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A FRACTAL ANALYSIS APPROACH TO IDENTIFICATION OF

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					   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
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    pp. 1035-1041, (1992).
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