Multi fractal analysis of human brain mr image

IJRET: International Journal of Research in Engineering and Technology eISSN: 2319-1163 | pISSN: 2321-7308
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Volume: 03 Issue: 03 | Mar-2014, Available @ http://www.ijret.org 470
MULTI FRACTAL ANALYSIS OF HUMAN BRAIN MR IMAGE
Tarun Patil M S1
, Madhukar Bhat2
1
UG Student, Department of ECE, R.V. College of Engineering, Bangalore, Karnataka, India
2
UG Student, Department of ECE, R.V. College of Engineering, Bangalore, Karnataka, India
Abstract
Brain MR Image can detect many abnormalities like tumor, cysts, bleeding, infection etc. Analysis of brain MRI using image
processing techniques has been an active research in the field of medical imaging. In this work, it is shown that MR image of brain
represent a multi fractal system which is described a continuous spectrum of exponents rather than a single exponent (fractal
dimension). Multi fractal analysis has been performed on number of images from OASIS database are analyzed. The properties of
multi fractal spectrum of a system have been exploited to prove the results. Multi fractal spectra are determined using the modified
box-counting method of fractal dimension estimation.
Keywords: Brain MR Image, Multi fractal, Box-counting
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1. INTRODUCTION
Magnetic resonance imaging and computed tomography are
the two common methods used to get image of brain for
diagnosis [1]. An active research has been going on in the
field of medical imaging for automated detection of brain
tumor using image processing techniques. With the advances
in the field of technology, brain tumor imaging has become an
essential component in diagnosis, treatment planning, and
monitoring treatment response. Brain tumors are classified
depending on the location of the tumor and the type of tissue
involved.
Considerable research and application of fractal geometry has
been done in image analysis and pattern recognition [2]. The
brain MRI typically has a degree of randomness associated
with it. This property of the image of brain is used for fractal
analysis on it. Based on the fractal dimension obtained the
image is analyzed for abnormalities. In this paper, brain is
proved to be a multi fractal system rather than a mono fractal
system.
1.1 Background Review
Few investigations in the past have tied to identify the fractal
dimensions [3]-[6]. The limitations of the previous works were
patient independent. Those limitations were due to the
assumption of brain MR images as a mono fractal system.
Thus we introduce multi fractal analysis which is a spectrum
analysis, so the error of computation is very less. The multi
fractal modelling and analysis done in our work show that the
brain MRI represents a multi fractal system This result
overcomes the limitations of mono fractal dimension concept
in previous works.
2. METHODOLOGY
The key feature of the proposed method is the implementation
of the algorithm in a DSP module. Fig-1 shows the schematic
view of the proposed method. Code composer studio is used as
an interface between PC and the TMS320C6416 DSP board.
Brian MR images have been obtained from OASIS database.
Histogram matching is applied to MR images to enhance the
dynamic range and increase contrast. The advantage of the
histogram matching is that it can have variable dynamic range
of our choice. Then multi fractal modelling using modified
box counting method is applied. The analysis of the fractal
spectrum has been conducted extensively to prove that brain is
a multi fractal system rather than a mono fractal system which
is represented by a single exponent.
Fig -1: Schematic overview of the proposed method

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3. FRACTAL MODELLING
The fractal geometry concept was first proposed by
Mandelbrot. A fractal is a set of iterative self similar patterns.
Fractal dimension is a real number that describes the fractal
property of the system. Unlike Euclidian geometry, fractal
dimension can take a real value rather than just an integer.
Fractal dimension value depends on the properties of the
fractal system. The different methods used to calculate the
fractal dimension of a system are box-counting, modified box-
counting, piecewise modified box-counting and piecewise
triangular prism surface area (PTPSA). The famous fractal von
Koch curve [7] shown in Fig-2 has a fractal dimension of
1.2618.
Fig -2: von Koch curve
3.1 Fractal Dimension
The notion of dimension is central to fractal geometry.
Roughly, dimension indicates how much space a set occupies
near to each of its points [8]. Fundamental to most definitions
of dimension is the idea of ‘measurement at scale ‘δ’. For
each ‘δ’, we measure a set in a way that ignores irregularities
of size less than δ, and we see how these measurements
behave as δ → 0. If H is a plane curve, then our
measurement, Mδ(H ), might be the number of steps required
by a pair of dividers set at length δ to traverse H. A
dimension of H is then determined by the power law obeyed
by Mδ(H ) as δ → 0. The dimension ‘s’ is given by,
s = lim (logMδ(H ))/( −log δ)
δ→0
3.2 Box Counting Dimension
Box counting [9] is the most commonly used method to find
the dimension of fractal system as it is relatively easy for
mathematical calculations and empirical estimations. Let H be
any non-empty bounded subset of Rn
and let Nδ(H ) be the
smallest number of sets of diameter at most δ which can
cover H. The box-counting dimension of H respectively are
defined as,
dimBH = lim (logNδ(H ))/( −log δ)
δ→0
The algorithm implemented in our work to determine the
fractal dimension is,
• Cover given shape with boxes and find the pattern of
change of number of boxes with the change in size of
the boxes.
• If the object is 1-dimensional, we expect N(boxes) =
1/δ.
• If the object is 2-dimensional, we expect N(boxes) =
(1/δ)2
.
• For complicated shapes, the relation between
N(boxes) and 1/δ may be a power law, N(boxes) =
k(1/δ)s
.
• Taking Log of both sides, we obtain
Log(N(boxes)) = Log(k) + Log((1/δ)s
) = s⋅Log(1/δ) +
Log(k)
Dimension or s = lim (logN(boxes)/( log 1/δ)
δ→0
3.3 Modified Box Counting Dimension
This is the method used in our work to compute fractal
dimension. There are few disadvantages of traditional box
counting method. The box-counting dimension of each
rational number regarded as a one-point set is clearly zero, but
the countable union of these singleton sets has dimension 1.
This severely limits the usefulness of box dimension. Thus,
simple modifications to suit our work has been done. For H a
subset of Rn
we can try to decompose H into a countable
number of pieces H1, H2,... in such a way that the largest
piece has as small dimension as possible.
4. MULTI FRACTAL MODELLING
Multi fractal patterns scale with multiple scaling rather than
one scaling rule. Single parameter like fractal dimension is not
enough to describe the dynamics of multi fractal system. A
continuous spectrum of exponents is required to describe the
system which is called the multi fractal spectrum. Multi fractal
analysis was first introduced for turbulence data analysis and
modelling. Now, its application in medical imaging is huge.
In multi fractal system H, its dynamics around any point is
given by,
H(y+δ) - H(y) = bh(y)
The exponent h(y) is called the singularity exponent. There are
many methods to obtain the multi fractal spectrum like DQ vs
Q method, dimensional ordering, f(α) vs α. Multi fractal

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spectrum is calculated using modified box counting in our
work. Firstly, modified box counting scan is done to
determine the pixels distribution. Then, this pixel distribution
becomes the basis for a series of calculations. In the f(α) vs α
method, the multi fractal spectrum show a parabola for the
multi fractal system and random shape for non – multi fractal
system. This property has been exploited to show the results.
5. RESULTS
In our work we have enhanced the MR Image using histogram
matching method. The results are shown from Fig-3 to Fig-4.
Fig -3: Original Image Fig -4: Enhanced Image
Fig -5: Original Image Fig -6: Enhanced Image
Table -1: Generalized dimension Dx for x=-10,10 and
standard deviation βx for x=-10,10 of 10 images from OASIS
database (status: N=normal; P=pathological)
Image
name
status
D-10 β-10 D10 β10
Img001.jpg N 1.942 0.015 1.431 0.003
Img002.jpg N 1.967 0.017 1.478 0.003
Img003.jpg N 1.939 0.021 1.440 0.002
Img007.jpg N 1.922 0.012 1.418 0.003
Img011.jpg P 1.897 0.018 1.396 0.001
Img025.jpg P 1.834 0.024 1.346 0.002
Img030.jpg N 1.926 0.022 1.432 0.002
Img040.jpg P 1.847 0.025 1.371 0.002
Img057.jpg P 1.804 0.019 1.339 0.003
Img067.jpg N 1.968 0.022 1.450 0.003
Enhanced Images are then modelled using modified multi
fractal analysis. As shown in Fig-7, alpha (q) is the range of
variation in the scale of fractal measure and f (q) is the fractal
dimension that we have obtained as a raw scale calculation of
alpha (q). Fig-7 is the multi fractal model for the enhanced
image shown in Fig-4.
Fig -7: Variation of alpha (q) and f (q) for image in Fig-4
Fig -8: Multi fractal spectrum of enhanced image of Fig-4

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The above is the multi fractal spectrum of the enhanced image
shown in Fig-4. It is obtained from the raw scale calculation of
Fig-7. The variation of f (q) with respect to the range of
variation in scale of fractal dimension is plotted in Fig-8. The
multi fractal spectrum shows a parabola for a multi fractal
system. A best fit parabola to the multi fractal spectrum is
plotted in Fig-9. Thus from the multi fractal spectrum shown
in Fig-8-9, it can be concluded that the brain MR image shown
in Fig-4 is a Multi fractal system.
Fig -9: Best parabola fit for the multi fractal spectrum of Fig-8
6. CONCLUSIONS
In conclusion, we show in our work that the structure of
human brain captured using MRI represent a multi fractal
structure characterized by a continuous spectrum of exponents
rather than a single fractal dimension. We have analyzed 40
images from the OASIS database, in which half correspond to
normal brain and half correspond to abnormal brain. In all the
analyzed cases, we observe a clear multi fractal behavior.
ACKNOWLEDGEMENTS
The authors would like to thank the OASIS project group for
making the database publicly available.
REFERENCES
[1]. Medicine net: Magnetic Resonance Imaging (MRI scan)
website: http://www.medicinenet.com/ mri_scan/article.htm
[2]. Buades, Antoni, Bartomeu Coll, and Jean-Michel Morel.
"A review of image denoising algorithms, with a new one"
Multiscale Modeling & Simulation 4.2 (2005): 490-530.
[3]. Tang, Min, and Hui Nan Wang. "Feature analysis of brain
MRI images based on fractal dimension." Engineering in
Medicine and Biology Society, 2005 IEEE-EMBS 2005 27th
Annual International Conference of the IEEE, 2006
[4]. Iftekharuddin, Khan M., et al. "Fractal-based brain tumor
detection in multimodal MRI." Applied Mathematics and
Computation 207.1 (2009): 23-41.
[5]. Iftekharuddin, Khan M. "Texture Models for Brain Tumor
Segmentation."Quantitative Medical Imaging. Optical Society
of America, 2013
[6]. Roy, Sudipta, et al. "A Review on Automated Brain
Tumor Detection and Segmentation from MRI of Brain."
arXiv preprint arXiv: 1312.6150 (2013).
[7]. Dorfman, J. R., and Thomas Gilbert. "Statistical properties
of γ-adic processes and their connections to families of
popular fractal curves (Perspectives of Nonequilibrium
Statistical Physics-The Memory of Professor Shuichi Tasaki-
).97.3 (2011): 357-369.
[8]. Ivanovici, Mihai, and Noël Richard. "Fractal dimension of
color fractal images."Image Processing, IEEE Transactions on
20.1 (2011): 227-235.
[9]. Ostwald, Michael J. "The fractal analysis of architecture:
calibrating the box-counting method using scaling coefficient
and grid disposition variables."Environment and Planning B:
Planning and Design 40.4 (2013): 644-663.
BIOGRAPHIES
Tarun Patil M S is currently in his 8th
Semester at R. V. College of Engineering,
Bangalore. His research interests include
medical imaging, embedded systems,
Computer vision and FPGA
Madhukar Bhat is currently in his 8th Semester
at R. V. College of Engineering, Bangalore.
His research interests include medical imaging,
Computer vision multimedia and photonic
signal processing and communication systems.

Multi fractal analysis of human brain mr image

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Multi fractal analysis of human brain mr image