Almanzor Sapena

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.

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.

On a class of completable fuzzy metric spaces

In this paper we study some properties of a class of fuzzy metric spaces, in the sense of George and Veeramani, called strong. This class includes the class of stationary fuzzy metrics, and in particular, when the fuzzy metric is principal, we obtain a family of metrics which are compatible with the topology induced by the fuzzy metric. Also, we study some aspects of the completion of strong fuzzy metrics and we find a class of completable stationary fuzzy metrics which includes the class of stationary fuzzy ultrametrics.

New adaptive vector filter using fuzzy metrics

Classical nonlinear vector median-based filters are well-known methods for impulsive noise suppression in color images, but mostly they lack good detail-preserving ability. We use a class of fuzzy metrics to introduce a vector filter aimed at improving the detail-preserving ability of classical vector filters while effectively removing impulsive noise. The output of the proposed method is the pixel inside the filter window which maximizes the similarity in color and spatial closeness. The use of fuzzy metrics allows us to handle both criteria simultaneously. The filter is designed so that the importance of the spatial criterion can be adjusted. We show that the filter can adapt to the density of the contaminating noise by adjusting the spatial criterion importance. Classical and recent filters are used to assess the proposed filtering. The experimental results show that the proposed technique performs competitively.

Fuzzy bilateral filtering for color images

Bilateral filtering is a well-known technique for smoothing gray-scale and color images while preserving edges and image details by means of an appropriate nonlinear combination of the color vectors in a neighborhood. The pixel colors are combined based on their spatial closeness and photometric similarity. In this paper, a particular class of fuzzy metrics is used to represent the spatial and photometric relations between the color pixels adapting the classical bilateral filtering. It is shown that the use of these fuzzy metrics is more appropriate than the classical measures used.

Two-step fuzzy logic-based method for impulse noise detection in colour images

We present a novel fuzzy noise detector based on a fuzzy metric specifically designed to detect impulses. The fuzzy detector is inspired on the recent rank-ordered differences (ROD) statistic. We propose a noise detection process performed in two steps followed by noise filtering using the vector median filter. Experimental results show that the method is among the best-performing methods for impulse noise reduction and computational analysis indicates that the method is pretty efficient.

A Simple Fuzzy Method to Remove Mixed Gaussian-Impulsive Noise From Color Images

Mixed impulsive and Gaussian noise reduction from digital color images is a challenging task because it is necessary to appropriately process both types of noise that in turn need to be distinguished from the original image structures such as edges and details. Fuzzy theory is useful to build simple, efficient, and effective solutions for this problem. In this paper, we propose a fuzzy method to reduce Gaussian and impulsive noise from color images. Our method uses one only filtering operation: a weighted averaging. A fuzzy rule system is used to assign the weights in the averaging so that both noise types are reduced and image structures are preserved. We provide experimental results to show that the performance of the method is competitive with respect to state-of-the-art filters.

Some improvements for image filtering using peer group techniques

An image pixel peer group is defined as the set of its neighbor pixels which are similar to it according to an appropriate distance or similarity measure. This concept has been successfully used to devise algorithms for detection and suppression of impulsive noise in gray-scale and color images. In this paper, we present a novel peer group-based approach intended to improve the trade-off between computational efficiency and filtering quality of previous peer group-based methods. We improve the computational efficiency by using a modification of a recent approach that can only be applied when the distance or similarity measure used fulfills the so-called triangular inequality property. The improvement of the filtering quality is achieved by the inclusion of a refinement stage in the noise detection. The proposed method performs according to the following steps: First, we partition the image into disjoint blocks and we perform a fast classification of the pixels into three types: non-corrupted, non-diagnosed and corrupted; second, we refine the initial findings by analyzing the non-diagnosed pixels and finally every pixel is classified either as corrupted or non-corrupted. Then, only corrupted pixels are replaced so that uncorrupted image data is preserved. Experimental results suggest that the proposed method is able to outperform state-of-the-art methods both in filtering quality and computational efficiency.