Reconstruction in CT and relation to other imaging modalities

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1 Reconstruction in CT and relation to other imaging modalities Jørgen Arendt Jensen November 16, 2015 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 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

2 Modern CT system generations From: W. A. Kalender; Computed Tomography, Publicis, 2005 Fan beam scan From: W. A. Kalender; Computed Tomography, Publicis,

3 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

4 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

5 MR scanner 9/x Magnetic Resonance (MR) scanning Larmor frequency: ω 0 = γ B 0 γ Gyromagnetic ratio, MHz/Tesla B 0 Magnetic field in Tesla Typically T From Cho et al (1993), Foundations of Medical Imaging, Wiley 10/x 5

6 11/x Gradient coils From Cho et al (1993), Foundations of Medical Imaging, Wiley 12/x 6

7 MR measurement and reconstruction 13/x MR images 14/x 7

8 MR images 15/x MR overview image 16/x 8

9 Algebraic reconstruction From: W. A. Kalender; Computed Tomography, Publicis, /x PET and PET/CT scanning 18/x 9

10 Images CT PET PET/CT 19/x HRRT PET scanner with ART HRRT scanner Conventional PET scanner 20/x 10

11 Reconstruction methods From Cho et al (1993), Foundations of Medical Imaging, Wiley 21/x Advise for the assignments 3 Ideal Shepp logan phantom, 512 x 512 pixels, Range: [ ] 2 1 Relative y coordinate Relative x coordinate 22/x 11

12 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 23/x Fourier slice theorem 24/x 12

13 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 25/x Transfer function of filters Ram-Lak 26/x 13

14 Shepp_Logan filter Transfer function: " $ sin 2πρ $ ρ 4B, ρ B h(ω) = # 2πρ $ 4B $ % 0 else Impulse response 8B 2 h(k) = π (4k 2 1) 27/x Shepp-Logan filter 28/x 14

15 Hanning weighted filter 29/x Hanning weighted filter 30/x 15

16 Filter transfer functions and impulse responses 31/x Comparison between filters 32/x 16

17 Quantitative comparison 33/x Circular convolution Filter 0.02 h(n) 0.01 g 1 (n) Periodic time signal Resulting periodic time signal with overlap g 1 (n) /x 17

18 Circular convolution Filter 0.02 h(n) 0.01 g 1 (n) Periodic time signal Periodic time signal without overlap g 2 (n) Sample number (n) 35/x Circular convolution Shepp-Logan 3 Ideal Shepp logan phantom, 512 x 512 pixels, Range: [ ] 2 1 Relative y coordinate Relative x coordinate 36/x 18

19 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 37/x Influence from number of projections 38/x 19

20 Parallel beam projection geometry y y CT coordinate system Point ψ Φ φ x x Patient coordinate system Physical coordinates Versus Matlab indices 39/x 40/x 20

21 41/x 42/x 21

22 43/x 44/x 22

23 Ideal sinogram for Shepp-Logan phantom 45/x Shepp-Logan phantom 46/x 23

24 Influence from number of projections 47/x Backprojection Patient grid 48/x 24

25 Quantitative comparison 49/x Reconstruction Filtered backprojection algorithm and choices Fan-beam geometry and reconstruction Overview of other reconstruction methods MR, PET, PET/CT Next time: MR with Axel Thielscher Advise for the assignments Reading material: MR notes on the web site 25

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