Image Reconstruction. R.D. Badawi. Department of Radiology Department of Biomedical Engineering. Page 1

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1 First ever X-ray iage Iage Reconstruction R.D. Badawi Departent of Radiology Departent of Bioedical Engineering X-ray Projection Iaging Requires straight line trajectory of unattenuated x-rays Nuclear Projection Iaging 3D inforation copressed to 2D 3D inforation copressed to 2D Page

2 Toographic Acquisition Toographic Acquisition Toographic Acquisition Toographic Acquisition Toographic Reconstruction Filtered Backprojection Page 2

3 The Anger (Gaa) Caera Single-photon Eission Coputed Toography (SPECT) Single-photon Eission Coputed Toography (SPECT) Single-photon Eission Coputed Toography (SPECT) Page 3

4 Single-photon Eission Coputed Toography (SPECT) Single-photon Eission Coputed Toography (SPECT) Single-photon Eission Coputed Toography (SPECT) Single-photon Eission Coputed Toography (SPECT) Page 4

5 Single-photon Eission Coputed Toography (SPECT) Planar Iages or 2D Projections Single-photon Eission Coputed Toography (SPECT) Planar Iages or 2D Projections Single-photon Eission Coputed Toography (SPECT) Planar Planar Iages or Projection or 2D Projections Iages Toographic Iages The Transfor - You can approxiate any sensible shape by adding a ixture of sine and cosine waves together. Deo (Paul Falstad) Page 5

6 Transfor Exaples Function (real space) Transfor (frequency space) Transfor Exaples Function (real space) Transfor (frequency space) Boxcar Sinc function Gaussian Another Gaussian Triangle Sinc 2 function Inverse function, /r Inverse function, /k The 2D Transfor The 2D Transfor Picture fro Sieens Medical Picture fro Sieens Page 6

7 The Transfor The Transfor Picture fro Sieens Medical Picture fro Sieens Medical The Transfor The Transfor Picture fro Sieens Medical Picture fro Sieens Medical Page 7

8 Phew! Toographic Acquisition - You can ake iages out of sines and cosines - Low frequency (long wavelength) waves give broad brushstrokes - High frequency (short wavelength) waves give the details Toographic Acquisition Toographic Acquisition 22.5 (Iages courtesy of Andrew Goertzen, Ph.D., University of Mannitoba) (Iages courtesy of Andrew Goertzen, Ph.D., University of Mannitoba) Page 8

9 Toographic Acquisition Toographic Acquisition (Iages courtesy of Andrew Goertzen, Ph.D., University of Mannitoba) (Iages courtesy of Andrew Goertzen, Ph.D., University of Mannitoba) Toographic Acquisition Toographic Acquisition (Iages courtesy of Andrew Goertzen, Ph.D., University of Mannitoba) (Iages courtesy of Andrew Goertzen, Ph.D., University of Mannitoba) Page 9

10 Toographic Acquisition Toographic Acquisition (Iages courtesy of Andrew Goertzen, Ph.D., University of Mannitoba) (Iages courtesy of Andrew Goertzen, Ph.D., University of Mannitoba) Toographic Acquisition Toographic Acquisition (Iages courtesy of Andrew Goertzen, Ph.D., University of Mannitoba) (Iages courtesy of Andrew Goertzen, Ph.D., University of Mannitoba) Page

11 Toographic Acquisition Toographic Acquisition Toographic Reconstruction Object The Radon Transfor (forward projection) Sinogra (Iages courtesy of Andrew Goertzen, Ph.D., University of Mannitoba) Backprojection Forward Projection (The Radon Transfor ) Back Projection Three projections Six projections Many projections Page

12 Back Projection Three projections Six projections Many projections Back-projection results in a convolution Object Scanning, then Back projection Iage In back-projection, every single point in the object is turned into the inverse function (/r): -4 So the back-projected iage consists of a whole load of such functions (one for each point), overlaid on each other This is known as a CONVOLUTION. Scanning Back projection (forward projection) Inverse function, or /r (Iages courtesy of Andrew Goertzen, Ph.D., University of Page 2

13 Filtered Back-projection Filtered Back-projection = * = * Back-projection (object) = (true iage) * (/r) Back-projection (object) = (true iage) * (/r) Note: Convolution is NOT the sae as ultiplication: X = Theore ( A * B ) = ( A) X ( B ) Filtered Back-projection Filtered Back-projection Back-projection (object) = = ( ) * Back-projection (object) = = ( ) * ( * ) ( ) = ( * ) ( ) = ( ) X ( ) ( * ) ( ) = ( ) = ( ) ( ) X ( ) Page 3

14 Filtered Back-projection The Rap Filter ( ) = 5 4 = ( ) = ( / ) /k r = k = The Rap Filter Filtered Back-projection ( * ) ( ) = ( ) Corresponds to Nyquist Frequency - half the sapling Page 4

15 Filtered Back-projection ( ) = ( ) ( ) ( ) Back- ( ) projected Object X X Ra p filter = = ( ) ( Iage) Object Iage Filtered Back-projection Scanning (forward projection) Inverse Transfor Sinogra Back projection Multiply by Rap filter (Iages courtesy of Andrew Goertzen, Ph.D., University of Mannitoba, and Sieens Transfor High-frequency coponents and Noise High-frequency coponents and Noise Deo (Paul Falstad) (Iages courtesy of Andrew Goertzen, Ph.D., University of Mannitoba) Page 5

16 High-frequency coponents and Noise Effects of Filtering Rap, cut-off =. x Nyquist Rap, cut-off =.4 x Nyquist (Iages courtesy of Andrew Goertzen, Ph.D., University of Mannitoba) Effects of Filtering Rap, cut-off =.4 x Nyquist Shepp-Logan, cut-off =.5 x Nyquist Hanning, cut-off =.5 x Nyquist Rap Shepp-Logan Hanning Page 6

17 SUVax SUVax SUVax Liited Angle Reconstruction soothing () soothing () soothing () Positron Eission Maography Rap Shepp-Logan Hanning (Naviscan ) Liited Angle Reconstruction Liited Angle Reconstruction Page 7

18 Liited Angle Reconstruction Iterative Reconstruction Guess the iage, forward project it, calculate the ratio between the resu and the easured data. Now do a weighted backprojection of the ratio and ultiply your guess by the result. That s the next iteration When difference between iterations is sall, stop. Iterative Reconstruction Guess the iage, forward project it, calculate the ratio between the resu and the easured data. Now do a weighted backprojection of the ratio and ultiply your guess by the result. That s the next iteration When difference between iterations is sall, stop. Statistical odel gives ore weight to stronger data Iterative Reconstruction Guess the iage, forward project it, calculate the ratio between the resu and the easured data. Now do a weighted backprojection of the ratio and ultiply your guess by the result. That s the next iteration When difference between iterations is sall, stop. Statistical odel gives ore weight to stronger data Syste odel can iprove resolution Page 8

19 Positron Eission and Annihilation Positron Eission and Annihilation ~.5 degree 5 kev e+ e- e+ e- P N P N N P P PN e+ N ν N P P N P N N P P PN N N P 5 kev Positron Eission and Annihilation 5 kev Positron Eission and Annihilation ~.5 degree 5 kev 8 e+ e- e+ e- P N P N N P P PN N N P 5 kev P N P N N P P PN N N P 5 kev Page 9

20 Electronic colliation and intrinsic resolution Depth of interaction and intrinsic resolution Detector Interaction point Resolution degrades as the radial distance increases Field of View Resolution depends on size of detector eleents Incidence point Interaction point Incidence point Positioning error Resolution Recovery with FDG PET/CT OSEM+LOR 4 iterations, 28 subsets 7 Post Filter Transaxial Profile through Rib Iterative Reconstruction Guess the iage, forward project it, calculate the ratio between the resu and the easured data. Now do a weighted backprojection of the ratio and ultiply your guess by the result. That s the next iteration When difference between iterations is sall, stop. Statistical odel gives ore weight to stronger data OSEM+LOR+PSF 4 iterations, 28 subsets Syste odel can iprove resolution Observation: Even with 7 post-soothing, there is enhanceent with addition of PSF (% higher) (Ada Alessio University of Washington, Seattle) Page 2

21 Iterative Reconstruction Guess the iage, forward project it, calculate the ratio between the resu and the easured data. Now do a weighted backprojection of the ratio and ultiply your guess by the result. That s the next iteration When difference between iterations is sall, stop. Statistical odel gives ore weight to stronger data Syste odel can iprove resolution Can speed it up using subsets As with FBP, can sooth (filter) the data after reconstruction One subset Ordered Subsets Expectation Maxiization (OSEM) Two subsets Four subsets Ordered Subsets Expectation Maxiization (OSEM) - subsets Ordered Subsets Expectation Maxiization (OSEM) - iterations One subset Six subsets Thirty subsets SUVax =.5 SUVax = 2.8 SUVax = 4. (One iteration) SUVax iteration = 4. 2 SUVax iterations = 5. 4 SUVax iterations = 6. 8 (3 subsets) SUVax iterations = 6.6 Page 2

22 Ordered Subsets Expectation Maxiization (OSEM) postreconstruction filtering Effect of reconstruction ethod- OSEM (an iterative algorith) vs Filtered Back-projection (an analytic algorith) 2 filter SUVax = 5. 6 filter SUVax = 4.4 filter SUVax = 3.4 (2 iterations, 3 6 filter SUVax = 2.7 Attenuation ap FDG distribution - FBP FDG distribution - OSEM FDG distribution - FBP, no attenuation correction Scintillation Light (Martin Lodge, Ph.D. Johns Hopkins) BGO crystals photographed with roo lights on. Departent of Radiology Departent of Bioedical Engineering BGO crystals with roo lights off. Pulsed x-rays give rise to scintillation light in the visible range. Page 22