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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):

W (a, b)  W [ f ]( a, b)   a ,b f   a ,b ( x) f ( x)dx
 x b

 a 
 a ,b ( x)  | a |1/ 2 

1
f ( x) 
C
1
a
2
 (x)
W (a, b) a ,b ( x)dadb
  
W [ f ]
f (x)
L2(R)
 
W [ f ](a, b)
 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
f ( x) 
C
 
1
a
W (a, b) a ,b ( x)dadb
2
  

C 


 ( )

2
d
0C 
DWT
a  a0
m
a  2m
b  nb0 a0
a0  21/ v
m
a0  2
b0  1
Binary dilation
b  n 2  m Dyadic translation
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
2
 ( x)  e  x cos 
 x 
 ln 2 
Wavelet Transform
Morlet Wavelet - Non-visible Oscillation
Scalogram
f1 (x)
f1 (x)  1000e
0.01(x10)2
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

Ψ a,b ( r ) 
1
2π a x a y
Wavelet transformed
 T    1 r b T P r b 

2  r  b P r  b  e 2



 

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