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Data processing. (a) Microphotograph of a peridotite thin section. (b) 2D distribution (in black) of orthopyroxene drawn from this thin section. (c) Fractal Dimension (FD) computed at fixed observation window, by varying the resolution (apparent pixel size) ε, and counting the number N(ε) of ε-pixels occupied by the orthopyroxene structure limited by the observation window. The red line represents the theoretical line; the blue line corresponds to the sample analysis. This figure is available in colour online at wileyonlinelibrary.com/journal/gj
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As rock textures reflect the physical conditions and the mechanisms of formation of the rocks, new approaches are used for improving texture analyses, both qualitatively and quantitatively. Pioneer work has recently boosted interest in fractal analysis for quantifying and correlating patterns. Fractal-like patterns relate to a degree of multiscale...
Contexts in source publication
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... of natural and cross-polarized light digital microphotographs. Images of the thin sections were acquired with a digital camera coupled with an optical polarizing microscope to get com- posite photographs of each section through an image editing software. The crystal boundaries have been extracted and/or redrawn through photo-processing softwares (Fig. 2a); the spatial regions occupied by orthopyroxene in a thin section have been delineated (Fig. 2b). The obtained digital images were used for fractal analysis of each sample. ...
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... acquired with a digital camera coupled with an optical polarizing microscope to get com- posite photographs of each section through an image editing software. The crystal boundaries have been extracted and/or redrawn through photo-processing softwares (Fig. 2a); the spatial regions occupied by orthopyroxene in a thin section have been delineated (Fig. 2b). The obtained digital images were used for fractal analysis of each sample. ...
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... is the relationship N(r) ~ r D , where r is a variable radius, N(r) is the number of particles or the mass contained in a ball of radius r, and D is the fractal di- mension (FD). In our case, the texture of a given sample will be characterized by the fractal dimension of the distribution of orthopyroxene grains in the thin section of this sample (Fig. 2b). The FD (Fig. 2c) has been determined using the Fractalyse 2.4.1 software developed within the team 'City, mobility, territory' of ThéMA (Théoriser et Modéliser pour Aménager) of University of Franche-Comté in Bourgogne, France (Frankhauser et al., 2007(Frankhauser et al., , 2008Tarnier et al., 2008aTarnier et al., , 2008b). This ...
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... N(r) ~ r D , where r is a variable radius, N(r) is the number of particles or the mass contained in a ball of radius r, and D is the fractal di- mension (FD). In our case, the texture of a given sample will be characterized by the fractal dimension of the distribution of orthopyroxene grains in the thin section of this sample (Fig. 2b). The FD (Fig. 2c) has been determined using the Fractalyse 2.4.1 software developed within the team 'City, mobility, territory' of ThéMA (Théoriser et Modéliser pour Aménager) of University of Franche-Comté in Bourgogne, France (Frankhauser et al., 2007(Frankhauser et al., , 2008Tarnier et al., 2008aTarnier et al., , 2008b). This software, originally ...
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... structure (box counting method); the log-log plot of N(ε) vs ε would then display a straight line of opposite slope ÀD. The difference between the FD values obtained by the two methods (1) and (2) lies between 0.01 and 0.05, which is very small, so in this paper we con- sider only the values derived by applying the second calcu- lation option (Fig. 2c). Because a rock texture (or the photomicrograph of a thin section) is not a pure fractal (it is not a continuous structure but a discrete one spanning a finite range of scales), it is only possible to get an approximate fractal law, that is, the log- log plot of N(ε) vs ε displays only a roughly straight region, moreover in a finite ...
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... one spanning a finite range of scales), it is only possible to get an approximate fractal law, that is, the log- log plot of N(ε) vs ε displays only a roughly straight region, moreover in a finite range of scales. The slope value D is ob- tained from the best fit, its quality is appreciated by the cor- relation coefficient R (here R 2 = 0.90, Fig. 2c). For more information on the mathematical basis of the method, see the original papers of the ThéMA laboratory (Frankhauser et al., 2007(Frankhauser et al., , 2008Tarnier et al., 2008aTarnier et al., , ...
Citations
... Nevertheless, establishing a relationship between fractal dimension (FD) (a ratio providing a statistical index of complexity) and various geological conditions that control textural pattern remains a challenge. Fractal studies have mostly been limited to silicate minerals [24][25][26][27][28][29] . Not much work has been done on the fractal nature of sulphides and specifically their textures [30][31][32] , in spite of the fact that they reveal a lot of information about the geological setting and conditions of formation of a mineral deposit. ...
... Recently we have noticed the renascence of application of the fractal dimension: in geology (Nkono et al., 2015), materials science (Lashgari et al., 2015), novel pharmaceutics Demetzos, 2014, 2015), medicine (Nakatsuka et al., 2015;Dedović et al., 2015;Lennon et al., 2015;Gokilavani and Vanitha, 2015;Smitha and Narayanan, 2015;Lawrence et al., 2015), etc. Examples of such recent applications are described in detail below. ...
The estimation of intrinsic dimensionality of high-dimensional data still remains a challenging issue. Various approaches to interpret and estimate the intrinsic dimensionality are developed. Referring to the following two classifications of estimators of the intrinsic dimensionality local/global estimators and projection techniques/geometric approaches we focus on the fractal-based methods that are assigned to the global estimators and geometric approaches. The computational aspects of estimating the intrinsic dimensionality of high-dimensional data are the core issue in this paper. The advantages and disadvantages of the fractal-based methods are disclosed and applications of these methods are presented briefly.
This work proposes a new model based on fractal descriptors for the classification of grayscale texture images. The method consists of scanning the image with a sliding box and collecting statistical information about the pixel distribution. Varying the box size, an estimation of the fractality of the image can be obtained at different scales, providing a more complete description of how such parameter changes in each image. The same strategy is also applied to a especial encoding of the image based on local binary patterns. Descriptors both from the original image and from the local encoding are combined to provide even more precise and robust results in image classification. A statistical model based on the theory of sliding window detection probabilities and Markov transition processes is formulated to explain the effectiveness of the method. The descriptors were tested on the identification of Brazilian plant species using scanned images of the leaf surface. The classification accuracy was also verified on three benchmark databases (KTH-TIPS2-b, UIUC and UMD). The results obtained demonstrate the power of the proposed approach in texture classification and, in particular, in the practical problem of plant species identification.
This work proposes the study of a simple yet efficient strategy to accomplish multiscale analysis of gray level images. The method is applied to the classification of texture/material images. The proposal employs histograms of down-sampled versions of the initial image to compose a feature vector. A statistical model based on Markov random fields is also developed and employed to explain how the information conveyed by the proposed descriptors can express degrees of homo/heterogeneity in the image. This is widely known to be a fundamental feature in texture analysis. The accuracy of our descriptors is compared to several state-of-the-art approaches and the achieved results confirm our expectation that a straightforward method can be efficiently employed even in the recognition of objects in large and complex databases.
This work proposes a method for supervised classification of grayscale texture images using the numerical computation of the box counting fractal dimension. Each pixel is mapped onto a point in a three-dimensional cloud, where the normalized gray level of each pixel is the third coordinate, and we analyze the distribution of points inside a mesh of boxes. Information at different resolutions are captured by varying the size of each box in the mesh. The texture descriptors are provided by a measure of organization of the points in the mesh, which is the entropy, and other statistical measures of this distribution, namely, mean, deviation and energy. We also propose a mathematical analysis of the model, which is accomplished here by employing techniques from Statistics and Combinatorics, quantifying the relation between the distribution of points and attributes classically associated to textures such as homogeneity and scale dependence. The proposed descriptors are applied to the classification of three well-known texture databases for benchmark purposes. In a comparison with other texture descriptors in the literature, the proposal demonstrated to be competitive, confirming the potential of a combination of box counting fractal dimension and statistics.
This work investigates the use of Dirichlet series in the modeling of texture images, with application in image classification. The proposed model is based on a strategy that associates each pixel with its corresponding color (gray level in our case) to a vertex of a complex network and the gray level dissimilarity within neighbor pixels with edge weights. The degree distribution of such network is known to be very effective in providing image descriptors. Here, we propose an improvement over this technique, by working on this distribution as a Dirichlet (exponential) series and varying the exponential parameter. A family of statistical measures are extracted from the series and compose a feature vector employed here for texture image classification. In our tests, the achieved accuracy is promising when compared with other state-of-the-art approaches in different databases classically used for benchmark purposes.
