CT
Image Reconstruction
CT
• Please read Ch 13.
• Homework is due 1 week from today at
4 pm.
Tomographic reconstruction
detectors
= 0o
The main idea
detectors
= 90o
The main idea
Reconstruct the image of a non uniform sample using
its x-ray projection at different angles
The Sinogram

 


Detectors position
Image reconstruction
• Back projection
• Filtered Back projection
• Iterative methods (CH 22)
Back-projection
• Given a sample with 4 different spatial
absorption properties
A
B
D1= A+B=7
C
D
D2=C+D=7
 =0o
Back-projection
A
B
C
D
 = 90o
D3= A+C=6
D4= B+D=8
Back-projection
9
A
B
C
D
6
8
7
A+B=7
A+C=6
A+D=5
B+C=9
B+D=8
C+D=7
7
5
2
5
4
3
Real back-projection
• In a real CT we have at least 512 x 512
values to reconstruct
• We don’t know where one absorber
ends where the next begins
• ~ 800,000 projections
Back projection

   
p x
'
f  x , y   x cos   y sin   x 'dxdy    f 

The projection of a function is the radon transform of that function
Projections
• Are periodic in +/- 
• The radon transform of an image
produces a sinogram
Central Slice Theorem
• Relates the 1 D Fourier transform of a
projection of an object
– F(p(x’)) at a given angle 
• To a line through the center of the 2D
Fourier transform of the object at a
given angle 
Central Slice Theorem

   
p x
'
f  x , y   x cos   y sin   x 'dxdy    f 

     f ( x , y )
 p x
'

p  ( )  F  cos    sin  
Why is it important?
• If you compute the 1D Fourier transform
of all the projection (at all angles f) you
can “fill” the 2 D Fourier transform of the
object.
• The object can then be reconstructed by
a simple 2D Fourier transform.
FILTERED back-projection
• If only the 2D inverse Fourier transform
is computed you will obtain a “blurry”
image. (it is intrinsic in inverse Radon)
• The blur is eliminated by deconvolution
• In filtered back projection a RAMP filter
is used to filter the data
Homework
• Prove the center slice theorem.
• Use imrotate
Imaging in Matlab
• An image is a 2D matrix of numbers
• imread - reads an image file
• imwrite - writes an image to file