# reconstruction

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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 
                 


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.
– 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
(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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