Radon Transform and Filtered Backprojection

Transcription

1 Radon Transform and Filtered Backprojection Jørgen Arendt Jensen October 13, 2016 Center for Fast Ultrasound Imaging, Build 349 Department of Electrical Engineering Center for Fast Ultrasound Imaging Department of Electrical Engineering CT reconstruction - outline CT scanners Projection and Radon transform Projection demo Fourier slice theorem Inverse Radon transform filtered backprojection Selection of filters Filtered backprojection algorithm Exercise 5 Questions for the assignments Reading material: Prince & Links chapter 6 2/x 1

2 Question for 2D signal processing: Is phase or amplitude most important? Demo in: for_13/matlab_demo/phase_demo 3/x Modern CT system generations 4/x From: W. A. Kalender; Computed Tomography, Publicis,

3 What do we measure? Intensity measured by detector: I = Io exp( µ x) Conversion to attenuation: µ = 1 ln x I I Attenuation values µ are scaled relative to water: µ HU = 0 µ µ tissue water water /x Hounsfield units 6/x From: W. A. Kalender; Computed Tomography, Publicis,

4 Measurement of attenuation 7/x From: W. A. Kalender; Computer Tomography, Publicis, 2005 Parallel beam projection geometry y y CT coordinate system Point ψ θ φ x x Patient coordinate system 8/x 4

5 Sinogram for point 9/x 10/x 5

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8 15/x 16/x 8

9 Shepp-Logan phantom 17/x Demo in: for_13/matlab_demo/proj_demo 18/x 9

10 Fourier slice theorem Demo in: for_13/matlab_demo/cd_demo 19/x Filtered backprojection Perform for all projection: 1. Make Fourier transform of projected data 2. Apply filter in Fourier domain 3. Make invers Fourier transform 4. Backproject and sum with previous image 20/x 10

11 Influence from number of projections 21/x Transfer function of filters - Ideal 22/x 11

12 Hanning weighted filter 23/x Shepp-Logan filter 24/x 12

13 Filter transfer functions and impulse responses 25/x Comparison between filters 26/x 13

14 Filtered backprojection Perform for all projection: 1. Make Fourier transform of projected data 2. Apply filter in Fourier domain 3. Make invers transform 4. Backproject and sum with previous image 27/x Summary Parallel beam projection and Radon transform Fourier slice theorem Filtered backprojection reconstruction and choices P & L: chapter 6 Now: Exercise 5 questions Questions for assignments 28/x 14

15 Exercise 5: Image processing 29/x 1. Show Shepp-Logan phantom images. 2. Shepp-Logan phantom gray scale mapping. 3. Clinical images of the brain and its gray scale mapping. 4. Make a two-dimensional Fourier transform of the sh_black image, and make a mesh plot of the amplitude spectrum with the command mesh. Plot the spectrum with the correct spatial frequency axis. Study the symmetry relations for the Fourier transform. 5. Make a low-pass filter with a circularly symmetric transfer function that removes all frequencies above a value of 116 m Use an edge enhancement filter given as [ ; ; ] to enhance the edges in the image sh_black. 7. Try the above mentioned image processing on the clinical images downloaded previously. 30/x 15