Reconstruction - outline Reconstruction in CT and relation to …bme.elektro.dtu.dk/31545/notes/reconstruction_ct_pet_mr_ver_2017_… · Reconstruction in CT and relation to other

1

Center for Fast Ultrasound Imaging Department of Electrical Engineering

Reconstruction in CT and relation to other imaging modalities Jørgen Arendt Jensen November 2, 2017 Center for Fast Ultrasound Imaging, Build 349 Department of Electrical Engineering Technical University of Denmark

Center for Fast Ultrasound Imaging, Department of Electrical Engineering Technical 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

•  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


Modern CT system generations

From

: W. A

. Kal

ende

r; C

ompu

ted

Tom

ogra

phy,

Pub

licis

, 200

5


Fan beam scan

From

: W. A

. Kal

ende

r; C

ompu

ted

Tom

ogra

phy,

Pub

licis

, 200

5

2


Fan beam reconstruction geometry

5/x

From Cho et al (1993), Foundations of Medical Imaging, Wiley


Reconstruction methods

6/x

From

Cho

et a

l (19

93),

Fo

unda

tions

of M

edic

al Im

agin

g, W

iley


Fourier slice theorem

7/x Center for Fast Ultrasound Imaging, Department of Electrical Engineering Technical University of Denmark

Reconstruction in the Fourier domain

8/x

From

Mag

nuss

on (1

993)

, Li

nogr

am a

nd o

ther

dire

ct F

ourie

r met

hods

fo

r tom

ogra

phic

reco

nstru

ctio

n, L

indk

öpin

g

3


MR scanner


Magnetic Resonance (MR) scanning

Larmor frequency:

ω0 = γ B0

γ – Gyromagnetic ratio, 42.58 MHz/Tesla B0 – Magnetic field in Tesla Typically 1.5 – 3 T

10/x


Center for Fast Ultrasound Imaging, Department of Electrical Engineering Technical University of Denmark 11/x


Gradient coils

12/x


4


MR measurement and reconstruction


MR images

14/x


MR images


MR overview image

16/x

5


Algebraic reconstruction

17/x

From

: W. A

. Kal

ende

r; C

ompu

ted

Tom

ogra

phy,

Pub

licis

, 200

5


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


Positron Emission Tomography

19/x

From Prince & Links, 2015


Images CT PET PET/CT

20/x

6


HRRT PET scanner with ART

21/x

HRRT scanner

Conventional PET scanner


Reconstruction methods

22/x

From

Cho

et a

l (19

93),

Fo

unda

tions

of M

edic

al Im

agin

g, W

iley


Exercise 5: Shepp-Logan image


Clinical images

24/x

7


Filtration


Filter

26/x


Clinical low-pass


High-pass filter

28/x

8


Advise for the assignments

29/x

Ideal Shepp−logan phantom, 512 x 512 pixels, Range: [0.95 1.1]

Relative x−coordinate

Rel

ativ

e y−

coor

dina

te

−3 −2 −1 0 1 2

−3

−2

−1

0

1

2


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


Parallel beam projection geometry

x’

y’

x

y

Patient coordinate system

CT coordinate system

Point

φ

ψ

Φ

31/x

Physical coordinates Versus Matlab indices


Backprojection

32/x

Patient grid

Projec

tion

9



Influence from number of projections

34/x



Ideal sinogram for Shepp-Logan phantom

36/x

10


Ram-Lak filter

Transfer function: Impulse response

37/x

h(ω) =ρ , ρ ≤ B

0 else

"#$

%$

h(k) =

B2 k = 0

−B2

π2k

"

#$

%

&'2 k odd

0 k even

(

)

****

+

****


Transfer function of filters – Ram-Lak

38/x


Filter transfer functions and impulse responses


Comparison between filters

40/x

11


Quantitative comparison


Circular convolution

42/x

−100 −80 −60 −40 −20 0 20 40 60 800

0.01

0.02

h(n)

Filter

−100 −80 −60 −40 −20 0 20 40 60 800

5

10

15

20

g 1(n)

Periodic time signal

−100 −80 −60 −40 −20 0 20 40 60 800

2

4

g 1(n)

Resulting periodic time signal with overlap


Circular convolution

43/x

−100 −80 −60 −40 −20 0 20 40 60 800

2

4

Sample number (n)

g 2(n)

Periodic time signal without overlap

−100 −80 −60 −40 −20 0 20 40 60 800

0.01

0.02

h(n)

Filter

−100 −80 −60 −40 −20 0 20 40 60 800

5

10

15

20

g 1(n)

Periodic time signal


Circular convolution – Shepp-Logan

44/x

Ideal Shepp−logan phantom, 512 x 512 pixels, Range: [0.95 1.1]

Relative x−coordinate

Rel

ativ

e y−

coor

dina

te

−3 −2 −1 0 1 2

−3

−2

−1

0

1

2

12


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


Assignment data

• DICOM data from female patient

• All data available on the web • Task is to find which slice it is

46/x


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

Documents

Reconstruction - outline Reconstruction in CT and relation to …bme.elektro.dtu.dk/31545/notes/reconstruction_ct_pet_mr_ver_2017_… · Reconstruction in CT and relation to other