 shows some similarity measures derived from 2. Here, the fuzzification schemes described by 2. We would like to conclude this section by pointing out that some similarity measures given by Table 1 satisfy the following properties 4 :. S 17 , S 18 , and S 19 are locally reflexive,.
In this section, we present the generalized exponential bidirectional fuzzy associative memories GEB-FAMs , which have been recently proposed by us in the conference paper In fact, Table 2 , extracted from 21 , gives the accuracy obtained by several state-of-the-art approaches to the face recognition problem using the AR database as explained below. In the face recognition problem, we must identify a person from a face image using a set of labeled images, called the training set. Such as Luo et al. We considered 8 gray-scale images of each individual from a group of people as the training set.
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Figure 1 shows the 8 gray-scale images from a certain individual of the training set. Two experiments were conducted to evaluate the performance of an approach to the face recognition problem:.
A test set composed by 4 images from each individual with sunglasses and different illumination conditions. In this section, we present the generalized exponential bidirectional fuzzy associative memories GEB-FAMs , which have been recently proposed by us in the conference paper In fact, Table 2 , extracted from 21 , gives the accuracy obtained by several state-of-the-art approaches to the face recognition problem using the AR database as explained below.
In the face recognition problem, we must identify a person from a face image using a set of labeled images, called the training set. Such as Luo et al. We considered 8 gray-scale images of each individual from a group of people as the training set. Figure 1 shows the 8 gray-scale images from a certain individual of the training set.
Two experiments were conducted to evaluate the performance of an approach to the face recognition problem:. A test set composed by 4 images from each individual with sunglasses and different illumination conditions.
Self Organization And Associative Memory
A test set consisting of 4 images from each individual with scarf and different illumination conditions. In our computational experiments, we considered all the similarity measures listed in Table 1. As a consequence, we have 50 fuzzy similarity measures, namely, the three fuzzy versions of each of the similarity measures R 1 , R 2 , The GEB-FAM based on the locally-reflexive non T-transitive similarity measure S 17 with the minimum and the product t-norms yielded accuracy rates of In the light of these remarks, let us focus on T-transitive similarity measures.
Excluding the GEB-FAM based on the similarity measure R 13 , which correspond to the two outliers shown in the boxplots on Figure 4 , the GEB-FAM models derived from the minimum and the product achieved performance competitive to other models from the literature.
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In this paper, we investigated the role of a fuzzy similarity measure in a generalized exponential bidirectional fuzzy associative memory GEB-FAM. Precisely, we first revised the cardinality-based similarity measures and the fuzzification schemes proposed by De Baets and De Meyer 5 , 4.
Using the AR database and different GEB-FAM models, we concluded that the memories based on non T-transitive similarity measures usually produce poor performance, i. The only exception we found is the GEB-FAM based on the locally reflexive non T-transitive measure S 17 with minimum t-norm, which achieved an accuracy rates of In contrast, GEB-FAMs defined using T-transitive measures with either minimum or product t-norms achieved competitive results in comparison with others models from literature.
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In particular, the best results in this face recognition problem were obtained by considering T-transitive fuzzy similarity measures based on the minimum t-norm. Computer Methods and Programs in Biomedicine, 3 , Vinh, N. Hung, N. Springer Berlin Heidelberg , pp. Recurrent Correlation Associative Memories. De Meyer. Transitivity-preserving fuzzification schemes for cardinality-based similarity measures. European Journal of Operational Research, 3 , - De Baets, H. A class of rational cardinality-based similarity measures.
Journal of Computational and Applied Mathematics, 1 , Esmi, P. Sussner, H. Sussner, M. Valle, F.
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