The prediction of the generalization error in deep neural networks is a fundamental task with important implications, both in theoretical and practical terms. Recently, inspired by the intuitive conjecture that deep neural networks possess some low-dimensional embedding, which is the main responsible for the learning progress, fractal dimension has been presented as a promising measure to describe the intrinsic dimension of this embedding and, as an immediate consequence, also providing more reliable prediction of generalization. Nevertheless, the investigated approaches rely on the indirect modeling of the training dynamics and involves an analysis of the entire trajectory of the parameter space, which is prohibitively expensive in computational terms. Based on this, here we propose a straightforward and efficient numerical method to calculate the fractal dimension of the parameter distribution and use it as a predictor of the generalization error. The method takes inspiration from the variogram method to estimate the fractal dimension of a function profile and relies on the empirical power-law scaling between a small perturbation added to the trained parameters and the corresponding displacement in the loss function. The proposed measure is compared with other classical and state-of-the-art generalization measures in the literature using different hyperparameter configurations and the results suggest its promising potential, achieving competitive performance in all the analyzed datasets. Our study also proposes an adapted loss function that accounts for the fractal dimension and a theoretical generalization bound.

Texture recognition is an important task in computer vision and, as most problems in the area nowadays, has benefited from the use of deep convolutional neural networks . Nevertheless, typical architectures designed for object recognition usually do not perform optimally in such tasks. One of the most important elements of deep architectures in this application concerns the pooling operation. In particular, the well established global average pooling fails to capture more complex, multiscale and non-linear relation among the output activations . In this context, we propose fractal pooling , where the average is replaced by fractal dimension of the feature map. The new module is coupled with a convolutional backbone and a trainable residual block. The method is evaluated on texture classification , both on benchmark databases and on a real-world problem in botany. Our results are competitive with the state-of-the-art in texture classification , outperforming several modern deep learning approaches in terms of classification accuracy with the addition of minimal computational burden. The results suggest the potential of the proposed methodology for texture recognition in general and that using complexity measures is a promising strategy to perform pooling of deep features for this type of image.

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.

This work proposes a method to extract features from texture images by applying a Gaussian pyramid multiscale approach to the Bouligand–Minkowski fractal descriptors. The proposal starts from the texture image and computes the stack of multi-resolution images that compose the pyramid, in both directions, of reduction and expansion. In the following, each image in the stack is mapped onto a surface, which is dilated by spheres with variable radii and the dilation volumes are used to compute the Bouligand–Minkowski fractal descriptors for each level. Both the descriptors of each level and combinations with descriptors from the original image are verified in the classification of well-known databases of textural images. The proposed method outperformed other classical and state-of-the-art descriptors with a significant advantage in most cases, including situations where random noise is added to the images.

This work proposes the use of Singular Spectrum Analysis (SSA) for the classification of texture images, more specifically, to enhance the performance of the Bouligand-Minkowski fractal descriptors in this task. Fractal descriptors are known to be a powerful approach to model and particularly identify complex patterns in natural images. Nevertheless, the multiscale analysis involved in those descriptors makes them highly correlated. Although other attempts to address this point was proposed in the literature, none of them investigated the relation between the fractal correlation and the well-established analysis employed in time series. And SSA is one of the most powerful techniques for this purpose. The proposed method was employed for the classification of benchmark texture images and the results were compared with other state-of-the-art classifiers, confirming the potential of this analysis in image classification.

This work presents a new procedure to extract features of grey-level texture images based on the discrete Schroedinger transform. This is a non-linear transform where the image is mapped as the initial probability distribution of a wave function and such distribution evolves in time following the Schroedinger equation from Quantum Mechanics. The features are provided by standard deviation of the distribution measured at different times. The proposed method is applied to the classification of three databases of textures used for benchmark and compared to other well-known texture descriptors in the literature, such as textons, local binary patterns, multifractals, among others. All of them are outperformed by the proposed method in terms of percentage of images correctly classified. The proposal is also applied to the identification of plant species using scanned images of leaves and again it outperforms other texture methods. A test with images affected by Gaussian and "salt & pepper" noise is also carried out, also with the best performance achieved by the Schroedinger descriptors.

This work presents a method for color texture analysis based on fractal geometry. The method is based on its predecessor [4] and consists of mapping each color channel onto a surface and dilating such surface by spheres with a variable radius. The descriptors are obtained from the relation between the volumes of the dilated surfaces and the dilation radii. The dilation process creates a mutual interference among the color channels. The proposed descriptors measure the degree of such interference as well as the complexity of pixel intensity arrangements. This combination provides a robust and precise texture description. The efficiency of the method is assessed in a classification task of well-known texture data sets and the results demonstrate that it outperforms the best approaches described in the literature.

This work proposes a new method of texture analysis for grey-level images based on the distribution of connectivity indexes in local neighbourhoods. The connectivity index acts as a measure of homogeneity of textures and its distribution is computed at various local neighbourhood sizes. The resulting descriptors provide an efficient multiscale representation of connectivity at different scales. The method was tested in the classification of UIUC, Outex, and KTH-TIPS2b databases and outperformed several state-of-the-art approaches, including such as LBP, LBP+VAR, MR8, multifractals among others.