Fractal Dimension Box-Counting Algorithm Optimization Through Integral Images

2022 ◽  
pp. 95-101
Author(s):  
Mircea-Sebastian Șerbănescu
Keyword(s):  
Fractal Dimension ◽  
Algorithm Optimization ◽  
Box Counting ◽  
Integral Images ◽  
Counting Algorithm

3D box-counting algorithm for calculating fractal dimension of cities

2010 ◽  
Vol 30 (8) ◽  
pp. 2070-2072
Author(s):  
Le-shan ZHANG ◽  
Ge CHEN ◽  
Yong HAN ◽  
Tao ZHANG
Keyword(s):  
Fractal Dimension ◽  
Box Counting ◽  
Counting Algorithm

scholarly journals Asymmetry of Information in the Analysis of Socio-Economic Processes

Vestnik NSUEM ◽  
2020 ◽  
pp. 64-75
Author(s):  
A. N. Kislyakov
Keyword(s):  
Fractal Dimension ◽  
Shannon Entropy ◽  
The State ◽  
Actual Problem ◽  
Asymmetry Of Information ◽  
Market Participants ◽  
Box Counting ◽  
Cosine Distance ◽  
Minkowski Fractal ◽  
Counting Algorithm

The work is aimed at solving the actual problem of analyzing the interaction of market participants. The degree of unpredictability of market participant’s behavior determines economic risks and manifests itself as a violation of information symmetry. Asymmetry is expressed in different degrees of awareness of groups of sellers and groups of buyers-users of the product about the state of the market, which determines the different behavioral moods and intentions of market participants. The possibility of using the Shannon entropy and fractal dimension indicators to assess the degree of ordering of relationships between groups of buyers and the results of their behavior is considered.This allows us to draw conclusions about the logic of relationships between the behavior of different clients. An iterative box-counting algorithm is used to determine the approximate value of the Minkowski fractal dimension.As a metric of distances between the signs of transactions of pairs of clients, the cosine distance can be used for the case of sparse data.It is shown how the fractal dimension will change in the case of observation of more stable relationships between groups of clients.

Fractal Estimation Using Extended Triangularisation and Box Counting Algorithm for any Geo-Referenced Point Data in GIS

2013 ◽  
pp. 1988-2005 ◽  
Author(s):  
R. Sridhar ◽  
S. Balasubramaniam
Keyword(s):  
Fractal Dimension ◽  
Linear Model ◽  
Crime Data ◽  
Cancer Data ◽  
Box Counting ◽  
Point Event ◽  
Point Data ◽  
Counting Algorithm

Fractal dimension is often used as a measure of how fast length, area, or volume increases or decreases with increase or decrease in scale, or as a measure of complexity of a system. In this paper, input depends only on the Geo-referenced point data where the point event has occurred. An Extended Triangularisation Algorithm is developed to cover the area of point data as a polygon and its perimeter is calculated. Box Counting Algorithm is applied on those point data to calculate the Fractal values, which in turn work as an input to Prediction Plot Linear Model, to show that fractal value increases or decreases as perimeter of Polygon increases or decreases. To validate this model, Crime data was used and its results were analyzed. It provides information to police officials about the intensity of crime, area of patrolling and deputation of police in the sensitivity area. This model could be applied for any Geo-referenced point data such as cancer data, hypertension data and so on.

OPTIMAL DYNAMIC BOX-COUNTING ALGORITHM

2010 ◽  
Vol 20 (12) ◽  
pp. 4067-4077 ◽  
Author(s):  
PANAGIOTIS D. ALEVIZOS ◽  
MICHAEL N. VRAHATIS
Keyword(s):  
Fractal Dimension ◽  
Time Complexity ◽  
Dimensional Space ◽  
Computational Cost ◽  
Space Complexity ◽  
Counting Problem ◽  
Counting Approach ◽  
Box Counting ◽  
Optimal Dynamic ◽  
Counting Algorithm

An optimal box-counting algorithm for estimating the fractal dimension of a nonempty set which changes over time is given. This nonstationary environment is characterized by the insertion of new points into the set and in many cases the deletion of some existing points from the set. In this setting, the issue at hand is to update the box-counting result at appropriate time intervals with low computational cost. The proposed algorithm tackles the dynamic box-counting problem by using computational geometry methods. In particular, we use a sequence of compressed Box Quadtrees to store the data points. This storage permits the fast and efficient application of our box-counting approach to compute what we call the "dynamic fractal dimension". For a nonempty set of points in the d-dimensional space ℝd (for constant d ≥ 1), the time complexity of the proposed algorithm is shown to be O(n log n) while the space complexity is O(n), where n is the number of considered points. In addition, we show that the time complexity of an insertion, or a deletion is O( log n), and that the above time and space complexity is optimal. Experimental results of the proposed approach illustrated on the well-known and widely studied Hénon map are presented.

Fractal Estimation Using Extended Triangularisation and Box Counting Algorithm for any Geo-Referenced Point Data in GIS

2012 ◽  
Vol 3 (3) ◽  
pp. 88-108 ◽  
Author(s):  
R. Sridhar ◽  
S. Balasubramaniam
Keyword(s):  
Fractal Dimension ◽  
Linear Model ◽  
Crime Data ◽  
Cancer Data ◽  
Box Counting ◽  
Point Event ◽  
Point Data ◽  
Counting Algorithm

Fractal dimension is often used as a measure of how fast length, area, or volume increases or decreases with increase or decrease in scale, or as a measure of complexity of a system. In this paper, input depends only on the Geo-referenced point data where the point event has occurred. An Extended Triangularisation Algorithm is developed to cover the area of point data as a polygon and its perimeter is calculated. Box Counting Algorithm is applied on those point data to calculate the Fractal values, which in turn work as an input to Prediction Plot Linear Model, to show that fractal value increases or decreases as perimeter of Polygon increases or decreases. To validate this model, Crime data was used and its results were analyzed. It provides information to police officials about the intensity of crime, area of patrolling and deputation of police in the sensitivity area. This model could be applied for any Geo-referenced point data such as cancer data, hypertension data and so on.

scholarly journals PARALLEL IMPLEMENTATION OF THE BOX COUNTING ALGORITHM IN OPENCL

Fractals ◽  
2015 ◽  
Vol 23 (03) ◽  
pp. 1550023 ◽  
Author(s):  
RAMAKRISHNAN MUKUNDAN
Keyword(s):  
Fractal Dimension ◽  
Efficient Method ◽  
Parallel Implementation ◽  
Single Pass ◽  
Highly Efficient ◽  
Box Counting ◽  
Parallel Implementations ◽  
Multiple Levels ◽  
Counting Algorithm

The box counting algorithm is a well-known method for the computation of the fractal dimension of an image. It is often implemented using a recursive subdivision of the image into a set of regular tiles or boxes. Parallel implementations often try to map the boxes to different compute units, and combine the results to get the total number of boxes intersecting a shape. This paper presents a novel and highly efficient method using Open Computing Language (OpenCL) kernels to perform the computation on a per-pixel basis. The mapping and reduction stages are performed in a single pass, and therefore require the enqueuing of only a single kernel. Each instance of the kernel updates the information pertaining to all the boxes containing the pixel, and simultaneously increments the box counters at multiple levels, thereby eliminating the need for another pass to perform the summation. The complete implementation and coding details of the proposed method are outlined. The performance of the method on different processors are analyzed with respect to varying image sizes.

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