DATA PROCESSING
Data Processing Options for Box Counting and Multifractal Scans
The options on this page are for standard box counting and multifractal scans.
OPTIMIZERS AND FILTERS
How Optimization Works
Optimizers use a set of parameters characterizing
typical
multifractal spectra
to choose one dataset over others. Each
grid orientation
has its own dataset, and these are what the optimizer compares.
The optimizer runs if either of the 2 optimizing options
is selected from the
optimizer dropdown
on the
DATA PROCESSING panel. Box counting
data are gathered normally,
then an optimal sampling orientation is selected from the
multiple grid orientations.
The difference between the 2 optimizing options
is in what each shows;
Show Optimal Sample Only
discards all of the data and graphs except for
the one grid orientation deemed optimal;
Mark Optimal But Show All
shows them all and indicates which was considered optimal.
The
number of grid orientations
needs to be more than 1 and
relatively high (e.g., more than 4, dependent
on the image) for the optimizer to be effective.
The optimal sample is selected by going down a hierarchical decision tree. The decision tree chooses the sample for which the maximum and the value at Q = 0 for ƒ(α) were closest, then it compares in random order and selects on the basis of curving for ƒ(α) vs Q, as shown in the illustration.
The Decision is Based on Curving
The next selection is made according to α vs Q decreasing, as illustrated in the figure, then, similarly, the generalized dimension, D(Q) vs Q decreasing and dimensional ordering according to:
the Dcapacity ≥ DInformation ≥ Dcorrelation
Optimizing Based on Dimensional Ordering
If the decision tree is traversed this far, the final steps use the sum of the values for positive Q and then the highest CV for D(Q).
The optimum sample selected by the algorithm is graphed with the word "Optimized" in the graph's title, and the x,y coordinates of that grid orientation are printed on the graphic and recorded at the end of the results for the scan in the results file. If the option to show only the optimized sample is selected, then no other multifractal data are shown graphically and in the multifractal results file (box counting data will be shown for the other orientations in the data file, though.)
Optimizing improves the likelihood that an appropriate sample was taken; see tips for more on this topic.
Filters
Fractal analysis samples an image to detect
scaling,
usually without prior knowledge of the scaling to detect.
Thus, to ensure scaling is neither missed nor exaggerated,
a broad range of relevant sampling sizes should be used (e.g.,
a linear series).
Then, the
series
of box sizes generated for such an investigation can be filtered
to remove artefacts.
There are 2 types of
filters in FracLac: smoothing and
minimum cover. Both can be
applied to multifractal scans
and regular box counting
scans.
Overview of Smoothing Filters
Smoothing filters are one way to improve sampling from
box counting.
Select a smoothing filter to find the smoothed as well
as the standard
Dʙ.
The program
calculates the most-efficient smoothed Dʙ
if the
Minimum Cover
option is also selected.
The
DB smoothed
is calculated from data transformed
by removing horizontal periods, i.e., where there is no change,
in the
regression data
from which the fractal dimension is taken as the slope.
Horizontal slope periods arise spuriously in plots of box size and count because the scaling factor is not known ahead of time. To illustrate, when FracLac uses a linear series of box sizes to maximally capture scaling in an image, after a point, as box size increases relative to image size, the number of boxes required to cover an image stays the same over a long interval of change in size. These plateaus affect the final slope and therefore the Dʙ, but do not necessarily reflect actual features of complexity in a pattern.
FracLac removes data points arising from such plateaus using 2 simple algorithms:
- Smoothed DB(biggest): a filter that shrinks the series by starting at the smallest box size and keeping only counts greater than the successor going from smallest to largest size. This filter obscures scaling and tends to bring the D closer to 1 when relatively large box sizes are used, but the effect depends on the image and the series of box sizes used.
- Smoothed DB(small): a filter that shrinks the series by starting at the largest size and keeping only counts smaller than the successor, which assumes that increases in count with increases in size should be ignored and that the smallest possible box for a given count holds density most efficiently. Thus, the smoothed DB(small), in essence, can be used to correct for box sizes that are too large (see discussion of limits in box counting options)
The smoothing filter is especially useful in Multifractal Analysis
Overview of Minimum Cover Filters
Finding a minimum cover means filtering a dataset from
box counting
over multiple
grid positions
to find the set that used the lowest
number of boxes
to cover the
foreground pixels of an image.
That is, for each
grid size,
the box count that
was most efficient is selected from all of the
grid positions tried.
It is thus assumed to be most efficient inasmuch
as it is the
covering of all coverings tried that needed the least boxes
to cover all of the pixels.
FracLac finds minimum cover fractal dimensions for
multifractal data and for
regular box counting data.
If the number of grid positions is set to 1, there is only one set per size so this value is the same as the Mean Dʙ.
For multifractal scans, select this option to first filter the pixel masses prior to calculating the multifractal spectra. Be aware that this is a very limiting filter that can introduce problems for sampling in multifractal analysis (e.g., it does not do an exhaustive search for the most efficient covering; see optimizing in this regard).
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