The secondary membership functions (MFs) of type-2 (T2) fuzzy sets (FSs) are generally assumed to be convex, normal, and upper semicontinuous. Although not explicitly stated, this fact can be detected by taking a look at the$\alpha$-cuts of their join and meet in vertical-slice format. These two operations induce$[0,1]_{\mathcal {F}}$, i.e., the complete lattice of the convex normal upper semicontinuous FSs on$[0, 1]$together with the so-called convolution order. In this article, we deduce the$\alpha$-cut representations of suprema and infima over finite and infinite sets that occur in the inference mechanisms of general type-2 (GT2) fuzzy inference systems (FISs). We continue by developing a complete, albeit not distributive, lattice structure for the class of all convex upper semicontinuous FSs on$[0, 1]$, denoted${\mathcal {F}}_{CU}$, which permits designing T2 FISs involving T2 FSs whose secondary MFs are not necessarily normal. Using our formulae of the$\alpha$-cuts of suprema and infima in$[0,1]_{\mathcal {F}}$and${\mathcal {F}}_{CU}$, whose elements we describe using so-called anti-dilations, we prove that$[0,1]_{\mathcal {F}}$is a closed sublattice of${\mathcal {F}}_{CU}$. We also expand on the lattice-theoretical relations between$[0,1]_{\mathcal {F}}$and the complete lattices of the convex (strictly) normal FSs. In addition, we show that$[0,1]_{\mathcal {F}}$and${\mathcal {F}}_{CU}$contain complete sublattices of (extended) interval type-2 (IT2) FSs. With these results at hand, it becomes easy to perform the transitions FISs under the minimum t-norm$\to$(closed) IT2 FISs$\to$GT2 FISs currently used in practice$\to$GT2 FISs having not necessarily normal secondary MFs.

Morphological perceptrons (MPs) belong to the class of morphological neural networks (MNNs) whose neuronal aggregation functions are drawn from mathematical morphology (MM). Most MNN models including MPs employ operators of gray-scale mathematical morphology as aggregation functions. Recently, a hybrid morphological/linear perceptron (HMLP) appeared in the literature. This neural network model combines the approximation capabilities of the two-layer perceptron having sigmoid activation functions with the capability of the MP to represent non-differentiable functions. For a number of reasons, that include the non-differentiability of morphological operators, it is advantageous to train HMLPs using extreme learning machine (ELM). The fact that gray-scale MM is closely related to fuzzy MM based on Lukasiewicz operators motivated us to introduce hybrid fuzzy morphological/linear perceptrons (FMLPs) and train them using ELM in this paper. We compare the performances of the HMLP and the FMLP with the ones of some related models in a number of well-known classification problems.

In this manuscript we focus on a system of differential equations that describes the trajectory of a particle that moves in the plane according to a hypocycloid curve. More precisely we consider a coupled system composed of two ordinary second order differential equations with initial conditions given by interactive fuzzy numbers. The concept of interactivity is given by means of joint possibility distribution (J). We provide a fuzzy solution for this system via sup-J extension, which can be seen as a generalization of the Zadeh's extension principle. We present an example in order to compare the proposed solution with the solution via Zadeh's extension principle.

A (weighted) Θ-fuzzy associative memory can be viewed as single hidden layer neural network whose inputs are drawn from an arbitrary bounded lattice L. The ξth hidden node applies a function Θξ: L → [0, 1] to an input A ∈ L. In this paper, we present a new algorithm for designing Θ-FAMs. Roughly speaking, this algorithm consists of the following two stages: 1) Construction of a set of functions Θξ: L → [0, 1] where ξ= 1,..., p; 2) Optimization of the weights. Our new algorithm differs from the previous algorithm for tunable equivalence fuzzy associative memories. Instead of merely extracting a subset of functions Θξ: L → [0, 1] from a given set, we generate a new set of functions on the basis of the partial ordering of L. The paper includes some experimental results in a set of benchmark classification problems. We also apply a combination of the resulting Θ-FAM and a deep convolutional neural network to a problem of image texture classification and compare the classification performance of our approach with the ones of some state-of-the-art methods from the literature.

The linear fuzzy transform (FT) has found a great variety of applications whereas the two versions of the lattice FT seem to have found almost no applications at first glance. However, a closer look at lattice FTs reveals their relationship to operators of mathematical morphology which are widely used in image processing and analysis including image noise filtering. This paper compares linear and lattice FTs in terms of some properties and experimental results in image noise reduction. Moreover, we show that both linear as well as lattice FTs can be expressed in terms of image-template products in the mathematical theory of image algebra.

In this manuscript we deal with non-homogeneous n-th order linear differential equations that have fuzzy initial conditions. We assume that the initial conditions are given by fuzzy numbers that satisfying a special type of relationship namely interactivity. In particular, we assume that the initial conditions are completely correlated fuzzy numbers. Based on this notion, we obtain solutions for linear fuzzy initial value problems by means of the extension principle. Finally, we show that endpoints of each α-level of the proposed solution can be determined by solving two associated classical initial value problems.

Arithmetic operations on fuzzy numbers are usually performed using Zadeh's extension principle. The amount of uncertainty expressed in the resulting fuzzy number tends to be much higher than the amounts of uncertainty in the operands. This effect is often undesired and does not reflect reality if there is interactivity between the operands. We recently presented an approach for adding fuzzy numbers using an extension principle based on a parametrized family of joint possibility distributions. In this paper, we employ this approach in order to derive a method for controlling the uncertainty given by the width, i.e., the length of the support, of the resulting sum of fuzzy numbers.

Most fuzzy associative memories (FAMs) in the literature correspond to neural networks with a single layer of weights that distributively contains the information on associations to be stored. The main applications of these types of associative memory can be found in fuzzy rule-based systems. In contrast, Θ-fuzzy associative memories ( Θ-FAMs) represent parametrized fuzzy neural networks with a hidden layer and these FAM models extend (dual) S-FAMs and SM-FAMs based on fuzzy subsethood and similarity measures. In this paper, we provide theoretical results concerning the storage capacity and error correction capability of Θ-FAMs. In addition, we introduce a training algorithm for Θ-FAMs and we compare the error rates produced by Θ-FAMs and some well-known classifiers in some benchmark classification problems that are available on the internet. Finally, we apply Θ-FAMs to a problem of vision-based self-localization in mobile robotics.

In this paper, we present a new class of fuzzy associative memories (FAMs) called tunable equivalence fuzzy associative memories, for short tunable E-FAMs or TE-FAMs, that belong to the class Θ-fuzzy associative memories (Θ-FAMs). Recall that 0-FAMs represent fuzzy neural networks having a competitive hidden layer and weights that can be adjusted via a training algorithm. Like any associative memory model, Θ-FAMs depend on the specification of a fundamental memory set. In contrast to other Θ-FAM models, TE-FAMs make use of parametrized fuzzy equivalence measures that are associated with the hidden nodes and allow for the extraction of a fundamental memory set from the training data. The use of a smaller fundamental memory set than in previous articles on Θ-FAMs reduces the computational effort involved in deriving the weights without decreasing the quality of the results.

In the mid 1990's, the morphological associative memory (MAM) was introduced as a distributive associative memory model. Since then several extensions of MAMs as well as applications in different domains have appeared in the literature. Just like other morphological neural network models, a MAM performs an elementary operation of mathematical morphology, possibly followed by an activation function, at every node. Generally speaking, a common trait of all distributive MAM models is their foundation in mathematical morphology on complete lattices. Morphological operators in the complete lattice framework come in dual pairs such as dilation/erosion, opening/closing, etc.. Therefore, MAM models also have two versions (denoted using the symbols W and M) that are tolerant to different types of noise in the input patterns. To overcome this drawback for MAM models, we resort to the more recent theory of mathematical morphology on inf-semilattices whose elementary operators are self-dual. This paper represents a first attempt at formulating an associative memory (AM) in this framework.

Mathematical morphology (MM) on complete lattices provides the theoretical basis for certain lattice computing models called morphological neural networks (MNNs). Morphological associative memories (MAMs) that are among the most widely known MNN models usually perform operations in the extended integer or real numbers or in the class of fuzzy sets (in which case we speak of fuzzy morphological associative memories) but can also be defined in more general complete lattice structures. Moreover, the relatively recent theory of MM on complete semilattices can be employed as a mathematical framework for novel (morphological) associative memory models. In this paper we introduce a sparsely connected associative memory model based on complete semilattices capable of dealing with the computational requirements for storing multi-valued and large-scale patterns.We finish with some experimental results concerning the problem of storing and reconstructing gray-scale as well as color images.

Mathematical morphology is a theory to extract specific information such as edges and patterns from images. The original binary morphology, for binary black and white images, was extended to greyscale images, amongst others, by a fuzzy approach known as fuzzy mathematical morphology. This approach was based on the observation that greyscale images and fuzzy sets can be modelled in the same way. Recently, fuzzy mathematical morphology has been further extended based on extensions of classical fuzzy set theory. In this paper, we focus on the extension based on interval-valued fuzzy set theory, i.e., interval-valued fuzzy morphology, and we give an overview of the basic properties that hold in this model.

Financial time series exhibit a mixture of linear and nonlinear components as indicated by the corresponding lagplots. As we will explain in this paper, certain financial time series can be approximated by increasing functions of a fixed number of time lags or antecedents. This work presents a suitable model for dealing with financial prediction problems, called increasing hybrid morphological-linear perceptron (IHMP). A pseudo-gradient steepest descent method is presented to design the IHMP (learning process), using the back-propagation algorithm and a systematic approach to overcome the problem of nondifferentiability of morphological operations. The learning process includes an automatic phase correction step that is geared at eliminating the time phase distortions that typically occur in financial time series prediction (“random walk dilemma”). Furthermore, we compare the proposed IHMP with other neural and statistical models using three complex nonlinear problems of financial time series prediction.

We have recently proven that many well-known fuzzy associative memory (FAM) models can be classified as (fuzzy) morphological neural networks (MNNs) because they perform an operation of (fuzzy) mathematical morphology at every node, possibly followed by the application of an activation function. One of the basic (fuzzy) morphological operators called (fuzzy) erosion is defined in terms of a (fuzzy) inclusion measure. In this paper, we take advantage of these considerations in order to derive a new non-distributive fuzzy morphological associative memory model on the basis of the Kosko subsethood measure that we named Kosko subsethood fuzzy associative memory (KS-FAM). After a brief discussion of the properties of the KS-FAM we compare the error correction capabilities of the KS-FAM and other fuzzy and gray-scale associative memories in terms of some experimental results concerning gray-scale image reconstruction.

Morphological neural networks grew out of a merger of ideas from artificial neural networks and mathematical morphology. The morphological perceptron, one of the first morphological neural networks that appeared in the literature, was originally developed as a simple model for solving binary classification problems. The morphological perceptron has not received much attention due to its simplicity and limited applicability. In this paper, we introduce a new version of the morphological perceptron called morphological perceptron with competitive learning including an appropriate algorithm for training this model. Instead of a single binary output neuron like the original morphological perceptron, the new model as a winner-take-all output layer and the decision surface after training does not depend on the order in which the patterns are presented to the network. Finally, the paper includes some experimental results on two well-known datasets that indicate the utility of the morphological perceptron with competitive learning in classification problems.

Fuzzy morphological associative memories (FMAMs) extend morphological associative memories (MAMs) to the fuzzy domain. FMAMs were derived using concepts of fuzzy set theory and mathematical morphology. In general, FMAMs either perform a maximum of conjunctions or a minimum of disjunctions at every node. We recently introduced a general class of learning strategies for FMAMs called fuzzy leaning by adjunction (FLA). The focus of this paper is on FMAMs that are based on (conjunctive or disjunctive) uninorms. We show that uninorm-based FMAMs generalize implicative fuzzy associative memories (IFAMs) and their dual versions. Moreover, we provide an isomorphism between FMAMs based on representable uninorms and MAMs. Finally, we point out that auto-associative uninorm-based FMAMs exhibit optimal absolute storage capacity and one-step convergence.

Mathematical morphology (MM) is a theory concerned with the processing and analysis of objects using operators based on topological and geometrical concepts. We speak of a fuzzy morphological associative memory (FMAM) when a fuzzy associative memory (FAM) model is equipped with neurons that correspond to an operator of mathematical morphology. This paper shows that several FAM models, including the FAMs of Kosko, most generalized FAMs of Chung and Lee, the FAM of Junbo et al., the max-min FAM with threshold, the fuzzy logic bidirectional associative memories, and the implicative fuzzy associative memories, belong to the FMAM class. Moreover, we present two strategies for deriving a new FMAM model from a given FMAM. These strategies are based on two duality relationship of mathematical morphology: duality with respect to negation and duality with respect to adjunction.

Morphological associative memories (MAMs) are based on a lattice algebra known as minimax algebra. In previous papers, we gained valuable insight into the storage and recall phases of gray-scale autoassociative memories. This paper extends these results to the heteroassociative and to the fuzzy case in view of the fact that a gray-scale MAM model can be converted into a fuzzy MAM model that coincides with the Lukasiewicz IFAM by applying an appropriate threshold. The paper includes some experimental results concerning applications of MAM and fuzzy MAM models in classification and prediction.

Associative neural memories are models of biological phenomena that allow for the storage of pattern associations and the retrieval of the desired output pattern upon presentation of a possibly noisy or incomplete version of an input pattern. In this paper, we introduce implicative fuzzy associative memories (IFAMs), a class of associative neural memories based on fuzzy set theory. An IFAM consists of a network of completely interconnected Pedrycz logic neurons with threshold whose connection weights are determined by the minimum of implications of presynaptic and postsynaptic activations. We present a series of results for autoassociative models including one pass convergence, unlimited storage capacity and tolerance with respect to eroded patterns. Finally, we present some results on fixed points and discuss the relationship between implicative fuzzy associative memories and morphological associative memories

Neural models of associative memories are usually concerned with the storage and the retrieval of binary or bipolar patterns. Thus far, the emphasis in research on morphological associative memory systems has been on binary models, although a number of notable features of autoassociative morphological memories (AMMs) such as optimal absolute storage capacity and one-step convergence have been shown to hold in the general, gray-scale setting. In this paper, we make extensive use of minimax algebra to analyze gray-scale autoassociative morphological memories. Specifically, we provide a complete characterization of the fixed points and basins of attractions which allows us to describe the storage and recall mechanisms of gray-scale AMMs. Computer simulations using gray-scale images illustrate our rigorous mathematical results on the storage capacity and the noise tolerance of gray-scale morphological associative memories (MAMs). Finally, we introduce a modified gray-scale AMM model that yields a fixed point which is closest to the input pattern with respect to the Chebyshev distance and show how gray-scale AMMs can be used as classifiers.

In morphological neural networks, the operations erosion or dilation of mathematical morphology are performed at each node. Alternatively, the total input effect on a morphological neuron can be expressed in terms of lattice induced matrix operations in the mathematical theory of minimax algebra. Morphological associative memories employ a recording strategy that resembles the widely known correlation strategy. The binary autoassociative morphological memory (AMM) can be viewed as the minimax algebra counterpart of the correlation-recorded discrete Hopfield net. In contrast to the Hopfield net, AMM's are not limited to the storage and retrieval of binary or bipolar patterns and exhibit attractive properties such as one-step convergence and an optimal absolute storage capacity. Heteroassociative morphological memories (HMMs) have yet to be studied extensively and only a few theorems on HMM's have been proven. This paper proves a number of theorems that yield an exact characterization of the recall phases of binary AMM's as well as binary HMM's.

Associative neural memories are models of biological phenomena that allow for the storage of pattern associations and the retrieval of the desired output pattern upon presentation of a possibly noisy or incomplete version of an input pattern. In this paper, we introduce implicative fuzzy associative memories (IFAM's), a class of associative neural memories models based on fuzzy set theory. An IFAM consists of a network of completely interconnected Pedrycz logic neurons whose connection weights are determined by the minimum of implications of presynaptic and postsynaptic activations. We present a series of results for autoassociative models including one pass convergence, unlimited storage capacity and tolerance with respect to eroded patterns.

Autoassociative morphological memories (AMM) are a construct similar to hopfield autoassociatived memories defined on the (R, +, v, /spl and/) lattice algebra. Unlimited storage and perfect recall of noiseless real valued patterns has been proved for AMMs. However AMMs suffer from sensitivity to specific noise models, that can be characterized as erosive and dilative noise. On the other hand, spectral unmixing of hyperspectral images needs the prior definition of a set of endmembers, which correspond to material spectra lying on vertices of the minimum convex region covering the image data. These vertices can be characterized as morphologically independent patterns. We present a procedure based on the AMM noise sensitivity for endmember detection based on this characterization.

Morphological associative memories (MAMs) belong to the class of morphological neural networks. The recording scheme used in the original MAM models is similar to the correlation recording recipe. Recording is achieved by means of a maximum (M/sub XY/ model) or minimum (W/sub XY/ model) of outer products. Notable features of autoassociative morphological memories (AMMs) include optimal absolute storage capacity and one-step convergence. The fixed points of AMMs can be characterized exactly in terms of the original patterns. Unfortunately, AMM fixed points include a large number of spurious memories. A combination of the M/sub XX/ model and the kernel method yields another binary AMM model. In this paper, we also introduce a dual kernel method. A new, dual model is given by a combination of the W/sub XX/ and the dual kernel method. The new AMM models exhibit better error correction capabilities than M/sub XX/ and W/sub XX/ and a reduced number of spurious memories, which can be easily described in terms of the fundamental memories. Finally, we present yet another pair of AMMs with very similar properties. Although these models are also derived from the kernel or dual kernel methods, their construction depends on less restrictive conditions.

Morphological associative memories are among several types of morphological neural network models which have been proposed over the course of the last few years. A neural network is called morphological if one of the fundamental operations of mathematical morphology, a dilation or an erosion, is performed at each node. These operations can be expressed as a max product or a min product in the mathematical theory of minimax algebra. This paper employs fuzzy set theory to generalize the operations "max product" and "min product" as used in binary autoassociative morphological memory (AMM) models. Replacing the original operations by new operations "fuzzy max product" and "fuzzy min product" in this setting yields a fuzzy AMM with crisp input patterns and fuzzy output patterns. A thresholding procedure can be applied to obtain crisp output patterns. This new approach significantly improves the error correction capability of binary autoassociative morphological memories.

Morphological neural networks (MNN) have been proposed as associative memories (with its two cases: autoassociative and heteroassociative). In this paper we are involved with heteroassociative MNN (HMNN). We propose their use for self-localization in a vision-based navigation framework for mobile robots. HMNN can be trained in a single computation step. Their storage capacity bound is the dimension of the patterns, and they have perfect recall of the patterns under very mild conditions. Recall is also very fast, because the MNN recall does not involve the search for an energy minimum.

We recently introduced a class of highly nonlinear associative memories called morphological associative memories. We have previously shown that autoassociative morphological memories (AMMs) exhibit many desirable characteristics, including optimal absolute storage capacity and one-step convergence (G.X. Ritter et al., 1998). Other aspects of AMM performance still require more detailed analysis and/or improvement. The paper provides considerable insight into the functionality of AMMs by giving necessary and sufficient conditions for fixed points. We then show that the output generated upon presentation of an input pattern x is either the smallest fixed point /spl ges/x or the largest fixed point /spl les/x.

Methods for matrix decomposition have found numerous applications in image processing, in particular for the problem of template decomposition. Since existing matrix decomposition techniques are mainly concerned with the linear domain, we consider it timely to investigate matrix decomposition techniques in the nonlinear domain with applications in image processing. The mathematical basis for these investigations is the new theory of rank within minimax algebra. Thus far, only minimax decompositions of rank 1 and rank 2 matrices into outer product expansions are known to the image processing community. We derive a heuristic algorithm for the decomposition of matrices having arbitrary rank.

The theory of artificial neural networks has been successfully applied to a wide variety of pattern recognition problems. In this theory, the first step in computing the next state of a neuron or in performing the next layer neural network computation involves the linear operation of multiplying neural values by their synaptic strengths and adding the results. A nonlinear activation function usually follows the linear operation in order to provide for nonlinearity of the network and set the next state of the neuron. In this paper we introduce a novel class of artificial neural networks, called morphological neural networks, in which the operations of multiplication and addition are replaced by addition and maximum (or minimum), respectively. By taking the maximum (or minimum) of sums instead of the sum of products, morphological network computation is nonlinear before possible application of a nonlinear activation function. As a consequence, the properties of morphological neural networks are drastically different than those of traditional neural network models. The main emphasis of the research presented here is on morphological associative memories. We examine the computing and storage capabilities of morphological associative memories and discuss differences between morphological models and traditional semilinear models such as the Hopfield net.

Convolutions are a fundamental tool in image processing. Nonlinear convolutions are used in such operations as the median filter, the medial axis transform, and erosion and dilation as defined in mathematical morphology. For large convolution masks or structuring elements, the computation cost resulting from implementation can be prohibitive. However, in many instances, this cost can be significantly reduced by decomposing the templates representing the masks or structuring elements into a sequence of smaller templates. In addition, such decomposition can often be made architecture specific and, thus, resulting in optimal transform performance. In this paper we provide methods for decomposing morphological templates which are analogous to decomposition methods used in the linear domain. Specifically, we define the notion of the rank of a morphological template which categorizes separable morphological templates as templates of rank one. We establish a necessary and sufficient condition for the decomposability of rank one templates into 3/spl times/3 templates. We then use the invariance of the template rank under certain transformations in order to develop template decomposition techniques for templates of rank two.

The theory of artificial neural networks has been successfully applied to a wide variety of pattern recognition problems. In this paper we introduce a novel class of neural networks, called morphological neural networks, in which the operations of multiplication and addition are replaced by addition and maximum (or minimum), respectively. By taking the maximum (or minimum) of sums instead of the sum of products, morphological network computation is nonlinear before thresholding. As a consequence, the properties of morphological neural networks are drastically different than those of traditional neural network models. In this paper we consider some of these differences and examine the computing capabilities of morphological neural networks. As particular examples of a morphological neural network we discuss morphological associative memories and morphological perceptrons.