Multifractal Descriptors Research Articles

Ergodic characterization of nonergodic anomalous diffusion processes

Anomalous diffusion in various complex systems abounds in nature and spans multiple space and time scales. Canonical characterization techniques that rely upon mean squared displacement break down for nonergodic processes, making it challenging to characterize anomalous diffusion from an individual time-series measurement. Nonergodicity reigns when the time-averaged mean square displacement differs from the ensemble-averaged mean squared displacement even in the limit of long measurement series. In these cases, the typical theoretical results for ensemble averages cannot be used to understand and interpret data acquired from time averages. The difficulty then lies in obtaining statistical descriptors of the measured diffusion process that are not nonergodic. We show that linear descriptors such as the standard deviation, coefficient of variation, and root mean square break ergodicity in proportion to nonergodicity in the diffusion process. In contrast, time series of descriptors addressing sequential structure and its potential nonlinearity: multifractality change in a time-independent way and fulfill the ergodic assumption, largely independent of the time series' nonergodicity. We show that these findings follow the multiplicative cascades underlying these diffusion processes. Adding fractal and multifractal descriptors to typical linear descriptors would improve the characterization of anomalous diffusion processes. Two particular points bear emphasis here. First, as an appropriate formalism for encoding the nonlinearity that might generate nonergodicity, multifractal modeling offers descriptors that can behave ergodically enough to meet the needs of linear modeling. Second, this capacity to describe nonergodic processes in ergodic terms offers the possibility that multifractal modeling could unify several disparate nonergodic diffusion processes into a common framework.

Fractal and multifractal descriptors restore ergodicity broken by non-Gaussianity in time series

Ergodicity breaking is a challenge for biological and psychological sciences. Ergodicity is a necessary condition for linear causal modeling. Long-range correlations and non-Gaussianity characterizing various biological and psychological measurements break ergodicity routinely, threatening our capacity for causal modeling. Long-range correlations (e.g., in fractional Gaussian noise, a.k.a. “pink noise”) break ergodicity—in raw Gaussian series, as well as in some but not all standard descriptors of variability, i.e., in coefficient of variation (CV) and root mean square (RMS) but not standard deviation (SD) for longer series. The present work demonstrates that progressive increases in non-Gaussianity conspire with long-range correlations to break ergodicity in SD for all series lengths. Meanwhile, explicitly encoding the cascade dynamics that can generate temporally correlated non-Gaussian noise offers a way to restore ergodicity to our causal models. Specifically, fractal and multifractal properties encode both scale-invariant power-law correlations and their variety, respectively—features that index the underlying cascade parameters. Fractal and multifractal descriptors of long-range correlated non-Gaussian processes show no ergodicity breaking and hence, provide a more stable explanation for the long-range correlated non-Gaussian form of biological and psychological processes. Fractal and multifractal descriptors offer a path to restoring ergodicity to causal modeling in these fields.

New Class of Topological Textural Multifractal Descriptors and Their Application for Processing Low-Contrast Radar and Optical Images

New Class of Topological Textural Multifractal Descriptors and Their Application for Processing Low-Contrast Radar and Optical Images

The Classification of Inertinite Macerals in Coal Based on the Multifractal Spectrum Method

Considering the heterogeneous nature and non-stationary property of inertinite components, we propose a texture description method with a set of multifractal descriptors to identify different macerals with few but effective features. This method is based on the multifractal spectrum calculated from the method of multifractal detrended fluctuation analysis (MF-DFA). Additionally, microscopic images of inertinite macerals were analyzed, which were verified to possess the property of multifractal. Simultaneously, we made an attempt to assess the influences of noise and blur on multifractal descriptors; the multifractal analysis was proven to be robust and immune to image quality. Finally, a classification model with a support vector machine (SVM) was built to distinguish different inertinite macerals from microscopic images of coal. The performance evaluation proves that the proposed descriptors based on multifractal spectrum can be successfully applied in the classification of inertinite macerals. The average classification precision can reach 95.33%, higher than that of description method with gray level co-occurrence matrix (GLCM; about 7.99%).

Multifractal analysis of the interstellar medium: first application to Hi-GAL observations

The multifractal geometry remains an under-exploited approach to describe and quantify the large-scale structure of interstellar clouds. In this paper, the typical tools of multifractal analysis are applied to Herschel far-infrared (70-500 $\mu$m) dust continuum maps, which represent an ideal case of study. This dust component is a relatively optically thin tracer at these wavelengths and the size in pixel of the maps is well suitable for this statistical analysis. We investigate the so-called multifractal spectrum and generalised fractal dimension of six Hi-GAL regions in the third Galactic quadrant. We measure systematic variations of the spectrum at increasing wavelength, which generally correspond to an increasing complexity of the image, and we observe peculiar behaviours of the investigated fields, strictly related to the presence of high-emission regions, which in turn are connected to star formation activity. The same analysis is applied to synthetic column density maps, generated from numerical turbulent molecular cloud models and from fractal Brownian motion (fBm), allowing for the confrontation of the observations with models with well controlled physical parameters. This comparison shows that, in terms of multifractal descriptors of the structure, fBm images exhibit a very different, quite unrealistic behaviour when compared with Hi-GAL observations, whereas the numerical simulations appear in some cases (depending on the specific model) more similar to the observations. Finally, the link between mono-fractal parameters (commonly found in the literature) and multifractal indicators is investigated: the former appear to be only partially connected with the latter, with the multifractal analysis offering a more extended and complete characterization of the cloud structure.

Three-dimensional multifractal analysis of trabecular bone under clinical computed tomography.

An adequate understanding of bone structural properties is critical for predicting fragility conditions caused by diseases such as osteoporosis, and in gauging the success of fracture prevention treatments. In this work we aim to develop multiresolution image analysis techniques to extrapolate high-resolution images predictive power to images taken in clinical conditions. We performed multifractal analysis (MFA) on a set of 17 exvivo human vertebrae clinical CT scans. The vertebrae failure loads (FFailure) were experimentally measured. We combined bone mineral density (BMD) with different multifractal dimensions, and BMD with multiresolution statistics (e.g., skewness, kurtosis) of MFA curves, to obtain linear models to predict FFailure. Furthermore we obtained short- and long-term precisions from simulated invivo scans, using a clinical CT scanner. Ground-truth data - high-resolution images - were obtained with a High-Resolution Peripheral Quantitative Computed Tomography (HRpQCT) scanner. At the same level of detail, BMD combined with traditional multifractal descriptors (Lipschitz-Hölder exponents), and BMD with monofractal features showed similar prediction powers in predicting FFailure (87%, adj. R2 ). However, at different levels of details, the prediction power of BMD with multifractal features raises to 92% (adj. R2) of FFailure. Our main finding is that a simpler but slightly less accurate model, combining BMD and the skewness of the resulting multifractal curves, predicts 90% (adj. R2) of FFailure. Compared to monofractal and standard bone measures, multifractal analysis captured key insights in the conditions leading to FFailure. Instead of raw multifractal descriptors, the statistics of multifractal curves can be used in several other contexts, facilitating further research.

Music Performers Classification by Using Multifractal Features: A Case Study

Abstract In this paper, we investigated the possibility to classify different performers playing the same melodies at the same manner being subjectively quite similar and very difficult to distinguish even for musically skilled persons. For resolving this problem we propose the use of multifractal (MF) analysis, which is proven as an efficient method for describing and quantifying complex natural structures, phenomena or signals. We found experimentally that parameters associated to some characteristic points within the MF spectrum can be used as music descriptors, thus permitting accurate discrimination of music performers. Our approach is tested on the dataset containing the same songs performed by music group ABBA and by actors in the movie Mamma Mia. As a classifier we used the support vector machines and the classification performance was evaluated by using the four-fold cross-validation. The results of proposed method were compared with those obtained using mel-frequency cepstral coefficients (MFCCs) as descriptors. For the considered two-class problem, the overall accuracy and F-measure higher than 98% are obtained with the MF descriptors, which was considerably better than by using the MFCC descriptors when the best results were less than 77%.

Revision of multifractal descriptors for texture classification based on mathematical morphology

This paper examines 2D multifractal model based on operations taken from mathematical morphology. Firstly, there is given an extension of the existing multifractal model based on Legendre transform and dilation and complementary model using erosion is investigated. Two additional methods are proposed by introducing a new definition of local multifractal measures and two realizations of coverings of local dimensions images used for calculation of global dimensions. One of local measures is implemented using dilation and the other using erosion. Derived multifractal spectra are used as descriptors of textural images. Experiments conducted on referent textural datasets and in comparison to state of the art descriptors provide classification results of proposed descriptors that exceed performance of both multifractal and non-fractal descriptors from literature.

Local multifractal detrended fluctuation analysis for non-stationary image's texture segmentation

Feature extraction plays a great important role in image processing and pattern recognition. As a power tool, multifractal theory is recently employed for this job. However, traditional multifractal methods are proposed to analyze the objects with stationary measure and cannot for non-stationary measure. The works of this paper is twofold. First, the definition of stationary image and 2D image feature detection methods are proposed. Second, a novel feature extraction scheme for non-stationary image is proposed by local multifractal detrended fluctuation analysis (Local MF-DFA), which is based on 2D MF-DFA. A set of new multifractal descriptors, called local generalized Hurst exponent (Lhq) is defined to characterize the local scaling properties of textures. To test the proposed method, both the novel texture descriptor and other two multifractal indicators, namely, local Hölder coefficients based on capacity measure and multifractal dimension Dq based on multifractal differential box-counting (MDBC) method, are compared in segmentation experiments. The first experiment indicates that the segmentation results obtained by the proposed Lhq are better than the MDBC-based Dq slightly and superior to the local Hölder coefficients significantly. The results in the second experiment demonstrate that the Lhq can distinguish the texture images more effectively and provide more robust segmentations than the MDBC-based Dq significantly.

Local multifractal detrended fluctuation analysis for tea breeds identification

In recent years, the popular multifractal detrended fluctuation analysis (MF-DFA) is extended to two-dimensional (2D) version, which has been applied in some field of image processing. In this paper, based on the 2D MF-DFA, a novel multifractal estimation method for images, which we called the local multifractal detrended fluctuation analysis (LMF-DFA), is proposed to recognize and distinguish 20 types of tea breeds. A set of new multifractal descriptors, namely the local multifractal fluctuation exponents is defined to portray the local scaling properties of a surface. After collecting 10 tea leaves for each breed and photographing them to standard images, the LMF-DFA method is used to extract characteristic parameters for the images. Our analysis finds that there are significant differences among the different tea breeds' characteristic parameters by analysis of variance. Both the proposed LMF-DFA exponents and another classic parameter, namely the exponent based on capacity measure method have been used as features to distinguish the 20 tea breeds. The comparison results illustrate that the LMF-DFA estimation can differentiate the tea breeds more effectively and provide more satisfactory accuracy.

MULTIFRACTAL DETRENDED FLUCTUATION ANALYSIS FOR IMAGE TEXTURE FEATURE REPRESENTATION

Multifractal theory has been widely used in different kinds of fields. In this paper, methods were proposed to extract two kinds of multifractal descriptors of gray series and two-dimensional surfaces for gray image based on the multifractal detrended fluctuation analysis. The proposed multifractal parameters can be well described by texture feature through the test of some textures. Three aspects of experiments have been conducted to verify the robustness of the proposed parameters, which include noise immunity, degree of image blurring and compression ratio. Comparisons were conducted between the proposed parameters and other kinds of texture feature parameters calculated by the standard multifractal analysis, the method of differential box counting and the methods of gray level co-occurrence matrix. Results demonstrate that the proposed exponents of H(2) and h(2) have great noise immunity and are robust to image compression and blurring.

A 2D wavelet-based multiscale approach with applications to the analysis of digital mammograms

A wavelet-based multifractal spectrum (MFS) for the analysis of images that possess an erratically changing oscillatory behavior at various scales is constructed and estimated. The methodology is applied to the analysis of mammograms. The key contribution is that the analysis is not focused on microcalcifications, but on the background of the image, thus presenting a new modality to be combined with other diagnostic tools. Differences in image backgrounds between malignant and normal cases are found, in terms of multifractal descriptors. The new tool is compared with another spectral method, based on monofractal descriptors.11All codes utilized in this work are available as a supplementary material with the electronic version of the paper.

Morphology-based multifractal estimation for texture segmentation

Multifractal analysis is becoming more and more popular in image segmentation community, in which the box-counting based multifractal dimension estimations are most commonly used. However, in spite of its computational efficiency, the regular partition scheme used by various box-counting methods intrinsically produces less accurate results. In this paper, a novel multifractal estimation algorithm based on mathematical morphology is proposed and a set of new multifractal descriptors, namely the local morphological multifractal exponents is defined to characterize the local scaling properties of textures. A series of cubic structure elements and an iterative dilation scheme are utilized so that the computational complexity of the morphological operations can be tremendously reduced. Both the proposed algorithm and the box-counting based methods have been applied to the segmentation of texture mosaics and real images. The comparison results demonstrate that the morphological multifractal estimation can differentiate texture images more effectively and provide more robust segmentations.

Multifractal analysis and feature extraction in satellite imagery

In the last 20 years, the use of fractal dimension indices has found wide application in various disciplines, including remote sensing. Recent studies have improved our understanding of both scaling and statistical properties of natural images and it is now broadly accepted that the fractal dimension, used as a sole texture descriptor, does not provide complete textural information. Natural images are characterized by a set of fractal dimensions (i.e. multifractal indices) rather than by a single fractal dimension. This paper aims to give a brief introduction to the theory of fractals and multifractals and to confirm, by means of examples, the inadequacy of using a single fractal dimension in image analysis. In addition, it will show how multifractal indices can provide better textural description. The main purpose of this paper is to prove that multifractal descriptors can substitute or complement classical textural descriptors by presenting a method to calculate the multifractal spectrum of pixels inside images and to extract multifractal indices that can be used as descriptors of textural features. In particular, the paper will propose a multifractal index named Spectrum Range and will introduce some of its properties. To assess the efficacy of the method and of the derived multifractal indices, three examples will be presented. The first two images are artificial images generated by a pseudo-random generator and partly by a deterministic function. The third image is an AVHRR image taken above Scotland.