A simple method for estimating the fractal dimension from digital images: The compression dimension

Chamorro‐Posada, Pedro

doi:10.1016/j.chaos.2016.08.002

Cited by 22 publications

(12 citation statements)

References 20 publications

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“…Some of the fractal image compression algorithms use the "fractal dimension" to cluster the original picture and to find the best fitting maps F i , see [44] and [17]. Usually, the box counting dimension is used in the applied papers on the fractal image compression, however, from the theoretical point of view, the Hausdorff dimension plays important role.…”

Section: Question What Is the Hausdorff Dimension Of The Set λ?mentioning

confidence: 99%

“…We give here every type results, namely, under some mild conditions, the dimension can be calculated. This gives a feedback for the algorithm given for example in [17], how the chosen parameters approximate the original picture in sense of dimension. On the other hand, if we want to find a certain picture in a large collection of pictures coded with fractal image compression then it helps if we know the value of the dimension of the corresponding repeller.…”

Section: Question What Is the Hausdorff Dimension Of The Set λ?mentioning

confidence: 99%

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Dimension of the repeller for a piecewise expanding affine map

Bárány¹,

Rams²,

Simon³

2020

Ann. Acad. Sci. Fenn. Math.

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In this paper, we study the dimension theory of a class of piecewise affine systems in euclidean spaces suggested by Barnsley, with some applications to the fractal image compression. It is a more general version of the class considered in the work of Keane, Simon and Solomyak [42] and can be considered as the continuation of the works [5, 6] by the authors. We also present some applications of our results for generalized Takagi functions and fractal interpolation functions.

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Section: Question What Is the Hausdorff Dimension Of The Set λ?mentioning

confidence: 99%

Section: Question What Is the Hausdorff Dimension Of The Set λ?mentioning

confidence: 99%

Dimension of the repeller for a piecewise expanding affine map

Bárány¹,

Rams²,

Simon³

2020

Ann. Acad. Sci. Fenn. Math.

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“…The fractal dimension can quantitatively describe the complexity and heterogeneity of the pore structure, 31 and its determination methods mainly include the mercury intrusion L. Zhang et al method, imaging analysis method and gas adsorption method. [35][36][37] The range of pore size measured by nitrogen adsorption method is 1.7-300 nm. While the image method uses SEM image to count the pore size, the smallest pixel is the minimum pore size that can be recognized.…”

Section: Fractal Characteristics Of Pore Structurementioning

confidence: 99%

Pore Structure Characterization for a Continental Lacustrine Shale Parasequence Based on Fractal Theory

Zhang

Chai

et al. 2019

Fractals

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Fractal dimension is an important parameter in the evaluation of tight reservoirs. For an outcrop section of the Nenjiang formation in the Songliao Basin, China, the pore structure and pore fractal characteristics of shale parasequences were investigated using fractal theory. In addition, factors causing pore structure changes were analyzed using the results of low-temperature nitrogen adsorption and scanning electron microscope (SEM) experiments. Conducive to gas migration and secondary pores development such as dissolution, results showed that nanoscale pores dominated by fracture-like morphology and consequent good internal connectivity were observed in each pore size section within the target layer. Each parasequence is characterized by a sequential upward decrease of average pore size and an upward increase of total pore volume, with an increasing number of pores from 2[Formula: see text]nm to 50[Formula: see text]nm. Pores are isolated from each other, with poor connectivity and relatively complex composition of brittle minerals and clay minerals. Main components of the brittle minerals, quartz and feldspar, occur in 20–50% and higher clay mineral content ranging from 50% to 70%. In the parasequence cycle, clay mineral gradually decreases while the brittle mineral content increases. Fractal dimension is negatively correlated with clay mineral content and positively correlated with brittle mineral (quartz and feldspar) content. The fractal dimension calculated by the imaging method and the FHH method shows an upward increasing tendency in each of the parasequence cycles. This is as a result of different phenomena, varied sediment hydrodynamic forces leading to particle size differences and increased brittle minerals resulting in microcracks, therefore, the fractal dimension of the large pores (imaging method) increases upward in the parasequence. Simultaneously, with increased content and accompanied dissolution of brittle minerals causing an increase of small pores from base to top of the parasequence, the fractal dimension of the small pores (FHH method) grows.

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“…However, the fractal dimension, affine transformation and its parameters of fractal compression are still the key factors of fractal image compression. Chamorro-Posada 21 proposed the compression fractal dimension to achieve high compression ratio by improving the search accuracy; Omari et al 22 proposed a new image compression mechanism, which constructs a compression map by using the relationship between rational numbers and corresponding quotients and achieves a high compression ratio while maintaining the image quality. Swalpa proposed a new method for calculating the fractal encoding affine parameters to reduce the computational complexity of fractal encoding.…”

Section: Fractal Image Encodingmentioning

confidence: 99%

Research on fractal image compression hybrid algorithm based on convolutional neural network and gene expression programming

Pan

Liang

et al. 2019

Journal of Algorithms & Computational Technology

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Regarding the problems of insufficient image segmentation intelligence, low compression rate, slow speed for global searching to find the optimal fractal image compression encoding, and bad decoding effect, this article proposes the fractal image compression hybrid algorithm based on convolutional neural network and gene expression programming. Firstly, according to the accurate and fast image classification of deep convolutional neural network and the fast search and matching encoding advantages of gene expression programming, it realizes theoretically the action mechanism of fractal image compression hybrid encoding by combining the convolutional neural network and the gene expression programming; then, it uses the deep convolutional neural network to train and classify the image, and uses the adaptive quadtree segmentation method to segment the classified image, thus generating the domain block and range block classification set. According to the action mechanism of gene expression programming in fractal image compression encoding, it then quickly obtains the optimal solution of fractal image compression encoding by searching and encoding the sub-blocks of range block classification set and the classification set corresponding to the domain. Finally, in the CPU/GPU environment, it conducts the comparative experiment with basic fractal image compression algorithm and fractal image compression algorithm based on convolutional neural network. The experimental results show that this proposed algorithm outperforms similar algorithms in terms of image segmentation speed and accuracy as well as fractal compression encoding speed and compression ratio. Therefore, this algorithm is a fractal image compression algorithm with intelligent segmentation, fast encoding and high compression ratio.

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A simple method for estimating the fractal dimension from digital images: The compression dimension

Cited by 22 publications

References 20 publications

Dimension of the repeller for a piecewise expanding affine map

Dimension of the repeller for a piecewise expanding affine map

Pore Structure Characterization for a Continental Lacustrine Shale Parasequence Based on Fractal Theory

Research on fractal image compression hybrid algorithm based on convolutional neural network and gene expression programming

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