Tomography

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					Tomography
            Tomography

 Introduction
 The Radon Transform
 Projection Slice Theorem
 Inversion of Radon Transform
     Theory
     Aplication
 Cone Beam Transform
 Various problems in tomography
What is Tomography?
 Wikipedia:
  Tomography is imaging by sections or sectioning,
  through the use of wave of energy.
  A device used in tomography is called a tomograph,
  while the image produced is a tomogram.
  The method is used in medicine, archaeology,
  biology, geophysics, oceanography, materials
  science, astrophysics and other sciences.
 A. Cormack and G. Hounseld built the first
  computed tomography scanners in 1960s,
  won 1979 Nobel prize in medicine.
How is it done?
‫תמונות‬
 CT – Computerized Tomography
 CAT – Comp. Axial Tomography
 SPECT – Single Particle Emission Tomo.
 PET – Positrion Emission Tomography
 MRI – Magnetic Resonance Tomography
 Optical Tomography
 Thermal Tomography
 Acoustic Tomography
Radon Transform
f: n ,                  Sn - 1 ,  
R                                   
                                     
       f     ,   f     ,   f  , 
              
:      f  x  dx     x     f  x  dx   f    y  dy
     x                  Rn                      




                    S n-1
                                 x   
                                                
                                            ρ
Radon Transform in Freq. Domain
Projection Slice Thm.
F  f  ,     f  ,     f  ,  
            
     1
     2                   
            e -i f  ,  d      
 1
2    e -i                f  x  dx d         1
                                                     2         e -i f  x  dx d 
                  
                x                                          x 


                           f  x  dx  2                         
                                                           f    
                                                     n-1


          e
                -i  x
     1
     2
                                                      2

           n




      
 f  ,     2                                
                                              f    
                                        n-1
                                         2
Radon Transform in Freq. Domain
Projection Slice Thm.
  
f  ,     2                 
                              f    
                        n-1
                         2




Space Domain                                   Freq. Domain

                                    F


                                           F

   Radon Domain
   The Backprojection Operator
 R was defined on Cn. It can be extended to L2 (as a
  bounded operator). Thus there exist an adjoint:

     R         L S  
         : L2       n
                                2
                                        n 1




     R : L S    L  
      *
                2
                         n 1
                                               2
                                                   n




              g   ,    x       g  , x   d
         *
     R                    
                                       S n1
The Backprojection Operator

         g   ,    x       g  , x   d
    *
R                    
                                  S n1




              S n-1                            

                                          x



                                                    g  , x  
 So, does it work?
                       
f  ,     f  ,   2                                           
                                                             n-1
                                                              2
                                                                       f 


                                                                                        n 
                                                                                                                  
                                                                                                         n-1
                          2   2 f         2  2  2  2 e i  f  d
                                           n-1
                     1
f  ,  F
                          
                                         
                                                             
                                                             




                                               *                  n          n-1                              
R
    *
         f   ,    x   R  2                      
                                                                   2    2   e
                                                                               2
                                                                                        i  
                                                                                                   f    d    x  
                     
                                                                                                             
                        n             
                                                        
              n-1

 1   2   2  2  2
                       
                             e f  d d 
                               ix

Sn                                   
                                                                                                                              1            
                                                                        
           n-1           n                                                                                     n-1
                                                      1
2  2           2    e                                                          d d  2  2                         n 1 f      x  
                                          ix                                                                          1
            2            2                                     f             n 1
                                                                                                                2
                                                                                                                     F
                             S   n1   0
                                                     n 1                                                                    
                                                                                                                                            
                                                                                                                                             
           n-1    1  1                             1          
2  2     2    F  n 1   x   f  x    2 2   f  x  
                                                n2

                                                    x         
                                                               
So, does it work?
                                        1          
    R
        *
             f   ,   x   2 2   f  x  
                                       x         
                                                   




                      Blurring!
Inversion Formula
                      *
f x              nR
                                              n 1       n1
                                                                       f   ,     ,    x 
              2      H
                                                     
                                                     
                                                                n 1
                                                                                    
                                                                                               
                                                                                                


                                     f t 
H    f     s    
            
                       1
                                      s t
                                              dt  f  t   1t                                H    f       i sgn   f  
                                                                                                            


    n 1 
                            
                          f  ,       i sgn     n1 f  ,                     
              n1                                        n 1  n 1

H          
        
                   n 1
                                  
                                                              

  i sgn                                                                                 sgn                  
                                                                                                                       f  ,    2                    
                          n 1                                                                                  n 1                        n 1
                                  2 i            f  ,      2 
                                              n 1                                       n 1
                                                                                                                                                   f  ,



H
    n 1 
           
        
              n1
                   n 1      
                          f  ,    , s   2  b  s   f  , s
                                  
                                  
                                                  n 1
                                                                                            
                                            n 2  n
                                   s     sn
                                            i                         n even
bs F              
              1          n 1
                                         
                                           i n 1 2  ( n 1)  s  n odd
                                           


Sharpening!  ill-posed problem
Inversion for 2D and 3D
                                
                  1            
                                    
                                    f ,   
  f 2D  x                            d  d
                 4 2     S1
                                 x   
                        2
                      1
  f3 D  x    2  2 f  , x  d
                8 S 2 
                                               
 For 3D inversion, only integrals of plains through x0
  are needed to reconstruct f(x0)
 For 2D inversion, the entire data of the Radon
  Transform is needed.
Reconstruction algorithms
 Filtered Backprojection
 Fourier methods
Filtered Backprojection
                    *
f x            nR
                                     n 1     n1
                                                          f   ,     ,    x 
            2      H
                                         
                                         
                                                    n1
                                                                       
                                                                                  
                                                                                   



H
    n 1 
            
         
              n1
                    n1      
                          f  ,    , s   2  b  s   f  , s
                                  
                                  
                                                  n 1
                                                                               
                                n 2  n
              n 1   s     s n
                                i                         n even
b s F
         1
                             
                               i n 1 2  ( n 1)  s  n odd
                               
Filtered Backprojection
         g   ,    x   f  x   R  g   ,    f   ,     x 
    *                                                 *
R                                                                        

                                                n-1             1            
R
    *
         g   ,    x   2  2          2
                                                      F
                                                          1
                                                                n 1 g      x 
                     
                                                               
                                                                             
                                                                              

R  g   ,    x     x  
    *
                
             n-1                           n
                    1
 2  2  2              g     2  2 
                                         
                     n 1
                      
                              1                                               1
                     2                                             2 
                                n                                              n
                              2                                               2
  g                                         g                                          low _ pass _ filter  
                                         n 1                                            n 1
                                                                                    
                          2                                               2
 Fourier Methods

       
     f  ,     2                 
                                   f    
                             n-1
                              2




                     
f  ,   f  ,                      2                 
                                                          f        f  x 
                                                    n-1

             fft                                    2                  ifft
                                                                             
                                                 interpolation
                                                f    u, v          f  x 
                                                                        ifft
                                                                             
But this is not how it is realy done!
Cone-Beam Transform
                           
    C  f  x   a,    f  a  t  dt
                                              S2
                           0


                                a
A
Our work was not in vain…

                          
    
               
       f  , a   
                          
                                           
                             C  f  a, d
                     S2



                             

                a                                 a
                                      
                                                      
                                              2
                                         S
Kirilov-Tuy Condition
                      

              
   f  , a   
                      
                                   
                         C  f  a, d
                 S2
                      2
           1
                             
f3 D  x    2  2 f  , x  d
              8 S 2 

 If each plane through supp(f) intersects the
  source curve transversally, then f is uniquely
  (and stably) determined by the Cone-Beam
  Transform.
Various Problems in Tomography
 Limited Data
    Angular
    Radial
    Local Tomography
        Inner

        Outer

 Noise
    Motion
    reflection
 Other technologies
    MRI
    PET
    Thermal \ Acoustic Tomo.
Inner Tomography
 Reducing radiation dosage
     And other costs (money, time) etc.
Outer Tomogrpahy
 Metal blocks X-Rays

				
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posted:10/25/2011
language:English
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