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					Image Reconstruction

 T-61.182, Biomedical Image
           Analysis
Seminar Presentation 7.4.2005
Seppo Mattila & Mika Pollari
                    Overview
• Reconstruction from projections (general)
   – projection geometry and radon transform
• Reconstruction methods
   – Backprojection, (Fourier slice theorem), Filtered
     Backprojection, and Algebraic Reconstruction Technique
• Reconstruction examples
• MRI reconstruction
   – examples
                   Introduction
• Only photography (reflection) and planar x-ray
  (attenuation) measure spatial properties of the
  imaged object directly.
• Otherwise, measured parameters are some how
  related to spatial properties of imaged object.
   – US (amplitude and time), CT, SPECT and PET
     (integral projections of parallel rays), MRI (amplitude,
     frequency and phase of the NMR signal) etc...
• Construct the object (image) which creates the
  measured parameters.
Reconstruction From the Projections
• Projection is a line integral along the path:

   p (t)   f(x, y)ds where s   x sin  y cos
          AB

• CT: measure the projections of passed photons,
  with different angles.
• MRI: measure projection of NMR signal with
  different magnetic gradients (projection based
  MRI not used anymore).
• Assumption: no notable diffraction.
 Projection Geometry and Radon
         Transformation
• Co-ordinate
  transformation:
   t   cos sin   x 
  s     sin cos   y 
                   

• Radon transformation
             
  p (t )     f (t , s )ds.
             
            Backprojection (BP)
• Simplest reconstruction method: Integrate
  all possible rays that pass through the same
  point.
            
  f ( x, y )   p ( t) d , where t  x cos  y sin
            0


• Cause smearing and blurring. Method has
  nowadays only historical importance.
Backprojection Graphically
FBP ramp vs. smooth filter
                        FST and FBP
• Starting from FST we end up to FBP
  without any approximations or assumptions
              
   f ( x, y )     F ( u, v) e[  j 2   ( ux vy )]
                                                    dudv
               

   .....
                                   
       P ( w) w e j 2      wt
                                    dwdw
     0                             
                                                        

   f(x, y)   q ( t )d where q (t )                   P ( w) w e j 2   wt
                                                                                dw
              0                                         
             Remarks of FBP
• From the previous equations it’s clear that
  the image is backprojection of filtered
  signal ( q (t )) and (|w|) is the ramp filter.
• FBP advantages:
   – Each projection may be filtered and
     backprojected while further projections are
     collected (on-line processing).
   – No need for 2D inverse Fourier transformation.
     Algebraic Reconstruction
        Techniques (ART)
• Each object entity (image pixel/voxel) has
  physical property (grey-level value) such as
  attenuation coefficient
• All pixel in the rays path contribute to sum
  an amount which equals pixel’s area along
  the path (weight) times pixels physical
  property (grey-level value)
• We end up set of simultaneous equations
                   ART Model
                                 N

• Each ray sum: p   w f ,
                         m
                                 n 1
                                        mn   n   m  1,..., M

• Set of simultaneous equations

 p1  w11 f 1  ...  w1 N f N


 p m  wM 1 f1  ...  wMN f N
 Kaczmarz Method – Solution to
            ART
• Each equation spans a hyperplane in n-
  dimensional space. If unique solution exist
  it is in intersection of the hyperplanes
• Solution is found iteratively by solving each
  ray equation at the time

                              f ( m 1) wm  pm 
  f   ( m)
             f   ( m 1)
                            
                             
                                                  wm
                                                 
                                     wm wm      
Reconstruction Examples –
  Number of Projection
  Reconstruction Examples –
      Projection Angle
PSF       BP   FBP (Butter)   ART
   Reconstruction Examples – BP,
          FBP, and ART
Phantom              BP         FBP (Butter)          ART




Reconstruction of phantom with different reconstruction
methods using 90 projections from interval 0-180 degree
    Display of CT Images Using
         Hounsfield Units
• Attenuation coefficients
  are normalized with
  respect of water
                  
  HU K 
                1
                      where K  1000
           w      

• Now mean and SD of of
  different tissues are
  known advanced
  (measured in HU)
MRI Reconstruction
MRI Signal
Reconstruction Examples
Effects of Sampling the K-space
                  Summary I
Reconstruct 2D CT image from 1D projections:
• Backprojection (BP)
 - Only historical importance
• Filtered backprojection (FBP)
 - Most widely used technique
 - Large number of projections over 0-180° required
• Algebraic Reconstruction Techniques (ART)
 - Better handling of sparse and non-uniform projections
 - Slow compared to FBP
                   Summary II
• Reconstruct 2D MR image from measured current (1D)
• Spatial (x,y) info encoded in frequency and phase
• Collect data to spatial frequency domain (k-space)
• Reconstruction by inverse 2D FT
+ non-invasive
+ imaging sequences (T1, T2, fMRi etc.) lots of possibilities
- image artifacts (distortions, ghosts, etc.)
- more expensive

				
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posted:12/2/2011
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