A Brief Software Description

In the presented software, the simulation of the fractional Brownian motion is based on the Cholesky decomposition of the correlation matrix (it is an exact but slow and storage-consuming method for Gaussian vectors).

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Simulation of random fields on the plane with different fractal dimensions (including one moving average scheme, fractional Brownian motion and sheet): MS Win binary - FDE_SimRF2D.zip

Program FBsim.EXE: Aim of the program is to simulate realizations of the fractional Brownian motion and sheet. The fractal dimension of the realizations (surfaces in 3D space) can vary from 2 to 3. In addition, the program will estimate fractal dimension of the simulated realization by 4 methods, according to formulas (12)-(15) from S.M.Prigarin, K.Hahn, and G.Winkler (2011).

The simulation algorithm is based on the Cholesky decomposition of the correlation matrix (it is an exact but slow and storage-consuming method for Gaussian distributions).

Below we present realizations of fractional Brownian motion:

and fractional Brownian sheet:

with the Hausdorff dimensions 2.2 (on the left) and 2.8 (on the right).

Program MA2iso.EXE: Aim of the program is to simulate realizations of an isotropic Gaussian random field on the plain by moving average scheme, see details and further references in S.M.Prigarin, K.Hahn, and G.Winkler (2010, 2011). The Hausdorff dimension of the realizations is 2.5. A realization of the field is presented below:

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Estimation of fractal dimension by the variance counting method for a random function defined on the plane: MS Win binary file - FDE_RF2D.zip

Program Get-vDim-2D.EXE is designed to estimate variance dimension of a surface, given as a matrix in an ASCII file. In addition, there are examples of input files FB.DAT (with a realization of the 2D fractional Brownian motion, dimension 2.75) and MA2iso.DAT (with a realization of a Gaussian isotropic homogeneous field, dimension 2.5). Files FB.DAT and MA2iso.DAT were generated by programs FBsim.EXE, MA2iso.EXE (see the above software description). Variance dimension is estimated by 4 methods, according to formulas (12)-(15) from S.M.Prigarin, K.Hahn, and G.Winkler (2011).

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Simulation of several 2D fractals (the Sierpinski carpet, the Cantor dust, the fractal percolation model, the Henon attractor), MS Win binary files - FDE_Sim2D.zip.

Sim-Sierpinski2D.EXE, Sierpinski2D_XY.EXE : the Sierpinski carpet, dimension = ln(8)/ln(3) = 1.893.

First 4 iterations of the Sierpinski carpet:

Sim-CantorDust2D.EXE : the Cantor dust, dimension = ln(4)/ln(3) = 1.262.

First 4 iterations of the Cantor dust:

Sim-StochCarpet2D.EXE : stochastic modification of the Sierpinski carpet. Input parameters: basic grid dimension (BGD) is a number of intervals on the square side for the first iteration (i.e. a square is divided into BGD*BGD subsquares), number M of nonempty subsquares in every square. The dimension of such a fractal is ln(M)/ln(BGD).

A realization of stochastic carpet with parameters (BGD=4, M=14, 3 iterations):

Sim-Percol2D.EXE : 2D fractal percolation model. The model is described, for example, in K.Falconer, “Fractal Geometry”, 2003 (see p.251)

A realization of the 2D fractal percolation model:

Henon_XY.EXE : the canonical Henon attractor with a Hausdorff dimension=~1.261+-0.003 and a correlation dimension =~1.25+-0.02

The canonical Henon attractor:

Henon_attractor.EXE : the Henon attractor with given parameters. The parameters and number of points to be simulated must be presented in file Henon_attractor.INP.

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Programs to estimate fractal dimension of 2D sets (MS Win binaries - FDE_2D.zip)

Program xDim_2D.EXE : computation of the fractal dimension of a 2D set by the extended counting method.

Program 012-Dim_2D.exe : computation of two-scale box, information, and correlation dimensions of a 2D set.

Program gen-Dim-Tab_2D.exe : computation of a table with the two-scale generalized dimensions of a 2D set.

Program GenDim2D.EXE (with input files GenDim2D.INP, GenDim2D.MAT and an example of command file GenDim2D.BAT): computation of a table with the generalized dimensions of a 2D set by least squares. Adjusted generalized dimensions are computed as well.

In addition, examples of 2D fractal sets are presented in files CANTORDUST2D.MAT, PERCOL2D.MAT, SIERPINSKI2D.MAT, STOCHCAPERT2D.MAT.

Utility VisuFish.EXE (with file VisuFishHelp.TXT) can be used to visualize MAT-files.

About extended counting, 2-scale and adjusted methods see S.M.Prigarin, K.Sandau, M.Kazmierczak, K.Hahn (2013) and references there.

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Programs to estimate fractal dimension of 3D sets. Simulation of the Menger sponge (3D analog of the Sierpinski carpet) and the fractal foam, MS Win binary files - FDE_3D.zip

Menger-sponge.EXE : the Menger sponge with dimension=ln(20)/ln(3)= 2.7268 (output will be in file Menger-sponge.XYZ with 3 columns of (x,y,z)-coordinates of the set points)

First 3 iterations of the Menger sponge:

Fractal-Foam.EXE : the fractal foam with dimension=ln(26)/ln(3)= 2.9656 (output will be in file Fractal-foam.XYZ with 3 columns of (x,y,z)-coordinates of the points), the construction of the fractal foam is similar to that for the Menger sponge.

Show3Dxyz.EXE : a simple utility to visualize XYZ-files

x-Dim3D.EXE : computation of fractal dimension by the extended counting method of a set presented in ASCII file x-Dim3D.XYZ

box-Dim3D.EXE : computation of box-dimensions (and adjusted box-dimensions) of a set presented in ASCII file box-Dim3D.XYZ. The results will be written in files box-Dim3D.REZ, box-Dim3D.LN

Additional details can be found in S.M.Prigarin, K.Sandau, M.Kazmierczak, K.Hahn (2013).

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▲ The software is free. Please, cite the software when you use it in your work.

▲ Please, inform us about successful application of the software.

▲ The software is still under development. Any feedback is highly appreciated. Please, send your remarks to Sergei M. Prigarin and Klaus Hahn

☼ The research was partially supported by RFBR (project 11-01-00641).