Valentín Gregori

On fixed-point theorems in fuzzy metric spaces

We give fixed-point theorems for complete fuzzy metric spaces in the sense of George and Veeramani, and also for Kramosil and Michaleks fuzzy metric spaces which are complete in Grabiecs sense.

A fast impulsive noise color image filter using fuzzy metrics

In this paper, the problem of impulsive noise reduction in multichannel images is addressed. A new filter is proposed on the basis of a recently introduced family of computationally attractive filters with a good detail-preserving ability (FSVF). FSVF is based on privileging the central pixel in each filtering window in order to replace it only when it is really noisy and preserve the original undistorted image structures. The new filter is based on a novel fuzzy metric and it is created by combining the mentioned scheme and the fuzzy metric. The use of the fuzzy metric makes the filter computationally simpler and it allows to adjust the privilege of the central pixel giving the filter an adaptive nature. Moreover, it is shown that the new filter outperforms the classical-order statistics filtering techniques and its performance is similar to FSVF, outperforming it in some cases.

Characterizing completable fuzzy metric spaces

Recently we showed the existence of a fuzzy metric space, in the sense of George and Veeramani, which is not completable (Fuzzy Sets and Systems 130 (2002) 399). Here we present an internal characterization of those fuzzy metric spaces that are completable. Some applications are derived and some illustrative examples are given.

On completion of fuzzy metric spaces

Completions of fuzzy metric spaces (in the sense of George and Veeramani) are discussed. A complete fuzzy metric space Y is said to be a fuzzy metric completion of a given fuzzy metric space X if X is isometric to a dense subspace of Y. We present an example of a fuzzy metric space that does not admit any fuzzy metric completion. However, we prove that every standard fuzzy metric space has an (up to isometry) unique fuzzy metric completion. We also show that for each fuzzy metric space there is an (up to uniform isomorphism) unique complete fuzzy metric space that contains a dense subspace uniformly isomorphic to it.

Fuzzy Peer Groups for Reducing Mixed Gaussian-Impulse Noise From Color Images

The <i>peer</i> <i>group</i> of an image pixel is a pixel similarity-based concept which has been successfully used to devise image denoising methods. However, since it is difficult to define the pixel similarity in a crisp way, we propose to represent this similarity in fuzzy terms. In this paper, we introduce the <i>fuzzy</i> <i>peer</i> <i>group</i> concept, which extends the <i>peer</i> <i>group</i> concept in the fuzzy setting. A <i>fuzzy</i> <i>peer</i> <i>group</i> will be defined as a fuzzy set that takes a <i>peer</i> <i>group</i> as support set and where the membership degree of each <i>peer</i> <i>group</i> member will be given by its fuzzy similarity with respect to the pixel under processing. The <i>fuzzy</i> <i>peer</i> <i>group</i> of each image pixel will be determined by means of a novel fuzzy logic-based procedure. We use the <i>fuzzy</i> <i>peer</i> <i>group</i> concept to design a two-step color image filter cascading a fuzzy rule-based switching impulse noise filter by a fuzzy average filtering over the <i>fuzzy</i> <i>peer</i> <i>group</i>. Both steps use the same <i>fuzzy</i> <i>peer</i> <i>group</i>, which leads to computational savings. The proposed filter is able to efficiently suppress both Gaussian noise and impulse noise, as well as mixed Gaussian-impulse noise. Experimental results are provided to show that the proposed filter achieves a promising performance.

A New Fuzzy Color Correlated Impulse Noise Reduction Method

A new impulse noise reduction method for color images is presented. Color images that are corrupted with impulse noise are generally filtered by applying a grayscale algorithm on each color component separately or using a vector-based approach where each pixel is considered as a single vector. The first approach causes artefacts especially on edge and texture pixels. Vector-based methods were successfully introduced to overcome this problem. Nevertheless, they tend to cluster the noise and to receive a lower noise reduction performance. In this paper, we discuss an alternative technique which gives a good noise reduction performance while much less artefacts are introduced. The main difference between the proposed method and other classical noise reduction methods is that the color information is taken into account to develop (1) a better impulse noise detection method and (2) a noise reduction method that filters only the corrupted pixels while preserving the color and the edge sharpness. Experimental results show that the proposed method provides a significant improvement on other existing filters.

Examples of fuzzy metrics and applications

In this paper we present new examples of fuzzy metrics in the sense of George and Veeramani. The examples have been classified attending to their construction and most of the well-known fuzzy metrics are particular cases of those given here. In particular, novel fuzzy metrics, by means of fuzzy and classical metrics and certain special types of functions, are introduced. We also give an extension theorem for two fuzzy metrics that agree in its nonempty intersection. Finally, we give an application of this type of fuzzy metrics to color image processing. We propose a fuzzy metric that simultaneously takes into account two different distance criteria between color image pixels and we use this fuzzy metric to filter noisy images, obtaining promising results. This application is also illustrative of how fuzzy metrics can be used in other engineering problems.

Fast detection and removal of impulsive noise using peer groups and fuzzy metrics

A novel approach to impulsive noise detection in color images is introduced. In the paper, the peer group concept is redefined by means of a certain fuzzy metric. This concept is employed for the fast detection of noisy pixels by taking advantage of the fuzzy metric properties. On the basis of the noisy pixel detection a switching filter between the arithmetic mean filter (AMF) and the identity operation is proposed. The proposed switching filter achieves a trade-off between noise suppression and signal-detail preservation and is faster than recently introduced switching filters based on the peer group concept.

Local self-adaptive fuzzy filter for impulsive noise removal in color images

This paper describes a new filter for impulsive noise reduction in color images based on a recently introduced family of vector filters with a good ability for preserving details. These filters use a reduced ordering of color vectors in order to detect and replace impulses. The proposed filter uses local statistics to adapt itself to simultaneously remove impulses and preserve image edges and details. The proposed filtering technique is robust and achieves a good balance between noise attenuation and detail reservation. In addition, it outperforms well-known vector filtering solutions and produces visually pleasing filtered color images.

Isolating impulsive noise pixels in color images by peer group techniques

A new method for removing impulsive noise in color images is presented. The fuzzy metric peer group concept is used to build novel switching vector filters. In the proposed filtering procedure, a set of noise-free pixels of high reliability is determined by applying a highly restrictive condition based on the peer group concept. Afterwards, an iterative detection process is used to refine the initial findings by detecting additional noise-free pixels. Finally, noisy pixels are filtered by maximizing the employed fuzzy distance criterion between the pixels inside the filter window. Comparisons are provided to show that our approach suppresses impulsive noise, while preserving image details. In addition, the method is analyzed in order to justify the necessity of the iterative process and demonstrate the computational efficiency of the proposed approach.