FractalNet: A biologically inspired neural networkapproach to fractal geometry
Ronald Marsh *
Department of Computer Science, University of North Dakota, Streibel Hall, Room 218 (box 9015), Grand Forks, ND 58201, USA
Received 9 July 2002; received in revised form 2 January 2003
Abstract
We present a multiply connected neural network to estimate the fractal dimension (Df ) using the Box-counting
method. Traditional methods used to estimate Df are sequential. A neural network implementation would be com-
putationally efficient and suggests a possible biological realization.
� 2003 Elsevier Science B.V. All rights reserved.
Keywords: Fractal geometry; Fractal analysis; Box-counting method
1. Introduction
Benoit Mandelbrot (1983) introduced a new
field of mathematics to the world. Mandelbrot
named this new field fractal (from the Latin term‘‘fractus’’) geometry. Mandelbrot claimed that
fractal geometry would provide a useful tool for
the explanation of a variety of naturally occurring
phenomena. A fractal is an irregular geometric
object with an infinite nesting of structure at all
scales (self-similarity). Many examples of fractal
objects can be found in nature, such as coastlines,
fern trees, snowflakes, clouds, mountains, andbacteria. Some of the most important properties
of fractals are self-similarity, chaos, and the non-
integer fractal dimension (Df ), which gives a
quantitative measure of self-similarity and scaling.
Fractal analysis has been used to characterize a
variety of naturally occurring complex objects as
well as many artificial objects (e.g. Comis, 1998)and a number of applications using fractal geom-
etry have been proposed. Fractal analysis has been
successful in measuring textural images (e.g. Sar-
kar and Chaudhuri, 1992) and surface roughness
(e.g. Davies and Hall, 1999). Fractal analysis has
proven useful in medicine as medical images typi-
cally have a degree of randomness associated with
the natural random nature of structure. Osmanet al. (1998) analyzed the fractal dimension of
trabecular bone to evaluate its potential structure.
Other studies successfully used the fractal dimen-
sion to detect micro-calcifications in mammo-
grams (e.g. Caldwell et al., 1990), predict osseous
changes in ankle fractures (e.g. Webber et al.,
1993), diagnose small peripheral lung tumors
*Tel.: +1-701-777-4013; fax: +1-701-777-3330.
E-mail address: [email protected] (R. Marsh).
0167-8655/03/$ - see front matter � 2003 Elsevier Science B.V. All rights reserved.
doi:10.1016/S0167-8655(03)00011-4
Pattern Recognition Letters 24 (2003) 1881–1887
www.elsevier.com/locate/patrec
(e.g. Mihara et al., 1998), distinguish breast tu-
mors in digitized mammograms (e.g. Pohlman
et al., 1996), and detect brain tumors in MR im-
ages (e.g. Jia, 1999). Studies have also shown that
the changes in the fractal dimension reflect alter-
ations of structural properties.Fractal geometry does not concern itself with
the explicit shape of objects. Instead, fractal geo-
metry identifies the value that quantifies the shape
of the object�s surface, the fractal dimension (Df )
of the object. For example, a line is commonly
thought of as 1-dimensional object, a plane as a 2-
dimensional object, and a cube as a 3-dimensional
object––objects whose dimensionality is integervalued. However, the surfaces of many objects
cannot be described with an integer value. These
objects are said to have a ‘‘fractional’’ dimension.
Consider the following example to illustrate the
concept of fractional dimension:
Consider a line (a 1-D object).Bend it.Bend it some more.Bend it infinitely many times.Eventually, the line is transformed into a plane (a2-D object).
Note that the 1-D line did not instantly become
a 2-D plane; there was a transition during which it
was 1.xx-D. Hence, fractal geometry can be con-
sidered an extension to classical geometry (e.g.
Voss, 1985).
The fractal theory developed by Mandelbrotand Van Ness (1968) was derived from the work
of mathematicians Hausdorff and Besikovich.
The Hausdorff–Besikovich dimension, DH, is de-
fined as:
DH ¼ lime!0þ
lnNe
ln 1=eð1Þ
where Ne is the number of elements of e diameter
required to cover the object. Mandelbrot defines a
fractal as a set for which the Hausdorff–Besikovich
dimension strictly exceeds the topological dimen-sion. When working with discrete data, one is
interested in a deterministic fractal and the asso-
ciated fractal dimension (Df ) which can be defined
as the ratio of the number of self-similar pieces (N )
with magnification factor (1=r) into which a figure
may be broken (e.g. Le M�eehaut�ee, 1990). Df is de-
fined as:
Df ¼lnNln 1=r
ð2Þ
Df may be a non-integer value, in contrast to
objects lying strictly in Euclidean space, which
have an integer value. However, Df can only be
directly calculated for a deterministic fractal. Ac-
cording to Falconer (1990) for any other object
(most physical objects and images) Df has to beestimated. There are a variety of applicable algo-
rithms for estimating Df , such as the Box-counting
algorithm, the pixel Dilation method (e.g. Smith
et al., 1989), the Calipher method (e.g. Rigaut
et al., 1983), and the Power Spectrum method
(e.g. Schepers et al., 1992).
1.1. A biological basis?
We were led to consider a neural network ap-
proach for calculating the fractal dimension by the
possible use of the Mellin transform by certain
biological visual processing systems. There is evi-
dence that the visual processing system of cats and
some species of primates uses a log-polar trans-
formation (e.g. Reitboeck and Altmann, 1984). Alog-polar transform followed by a Fourier trans-
form is a known method for scale and rotation
invariant pattern recognition (a Mellin transform).
Therefore, an obvious question to ask is: ‘‘Does
the biological visual processing system also employ
a Fourier transform?’’ That question has not yet
been answered. However, artificial neural net-
works have been developed which can perform aFourier transform (e.g. Culhane et al., 1989). The
evidence of the biological realization of the log-
polar transform and the development of the neural
network based Fourier transform, does open the
possibility that a Mellin transform is used by the
visual processing system for pattern recognition.
We now ask: ‘‘Does the biological visual pro-
cessing system employ more than one patternrecognition method and is that method fractal
geometry?’’ A neural network implementation
capable of estimating the fractal dimension does
open that possibility.
1882 R. Marsh / Pattern Recognition Letters 24 (2003) 1881–1887
This paper presents a multiply connected neural
network designed to estimate the fractal dimension
(Df ) using the Box-counting method (BCM) as
proposed by Block et al. (1990). Calculation of Df
is becoming a common data processing tool, and
in general, the faster one can do it, the better.Traditional techniques are sequential. Not only do
they not operate on the entire data vector simul-
taneously, they must revisit the data vector several
times, performing the sequential processing each
time. Hence, as the vector length increases so does
the processing time. Traditional techniques have a
computational cost of OðN 2 lnðNÞÞ. A parallel (or
neural network) method would be more efficientcomputationally and would open the possibility
of a biological realization.
In this paper, we present a review of BCM in
Section 2, our neural network implementation in
Section 3, our simulation results in Section 4, and
our concluding remarks in Section 5.
2. Background
The neural network architecture presented
performs a parallel computation of Df using the
BCM. BCM is one of the most popular and sim-
plest methods for the fractal analysis of image
data. With BCM, we divide an image containing
the object of interest into N equal-size pieces. Wethen count the number of pieces that contain a
portion of the object (ni). The process is repeated
for a user-defined sequence of Ni equal-size pieces.
We provide an example of the BCM process
using a dyatic increase of N in Fig. 1.
All four of the pieces in Fig. 1a contain parts of
the letter A, five pieces contain parts of the letter A
in Fig. 1b, 12 pieces contain parts in Fig. 1c, and35 pieces in Fig. 1d contain parts of the letter A.
We tabulate and plot ni versus the respective
piece size on a ln-ln plot. Linear regression is used
to find the best fitting line through the resulting
curve. Df is the slope of this line. For the above
example, Df is 1.139.
However, BCM has its criticisms, especiallywhen applied to digital images (e.g. Buczkowski
et al., 1998). In particular: A digital image has a
finite set of points, therefore there is an upper
limit (image size) and lower limit (a pixel) to e, andthe number of pieces (Ne) must be integer valued.
Furthermore, there is no standard method for
implementing BCM. In our example, we used a
dyatic progression of N ð4; 16; 64; . . .Þ. Other im-plementations use a linear progression ð3; 5; 7; . . .Þ.Hence, BCM (and most other digitally imple-
mented methods) are only estimates of Df (dB).
3. Fractal neural network (FractalNet)
FractalNet consists of two sections, a datasampling section and a linear regression section. A
FractalNet designed for 8 1-D inputs is shown in
Fig. 2. We discuss both sections in more detail
below.
3.1. Data sampling section
The data sampling section is designed to samplethe data in a dyatic manner. Three types of nodes
are required: thresholding input nodes (black
squares), which binarize the input data, summing
nodes (gray circles), and Heaviside step function
nodes (white circles). The alternation and layering
of summing nodes and Heaviside step function
nodes combine to provide the parallel dyatic
sampling. Fig. 3 provides a more detailed look atthe resulting dyatic sampling. We assume that the
Fig. 1. (a–d) BCM algorithm example.
R. Marsh / Pattern Recognition Letters 24 (2003) 1881–1887 1883
four inputs are from a gray scale source and that
the input nodes are set to threshold the data such
that any value >n ¼ 1 and any value 6 n ¼ 0
(where n ¼ 0 here). We assume that an input value
of 1 indicates a sample location that contains part
of the object of interest. We also assume that all
connection weights are þ1.
The middle layer contains alternating summingand Heaviside step function nodes. The output of
each summing node is the number of inputs in the
node�s input neighborhood that contain a portion
of the object. The (binary) output of each Heavi-
side step function node states whether or not the
node�s input neighborhood contains a portion ofthe object at all. The output layer contains two
summing nodes and one Heaviside step function
node. Output summing node A has inputs from
the two middle layer summing nodes and there-
fore ‘‘sees’’ each of the four original inputs as
four separate datum. Since three of the four
original inputs have a value of one, the output
value for A is 3. Output summing node B hasinputs from the two middle layer Heaviside step
function nodes and therefore ‘‘sees’’ the four
original inputs as if they were actually two inputs
(the input pairs in the dotted boxes). Since both
of the input node pairs contain portions of
the object, the output value for B is 2. Out-
put Heaviside step function node C has in-
puts from the two middle layer Heaviside stepfunction nodes and therefore ‘‘sees’’ the four
original inputs as if they were only one input
(note the double-lined box). Since the input
quad contains a portion of the object, the out-
put value for C is 1. As the data set grows,
the dyatic sampling pattern continues (see Sec-
tion 3.3).
Fig. 2. FractalNet for 8 1-D inputs.
Fig. 3. Dyatic sampling of FractalNet.
1884 R. Marsh / Pattern Recognition Letters 24 (2003) 1881–1887
3.2. Linear regression section
In keeping with the all-neural network archi-
tecture, we developed a neural network to com-
pute the regression line (linear regression) of the
data generated by the sampling section (note that
the box size term ‘‘xi’’ is the independent variable).The purpose of the nodes is given by the equa-
tions/parameters next to each set of nodes (Fig. 2).
Note that some of the connections are inhibitory(have a weight of )1) and others are weighted (1/
number of inputs to the linear regression section––
1/3 in Fig. 2). All other connections have a weight
of þ1. Finally, note that this section, unlike the
data sampling section, has some temporal depen-
dencies. Specific nodes (x-bar and y-bar calculatingnodes) must complete their processing before other
nodes can perform their processing.
3.3. Scaling
Our second item of interest is how well Fractal-
Net scales. As long one wishes to continue with the
dyatic sampling rate, FractalNet can easily be
scaled. Each doubling of the data set size requiresone additional hidden layer in the data sampling
section and four additional nodes in the linear re-
gression section. For example, compare the Frac-
talNet for 8 inputs (Fig. 2) with the FractalNet
for 16 inputs (Fig. 4). Note that the connec-
tion weights (for the fractional weighted connec-
tions) must also be adjusted accordingly (1/4 in
Fig. 4).
3.4. Two-dimensional data
The third item of interest is the ability for
FractalNet to process 2-D data such as binary
image data. Again, only minor changes are re-
quired. Fig. 5 shows the data sampling section of a
FractalNet that has 16 inputs (assuming imagedata––each black square would correspond to an
individual pixel). The black hexagonal nodes are
the inputs to the linear regression section. Note
that the sample rate for the subnet shown in Fig. 5
is 16, 4, and 1. The numbers next to each node
Fig. 4. FracalNet for 16 inputs.
R. Marsh / Pattern Recognition Letters 24 (2003) 1881–1887 1885
indicate the node�s value given the 16 input valuesshown.
4. Simulation and results
We implemented the BCM as proposed by
Block et al. (1990) and a realistic rendition (each of
the 1583 nodes and applicable connections in thenetwork were duplicated in code) of FractalNet to
process a 512� 512 image. Both algorithms were
implemented in C++ on a 333 Mhz. PC running
Linux. Obviously, our FractalNet simulation was
a serial implementation and not a parallel imple-
mentation.
Using four images of fractals, we compared the
results obtained with both of our implementationswith those obtained with the BCM option in the
HarFa software (Buchn�ıı�ccek et al., 2000; Ne�zz�aadalet al., 2001) and those published for the Modified
Box Counting Method (Buczkowski et al., 1998).
We selected image sizes and calculation parame-
ters to be as closely matched as possible. The re-
sults are shown in Table 1. We were not surprised
by the identical results obtained with the BCM and
FractalNet as FractalNet was designed to imple-
ment the same algorithm as the traditional BCM,
only able to do so as a neural net. We are also not
surprised that our methods (BCM and FractalNet)returned low estimates for Df as this is a known
characteristic of the BCM.
The computational-complexity of the BCM is
OðN 2 lnðNÞÞ. The time-complexity of BCM and
FractalNet, when implemented serially, are both
approximately T ðN 2 þPdlnðNÞe
i¼1 ½N=2i2Þ. Since our
implementations of the BCM and FractalNet were
serial, both have the same time-complexity andexhibited similar run times––approximately 37 ms
for a 512� 512 image on our machine. However,
when implemented parallelly the time-complexity
of FractalNet is significantly lower, approximately
T ðlnðNÞÞ.
5. Conclusion
This paper presents a multiply connected neural
network designed to estimate the fractal dimension
(Df ) using the BCM. Fractal analysis is a powerful
shape recognition tool and the BCM is one of the
most popular methods for estimating the fractal
dimension (Df ). Traditional BCM implementa-
tions are sequential. A parallel (or neural network)implementation would be more temporally effi-
cient and, more interestingly, would suggest a
possible biological realization. Our goal was to
develop a rendition of the BCM that was neural
network-based and not to improve upon the basic
BCM algorithm. While the neural network pre-
sented does generate an estimation (dB) of Df using
the BCM, it is not entirely biologically founded.
Table 1
Fractal dimension (Df ) comparison
Square Error (%) Sierpinski
carpet
Error (%) Sierpinski
gasket
Error (%) Vicsek
growth
Error (%)
Known 2.0000 1.8928 1.5849 1.4650
BCM 1.8510 8.05 1.8625 1.63 1.5350 3.25 1.3464 8.81
FractalNet 1.8510 8.05 1.8625 1.63 1.5350 3.25 1.3464 8.81
HarFa 1.8784 6.47 1.8960 )0.17 1.5491 2.31 1.3590 7.80
MBCM 1.9997 0.02 1.8814 0.61 1.5790 0.37 1.5089 )2.91
Fig. 5. FracalNet for 2-D (4� 4) data.
1886 R. Marsh / Pattern Recognition Letters 24 (2003) 1881–1887
Several of the activation functions are not bio-
logically plausible. Additionally, our network re-
quires that the data either be binary or easily
segmented (into binary data). These concerns
are the focus of future work on FractalNet.
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