Set Up a Multifractal Analysis

Multifractal Analysis Options

The MULTIFRACTAL OPTIONS panel

Use this panel for multifractal analysis.

1. After setting up the essential features for a scan, set up the Q range
2. Choose multifractal spectra to display
3. Select a data processing mode.
4. View data files and graphics.

Multifractal analysis can be done using standard sampling or random sampling. The option for random sampling is on the SUB SCAN panel. (See notes on random sampling for multifractal analysis.)

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Results

FracLac makes data files and graphics of the multifractal spectra for you to use when you do multifractal analysis. There are 5 main data types, each corresponding to one of the types of multifractal dimensions. FracLac generates files and graphics for each, including additional information such as the aperture, dimensional ordering, etc., that is essential to multifractal analysis. To sum it up, use the graphs to see the general nature of the multifractal scaling, and the data files to quantitate it.

Data Types

Data produced in a multifractal scan include data files and graphics for the following data types:

• DQ vs Q - Generalized Dimension Graph
• ƒ(α) vs. α - Multifractal Spectrum
• ƒ(α) vs Q
• α vs Q
• τ vs Q

Multifractal Data Files

The point of a multifractal analysis is to examine multifractal scaling. With FracLac, use the graphics to understand the general nature of multifractal scaling in your images and use the data files automatically generated with every scan to quantitate it.

Types of Multifractal Analysis Data Files

• Q(Min to Max) Multifractal Data Files
  These are the essential data sets containing the values of each type of dimension against Q, as can be used to plot the multifractal spectra. There is a file for each data type. Use these to plot multifractal spectra and to compare how each image varied for different values of the arbitrary exponent, Q. That is, each file contains a row for each image or roi. These files are named using the range of Q values, the increment, the name of the version of FracLac, and the data type (e.g., "Q(-10 to 10)x.25FracLac2015b2995 ƒ(α(Q)) Multifractal Data").
• Data Set Description Files
  Use these files along with the Q files to assess the multifractal scaling in an image. These include a description of various features of each dataset, such as the aperture length and slope. The information is also used to determine the optimal set of parameters when multiple orientations are used in a scan.
• Multifractal Data Files
  These files list all of the data types in one place. Use them to compare how an image or roi was described by each data type. The file also includes slopes of the data types against Q.
• A set of default box-counting files.
  Standard box-counting data are generated with multifractal analysis scans if the option to generate them is selected. This option is on the data files panel. Note that the scan may be set up improperly for a standard box-count if it is set up for multifractal analysis, because some of the options that maximize results with multifractal analysis may not be optimal for lacunarity and mono-fractal analysis. See box-counting for details of how to calibrate these scans.

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The Q SET Panel

Q SET - arbitrary exponents for calculating multifractal spectra

Use this sub-panel on the MULTIFRACTAL OPTIONS pane to set values for Q, part of the generalized dimension, DQ. This is a key variable in multifractal analysis. Q is a range of exponents used to calculate several quantities and make graphs for the variables in a multifractal analysis.

The default Q–range is 0±10, incremented by 0.25.

Use the sliders or number fields to set the
minimum,
maximum,
and increment.

Usually, the values of exactly -1, 0, and 1 should be included in the range. These are essential for assessing dimensional ordering and determining the aperture. If your range does not include them, these 2 important elements for quantitating your results cannot be generated.

Experiment with Q settings to see how the multifractal spectra are affected for different images and ranges. Be aware that negative and positive Qs contribute differently to the different multifractal spectra. The spectra are calculated using functions of the sums of the mass probability distribution raised to the values of Q and seeing how the function changes for different Qs. Higher values of Q affect denser parts of a probability distribution for the ƒα spectra, for instance. For D_Q, when Q is negative, the sum is amplified more by areas with smaller probabilities, but this reverses at 0. Note also that, in general, Q<−10 and Q>12 can cause calculation problems.

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inc - Increment between exponents in Q SET

Use this option on the Q SET panel on the MULTIFRACTAL OPTIONS pane to set the increment between Qs for the Generalized Dimension and other multifractal variables.

Use the slider or type a number for the change from one value of Q to the next. See minimum Q and maximum Q.

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min - Minimum exponent in Q SET

Use the slider and number field on the Q SET panel on the MULTIFRACTAL OPTIONS pane to set the minimum Q for the Generalized Dimension and other multifractal variables.

This sets the lower bound for the arbitrary exponent, "Q" used to determine multifractal scaling. This is generally a negative number, but should be set taking into consideration the value to be used for the maximum Q - you may wish to bracket 0 by making the minimum -20 and the maximum +20, for instance.

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max - Maximum exponent in Q SET

Use the slider and number field on the Q SET panel on the MULTIFRACTAL OPTIONS pane to set the maximum Q for the Generalized Dimension and other multifractal variables.

This sets the upper bound on the arbitrary exponent, "Q", used to determine multifractal scaling. This is generally a positive number set in accordance with the minimum Q (e.g., you may wish to bracket 0 by making the minimum -20 and the maximum +20).

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MF GRAPHS - choose the multifractal spectra graphs to view

Use the MF GRAPHS panel on the MULTIFRACTAL OPTIONS pane to select graphs to generate for multifractal spectra. You can select all or none. The selections do not affect the data files; all data are always included in the data files regardless of which graphs are generated. To read more about the choices, click on one of the datatypes.

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τ vs Q - Multifractal Spectra Graph

Select this option on the MF GRAPHS panel on the MULTIFRACTAL OPTIONS pane to generate a graph of τ vs Q.

Two graphs are generated, reflecting two methods of calculating τ (Click for Calculations). Note that the optimizer will affect your results if also selected. See graphing multifractal spectra for more information.

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ƒ(α) vs α - Multifractal Spectra Graph

Select this option on the MF GRAPHS panel on the MULTIFRACTAL OPTIONS pane to generate a graph of ƒ (α) vs α. The image shows a sample of the typical pattern for multifractals, with the green portion for values of Q ≥0 and red for Q≤0.

Aperture The aperture is used to quantitate the degree of multifractal scaling. Think of the aperture as the "cone" of the curve, but not necessarily a symmetrical slice. It is defined by a line that slices through the positive and negative sides around the maximum, where Q = 1 and -1, and by the lines from those outer points to Q = 0. It is drawn on the graphs if that option is selected from a popup that appears when you close the dialog and have selected this graph. A longer aperture suggests a more heterogeneous fractal.

See graphing multifractal spectra for more information. (Calculations).

Note that the optimizer will affect results if selected.

The maximum and value at Q = 0 are listed on the graph. At Q = 0, the generalized dimension is equal to the box counting dimension.

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ƒ(α) vs Q - Multifractal Spectra Graph

Select this option on the MF GRAPHS panel on the MULTIFRACTAL OPTIONS pane to generate a graph of ƒ(α) vs Q.

See graphing multifractal spectra for more information. (Click for Calculations).

Note that the optimizer will affect results if also selected.

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α(Q) vs Q - Multifractal Spectra Graph

Select this option on the MF GRAPHS panel on the MULTIFRACTAL OPTIONS pane to generate a graph of α(Q) vs Q.

See graphing multifractal spectra for more information. (Click for Calculations).

The optimizer will affect results if also selected.

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D_Q vs Q - Multifractal Spectra Graph of the Generalized Dimension

Select this option on the MF GRAPHS panel on the MULTIFRACTAL OPTIONS pane to generate a graph of DQ vs Q.

Two graphs are generated when this option is selected, one graph bounded by the actual minimum and maximum DQ values (see image) the other by 0 and 3, so that multiple images can be readily compared. (Click for Calculations).

To interpret your results using this graph, refer to the section on dimensional ordering.

To quantitate your image using this feature, use the data set description automatically generated with every scan.

Note that the optimizer will affect results if also selected.

See graphing multifractal spectra for more information.

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Calculations for Multifractal Spectra

The results of a multifractal analysis include a data file with columns for the variables used to make multifractal spectra. The values for each column are described below.

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The Q̇ column

This column lists the values that were used for the Q̇ Set, which is the range of exponents used in an analysis. The parameters used to generate the range are specified by the user from the Q̇ SET panel, but the actual values used may differ slightly from those displayed on the dialog.

In the results file, beside each exponent in the Q̇ Set is a row with a value for each column. The specifics for each column are explained after this section, but they all rely on the following:

• N[ε] = the number of boxes from the box count at some size or ε
• m[i,ε] = mass at any box, i, at ε
• M[ε] = ∑m[i,ε] for all i = 1 to N [ε]
• P[i,ε] = m[i,ε] / M[ε]
• S[ε](Q) = ∑(P[i,ε] ^Q) for all i = 1 to N [ε]
• μ[i,ε](Q) = (P[i,ε] ^Q) /S[ε](Q)

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The Generalized Dimension Column

• DQ = τ(Q) / (Q-1) (See graph)

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The τ columns

• τ(Q) = (Q × α(Q)) − ƒ(α (Q))
• The mean τ(Q) column:
  the slope of the regression line for ln τ [ε](Q) vs ln ε, where:
  τ [ε](Q) = N[ε]⁻¹ × (for i=1 to N [ε]) ∑ [(m[i,ε] / M[ε])^Q-1]

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The α(Q) Column

• The slope of the regression line for α [ε](Q) vs ln ε where
  α [ε](Q) = (for i=1 to N) ∑ μ[i,ε]_(Q) × ln P[i,ε]

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The ƒ(α (Q)) Column

• The slope of the regression line for ƒ(α(Q)) vs ln ε where
• ƒ(α(Q)) = (for i=1 to N)∑ μ[i,ε](Q) × ln μ[i,ε]_(Q)

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Filename:

Along with each set of these columns of data is a filename with numbers to identify the image, slice, ROI coordinates and size, and grid coordinates and size. This information is changed and printed before each set. If the optimizer is selected, the set includes rows explaining the selection criteria and how the sample at the specified coordinates met those criteria.

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See Multifractal Calculations

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Tips

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The defaults for FracLac's multifractal scan are set up for a Henon Map. This may not be an appropriate baseline for all images, however. Following are some pointers that may help if you experience unusual results with multifractal scans, including some workarounds for known bugs.

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All Origins are the Same

For perfectly square bounding boxes using exact multiples of the bounding box for the largest grid calibre, the first four scan origins will be identical. This is owing to the way FracLac calculates the first 4 in multifractal scans. A temporary workaround is to select more than 4 to find extra, random origins (e.g., select 8 to get 4 random origins in addition to 4 identical ones).

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Unusual Spectra

Sampling is usually the problem when your spectra are not quite right. If the graph of the Generalized Dimension, DQ, is not decreasing, or the two coloured parts of the red and green graph of ƒ(α) vs α do not meet in a continuous curve or appear to cross over as in the example, for instance, sampling may be inappropriate. This may reflect a density distribution that attributes too much importance to very small probabilities that appear at some, but not all, grid positions. Negative Qs are most relevant.

If the Q values are not obviously at fault, this problem may simply be a matter of needing to find a better grid orientation, for which the solution is to increase the number of grid orientations sampled, select to optimize the scan, and use a slope-correcting filter. Other potential causes and their fixes are listed below.

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Using the Multifractal Optimizer with Invalid Calculations

If the optimizer, incorrectly selects such a sample over a better one, there may be sampling or calculation errors. One solution that often works is to increase the number of grid positions. For some images, the usual 12 is not enough but 500 locations will work.

Another option to try is setting the check pix box on the OPTIONS panel, to avoid sampling areas with drastically different densities near the edges. Changing the sampling element to oval also corrects sampling errors in some patterns.

If a crossover is being selected over a smooth curve, the values for ± Q may be too high. If it is happening, there may be non-numbers in the results files, and non-numbers (i.e., unintelligible symbols) printed on the graph of ƒ (α), for instance. A workaround is to try setting the maximum and minimum to a smaller bracket around 0 (e.g., to 5 and -5 instead of 10 and -10, for example).

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Invalid Sampling Owing to Problems with Resolution and Grid Calibre in Multifractal Analysis

With multifractal scan problems, if there are no invalid number errors, increasing the sampling orientations does not correct the problem, and selecting to optimize is also unsuccessful, the minimum resolvable box size may need special attention. To correct this, try changing (usually increasing) the minimum grid size in pixels to a value appropriate for your image.

The problem may also be in the upper end of the series, in which case increasing the maximum size or changing the selection at the option to Use the Greater Dimension of Roi can help.

In some cases, selecting a different series (e.g., switching from scaled series to "default") will fix the problem. In addition, changing (usually decreasing) the number of box sizes may improve the result.

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Very Long Processing Time

The number of box sizes can affect results and processing time. To reduce processing time, it is often helpful to reduce the number of box sizes (e.g., 10 or less). Select a custom series or a power series, for instance, in the series option. If you are doing a random mass sample, using fewer samples or decreasing the sample size may help.

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Problems with Random Mass Sampling in Multifractal Analysis

Select the option on the SUB SCAN panel to do a multifractal analysis using random mass sampling.

If you get strange results, you may need to adjust the maximum box size as a percent to 100%, adjust the number of box sizes up or down, and ensure the minimum grid size in pixels is high enough (e.g., >5 or 10). You may also want to compare the results of using different numbers of samples and different sizes of samples.