reconstruction in ct and hints to the assignments

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Center for Fast Ultrasound ImagingDepartment of Health Technology

Reconstruction in CT and hints to the assignments

Jørgen Arendt Jensen

October 28, 2021

Center for Fast Ultrasound Imaging, Build 349Department of Health TechnologyTechnical University of Denmark

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Center for Fast Ultrasound Imaging, Department of Health TechnologyTechnical University of Denmark

CT reconstruction repetition & hints

• Filtered backprojection algorithm–Radon transform–Filtered backprojection–Filters and their impulse

responses

• Advise for the assignments

• Reading material: Prince & Links Chapter 6

From

: W. A

. Kal

ende

r; C

ompu

ted

Tom

ogra

phy,

Pub

licis

, 200

5

2

2


Modern CT system generations

From

: W. A

. Kal

ende

r; C

ompu

ted

Tom

ogra

phy,

Pub

licis

, 200

5

3


Measurement of attenuation

From

: W. A

. Kal

ende

r; C

ompu

ter T

omog

raph

y, P

ublic

is, 2

005

4

3


Parallel beam projection

From

: W. A

. Kal

ende

r; C

ompu

ted

Tom

ogra

phy,

Pub

licis

, 200

5

5


Parallel beam projection geometry

x’

y’

x

y

Patient coordinate system

CT coordinate system

Point

fy

q

6/x

6

4


Radon transform

7/x

p(x ',φ) = f (x 'cos−∞

+∞

∫ φ − y 'sinφ,x 'sinφ + y 'cosφ)dy '

f(x,y) – Attenuation imagex’, y’ – Gantry coordinate systemx, y – Patient coordinate system

7

Center for Fast Ultrasound Imaging, Department of Health TechnologyTechnical University of Denmark 8/x

8

5


9


10

6


11


12

7


13


Shepp-Logan phantom

14/x

Note that the Shepp-Logan phantom isnot in Hounsfield units, but the relativescaling is the same.

14

8


Demo in: for_13/matlab_demo/proj_demo

15


Fourier slice theorem

16/x

Demo in: for_13/matlab_demo/ct_demo

16

9


Fourier Slice Theorem

17/x

P(ρ,φ) = p(x ',φ)e− j2πρx ' dx '−∞

+∞

∫

= F (ρ cosφ,ρ sinφ)

F(u,v) – 2D Fourier transform of imagef – Gantry rotation

17


Filtered Back Projection (FBP)

18/x

!𝑓 x, y – Reconstructed imagef – Gantry rotationx’ – Detector position

!𝑓 x, y = '!

"

'#$

%$

𝜌 𝑃(𝜌, 𝜙)𝑒&'"()!𝑑𝜌𝑑𝜙

18

10


Filtered backprojection

• Perform for all projections:• Make Fourier transform of

projected data• Apply filter in Fourier domain• Make invers transform• Back-project and sum with

previous image

19/x

19


Influence from number of projections

20/x

20

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Ram-Lak filter

Transfer function:

ℎ 𝜌 = % ρ , ρ ≤ 𝐵0 𝑒𝑙𝑠𝑒

Impulse response

21/x

h(k) =

B2 k = 0

−B2

π2k

"

#$

%

&'2 k odd

0 k even

(

)

****

+

****

21


Impulse response for Ram-Lak filter

22/x

ℎ!"#$%& 𝑥 = 𝐴2𝐵sin 2𝜋𝐵 𝑥2𝜋𝐵𝑥

ℎ456758 𝑥 = 2𝐵𝐵 9:; '"<)'"<) - 𝐵' 9:;

. "<)("<)).

• Impulse response of square wave:

• Ram-Lak impulse response: square wave – triangle:

22

12


Sampling of impulse response for Ram-Lak filter

23/x

ℎ%$'($) 𝑘 = 2𝐵*+,- *./ !

"#

*./ !"#

- 𝐵*+,-" ./ !

"#

(./ !"#)

"

sin 2𝜋𝐵 )*/ =sin 𝜋𝑘 = 0

sin* 𝜋𝐵𝑘2𝐵 =sin

* 𝜋2 𝑘 = 1 𝑘 𝑜𝑑𝑑 𝑒𝑙𝑠𝑒 0

• Sampling at 234 = 2𝐵, 𝑥 = 𝑘 Δ𝑥 = )*/ :

• Gives:ℎ 0 = 𝐵*

ℎ 𝑘 = −𝐵*+,-" )$"()$")

" =− /"

()$")" (k odd)

ℎ 𝑘 = 0 else

23


Ram-Lak filter

Transfer function:

ℎ 𝜌 = % ρ , ρ ≤ 𝐵0 𝑒𝑙𝑠𝑒

Impulse response

24/x

h(k) =

B2 k = 0

−B2

π2k

"

#$

%

&'2 k odd

0 k even

(

)

****

+

****

24

13


Transfer function of filters – Ram-Lak

25/x

25


Shepp-Logan filter

Transfer function:

ℎ 𝜌 = ρ𝑠𝑖𝑛 2𝜋𝜌4𝐵2𝜋𝜌4𝐵

, ρ ≤ 𝐵

0 𝑒𝑙𝑠𝑒

Impulse response

26/x

h(k) = − 8B2

π (4k2 −1)

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14


Shepp-Logan filter

27/x

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Hanning weighted filter

28/x

28

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Hanning weighted filter

29/x

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Filter transfer functions and impulse responses

30/x

30

16


Comparison between filters

31/x

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Circular convolution

32/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

32

17


Circular convolution

33/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

0.02h(

n)

Filter

−100 −80 −60 −40 −20 0 20 40 60 800

5

10

15

20

g 1(n)

Periodic time signal

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Circular convolution – Shepp-Logan

34/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

34

18


Backprojection

35/x

Patientgrid

Projec

tion

35


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

36/x

36

19


Assignment data

• DICOM data from female patient

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

37/x

Data and program in: undervisning/k_22485_31545_billeder/ct_data/dicom_data

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Reading DICOM data

38/x

% Set overall parametersdir_name='DICOM/ST00001/SE00001/'; % Directory namestart_image=1; % First image in seriesend_image=747; % Last image in seriesframe_rate=50; % Frame rate for playing back the movie

% Set the dynamic range for the displayoff_set=100; % Offset [Hu]range=400; % Range to display [Hu]map_values=128; % Number of gray level valuesbone_off_set= -250; % Offset for showing the bonesbone_range=100; % Range for showing the bones

% Initialize figurecolormap(gray(map_values));dicom_movie(end_image+1-start_image) = struct('cdata',[],'colormap',[]);

% Read information for the first imagesfile_name='IM00001';info=dicominfo([dir_name, file_name]);dx=info.PixelSpacing(1);dy=info.PixelSpacing(2);Y = dicomread(info);[Nx,Ny]=size(Y);

% Make space for all the imagesY=zeros(Nx,Ny,end_image+1-start_image);z_positions=zeros(end_image+1-start_image,1);

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20


% Loop through the images and read and display them

for i=start_image:end_imagefile_name=['IM00',num2str(floor(i/100)),num2str(floor(rem(i,100)/10)),num2str(rem(i,10))];info=dicominfo([dir_name, file_name]);Y(:,:,i) = dicomread(info);image(((1:Nx)-Nx/2)*dx,((1:Ny)-Ny/2)*dy,(double(Y(:,:,i))+off_set)/range*map_values)xlabel('Distance [mm]')ylabel('Distance [mm]')pos=sprintf('%5.1f, %5.1f, %5.1f', info.ImagePositionPatient(1), …

info.ImagePositionPatient(2), info.ImagePositionPatient(3));z_positions(i)= info.ImagePositionPatient(3);text(-150, -150, ['Image ', num2str(i),', Pos. (x,y,z) = ', pos,' mm'], 'Color', [1 1 1])axis('image')drawnowdicom_movie(i)=getframe;

end

% Display the movie

movie(dicom_movie, 5, frame_rate);

Full script can be found at:

courses.healthtech.dtu.dk/22485/files/ct_data/dicom_data/display_dicom_images.m

on the page for the CT data: courses.healthtech.dtu.dk/22485/?ct_data/assign_data.html

Script in: ct_data/dicom_data

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https://courses.healthtech.dtu.dk/22485/files/ct_data/dicom_data/display_dicom_images.m

reconstruction in ct and hints to the assignments

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