The Filtered Backprojection Algorithm in MATLAB

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The Filtered Backprojection Algorithm in MATLAB

• Greg Gallardo
• 51185

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Filtered Backprojection Algorithm

• Measure the family of projections
• K of projections (angles), N of rays
  (detector size)
• Perform FFT
• Multiply FFT by FFT .
• Perform IFFT product from step 3
• Perform backprojection as

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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 to each row of R.

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1) Measure the family of projections

• Function parameters
• size() gets N and K
• nextpow2() get width for FFT

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2) Perform FFT

• MATLAB stands for MATrix LABoratory
• Matrix operations are faster than visiting each
  element in a loop.

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3) Multiply FFT by FFT

• Backprojection 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.

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3) Multiply FFT by FFT

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

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3) Multiply FFT by FFT

• FIR Filters. Commonly used Windows

Figure from Discrete-Time Signal Processing,
Oppenheim Schafer, Prentice-Hall
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3) Multiply FFT by FFT

• Rectangular
• Bartlett (triangular)
• Hanning (von Hann)
• Hamming
• Blackman

from Discrete-Time Signal Processing, Oppenheim
Schafer, Prentice-Hall
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3) Multiply FFT by FFT
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3) Multiply FFT by FFT
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3) Multiply FFT by FFT

• Filter step. Element by element multiplication

4) IFFTproduct from step 3
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5) Perform Backprojection as
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Results
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Results