In data stream (DS) learning, the system has to extract knowledge from data generated continuously, usually at high speed and in large volumes, making it impossible to store the entire set of data to be processed in batch mode. Hence, machine learning models must be built incrementally by processing the incoming examples, as data arrive, while updating the model to be compatible with the current data. In fuzzy DS clustering, the model can either absorb incoming data into existing clusters or initiate a new cluster. As the volume of data increases, there is a possibility that the clusters will overlap to the point where it is convenient to merge two or more clusters into one. Then, a cluster comparison measure (CM) should be applied, to decide whether such clusters should be combined, also in an incremental manner. This defines an incremental fusion process based on aggregation functions that can aggregate the incoming inputs without storing all the previous inputs. The objective of this article is to solve the fuzzy DS clustering problem of incrementally comparing fuzzy clusters on a formal basis. First, we formalize and operationalize incremental fusion processes of fuzzy clusters by introducing recursively extendable (RE) aggregation functions, studying construction methods and different classes of such functions. Second, we propose two approaches to compare clusters: 1) similarity and 2) overlapping between clusters, based on RE aggregation functions. Finally, we analyze the effect of those incremental CMs on the online and offline phases of the well-known fuzzy clustering algorithm d-FuzzStream, showing that our new approach outperforms the original algorithm and presents better or comparable performance to other state-of-the-art DS clustering algorithms found in the literature.

Overlap and grouping functions can be used to measure events in which we must consider either the maximum or the minimum lack of knowledge. The commutativity of overlap and grouping functions can be dropped out to introduce the notions of pseudo-overlap and pseudo-grouping functions, respectively. These functions can be applied in problems where distinct orders of their arguments yield different values, i.e., in non-symmetric contexts. Intending to reduce the complexity of pseudo-overlap and pseudo-grouping functions, we propose new construction methods for these functions from generalized concepts of additive and multiplicative generators. We investigate the isomorphism between these families of functions. Finally, we apply these functions in an illustrative problem using them in a time series prediction combined model using the IOWA operator to evidence that using these generators and functions implies better performance.

Restricted equivalence functions are well-known functions to compare two numbers in the interval between 0 and 1. Despite the numerous works studying the properties of restricted equivalence functions and their multiple applications as support for different similarity measures, an extension of these functions to an n-dimensional space is absent from the literature. In this article, we present a novel contribution to the restricted equivalence function theory, allowing to compare multivalued elements. Specifically, we extend the notion of restricted equivalence functions from$L$to$L^{n}$and present a new similarity construction on$L^{n}$. proposal is tested in the context of color image anisotropic diffusion as an example of one of its many applications.

Comparison and similarity measurement have been a key topic in computer vision for a long time. There is, indeed, an extensive list of algorithms and measures for image or subimage comparison. The superiority or inferiority of different measures is hard to scrutinize, especially considering the dimensionality of their parameter space and their many different configurations. In this work, we focus on the comparison of binary images, and study different variations of Baddeley’s Delta Metric, a popular metric for such images. We study the possible parameterizations of the metric, stressing the numerical and behavioural impact of different settings. Specifically, we consider the parameter settings proposed by the original author, as well as the substitution of distance transformations by regularized distance transformations, as recently presented by Brunet and Sills. We take a qualitative perspective on the effects of the settings, and also perform quantitative experiments on separability of datasets for boundary evaluation.

It is known that the human visual system performs a hierarchical information process in which early vision cues (or primitives) are fused in the visual cortex to compose complex shapes and descriptors. While different aspects of the process have been extensively studied, such as lens adaptation or feature detection, some other aspects, such as feature fusion, have been mostly left aside. In this work, we elaborate on the fusion of early vision primitives using generalizations of the Choquet integral, and novel aggregation operators that have been extensively studied in recent years. We propose to use generalizations of the Choquet integral to sensibly fuse elementary edge cues, in an attempt to model the behaviour of neurons in the early visual cortex. Our proposal leads to a fully-framed edge detection algorithm whose performance is put to the test in state-of-the-art edge detection datasets.

Blob detection and image denoising are fundamental, sometimes related tasks in computer vision. In this paper, we present a computational method to quantitatively measure blob characteristics using normalized unilateral second-order Gaussian kernels. This method suppresses non-blob structures while yielding a quantitative measurement of the position, prominence and scale of blobs, which can facilitate the tasks of blob reconstruction and blob reduction. Subsequently, we propose a denoising scheme to address high-ISO long-exposure noise, which sometimes spatially shows a blob appearance, employing a blob reduction procedure as a cheap preprocessing for conventional denoising methods. We apply the proposed denoising methods to real-world noisy images as well as standard images that are corrupted by real noise. The experimental results demonstrate the superiority of the proposed methods over state-of-the-art denoising methods.

Feature aggregation is a crucial step in many methods of image classification, like the Bag-of-Words (BoW) model or the Convolutional Neural Networks (CNN). In this aggregation step, usually known as spatial pooling, the descriptors of neighbouring elements within a region of the image are combined into a local or a global feature vector. The combined vector must contain relevant information, while removing irrelevant and confusing details. Maximum and average are the most common aggregation functions used in the pooling step. To improve the aggregation of relevant information without degrading their discriminative power for classification in this work we propose the use of Ordered Weighted operators. We provide an extensive evaluation that shows that the final result of the classification using OWA aggregation is always better than average pooling and better than maximum pooling when dealing with small dictionary sizes.

Blob detection and image denoising are fundamental, and sometimes related, tasks in computer vision. In this paper, we propose a blob reconstruction method using scale-invariant normalized unilateral second order Gaussian kernels. Unlike other blob detection methods, our method suppresses non-blob structures while also identifying blob parameters, i.e., position, prominence and scale, thereby facilitating blob reconstruction. We present an algorithm for high-ISO long-exposure noise removal that results from the combination of our blob reconstruction method and state-of-the-art denoising methods, i.e., the non-local means algorithm (NLM) and the color version of block-matching and 3-D filtering (CBM3D). Experiments on standard images corrupted by real high-ISO long-exposure noise and real-world noisy images demonstrate that our schemes incorporating the blob reduction procedure outperform both the original NLM and CBM3D.

Differentiation of interval-valued functions is an intricate problem, since it cannot be defined as a direct generalization of differentiation of scalar ones. Literature on interval arithmetic contains proposals and definitions for differentiation, but their semantic is unclear for the cases in which intervals represent the ambiguity due to hesitancy or lack of knowledge. In this work we analyze the needs, tools and goals for interval-valued differentiation, focusing on the case of interval-valued images. This leads to the formulation of a differentiation schema inspired by bilateral filters, which allows for the accommodation of most of the methods for scalar image differentiation, but also takes support from interval-valued arithmetic. This schema can produce area-, segment- and vector-valued gradients, according to the needs of the image processing task it is applied to. Our developments are put to the test in the context of edge detection.

There exist a significant number of benchmarks for evaluating the performance of boundary detection algorithms, most of them relying on some sort of comparison of the automatically-generated boundaries with human-labeled ones. Such benchmarks are composed of a representative image data set, as well as a comparison measure on the universe of boundary images. Despite many such data sets and measures have been proposed, there is no clear way of knowing which combinations of them are the most suitable for the task. In this paper, we introduce four criteria that allow for a sensible evaluation of the performance of a comparison measure on a given data set. The criteria mimic the way in which humans understand boundary images, as well as their ability to recognize the underlying scenes. These criteria can, as a final goal, quantify the ability of the boundary detection benchmarks to evaluate the performance of boundary detection methods, either edge-based or segmentation-based.

An important problem in working with fuzzy sets is the correct construction of the membership functions that represent the objects of the system. Different experts construct different membership functions to represent the same object. In this paper, we construct an interval type-2 fuzzy set (IT2FS) with different fuzzy sets such that the length of the (membership) interval represents the uncertainty of the expert with respect to the choice of the membership function. We analyze this problem in the context of image segmentation. We propose a new version of the classical fuzzy thresholding algorithm, in which an expert can select multiple membership functions, to avoid the problem of selecting only one to represent the image. From these membership functions, we construct an IT2FS, and by minimizing its entropy, we find a threshold with which to binarize the image. We present experimental results that show that it is advisable to use this methodology when it is not known which membership function is the most suitable.

The multiscale techniques for edge detection represent an effort to combine the spatial accuracy of small-scale methods with the ability to deal with spurious responses inherent to the large scale ones. In this work we introduce a multiscale extension of the Sobel method for edge detection based on Gaussian smoothing and fine-to-coarse edge tracking. We include examples illustrating the procedure and its results, as well as some quantitative measurements of the improvement obtained with the multiscale approach with respect to the original one.

In this paper, we present a new construction method for interval-valued fuzzy relations (interval-valued fuzzy images) from fuzzy relations (fuzzy images) by vicinity. This construction method is based on the concepts of triangular norm (t-norm) and triangular conorm (t-conorm). We analyze the effect of using differentt-norms andt-conorms. Furthermore, we examine the influence of different sizes of the submatrix around each element of a fuzzy relation on the interval-valued fuzzy relation. Finally, we apply our construction method to image processing, and we compare the results of our approach with those obtained by means of other, i.e., fuzzy and nonfuzzy, techniques.

The edge detection process can be broken down into four basic transformations, modifying the image from the original presentation to the final edges one. The adoption of this framework makes the process far more understandable, and offers an starting point for the combination and comparison of different edge detection methods. In this work we analyze the role of the third of the transformations, the blending, where the edge features are combined to obtain the edginess values. This work studies the use of quasi-aritmethic means for the combination of the edge features. Moreover, we show results obtained with different operators on real images, in order to illustrate the importance of the blending phase in the edge detection process. Results will show the impact of the function selection in the final results.

This work explores the possibilities of extracting edges using a t-conorm based gravity approach and its relation with the t-norm based one.