A FRACTAL ANALYSIS APPROACH TO
IDENTIFICATION OF TUMOR IN BRAIN MR
IMAGES
Khan M. Iftekharuddin, Wei Jia*, and Ronald Marsh*
ISIP lab, ECE Department, The University of Memphis, Memphis, TN
38152.
*Dept. of Computer Science, North Dakota State University, Fargo, ND
58105.
This research is partly supported by ND EPSCoR through a biomedical SEED grant.
Introduction
• The purpose of this study is to apply
fractal analysis to identify tumor in
brain MR images.
• Three models are developed to
detect tumor in MR brain images
using fractal dimension analysis.
• A multimedia web-based application
is developed for tumor detection
application
Introduction
• Three models are:
– Piecewise-threshold-box-counting (PTBC),
– Piecewise modified box-counting (PMBC),
– Piecewise triangular prism surface area (PTPSA).
Fractal
• Seminal works from Hilbert, Minkowski,
Cantor, Mandelbrot, (Hausdorff, Lyapunov,
Ken Wilson, …)
• VP A. Gore is “fascinated by fractals” –
Time Mag., 8/21/00, p. 41
What is Fractal Geometry?
• Fractal is an irregular geometric object
with an infinite nesting of structure at all
scales.
• Fractal objects can be found everywhere
in the nature, such as trees, ferns, clouds,
snow flakes, mountains, bacteria, and
coastlines.
Application of Fractal Analysis
• Identification of corn roots stressed by
nitrogen fertilizer,
• Determination of steers body
temperature fluctuations in hot and cool
chambers,
• Estimation of surface roughness of
textural images.
Application of Fractal Analysis
• Medical images:
– Detection of micro-calcifications in
mammograms,
– Prediction of osseous changes in ankle
fractures,
– Diagnosis of small peripheral lung tumors,
– Identification of breast tumors in digitized
mammograms.
Properties of Fractal Object
• Three most important properties of
fractals are:
– self-similarity,
– chaos,
– non-integer fractal dimension (FD).
Example Fractal Geometry – Self Similarity
Example Fractal Geometry – Chaos
Fractal Dimension
• The equation for fractal dimension (FD) is as
follows:
ln (number of self-similar pieces, N)
FD = lim
r->00+ ln (magnification factor, 1/r)
Non-integer Fractal Dimension
• The Fractal Dimension for Koch Curve is:
For N = 4, the
magnification
(height/width ) is
reduces by 1/3 ( = r)
• FD = ln 4 / ln 3 = 1.2618…
Non-integer Fractal Dimension
• The Fractal Dimension for Sierpinski triangle is:
• Each triangle is divided into 3 (= N) equal triangles for
each iteration and the height/width are reduced by ½ (
= r).
• FD = ln 3 / ln 2 = 1.5849…
Methods to Estimate Fractal Dimension
• There are a wide variety of computer
algorithms for estimating the fractal
dimension of a structure such as,
– Box-counting algorithm (BC).
– Modified Box-counting algorithm (MBC).
– Triangular Prism Surface Area (TPSA).
BC Method to Estimate Fractal Dimension
– Box-counting algorithm.
1. Box size r = 3, 5, 7, 9, 11, 13 pixels.
2. Number of boxes occupied (N).
3. A linear regression of the ln N versus ln 1/r to
find the slope (FD):
BC Method to Estimate Fractal Dimension
Box size(pixels) No. of Occupied boxes
r = 40, N = 16
r = 30, N = 24
…. ….
r = 20, N = 31
BC Method to Estimate Fractal Dimension
BC Method to Estimate Fractal Dimension
Test Clouds images
Box-Counting Algorithm Analyzing Clouds Images
The estimation the fractal dimensions of clouds of 2.3, 2.5, and
2.8 using box-counting algorithm for whole image.
BC D = 2.3 D = 2.5 D = 2.8
Whole 2.034 2.034 2.034
MBC Method to Estimate Fractal Dimension
– Modified box-counting (MBC) method for
measurement surface fractal dimension.
Image Intensity
max ( ri) - min (ri)
N = floor { }+1
r
MBC Algorithm Analyzing Clouds Images
The estimation the fractal dimensions of clouds of 2.3, 2.5,
and 2.8 using modified box-counting algorithm for whole
image.
M BC D = 2.3 D = 2.5 D = 2.8
W hole 2.17 2.27 2.40
TPSA Method to Estimate Fractal Dimension
– Triangular Prism Surface Area (TPSA).
• The connections of the pixels grayscale values p1,
p2, p3, p4 and pc produces four triangles.
• N = Sum of the top areas.
TPSA Algorithm Analyzing Clouds Images
The estimation the fractal dimensions of clouds of 2.3, 2.5, and 2.8 using
Triangular Prism Surface Area Procedure algorithm for whole image.
TPSA D = 2.3 D = 2.5 D = 2.8
Whole 2.44 2.60 2.81
Developed Algorithms
• Piecewise-threshold-box-counting (PTBC),
• Piecewise modified box-counting (PMBC),
• Piecewise triangular prism surface area (PTPSA).
PTBC Algorithm for Brain MRI Detection
Load .pgm MR image
Divide the image into sub-images
Histogram the sub-
Divide the sub-images into
images intensity different Intensity period
Count the occupied box
number (N) of box size (r)
Cumulative
histogram
Calculate FD using
ln(N)/ln(1/r)
No
No Is it the last Yes Last
threshold sub-images
Yes
Plot sub-image’s FD versus
cumulative histogram
Brain MR Images (source: Harvard med school web)
m35 m40 m45
m35b m40b m45b
m35w m40w m45w
PTBC Algorithm Analyzing MR Images
PMBC and PTPSA Algorithms for Brain MRI Detection
Load .pgm Load .pgm
normal MRI test MRI
Divide the image into Divide the image into
sub-images sub-images
PMBC or PTPSA PMBC or PTPSA
PMBC PTPSA PMBC PTPSA
Box Size Box Size Box Size Box Size
r = 3, 5,.. 13 r = 3, 5,.. 13 r = 3, 5,.. 13 r = 3, 5,.. 13
Count sub-image Count sub-image
N = floor { (max – min)/r } +1 N = floor { (max – min)/r } +1
Count sub-image Count sub-image
N = sum of top area N = sum of top area
Calculate sub-image Calculate sub-image
FD = ln N versus ln 1/r FD = ln N versus ln 1/r
Same
No Last sub-image Yes Yes Last sub-image No
Compare FD?
Not same
Record FD and position Plot tumor position
Illustration of the tumor positions and differences in FD between the
normal and tumor MR images using PMBC algorithm (8 x 8)
m35b m35 and m35b m35 and m35bw m35w
m40b m40 and m40b m40 and m40bw m40w
m45w
m45b m45 and m45b m45 and m45bw
Illustration of the tumor positions and differences in FD between the
normal and tumor MR images using PTPSA algorithm
m35b m35 and m35b m35 and m35bw m35w
m40b m40 and m40b M40 and m40bw m40w
m45w
m45b m45 and m45b m45 and m45bw
Real Brain MR Images (source: ACR CD)
306-a 306-b
401-a 401-b
503-a 503-b
The results of PTBC algorithm test on real MR image
306-a and 306-b, 401-a and 401-b, 503-a and 503-b divided into 2 x 2 pieces
Illustration of the tumor positions and differences in FD between MR
images using PMBC and PTPSA algorithms
306-a 306-b PMBC_306 PTPSA_306
401-a 401-b PMBC_401 PTPSA_401
503-a 503-b PMBC_503 PTPSA_503
Tumor Detection – Multimedia
Interface
http://engronline.ee.memphis.edu/iftekhar/ISIP_det.htm
Tumor Detection – Database
Support
Conclusion
• The original box-counting (BC) method offers correct
results for 1-D signal only.
• However, BC fails to yield correct result for 2-D image.
• The PTBC method can detect the tumor in MR images,
though it is hard to locate the exact position of tumor.
• The PMBC and PTPSA methods can detect and locate
the tumor correctly when applied to the brain tumor MR
images.
• The PMBC algorithm is more sensitive and offers better
result to detect and locate the tumor.
Conclusion
• This program is first developed in C on Unix OS. We then
develop an easy and friendly user interface in java.
• We automate the tumor identification process by building
a reference FD database to compare with test brain
images.
• We improve the algorithms such that we divide each MR
image into two halves – use one half as reference for the
other.
• The research is presented at the World Congress on
Medical Physics and Biomedical Engineering, Chicago,
June, 2000.
Future Work
• Develop a more robust algorithm for tumor
detection to detect hard-to-recognize feature.
• Combination of multiresolution-based wavelet to
fractional Brownian motion analysis?
• Extend to volume rendering/registration.
• Expand web-based approach to support remote
learning, surgery and research.
Reference
• 1. B. B. Mandelbrot, The fractal geometry of nature, Freeman, San Francisco, (1983).
• 2. D. Comis, “Fractals--A bridge to the future for soil science,” Agricultural Research Magazine. 46(4), pp. 10-13, (1998).
• 3. N. Sarkar and B. B. Chaudhuri, “An efficient approach to estimate fractal dimension of textural images,” Pattern Recognition. 23(9),
pp. 1035-1041, (1992).
• 4. S. Davies and P. Hall, “Fractal analysis of surface roughness by using spatial data,” Journal of The Royal Statistical Society Series,
B Statistical Methodology. 61(1), pp. 3-29, (1999).
• 5. D. Osman, D. Newitt, A. Gies, T. Budinger, V. Truong, and S. Majumdar, “Fractal based image analysis of human trabecular bone
using the box counting algorithm: impact of resolution and relationship to standard measures of trabecular bone structure,” Fractals. 6(3),
pp. 275-283, (1998).
• 6. C. B. Caldwell, S. J. Stapleton, D. W. Hodsworth, R. A. Jong, W. J. Weiser, G. Cooke, and M. J. Yaffe, “Characterisation of
mammographic parenchymal pattern by fractal dimension,” Physics in Medicine & Biology. 35(2), pp. 235-247, (1990).
• 7. R. L. Webber, T. E. Underhill, R. A. Horton, R. L. Dixon, T. L. Pope. Jr, “Predicting osseous changes in ankle fractures,” IEEE
Engineering In Medicine And Biology Magazine. 12(1), pp. 103-110, (1993).
• 8. N. Mihara, K. Kuriyama, S. Kido, C. Kuroda, T. Johkoh, H. Naito, and H. Nakamura, “The usefulness of fractal geometry for the
diagnosis of small peripheral lung tumors,” Nippon Igaku Hoshasen Gakkai Zasshi. 58(4), pp. 148-151, (1998).
• 9. S. Pohlman, K. A. Powell, N. A. Obuchowski, W. A. Chilcote, and S. Grundfest-Broniatowski, “Quantitative classification of breast
tumor in digitized mammograms,” Medical Physics. 23(8), pp. 1337-1345, (1996).
• 10. V. Swarnakar, R. S. Acharya, C. Sibata and K. Shin, “Fractal based characterization of structural changes in biomedical images.”
SPIE. 2709, pp. 444-455, (1996).
• 11. A. Bru, J. M. Pastor, I. Fernaud, I. Bru, S. Melle, C. Berenguer, “Super-rough dynamics on tumor growth,” Physical Review Letters.
81(18), pp. 4008-4011, (1998).
• 12. K. J. Falconer, Chapter 2: Hausdorff measure and dimension, In “Fractal Geometry Mathematical Foundations And Applications.”
Thomson Press Ltd, (1990).
• 13. K. C. Clarke, “Computation of the fractal dimension of topographic surfaces using the triangular prism surface area method.”
Computers and Geosciences 12( 5), pp. 713-722, (1986).
• 14. B. B. Mandelbrot and J. W. Van Ness, “Fractional Brownian motions, fractional noises and applications.” SIAM Review. 10(40), pp.
422-437, (1968).