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Center for Fast Ultrasound ImagingDepartment of Health Technology
Reconstruction in CT and relation to other imaging modalities
Jørgen Arendt Jensen
October 29, 2021
Center for Fast Ultrasound Imaging, Build 349Department of Electrical EngineeringTechnical University of Denmark
1
Center for Fast Ultrasound Imaging, Department of Health TechnologyTechnical University of Denmark
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
• Reading material: Prince & Links chapter 6 & 9
2
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Center for Fast Ultrasound Imaging, Department of Health TechnologyTechnical University of Denmark
Fan beam scan
From
: W. A
. Kal
ende
r; C
ompu
ted
Tom
ogra
phy,
Pub
licis
, 200
5
3
Center for Fast Ultrasound Imaging, Department of Health TechnologyTechnical University of Denmark
Fan beam reconstruction geometry
4/x
From Cho et al (1993), Foundations of Medical Imaging, Wiley
4
3
Center for Fast Ultrasound Imaging, Department of Health TechnologyTechnical University of Denmark
Fan-beam reconstruction algorithm
!𝑓 𝑥, 𝑦 = '!
"#𝑤" ['
$%!
%!𝑤&𝑝' 𝛽 𝑔 𝛽( − 𝛽 𝑑𝛽]𝑑𝛼
•Weight 1: 𝑤! = 𝑅" cos 𝛽
•Weight 2: 𝑤# =!#$
!%!
•Filter: 𝑔 𝛽 = &'() & ℎ 𝛽 , ℎ(𝛽)↔ |r|
•Definition of variables on previous slide
5/x
5
Center for Fast Ultrasound Imaging, Department of Health TechnologyTechnical University of Denmark
Reconstruction methods
6/x
From
Cho
et a
l (19
93),
Foun
datio
ns o
f Med
ical
Imag
ing,
Wile
y
6
4
Center for Fast Ultrasound Imaging, Department of Health TechnologyTechnical University of Denmark
Fourier slice theorem
7/x
7
Center for Fast Ultrasound Imaging, Department of Health TechnologyTechnical University of Denmark
Reconstruction in the Fourier domain
8/x
From
Mag
nuss
on (1
993)
, Li
nogr
aman
d ot
her d
irect
Fou
rier m
etho
ds
for t
omog
raph
ic re
cons
truct
ion,
Lin
dköp
ing
8
5
Center for Fast Ultrasound Imaging, Department of Health TechnologyTechnical University of Denmark
MR scanner
9/x
9
Center for Fast Ultrasound Imaging, Department of Health TechnologyTechnical University of Denmark
Magnetic Resonance (MR) scanning
Larmor frequency:
w0 = g B0
g – Gyromagnetic ratio, 42.58 MHz/Tesla
B0 – Magnetic field in TeslaTypically 1.5 – 3 T
10/x
From Cho et al (1993), Foundations of Medical Imaging, Wiley
10
6
Center for Fast Ultrasound Imaging, Department of Health TechnologyTechnical University of Denmark 11/x
11
Center for Fast Ultrasound Imaging, Department of Health TechnologyTechnical University of Denmark
Gradient coils
12/x
From Cho et al (1993), Foundations of Medical Imaging, Wiley
12
7
Center for Fast Ultrasound Imaging, Department of Health TechnologyTechnical University of Denmark
MR measurement and reconstruction
13/x
13
Center for Fast Ultrasound Imaging, Department of Health TechnologyTechnical University of Denmark
MR images
14/x
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8
Center for Fast Ultrasound Imaging, Department of Health TechnologyTechnical University of Denmark
MR images
15/x
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Center for Fast Ultrasound Imaging, Department of Health TechnologyTechnical University of Denmark
MR overview image
16/x
16
9
Center for Fast Ultrasound Imaging, Department of Health TechnologyTechnical University of Denmark
Algebraic reconstruction
17/x
From
: W. A
. Kal
ende
r; C
ompu
ted
Tom
ogra
phy,
Pub
licis
, 200
5
17
Center for Fast Ultrasound Imaging, Department of Health TechnologyTechnical University of Denmark
PET and PET/CT scanningPositron 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
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Center for Fast Ultrasound Imaging, Department of Health TechnologyTechnical University of Denmark
Positron Emission Tomography
19/x
From Prince & Links, 2015
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Center for Fast Ultrasound Imaging, Department of Health TechnologyTechnical University of Denmark
ImagesCT PET PET/CT
20/x
20
11
Center for Fast Ultrasound Imaging, Department of Health TechnologyTechnical University of Denmark
HRRT PET scanner with ART
21/x
HRRTscanner
ConventionalPET scanner
21
Center for Fast Ultrasound Imaging, Department of Health TechnologyTechnical University of Denmark
Reconstruction methods
22/x
From
Cho
et a
l (19
93),
Foun
datio
ns o
f Med
ical
Imag
ing,
Wile
y
22
12
Center for Fast Ultrasound Imaging, Department of Health TechnologyTechnical University of Denmark
Exercise 5 - Shepp-Logan phantom
23/x
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Center for Fast Ultrasound Imaging, Department of Health TechnologyTechnical University of Denmark
Hounsfield units
From
: W. A
. Kal
ende
r; C
ompu
ted
Tom
ogra
phy,
Pub
licis
, 200
5
Note that the in-vivo projected data on the website is off-set by 1000 HU (-1000 is off-set to 0)
24
13
Center for Fast Ultrasound Imaging, Department of Health TechnologyTechnical University of Denmark
From
: W. A
. Kal
ende
r; C
ompu
ter T
omog
raph
y, P
ublic
is, 2
005
25
Center for Fast Ultrasound Imaging, Department of Health TechnologyTechnical University of Denmark
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: Clinical CT imaging by PhD, MD Thomas Kristensen, Diagnostic Center, Radiology, Rigshospitalet
• Algebraic reconstruction with Professor Per Christian Hansen, DTU Compute
• Reading material: – Prince & Links chapter 6-9, 12 & 13
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