Pedro Manuel Martínez-Jiménez

A discussion on fuzzy cardinality and quantification. Some applications in image processing

In this paper we discuss on some different representations of the cardinality of a fuzzy set and their use in fuzzy quantification. We have considered the widely employed sigma-count, fuzzy numbers, and gradual numbers. Gradual numbers assign numbers to values of a relevance scale, typically 0,1. Contrary to sigma-count and fuzzy numbers, they provide a precise representation of the cardinality of a fuzzy set. We illustrate our claims by calculating the cardinality of the fuzzy set of pixels that match a certain fuzzy color in an image. For that purpose we consider fuzzy color spaces previously defined by the authors, consisting of a collection of fuzzy sets providing a suitable, conceptual quantization with soft boundaries of crisp color spaces. Finally, we show the suitability of our approaches to fuzzy quantification for different applications in image processing. First, the calculation of histograms. Second, the definition of the notion of dominant fuzzy color, and the calculation of the degree to which we can say that a certain color is dominant in an image.

A comparative study of texture coarseness measures

There are a wide variety of measures in the literature that capture the ¿coarseness¿ texture property. Some of them have better ability to represent coarseness than the others. Furthermore, some of them are more robust against the variation of other image features, like brightness, contrast, noise and size of the image. In this paper, we propose to study the robustness and the relationship with human coarseness perception of 17 classical measures of coarseness, in order to obtain a ranking of measures. This ranking can be used to identify those measures that have the highest relationship degree with perception and the least variation with the other image features.

Fuzzy Color Spaces: A Conceptual Approach to Color Vision

In this paper, we introduce formal definitions of the concepts of fuzzy color and fuzzy color space. First, we formalize the notion of fuzzy color for representing the correspondence between computational representation of colors and perceptual color categories identified by a color name. Second, we propose a methodology for learning fuzzy colors based on the paradigm of conceptual spaces, where prototypes are used for each category to be learnt. Since the conceptual space approach yields crisp categorizations, we introduce a novel methodology for defining fuzzy boundaries of color categories on the basis of a Voronoi tessellation of a color space. Finally, we also formalize the notion of fuzzy color space as the collection of fuzzy colors corresponding to the color categories employed in a certain context/application and/or for a specific user. Different typologies of fuzzy color spaces are proposed in order to be consistent with the nature of the categories we want to model. Our approach is illustrated by defining fuzzy color spaces using RGB with the Euclidean distance. Examples based on the well-known ISCC-NBS color naming system are presented, as well as others based on collections of color names and prototypes provided by users. The proposal is evaluated and compared with the most used approaches for color modeling. Additionally, a website located at http://www.jfcssoftware.com including all experimentation data, software implementing our models, and additional materials is available to researchers in color modeling.

JFCS: A Color Modeling Java Software Based on Fuzzy Color Spaces

This paper introduces JFCS (Java Fuzzy Color Space), an open source software for modeling colors on the basis of fuzzy color spaces, which is able to fill the semantic gap between the color representation in computers and the subjective human perception. Fuzzy colors allow introducing semantics in the description of color by using linguistic labels, taking into account the fuzzy boundaries between the representation of color terms. The methodology for building fuzzy color spaces implemented in JFCS was proposed by the authors in previous works, it is based on the paradigm of conceptual spaces including fuzzy boundaries and considering a collection of color names and corresponding crisp color representatives. In this sense, the necessary information for modeling fuzzy color spaces can be easily obtained from non-expert users, since the only information required is a representative crisp color for each linguistic color term to be modeled. The software is implemented in Java and it includes several graphical tools for building different types of spaces with different characteristics in an easy way. These are needed to be consistent with the different nature of the colors a user wants to model. Membership degrees of pixels in images to each fuzzy color in a certain fuzzy color space can be obtained. Tools for 3D visualization of fuzzy color spaces, as well as for describing colors and mapping pixels in images in term of linguistic labels, are also included in JFCS.

A fuzzy approach for modelling visual texture properties

Abstract In this paper, we address the problem of simulating the human perception of texture properties in images. In particular, we have focused our study on the properties of coarseness, contrast and directionality, that play a fundamental role in the human perception of texture. The objective is not to precisely identify, classify or discriminate between different textures as a whole, but to be able to assess the presence of each texture property in the image. For this purpose, fuzzy sets defined on the domain of different groups of measures are employed in order to model each property by using parametric membership functions. The corresponding parameters are obtained by learning a functional relationship between the computational values given by the measures and the human perception. The performance of each fuzzy set is analyzed and tested with the human assessments, and a ranking of subsets of measures is obtained according to their ability to represent the perception of the property, allowing us to identify the most suitable combination of measures.

Perception-based fuzzy sets for visual texture modelling

Texture is one of the most used low-level features for image analysis and, in addition, one of the most difficult to characterize. Although there is not an accurate definition for the concept of texture, it is usual for humans to describe visual textures according to some perceptual properties like coarseness, directionality, contrast, line-likeness or regularity. In this paper, we propose to model texture on the basis of its perceptual properties. To do this, fuzzy sets defined on the domain of some of the most representative measures of each property are employed. This approach achieves a double objective: first, to obtain models that allow to represent the imprecision related to texture properties, and second, to identify the most appropriate measure for each of these properties. In order to define the fuzzy models, parametric membership functions are proposed, where the corresponding parameters are obtained by learning a functional relationship between the computational values given by the measure and the human perception of the corresponding property. The performance of each fuzzy set is analyzed and checked with the human assessments, and a ranking of measures is obtained according to their ability to represent the perception of the property, allowing to identify the most suitable measure. In order to explain the proposed methodology, we focus our study on coarseness, contrast and directionality, that are considered the three most important texture properties.

Histograms for Fuzzy Color Spaces

In this paper we introduce two kinds of fuzzy histograms on the basis of fuzzy colors in a fuzzy color space and the notion of gradual number by Dubois and Prade. Fuzzy color spaces are a collection of fuzzy sets providing a suitable, conceptual quantization with soft boundaries of crisp color spaces. Gradual numbers assign numbers to values of a relevance scale, typically [0,1]. Contrary to convex fuzzy subsets of numbers (called fuzzy numbers, but corresponding to fuzzy intervals as an assignment of intervals to values of [0,1]), they provide a more precise representation of the cardinality of a fuzzy set. Histograms based on gradual numbers are particularly well-suited for serving as input to another process. On the contrary, they are not the best choice when showing the information to a human user. For this second case, linguistic labels represented by fuzzy numbers are a better alternative, so we define linguistic histograms as an assignment of linguistic labels to each fuzzy color. We provide a way to calculate linguistic histograms based on the compatibility between gradual numbers and linguistic labels. We illustrate our proposals with some examples.

An adaptive fuzzy approach for modeling visual texture properties

The analysis of the perceptual properties of texture plays a fundamental role in tasks like semantic description of images, content-based image retrieval using linguistic queries, or expert systems design based on low level visual features. The presence of these properties in images is very difficult to characterize due to their imprecision, and, moreover, because their perception may change depending on the user or the image context. In this paper, texture properties are modeled by means of an adaptive fuzzy approach that takes into account the subjectivity of the human perception. For this purpose, a methodology in two phases has been proposed. First, non-adaptive fuzzy models, that represent the average human perception about the presence of the texture properties, are obtained. For this modeling, we propose to learn a relationship between representative measures of the properties and the assessments given by human subjects. In a second phase, the obtained fuzzy sets are adapted in order to model the particular perception of the properties that a user may have, as well as the changes in perception influenced by the image context. For this purpose, the membership functions are automatically transformed on the basic of the information given by the user or extracted from the image context, respectively.

Fuzzy partitions for modelling texture properties: Coarseness, contrast and directionality

The analysis of the perceptual properties of texture plays a fundamental role in tasks like semantic description of images or content-based image retrieval using linguistic queries. In this paper, we propose to model these properties by means of fuzzy partitions defined on the domain of some representative measures. In particular, we have focused our study on the properties of coarseness, contrast and directionality, that play a fundamental role in the human perception of texture. In our approach, the number of fuzzy sets in the partitions, as well as the parameters of the membership functions, are calculated by taking into account the relationship between the computational values given by the measures and the human perception of the corresponding property. The performance of the proposed modelling has been analyzed by applying the obtained fuzzy partitions in several experiments.

Fuzzy sets on 2D spaces for fineness representation

The analysis of the perceptual properties of texture plays a fundamental role in tasks like semantic description of images, content-based image retrieval using linguistic queries, or expert systems design based on low level visual features. In this paper, we propose a methodology to model texture properties by means of fuzzy sets defined on bidimensional spaces. In particular, we have focused our study on the fineness property that is considered as the most important feature for human visual interpretation. In our approach, pairwise combinations of fineness measures are used as a reference set, which allows to improve the ability to capture the presence of this property. To obtain the membership functions, we propose to learn the relationship between the computational values given by the measures and the human perception of fineness. The performance of each fuzzy set is analyzed and tested with the human assessments, allowing us to evaluate the goodness of each model and to identify the most suitable combination of measures for representing the fineness presence. We propose to model the fineness property of texture by means of fuzzy sets defined on the domain of pairwise combinations of fineness measures.The membership functions of the proposed fuzzy sets are obtained by taking into account the human perception of fineness.The bidimensional models proposed in this paper are able to represent the presence degree of fineness, matching what a human would expect.