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					     Computer Tomography:
Computational theory and methods

           University of Bergen
        15 October 2004, Bergen.

            Borislav V. Minchev
           Borko.Minchev@ii.uib.no
        http://www.ii.uib.no/ borko

                                

        Department of computer science
         University of Bergen, Norway

                                     Computer Tomography: Computational theory and methods – p.1/28
Outline
     Introduction
 ¡




     Fields of applications
 ¡




     Principles of X-ray CT
 ¡




     Reconstruction techniques in 2D
 ¡




        - Transform based methods (FBP)
        - Algebraic reconstruction techniques

     Con-beam reconstructions
 ¡




         - Circular
         - Helical

     Conclusions
 ¡




                                                Computer Tomography: Computational theory and methods – p.2/28
Definition


    Tomography: technique for imaging cross-sections
    or slices from a set of external measurements of a
    spatially varying function.




                                    Computer Tomography: Computational theory and methods – p.3/28
Fields of applications
      X-ray CT
  ¡




          - medical imaging
          - industrial nondestructive evaluation
          - airport screening
          - microtomography




                                                   Computer Tomography: Computational theory and methods – p.4/28
Fields of applications
      X-ray CT
  ¡




          - medical imaging
          - industrial nondestructive evaluation
          - airport screening
          - microtomography
      Magnetic Resonance Imaging (MRI)
  ¡




         - electromagnetic waves




                                                   Computer Tomography: Computational theory and methods – p.4/28
Fields of applications
      X-ray CT
  ¡




          - medical imaging
          - industrial nondestructive evaluation
          - airport screening
          - microtomography
      Magnetic Resonance Imaging (MRI)
  ¡




         - electromagnetic waves
      Transmission electron microscopy
  ¡




          - an electron beem




                                                   Computer Tomography: Computational theory and methods – p.4/28
Fields of applications
      X-ray CT
  ¡




          - medical imaging
          - industrial nondestructive evaluation
          - airport screening
          - microtomography
      Magnetic Resonance Imaging (MRI)
  ¡




         - electromagnetic waves
      Transmission electron microscopy
  ¡




          - an electron beem
      CT principles are also applied in:
  ¡




          - oceanography
          - astronomy
          - geophysics
          - optics



                                                   Computer Tomography: Computational theory and methods – p.4/28
Some hisrory
     Radon 1917
 ¡




        - reconstruction of a function from its projections




                                                          Computer Tomography: Computational theory and methods – p.5/28
Some hisrory
     Radon 1917
 ¡




        - reconstruction of a function from its projections
     Hounsfield 1972
 ¡




        - patented the first CT scanner




                                                          Computer Tomography: Computational theory and methods – p.5/28
Some hisrory
     Radon 1917
 ¡




        - reconstruction of a function from its projections
     Hounsfield 1972
 ¡




        - patented the first CT scanner
     Cormack 1963 - 1964
 ¡




         - contributionto mathematics of X-ray CT




                                                          Computer Tomography: Computational theory and methods – p.5/28
Some hisrory
     Radon 1917
 ¡




        - reconstruction of a function from its projections
     Hounsfield 1972
 ¡




        - patented the first CT scanner
     Cormack 1963 - 1964
 ¡




         - contributionto mathematics of X-ray CT
     Hounsfield and Cormack shared the 1979 Nobel Prize for medicine
 ¡




                                                          Computer Tomography: Computational theory and methods – p.5/28
Some hisrory
     Radon 1917
 ¡




        - reconstruction of a function from its projections
     Hounsfield 1972
 ¡




        - patented the first CT scanner
     Cormack 1963 - 1964
 ¡




         - contributionto mathematics of X-ray CT
     Hounsfield and Cormack shared the 1979 Nobel Prize for medicine
 ¡




     since then the field is steadily evolving and givis rise to never ceasing flow of new
 ¡




     algorithms




                                                          Computer Tomography: Computational theory and methods – p.5/28
Principles of X-ray CT




                         Computer Tomography: Computational theory and methods – p.6/28
Principles of X-ray CT



     X-ray traverses an
 ¡




     object along stight line
     attenuated signals from
 ¡




     various directions are
     recorded
     reconstruction algorithms
 ¡




     are used to convert the
     projections




                                 Computer Tomography: Computational theory and methods – p.6/28
Projections
Let
                represents the density of a two-dimensional object
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                the attenuated intensity of a ray as it propagates throught the object
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                along the line

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                                                                Computer Tomography: Computational theory and methods – p.7/28
Projections
Let
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                  the attenuated intensity of a ray as it propagates throught the object
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              -   projection angle
        




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                                                                                          Computer Tomography: Computational theory and methods – p.7/28
Projections
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                                       Computer Tomography: Computational theory and methods – p.7/28
Projections
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The line integral can be written as
         Radon Transform
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                                                                                    Computer Tomography: Computational theory and methods – p.7/28
Fourier Slice Theorem
Consider the two-dimensional FT of the object function




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                                                                                   Computer Tomography: Computational theory and methods – p.8/28
Fourier Slice Theorem
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                                                                                                       Computer Tomography: Computational theory and methods – p.8/28
Fourier Slice Theorem
In general for any anglr



                       


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                                                                                  Computer Tomography: Computational theory and methods – p.8/28
Fourier reconstruction algorithm
     Take projection of an object function at angles
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                                                                                 Computer Tomography: Computational theory and methods – p.9/28
Fourier reconstruction algorithm
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                                                                                    Computer Tomography: Computational theory and methods – p.9/28
Fourier reconstruction algorithm
       Take projection of an object function at angles
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                                                                                                       Computer Tomography: Computational theory and methods – p.9/28
   Reconstruction for Parallel Projection
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                                                                                                             Computer Tomography: Computational theory and methods – p.10/28
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                                                                                                                                                                                                                                                                   Reconstruction for Parallel Projection




Computer Tomography: Computational theory and methods – p.10/28
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Computer Tomography: Computational theory and methods – p.10/28
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Computer Tomography: Computational theory and methods – p.10/28
The FBP algorithm
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The FBP algorithm
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                                                                                    Computer Tomography: Computational theory and methods – p.11/28
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                                                                                          Computer Tomography: Computational theory and methods – p.11/28
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                                                                                                       Computer Tomography: Computational theory and methods – p.11/28
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Reconstruction for Fan Projections
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                                                                                                                                                                                                                                                                                                                                  Reconstruction for Fan Projections




Computer Tomography: Computational theory and methods – p.12/28
Weighted Backprojection Algorithm
                                          Equiangular Rays
            is the sampling interval of the projection
      Š




Let
            are the angles at which projections are taken
       




Step 1 For each                           , generate the corresponding modified projection
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                                                                                  Computer Tomography: Computational theory and methods – p.13/28
Weighted Backprojection Algorithm
                                                Equiangular Rays
            is the sampling interval of the projection
      Š




Let
            are the angles at which projections are taken
       




Step 1 For each                                    , generate the corresponding modified projection
                           £

                                   ¨
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Step 2 Convolve           with                                                                           to generate the

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corresponding filtered projection (FFT)

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                                                                                                     Computer Tomography: Computational theory and methods – p.13/28
Weighted Backprojection Algorithm
                                                          Equiangular Rays
                    is the sampling interval of the projection
          Š




Let
                    are the angles at which projections are taken
           




Step 1 For each                                              , generate the corresponding modified projection
                                     £

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Step 2 Convolve           with                                                                                     to generate the

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corresponding filtered projection (FFT)

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where                   is a smoothing filter.
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                                                                                                               Computer Tomography: Computational theory and methods – p.13/28
Weighted Backprojection Algorithm
                                                                Equiangular Rays
                    is the sampling interval of the projection
          Š




Let
                    are the angles at which projections are taken
           




Step 1 For each                                                    , generate the corresponding modified projection
                                     £

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Step 2 Convolve           with                                                                                            to generate the

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corresponding filtered projection (FFT)

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Step 3 Perform a weighted backprojection
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where           is the angle of a ray that passes through the point                                                                   and                         .
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                                                                                                                      Computer Tomography: Computational theory and methods – p.13/28
Reconstruction for Fan Projections
                                              Equally Spaced Detectors

Let                be a fan projection
           £
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           #
      




    is the angle between the source
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     and a reference axis.

    is the distance along the detector
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      |SO| source to origin dist.
 ‚
     




   - the dist. source to projection
•




     of a pixel on the central ray
In polar coordinates
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                                                                   Computer Tomography: Computational theory and methods – p.14/28
                                                                                                                                         where
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                                                                                                                                                                                                                   "                                                              "
                                                                                                                                                                                                                                                                                     
                                                                                                                                                                                                                                                                                                                                 Reconstruction for Fan Projections




Computer Tomography: Computational theory and methods – p.14/28
Algebraic reconstruction techniques
                             Advantages

  - easy to handle different rays geometry
  - more adaptable to missing data - better image quality
  - metal artifacts are reduced
  - less radiation dose




                                             Computer Tomography: Computational theory and methods – p.15/28
Algebraic reconstruction techniques
                             Advantages

  - easy to handle different rays geometry
  - more adaptable to missing data - better image quality
  - metal artifacts are reduced
  - less radiation dose



                            Disadvantage

                    Higher computational cost!




                                             Computer Tomography: Computational theory and methods – p.15/28
Image and Projection Representation
The idea: Assume that the cross section consists of array of unknowns




                                             Computer Tomography: Computational theory and methods – p.16/28
Image and Projection Representation
The idea: Assume that the cross section consists of array of unknowns

     - total number of cells
 —




               is the value of


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                                       §
š˜




                                   ˜
                    
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™




        in the -th cell
                       ¨




     - the -th line integral ray-sum
               ¬
«©
ª




                                             Computer Tomography: Computational theory and methods – p.16/28
Image and Projection Representation
 The idea: Assume that the cross section consists of array of unknowns

         - total number of cells
     —




                   is the value of


                                                      ¤¥£¡


                                                              §
 š˜




                                                          ˜
                         
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 ™




            in the -th cell
                            ¨




         - the -th line integral ray-sum
                   ¬
 «©
 ª
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                                                              µ
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                                                     ´
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 where
   is the total number of rays
     µ




          represents the contribution
 ª±
     ™




                            -th                      -th
          of the                  cell to the              ray integral
                        ¨




                                                 ¬




                                                                          Computer Tomography: Computational theory and methods – p.16/28
Projection methods
Therefore we have to solve the following linear system of equations




                                e ¢




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                                                                               ¢
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                                                                  ˆ
                                                                     ˆ
for large values of       and       .
                      ¸




                                „




                                                                                       Computer Tomography: Computational theory and methods – p.17/28
Projection methods
Therefore we have to solve the following linear system of equations




                                          e ¢




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                                                                                                 ¢
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for large values of              and          .
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                                          „




Use iterative methods - projestions (Kaczmarz’37, Tanabe’71)
               y




                                                   ‚
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Let                                                     is the initial guess
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where
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                                                                                                         Computer Tomography: Computational theory and methods – p.17/28
Projection methods
Theorem   (Tanabe’71)




                                      ÇG
                                       S
If there exists a unique solution          to the system of equations, then




                                      ¢
                                    Æ




                                                          G

                                                                  S


                                                                          ÇG
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                                                              f
                                                                  ·



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                                                      1




                                                                                  Computer Tomography: Computational theory and methods – p.18/28
Projection methods
Theorem   (Tanabe’71)




                                      ÇG
                                       S
If there exists a unique solution          to the system of equations, then




                                      ¢
                                    Æ




                                                          G

                                                                  S


                                                                          ÇG
                                                                              S
                                                              f
                                                                  ·



                                                                          Æ
                                                          ¢




                                                                          ¢
                                               s9 %
                                               È




                                                                      
                                               Éf

                                                      1
The rate of convergence depends of the angles between the hyperplanes

      - Full orthogonalization is computationally not feasible

      - Pairwise orthogonalization scheme (Ramakrishnan’79)

      - Carefully choose the order of the hyperplanes (Hounsfield’72)




                                                                                  Computer Tomography: Computational theory and methods – p.18/28
Projection methods
Theorem    (Tanabe’71)




                                      ÇG
                                       S
If there exists a unique solution          to the system of equations, then




                                      ¢
                                    Æ




                                                              G

                                                                      S


                                                                              ÇG
                                                                                  S
                                                                  f
                                                                      ·



                                                                              Æ
                                                              ¢




                                                                              ¢
                                                   s9 %
                                                   È




                                                                          
                                                   Éf

                                                          1
The rate of convergence depends of the angles between the hyperplanes

      - Full orthogonalization is computationally not feasible

      - Pairwise orthogonalization scheme (Ramakrishnan’79)

      - Carefully choose the order of the hyperplanes (Hounsfield’72)

  When                   the process oscillate in the neighborhood of the intersections of the
                     „
            ¸Ë
Ê




hyperplanes

    When                 the process converges to a solution     , such that       ÆÇG
                                                                                      S
                                                                                    ¢†
            ¸


                     „
                 q
Ê




                                                    is minimized.
                                               G
                                               bS



                                                            ÆÇG
                                                              S
                                                             ¢†
                                           
                                               ¢


                                                        $




                                                                                         Computer Tomography: Computational theory and methods – p.18/28
                                                                                                         where
                                                                                                                                                                                                  where


                                                                                                                                      Therefore
                                                                                                                                                                                                                                            D ¢ G
                                                                                                                                                                                                                                                                                                                                                                  Consider again
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                                                                                                                                                                                                                                            D ¢ G                                                                                       ¢
                                                                                                                                                                                                                                                                                                                                             G
                                                                                                                                                                                                                                                                            It can also be written as                                          S
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                                                                  Ô WÒÓÑ                                                      2 eS                                                                                                                                                                        yV                                         yV
                                                                                                                                                                                                                                                                                                                                                                                   Projection methods




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                                                                                                                                                                                                                                       ¦
Computer Tomography: Computational theory and methods – p.19/28
Projection methods
Consider again




                                                                  ¹




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                                                                          y
It can also be written as


                                                  ¶


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where




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                                                                                                            U 2 S
                                                                                                                e
                                                                                          v
                                                   G

                                                           ˆ 2 S
                                                                e




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                                                                                                                     cf
                                                                                                                    
                                                                                         f
                                                                                               e

                                                                                                   f
                                                                                           a
Therefore
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The main computational diffuculty is in the calculation, storage and retrieval of

the weight coefficients                   .
                              U
                                  
                                    D




                                                                                                               Computer Tomography: Computational theory and methods – p.19/28
Algebraic Reconstruction Tech.(ART)
 Replace     by 0 or 1, depending wether the center of the              -th   cell is within the           -th   ray.




                                                                     Î




                                                                                                        Ã
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               U
                   
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Thus we can use




                                                 ¶


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for all cells whos centers are within the -th ray.




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                       E
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  „




                                                                                  Ã
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           f



                           f
                       
               a




                                                                 Computer Tomography: Computational theory and methods – p.20/28
Algebraic Reconstruction Tech.(ART)
 Replace     by 0 or 1, depending wether the center of the                                 -th   cell is within the           -th   ray.




                                                                                        Î




                                                                                                                           Ã
Ê




                 U
                     
                     D
Thus we can use




                                                                ¶


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                                                                     $
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for all cells whos centers are within the -th ray.




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    Superior approxiamtion is
Ê




                                                             ¶



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                                                                                ¦
                                                              



                                                                            „
where            is the lenght of the   -th   ray trought the reconstruction region.
            




                                    Ã




                                                                                    Computer Tomography: Computational theory and methods – p.20/28
Algebraic Reconstruction Tech.(ART)
 Replace     by 0 or 1, depending wether the center of the                                 -th   cell is within the           -th   ray.




                                                                                        Î




                                                                                                                            Ã
Ê




                 U
                     
                     D
Thus we can use




                                                                ¶


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for all cells whos centers are within the -th ray.




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           Í
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                 a




    Superior approxiamtion is
Ê




                                                             ¶



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                                                    S
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                                                                            „
where            is the lenght of the   -th   ray trought the reconstruction region.
            




                                    Ã




- ART suffer from salt and pepper noise, caused by the approximations used for                                            .




                                                                                                                 U
                                                                                                                     
                                                                                                                        D
                                                                                    Computer Tomography: Computational theory and methods – p.20/28
Algebraic Reconstruction Tech.(ART)
 Replace     by 0 or 1, depending wether the center of the                                      -th   cell is within the           -th   ray.




                                                                                             Î




                                                                                                                                 Ã
Ê




                 U
                     
                     D
Thus we can use




                                                                     ¶


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                                                        S




                                                                          $
                                                     ”
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for all cells whos centers are within the -th ray.




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             v




                   is the number of cells whose centers are within the                                             ray.
                         E
           Í
  „




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    Superior approxiamtion is
Ê




                                                                 ¶



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                                                      S
                                                    ”
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                                                                                     ¦
                                                                  



                                                                                 „
where            is the lenght of the   -th   ray trought the reconstruction region.
            




                                    Ã




- ART suffer from salt and pepper noise, caused by the approximations used for                                                 .




                                                                                                                      U
                                                                                                                          
                                                                                                                             D
- It is possible to reduce the noce by
                                                        D ¢ G
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       relaxation (i.e. update a pixel by                                        )
                                                      ”
                                                 Ö




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                                                                              u
                                                                          q
                                                                 ¦




       SIRT - change the cell values after going through all of the equations

                                                                                         Computer Tomography: Computational theory and methods – p.20/28
SART (Simultaneous ART)
First reported by Anderson and Kak ’84

    Reduce the error in the approximation of the ray integrals
Ê




                                                            Computer Tomography: Computational theory and methods – p.21/28
SART (Simultaneous ART)
First reported by Anderson and Kak ’84

  Reduce the error in the approximation of the ray integrals
Ê




Insted of solving




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We use again the Radon Transform


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where                          is the equation of the                -th   ray and              is the projection operator.
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                                                                                                    Computer Tomography: Computational theory and methods – p.21/28
SART (Simultaneous ART)
First reported by Anderson and Kak ’84

  Reduce the error in the approximation of the ray integrals
Ê




Insted of solving




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We use again the Radon Transform


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                                                                                                                         Computer Tomography: Computational theory and methods – p.21/28
SART (Simultaneous ART)
First reported by Anderson and Kak ’84

  Reduce the error in the approximation of the ray integrals
Ê




Insted of solving




                                         v



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                                                                                    c ¸
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                                              D

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                                                           inside the       -th   pixel
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                            D

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                                                  X
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- Superior reconstructions are obtained by using bilinear elements
                                  pyramid shaped basis

                                                                                  Computer Tomography: Computational theory and methods – p.21/28
SART (Simultaneous ART)
First reported by Anderson and Kak ’84

    Reduce the error in the approximation of the ray integrals
Ê




     bilinear elements insted of the traditional pixel basis




                                                            Computer Tomography: Computational theory and methods – p.21/28
SART (Simultaneous ART)
First reported by Anderson and Kak ’84

    Reduce the error in the approximation of the ray integrals
Ê




     bilinear elements insted of the traditional pixel basis

    Correction terms are simultaneously applied to all the rays
Ê




     as for the SIRT methods




                                                            Computer Tomography: Computational theory and methods – p.21/28
SART (Simultaneous ART)
First reported by Anderson and Kak ’84

    Reduce the error in the approximation of the ray integrals
Ê




     bilinear elements insted of the traditional pixel basis

    Correction terms are simultaneously applied to all the rays
Ê




     as for the SIRT methods

    Hamming window
Ê




     emphasizes the corrections applied near the middle of a ray relative to those
     applied near the ends.




                                                            Computer Tomography: Computational theory and methods – p.21/28
SART (Simultaneous ART)
First reported by Anderson and Kak ’84

    Reduce the error in the approximation of the ray integrals
Ê




     bilinear elements insted of the traditional pixel basis

    Correction terms are simultaneously applied to all the rays
Ê




     as for the SIRT methods

    Hamming window
Ê




     emphasizes the corrections applied near the middle of a ray relative to those
     applied near the ends.

The overall SART iterative scheme is
                                                              ·




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where        is the   -th   row of         and                are relaxation coefficients.
                                       h



                                                 Ö
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                                                     fË
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                                                                                            Computer Tomography: Computational theory and methods – p.21/28
More iterative algorithms
Censor and Elfving’02
The idea: Use generalized projections and diagonal weighting matrices to reflect the
sparsity of the matrix.
                       h

             A family           of diagonal        matrices with diagonal elements

                                        ·
Definition:                   à×
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                                                                                                                  ‡
                                                                                                                      
                                    

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is called Sparsity Pattern Oriented (SPO) with respect to          matix if:




                                                                                               h
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                                                                                „
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              Í

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                                                                               Computer Tomography: Computational theory and methods – p.22/28
More iterative algorithms
Censor and Elfving’02
The idea: Use generalized projections and diagonal weighting matrices to reflect the
sparsity of the matrix.
                       h

             A family           of diagonal        matrices with diagonal elements

                                        ·
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                                                                                                                                           ‡
                                                                                                                                               
                                    

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is called Sparsity Pattern Oriented (SPO) with respect to          matix if:




                                                                                                                       h
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                                                                                                      Computer Tomography: Computational theory and methods – p.22/28
More iterative algorithms
Censor and Elfving’02
The idea: Use generalized projections and diagonal weighting matrices to reflect the
sparsity of the matrix.
                       h

             A family           of diagonal        matrices with diagonal elements

                                         ·
Definition:                   à×
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where  is the number of the nonzero elements in column                                                          of      .




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                                                                                                           Computer Tomography: Computational theory and methods – p.22/28
More iterative algorithms
Censor and Elfving’02
The idea: Use generalized projections and diagonal weighting matrices to reflect the
sparsity of the matrix.
                       h

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                                         ·
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                                                                                                            Computer Tomography: Computational theory and methods – p.22/28
3D Reconstructions

     Slice by Slice
 ¡




                         Disadvantages
     - low speed
     - discrete nature
     - higher radiation dose




                                         Computer Tomography: Computational theory and methods – p.23/28
3D Reconstructions

     Slice by Slice
 ¡




                         Disadvantages
     - low speed
     - discrete nature
     - higher radiation dose
     Cone-Beam Reconstructions
 ¡




     - Circular
     - Helical




                                         Computer Tomography: Computational theory and methods – p.23/28
Circular Cone-Beam Reconstructions




                     Computer Tomography: Computational theory and methods – p.24/28
Circular Cone-Beam Reconstructions
       Apply fan-beam FBP to
The Idea:
       each plane in the cone
Feldkamp, Davis, Kress ’84 (FDK)




                                   Computer Tomography: Computational theory and methods – p.24/28
Circular Cone-Beam Reconstructions
       Apply fan-beam FBP to
The Idea:
       each plane in the cone
Feldkamp, Davis, Kress ’84 (FDK)

A ray from the cone is described by
                    




                                     
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     locate a ray in a fan
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4 £


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 - projection angle





 - fan angle
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                                                                                       Computer Tomography: Computational theory and methods – p.24/28
The FDK Algorithm
Based on the FBP Algorithm for equispatial rays

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                                                                                           Computer Tomography: Computational theory and methods – p.25/28
The FDK Algorithm
Based on the FBP Algorithm for equispatial rays

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The Algorithm
Step 1 Compute the modified projection



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Step 2 Convolve                            with
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                                                                                                                   Computer Tomography: Computational theory and methods – p.25/28
The Tuy–Smith sufficiency condition

    Exact reconstruction is possible if all planes intersecting the
    object also intersect the source trajectory at least once.




                                             Computer Tomography: Computational theory and methods – p.26/28
The Tuy–Smith sufficiency condition

    Exact reconstruction is possible if all planes intersecting the
    object also intersect the source trajectory at least once.


    Circular trajectory does not satisfay this condition since a
    plane parallel to the trajectory may also intersect the object.




                                             Computer Tomography: Computational theory and methods – p.26/28
The Tuy–Smith sufficiency condition

    Exact reconstruction is possible if all planes intersecting the
    object also intersect the source trajectory at least once.


    Circular trajectory does not satisfay this condition since a
    plane parallel to the trajectory may also intersect the object.


    Exact reconstruction form helical cone-beam data is
    possible. Tam’95 Kudo, Noo, Defrise ’98




                                             Computer Tomography: Computational theory and methods – p.26/28
Conclusions
                          The challenges:

     Three dimentional images
 ¡




     Better quality
 ¡




     Faster algorithms
 ¡




     Restricted in time
 ÷




                                            Computer Tomography: Computational theory and methods – p.27/28
References
     K. Tanabe, Projection Method for Solving a Singular Systems of Linear Equations
 ¡




     and its Applications, Numer. Math., 17, 203-214 (1971).
     H. Tuy, An Inversion Formula for Cone-Beam Reconstruction, SIAM J. Appl. Math.
 ¡




     42 546–552 (1983).
     L. Feldkamp, L. Davis and J. Kress, Practical Cone-Beam Algorithm, Journal of the
 ¡




     Optical Society of America 1, 612-619 (1984).
     K. Tam, Three-Dimensional Computerized Tomography Scanning Method and
 ¡




     System for Large Objects with Smaller Area Detectors, US Patent 5,390,112, Feb
     14.(1995).
     S. Nilsson, Fast Backprojection, Technical report LiTH-ISY-R-1865, Linköping
 ¡




     University (1996).
     A. Kak, M. Slaney, Principles of Computerized Tomographic Imaging, IEEE Press.
 ¡




     (1998).
     H.Kudo, F. Noo and M. Defris, Cone-Beam Filtered Backprojection Algorithm for
 ¡




     Truncated Helical Data, Physics in Medicine and Biology 43, 2885–2909 (1998).

                                                        Computer Tomography: Computational theory and methods – p.28/28
References
     G.Wang and M. Vannier, Computerized Tomography, (1998) available at:
 ¡




     http://dolphin.radiology.uiowa.edu/ge/Teaching/ct/ct.html
     Y. Saad, H. van der Vorst, Iterative Solution of Linear Systems in the 20th Century,
 ¡




     J. Comput. Appl. Math. 123, 1–33 (2000).
     F. Natterer and F. Wübbeling, Mathematical Methods in Image Reconstruction,
 ¡




     SIAM, (2001).
     H. Turbell, Cone-Beam Reconstruction Using Filtered Backprojectio, Dissertation
 ¡




     No.672, Linköping Sudies in Science and Technology (2001).
     Y. Censor and T. Elfving, Block-Iterative Algorithms with Diagonally Scaled Oblique
 ¡




     Projections for the Linear Feasibility Problems, Siam J. Matrix Anal. Appl., 24,
     40-58 (2002).
     M. Jiang and G. Wang, Convergence Studies on Iterative Algorithms for Image
 ¡




     Reconstruction, IEEE Transactions on Medical Imaging , 22, 569-579 (2003).




                                                         Computer Tomography: Computational theory and methods – p.28/28