Reconstruction in CT and relation to other imaging modalities Jørgen Arendt Jensen November 1, 2017 Center for Fast Ultrasound Imaging, Build 349 Department of Electrical Engineering Center for Fast Ultrasound Imaging Department of Electrical Engineering Reconstruction - outline Fan-beam geometry and reconstruction Overview of other reconstruction methods In the Fourier domain MR scanning Algebraic reconstruction PET and PET/CT scanning Exercise 5 solution Advise for the assignments Filtered backprojection algorithm Filters and their impulse responses Quantitative evaluation Programming in physical coordinates and backprojection Reading material: Prince & Links chapter 6 & 9 1
Modern CT system generations From: W. A. Kalender; Computed Tomography, Publicis, 2005 Fan beam scan From: W. A. Kalender; Computed Tomography, Publicis, 2005 2
Fan beam reconstruction geometry From Cho et al (1993), Foundations of Medical Imaging, Wiley 5/x Reconstruction methods From Cho et al (1993), Foundations of Medical Imaging, Wiley 6/x 3
Fourier slice theorem 7/x Reconstruction in the Fourier domain From Magnusson (1993), Linogram and other direct Fourier methods for tomographic reconstruction, Lindköping 8/x 4
MR scanner 9/x Magnetic Resonance (MR) scanning Larmor frequency: ω 0 = γ B 0 γ Gyromagnetic ratio, 42.58 MHz/Tesla B 0 Magnetic field in Tesla Typically 1.5 3 T From Cho et al (1993), Foundations of Medical Imaging, Wiley 10/x 5
11/x Gradient coils From Cho et al (1993), Foundations of Medical Imaging, Wiley 12/x 6
MR measurement and reconstruction 13/x MR images 14/x 7
MR images 15/x MR overview image 16/x 8
Algebraic reconstruction From: W. A. Kalender; Computed Tomography, Publicis, 2005 17/x PET and PET/CT scanning Positron Emission Tomography Radioactive FDG-18 injected Radioactive decay gives positron Annihilation with electron yields two 511 kev photons (gamma rays) Detected along line of response 18/x 9
Positron Emission Tomography From Prince & Links, 2015 CT Images PET 19/x PET/CT 20/x 10
HRRT PET scanner with ART HRRT scanner Conventional PET scanner 21/x Reconstruction methods From Cho et al (1993), Foundations of Medical Imaging, Wiley 22/x 11
Exercise 5: Shepp-Logan image 23/x Clinical images 24/x 12
Filtration 25/x Filter 26/x 13
Clinical low-pass 27/x High-pass filter 28/x 14
Advise for the assignments 3 Ideal Shepp logan phantom, 512 x 512 pixels, Range: [0.95 1.1] 2 1 Relative y coordinate 0 1 2 3 2 1 0 1 2 Relative x coordinate 29/x Filtered backprojection Perform for all projection: Make Fourier transform of projected data Apply filter in Fourier domain Make invers transform Backproject and sum with previous image 30/x 15
Parallel beam projection geometry y y CT coordinate system Point ψ Φ φ x x Patient coordinate system Physical coordinates Versus Matlab indices 31/x Backprojection Patient grid 32/x 16
33/x Influence from number of projections 34/x 17
35/x Ideal sinogram for Shepp-Logan phantom 36/x 18
Ram-Lak filter Transfer function: " $ h(ω) = # %$ ρ, ρ B 0 else Impulse response ( * * B 2 k = 0 * h(k) = ) B2 k odd 2 * " π $ # 2 k % * ' & * + * 0 k even 37/x Transfer function of filters Ram-Lak 38/x 19
Filter transfer functions and impulse responses 39/x Comparison between filters 40/x 20
Quantitative comparison 41/x Circular convolution Filter 0.02 h(n) 0.01 g 1 (n) 0 100 80 60 40 20 0 20 40 60 80 20 15 10 5 Periodic time signal 0 100 80 60 40 20 0 20 40 60 80 Resulting periodic time signal with overlap g 1 (n) 4 2 0 100 80 60 40 20 0 20 40 60 80 42/x 21
Circular convolution Filter 0.02 h(n) 0.01 g 1 (n) 0 100 80 60 40 20 0 20 40 60 80 20 15 10 5 Periodic time signal 0 100 80 60 40 20 0 20 40 60 80 Periodic time signal without overlap g 2 (n) 4 2 0 100 80 60 40 20 0 20 40 60 80 Sample number (n) 43/x Circular convolution Shepp-Logan 3 Ideal Shepp logan phantom, 512 x 512 pixels, Range: [0.95 1.1] 2 1 Relative y coordinate 0 1 2 3 2 1 0 1 2 Relative x coordinate 44/x 22
Data for testing and validation Use data sets on web site Circular phantom for geometry test Shepp-Logan for orientation and quantitative data In-vivo images for Hounsfield units 45/x Assignment data DICOM data from female patient All data available on the web Task is to find which slice it is 46/x 23
Reconstruction summary Filtered backprojection algorithm and choices Fan-beam geometry and reconstruction Overview of other reconstruction methods MR, PET, PET/CT Advise for the assignments Next time: Algebraic reconstruction with Professor Per Christian Hansen, DTU Compute 24