Filtered Backprojection Algorithm in MATLAB.ppt

The Filtered Backprojection  Algorithm in MATLAB Greg Gallardo 51:185

Filtered Backprojection Algorithm 1.

Measure the family of projections    1,2,..., K ; n  1,2,..., N 

 P  n  , i   i

 K  =

# of projections (angles),  N =# =# of rays (detector size)

2.

Perform FFT[ P  n , i  1,2,..., K  ]

3.

Multiply FFT[  P  n   ] by FFT[ hn   ].

4.

Perform IFFT[ product from step 3]

5.

Perform backprojection as:

 i

  i

 f   x, y  

 

 K 

sin n    Q  x cos   y si   K   i

i 1

i

i

1) Measure the family of projections 

[R,Xp] = RADON(...) returns two variables 



Matrix R  – columns are the Radon transform for the angles . Rows are detector position Vector Xp - containing the radial coordinates corresponding correspondi ng to each row of R.

1) Measure the family of projections 

Function parameters



size(): gets N and K



nextpow2(): get width for FFT

2) Perform FFT[  P  n , i  1,2,..., K  ]  i

 

MATLAB stands for MAT for MATrix rix LAB LABoratory oratory Matrix operations are faster than visiting each element in a loop.

3) Multiply FFT[



 P  n    



Backprojection Backprojec tion filters 







Shepp-Logan: Ram-Lak multiplied by sinc function (see iradon help) Cosine: Ram-Lak multiplied by cosine Hamming: Ram-Lak multiplied by Hamming window

Hanning, Blackman, etc.

i

]

by FFT[



h n 



]

3) Multiply FFT[



 P  n    



i

]

by FFT[



h n 

Ram-Lak, Shepp-Logan and Cosine filters in frequency domain



]

3) Multiply FFT[



 P  n    



i

]

by FFT[

FIR Filters. Commonly used Windows

Figure from “Discrete-Tim Timee Signal Processing”, Oppenheim & Schafer, Prentice-Hall



h n 



]

3) Multiply FFT[



]

w[n]

1,  0,

 P  n    











Rectangular 

Hanning (von Hann)

Blackman

w[n]

w[ n] 

by FFT[ 0

w[ n]

n



h n 

  M ,

otherwise

2n /  M ,  w[ n]  2  2n /  M  , 0, 

Bartlett (triangular)

Hamming

i

0

  M  / 2  M  / 2  n   M  n

otherwise

0.5  0.5 cos( 2  n /  M  ),  0,

0.54  0.46 cos( 2  n /  M  ),  0,

0

n

  M ,

otherwise

0

n

  M ,

otherwise

0.42  0.5 cos( 2  n /  M  )  0.08 cos( 4  n /  M  ), 0  n   M ,  otherwise 0,

from “Discrete-Time Signal Processing”, Oppenheim & Schafer, Sch afer, Prentice-Hall



]

3) Multiply FFT[



 P  n     i

]

by FFT[



h n 



]

3) Multiply FFT[



 P  n     i

]

by FFT[



h n 



]

3) Multiply FFT[



 P  n    



i

]

by FFT[

Filter step. Element by element multiplication

4) IFFT[product from step 3]



h n 



]

5) Perform Backprojection as:  f   x, y  

 

 K 

sin n    Q  x cos   y si   K   i

i 1

i

i

Results

Results