MULTIFRACTALS
The Details of Multifractal Analysis
Introduction
What are Multifractals?
One type of analysis FracLac is used for is multifractal analysis. Multifractals are a type of fractal, but they stand in contrast to monofractals, which have one definitive scaling rule, in that multifractals have multiple scaling rules.
Multifractal Scaling
The scaling rules (fractal dimensions) vary over the fractal, and they show up when we analyze an image using different degrees of distortion. So, instead of getting one value for an image as happens in standard monofractal analysis, we get a characteristic distribution of values gauged against the size of the distorting lens. There are several ways to quantitate the distributions.
Canonical Henon Map
This is an image of a canonical multifractal, an iterated Henon Map made using the Fractal Growth Models plugin for ImageJ.
As evidenced by the images of Henon multifractals shown on this page and the Multifractal Analysis page, it is not necessarily easy to determine on inspection whether a pattern is or is not multifractal. So, the question arises, "How do we know if a pattern is multifractal?".
One answer is that to explore multifractal scaling in digital images, you can do a multifractal analysis in FracLac.
What is Multifractal Analysis?
The point of a multifractal analysis is to detect the multi in multifractal scaling. How does multifractal analysis tie in with box counting? It all boils down to the distribution of pixel values in a digital image. That is, to do a multifractal analysis, FracLac uses the box counting grid technique to gather information about the distribution of pixel values (called the mass or probability distribution). This information becomes the basis for a series of calculations that reveal and explore the multiple scaling rules of multifractals.
Multifractal Spectra
From the mass
distribution, FracLac does calculations
and returns data files and multifractal spectra graphs.
Put simply, multifractal spectra show how
a pattern behaves if amplified in certain ways. (Click
here
to learn how to generate these graphs during
a multifractal analysis.)
To get an intuitive
grasp of what happens during a multifractal analysis,
you might compare the making of multifractal spectra to the
experience of zooming in on and distorting parts of an object as
you examine it, as in the illustration,
which shows a part of a multifractal
Henon Map magnified and distorted in two different ways.
In a multifractal analysis, the important difference is that it is all done in an orderly fashion, so you can interpret the results with specific rules.
In particular, the rules help us interpret the graphical spectra showing how certain variables behave. FracLac provides a lot of data to help you do your multifractal analysis. There are several data files to help you compare and quantitate various features of the multifractal spectra across images and within an image (e.g., aperture, slope, dimensional ordering, etc.).
Although understanding multifractal analysis requires that you understand the calculations for the variables, it also requires that you understand the graphical elements themselves. So, whereas the end of this page has a section devoted entirely to calculating the variables that the graphs represent, for now, you need only greet the variables you see in the images informally, starting with one known as the generalized dimension or D(Q), which is, in essence, one of the distorting lenses of multifractal analysis.
The Generalized Dimension
D(Q)
The generalized dimension, D(Q), is a distortion of the mean (μ) of the probability distribution for a pattern, and in FracLac, it is a distortion of μ for the distribution of pixel values at some ε from box counting.
What is meant by "a distortion"? To find D(Q), μ is exaggerated by being raised to the arbitrary exponent, Q, then compared again to how the exaggeration varies with ε. Thus, D(Q) basically addresses how mass varies with ε (resolution or box size) in an image, telling how it behaves when you scale or resolve or cut up the image into a series of ε-sized pieces and distort them by Q. (See calculations and Setting Options for Q in a multifractal scan).
Dimensional Ordering
The function, D(Q) vs Q is decreasing, sigmoidal around Q=0, where
The graphical spectrum D(Q) makes against Q is a marvelous feature of multifractal analysis that, as illustrated in the figure below, can help distinguish types of patterns. The figure shows D(Q) spectra from a multifractal analysis of non-, mono-, and multi-fractal images. FracLac generates graphs of this data using a simple plotting function and puts the values for D(Q) and all of the multifractal data types in data files that let you compare between images and inspect features of individual images.
The non-fractal was a binary contour (a circle) with box counting dimension around 1.0; the fractal, a quadric monofractal Cross with box counting dimension around 1.49, and the multifractal a Henon Map, shown above, with a box counting dimension around 1.29.
One thing in particular to note is that, as the comparison in the figure illustrates, non- and mono-fractals tend to have flatter D(Q) spectra than multifractals. Now, perhaps you have inspected the figure and noticed the similarity between the box counting dimensions and the values at Q=0, and all this has made you wonder how generalized this dimension is. In fact, one value of the generalized dimension,; D(Q=0), is equal to what we call the "Capacity Dimension", which you can understand as the box counting dimension here. D(Q=1) is equal to what we call the "Information Dimension", and D(Q=2) to the "Correlation Dimension".
These three you can make use of when interpreting multifractal analyses, inasmuch as we saw above that there is some natural ordering. We are not going to discuss the generalized dimension further here, except to put this all into perspective and make the point that it helps relate the "multi" in multifractal to the "mono" in monofractal—multi fractals have multiple dimensions in the D(Q) vs Q spectra but monofractals are rather flat in that area. Now that really is, marvelous, isn't it?
Other Variables
Some other very important variables are τ, α, and ƒ(α). A typical graph of τ vs Q is illustrated here for the multifractal Henon Map.
If you have been clicking on the links to the calculations area and studying assiduously, you may have come to know that all of these variables are themselves interrelated. The image comparing the graphs of D(Q) and α vs Q, for example, illustrates that these two variables hold very similar information.
It can be enlightening, nonetheless, to consider them separately, because they do hold unique information. Curves for ƒ(α) vs α, for instance, are illustrated in the image below. As shown in the figure, graphs of data from non- and mono-fractal patterns typically converge on certain values, whereas spectra from multifractal patterns are typically more broadly humped.
Differences in Multifractal and Monofractal Patterns
Non- and mono-fractal patterns converge, but multifractal patterns typically rise and fall relatively broadly in a hump.
Sampling
As was discussed earlier, box counting data depend on grid position. Because these multifractal spectra are based on pixel distributions determined from box counting, they depend on how the distribution is extracted from an image. Negative Q exponents, for instance, start to be very troublesome, indeed, especially with density distributions that attribute too much importance to very small probabilities that appear in the distribution at some, but not all, grid positions. This is illustrated in the example showing differences in the ƒ(α) vs α spectra at two grid orientations used for sampling the same image with the same set of grid calibres.
What can be done about this sampling issue? One approach to dealing with the problem of pixel distribution being dependent on grid position is to randomly sample a pattern to infer a distribution, which you can do with FracLac's random mass sampling function; but this is subject to several limitations in acquiring an adequate sample.
Another approach is to sample multiple locations fully then select an optimal sample. FracLac's default behaviour for a multifractal analysis is to scan with the grid anchored at each of the four corners of the rectangle enclosing an image, to provide four different spectra similar to what are obtained by rotating the image 90° and reapplying the same series of εs. Then FracLac can select an optimized sample for multifractal analysis based on several criteria (e.g., see the figure below). Another approach is to filter the data using minimum cover or smoothing options. Optimizing and filtering are explained in more detail in the Multifractal Analysis page.
Appendices
Calculations
Variables Used for Multifractal Spectra in FracLac
Important Note:
Please see the Results page for details of the actual methods used to calculate multifractal results.
D(Q) = limε→0 [ln I[Q,ε]/ ln ε-1]/(1-Q)
I[Q,ε] = for i=1 to N ∑[P(i, ε)] Q
For Q=1, let ε→1, and
D(Q)
=-lim[
for i=1 to N
∑P(i)ln[P(i)]
/ ln ε]
Finding the Probability Distribution
The probability distribution is found from the number of pixels, M, that were contained in each ith element of a size, ε, required to cover an object:
P(i,ε) = M(i,ε) / [for i=1 to N ∑ M(i,ε)]
Thus, P(i) is from the probability distribution of mass for all boxes, i, at this ε
where for i=1 to N ∑P(i,ε) Q=1 = 1
and (for i=1 to N) ∑ P(i,ε) Q=0 = N(ε), the number of boxes containing pixels
P(i,ε) = pixels(i,ε) / [for i=1 to N ∑pixels(i,ε)]
According to the method of Chhabra and Jensen (Phys. Rev . Lett. 62: 1327, 1989):
μ(I[Q,ε]) = P(i)Q/ [for i=1 to N ∑P(i)Q]
α[Q,ε] = [for i=1 to N ∑(μ(I[Q,ε]) × ln P(i))] / ln ε
ƒ(Q) = [ for i=1 to N ∑ (μ(I[Q,ε]) × ln (μ(I[Q,ε]))] / ln ε
τ(Q) = (Q-1) × D(Q)
and ƒ(α [Q,ε]) = Q × α (Q)-τ(Q)
References for Calculations
The method of calculating the multifractal spectrum:
A. Chhabra and R.V. Jensen, Direct Determination of the ƒ(α) singularity spectrum, Phys. Rev. Lett. 62: 1327, 1989. http://prl.aps.org/abstract/PRL/v62/i12/p1327_1
A. N. D. Posadas, D. Giménez, M. Bittelli, C. M. P. Vaz, and M. Flury, Multifractal Characterization of Soil Particle-Size Distributions, Soil Sci. Soc. Am. J.. 65:1361-1367, 2001. dx.doi.org/doi:10.2136/sssaj2001.6551361x
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