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					Master Thesis 2005
Wavelets (A relative new Mathematical method)
in Medical Image Processing
Wavelets
http://fag.grm.hia.no/fagstoff/perhh/htm/fag/matem/datwwww/wavelet.htm
         Interesting applications
         The subject of Wavelets is expanding at a tremendous rate



• Wavelet transform has been perhaps the most exciting development
  in the last decade to bring together researchers in several different fields:


  Seismic Geology
  Signal processing (frequency study, compression, …)
  Image processing     (image compression, video compression, ...)
  Denoising data
  Communications
  Computer science
  Mathematics
  Electrical Engineering
  Quantum Physics
  Magnetic resonance
  Musical tones
  Diagnostic of cancer
  Economics
  …
              Medical co-operation
              The Norwegian Radiumhospital in Oslo


-   Control of the Linear Accelerator
-   Databases (patient/employee/activity)
-   Computations of patientpositions
-   Mathematical computations
    of medical image information
    -   Different imageformat (bmp, dicom, …)
    -   Noise Removal
    -   Graylevel manipulation (Histogram, …)
    -   Convolution, Gradientcomputation
    -   Multilayer images
    -   Transformations (Fourier, Wavelet, …)
    -   Mammography
    -   ...




              Wavelet
     Co-operation




Mathematics - Computer Science - Medicine


      SINTEF Unimed Ultrasound in Trondheim

      The Norwegian Radiumhospital in Oslo

      Sørlandet hospital in Kristiansand / Arendal
International Co-operation




                    After an invitation to a mathematical conference
                    in Balticum 2003, I have had lessons in different countries
                    and the international co-operation now includes
                    Norway, Sweden, Scotland, Germany, Estonia and Russia
                                                                         f ,   L2 ( R) a, b  R a  0

             Definition of The Continuous Wavelet Transform
             CWT



The continuous-time wavelet transform (CWT)
of f(x) with respect to a wavelet (x):
                                                  
                                                                                                x b
W (a, b)  W [ f ]( a, b)   a ,b f   a ,b ( x) f ( x)dx          a ,b ( x)  | a |1/ 2     
                                                                                                a 
                                                 


                   
          1                1
f ( x) 
         C       a
                    
                           2
                               W (a, b) a ,b ( x)dadb




  f (x)            W [ f ]
                                W [ f ](a, b)
 L2(R)     (x)
                                                          1,0 ( x)        2, 0 ( x)                2,1 ( x)
                CWT - DWT


CWT
                                            
                                                                                           x b
W (a, b)  W [ f ]( a, b)   a ,b f   a ,b ( x) f ( x)dx  a ,b ( x)  | a |1/ 2  
                                                                                              
                                                                                        a 
                
          1             1
               a                                                      ( )
                                                                               2
f ( x)                   W (a, b) a ,b ( x)dadb
         C      
                        2                                  C    
                                                                  
                                                                         
                                                                                    d   0C 




DWT
         m
a  a0                                          a  2m        Binary dilation
              m
b  nb0 a0              a0  2   b0  1         b  n 2  m Dyadic translation
a0  21/ v    v number of voices per octave
v pieces of  m, n are processed as one group, called a voice

              m,n ( x)  2m / 2 2m x  n                   Dyadic Wavelets
Fourier transformation
                                                                           2    
                                                       ( x)  e  x cos     x
                                                                    2

                                                                         ln 2 
                                                                                
Wavelet Transform
Morlet Wavelet - Non-visible Oscillation



                                                                                     Scalogram


                                     0.01(x10)2
        f1 (x)
                  f1 (x)  1000e


                                                                                             1
                                                                                                Wψ [f1 ]
                                                                                             a2


                                                                                 Scalogram



        f 2 (x)              f1 (x)               if x   ,9  11, 
                  f 2 (x)  
                             f1 ( x)  5sin(2 x) if x  9,11

                                                                                             1
                                                                                                Wψ [f 2 ]
                                                                                             a2
       Master Thesis 2005
       Mathematical Image Processing in Medicine



The Norwegian Radiumhospital in Oslo (DNR)

   - Wavelets: Detecting Breath Cancer at an early stage




                   Mammography                     Wavelet transformed




                                                 T     1 r b  P r b 
                                                              
                                                                    T         
                                         2  r  b P r  b e 2
             Ψ a,b ( r ) 
               
                                1
                             2π a x a y   
                                                           
                                                            
      Definition DNR


The Norwegian Radiumhospital in Oslo (DNR)
       Master Thesis 2005
       Mathematical Image Processing in Medicine



SINTEF Unimed Ultralyd in Trondheim

   - Wavelets: Detecting Blood Vessel in Ultra Sound Image
       Master Thesis 2005
       Mathematical Image Processing in Medicine



Sørlandet hospital in Kristiansand
   - Wavelets:
     Compute Bone Thickness




                                                   1
                                                      Wψ [f]
                                                   a2




   -Wavelets:
     Compute Blood Vessel Thickness in Liver
     Schistosomiasis
       Master Thesis 2005
       Mathematical Image Processing in Medicine



Sørlandet hospital in Arendal

   - Wavelets: IR Diagnostics
      Master Thesis 2005
      Mathematical Image Processing in Medicine



HiA – German Company
  - Wavelets: Video Compression
      Master Thesis 2005
      Mathematical Image Processing in Medicine



HiA


 -Wavelet Program system in Mathematica, Mathlab, Mathcad, Java
   Different Wavelets
   Modulus and Phase Representation in Complex Wavelets
         Scalogram
                                    Different advanced analyse tools.
                                    Prepare different suitable program system.
                                    Special study of phase representation.




                     1
                        Wψ [f 2 ]
                     a2
       Master Thesis 2005
       Mathematical Image Processing in Medicine



Sørlandet Hospital

 - Continuous Wavelet Tranform
   ECG Recognition Based on Phase and Modulus Representation
   and Hidden Markov Models
                                   Wavelet transforms have been applied (CWT/DWT) to ECG for
                                   - ehancing late potentials
                                   - reducing noise
                                   - QRS detection
                                   - normal and abnormal beat recognition

                                   CWT of complex wavelets
                                   Phase representation study of local symmetry / asymmetry
       Master Thesis 2005
       Mathematical Image Processing in Medicine



Sørlandet Hospital

 - Statistical Analysis of Image Differences by Wavelet Decomposition
       Master Thesis 2005
       Mathematical Image Processing in Medicine



Sørlandet Hospital


 - Application of Wavelet Transforms to Blood Flow Velocimetry
   General: Velocity in Fluid / Gas




                     Ultrasound high fractional bandwidth and large time-bandwidth signals
                     Doppler Effect  Wideband / Wavelet Transform
       Master Thesis 2005
       Mathematical Image Processing in Medicine



Sørlandet Hospital

 - Wavelets Transform for speaker identification
       Master Thesis 2005
       Mathematical Image Processing in Medicine



Sørlandet Hospital

 - Wavelets to Suppress Noise in Biomedical Images
       Master Thesis 2005
       Mathematical Image Processing in Medicine



Sørlandet Hospital


 - Wavelets Technicques for Encoding Magnetic Resonance Images
      Master Thesis 2005
      Mathematical Image Processing in Medicine



Sence Technology


 - Wavelets Technicques for Detecting Pipe Connection in Video Image
Mathcad Program
Wavelet Transform
End

				
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posted:10/3/2012
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