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3D Cone beam CT (CBCT) projection backprojection FDK, iterative reconstruction Matlab examples

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3D Cone beam CT (CBCT) projection backprojection FDK, iterative reconstruction Matlab examples


Kyung Sang Kim (view profile)

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10 Mar 2012 (Updated )

3D Cone beam CT (CBCT) projection backprojection FDK, iterative reconstruction Matlab examples

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My name is Kyungsang Kim.
This program is about 3D cone-beam CT.
This is made for students who learn the medical imaging.
I provide the projection, back-projection codes using built-in GPU functions (semi-GPU codes)
(above 2013b with parallel computing toolbox; you can switch on/off the usage of GPU; The computational time of this code is several times faster than CPU-based code. Speed depends on your GPU.)
I also provide several reconstruction examples such as FDK, MLEM (Maximum Likelihood Expectation Maximization), SART (Simultaneous Algebraic Reconstruction Technique) and SQS (Separable Quadratic Surrogates).

Please let me know if the program has problems.
(E-mail: kssigari(at)gmail)

Thank you

Required Products Parallel Computing Toolbox
Statistics Toolbox
MATLAB release MATLAB 8.2 (R2013b)
MATLAB Search Path
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Comments and Ratings (21)
11 Mar 2015 Seta

Seta (view profile)

Thank you Kyung :-)

10 Mar 2015 Kyung Sang Kim

Kyung Sang Kim (view profile)

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Hi Seta,

Please do as following:

1) Open "ParamSetting.m"
2) Change to "param.gpu = 0"

Then, please try again.
This option needs "Parallel Computing toolbox" and "GPU (Hardware)",
If you have a GPU, please check memory of GPU and number of cores.
Also, it would be useful above 2013b version.

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09 Mar 2015 Seta

Seta (view profile)

Thank you for the code. Is there a description file to explain what each file does and how to run the code please?

When I run Recon1_FDK or MeasurmentGen, they both come up with errors saying some file or function is missing or undefined. More specifically, when running Recon1_FDK error suggests proj.mat does not exist and when I run MeasurementGen the error suggest gpuArray (called in Projection.m) is undefined.

Amny thanks for assistance in advance.

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12 Jun 2014 mat

mat (view profile)

29 May 2014 Hamdi Mahdjoub  
28 May 2014 Bella

Bella (view profile)

Would it be possible to use this program to forward project a 3D image where dx=dy, but dx=dy!=dz? Right now, my parameters are:

%% Parameter setting %%

param.nx = 400;
param.ny = 400; = 66;

%The real detector panel pixel density (number of pixels) = 400;
param.nv = 400;

% Detector setting, according to Varian Trilogy OBI (real size) = 350; % mm = 350; % mm

% X-ray source and detector setting
param.DSD = 1500; % Distance source to detector
param.DSO = 1000; % X-ray source to object axis distance

% This confusing stuff
% (1000/1500)*(350/400) = param.nx*res; % mm = param.ny*res; % mm =*res; % mm

but this doesn't create the correct forward projections, so far as I have been able to check. Am I trying to do something that is impossible in image processing?

Thank you!

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24 May 2014 Rama Teja A S

Please include analytical equation for better understanding. Or please explain the algorithm in projection.m. If possible please include analytical equation references

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11 Nov 2013 Kyung Sang Kim

Kyung Sang Kim (view profile)

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Hi, Erick,

That's a good point.

This factor is used in the weighted backprojection for fan-beam and cone-beam geometries.
(see Jiang Hsieh 2nd edition, Figs 3.35 (reason) and 3.37(geometry))

Result is similar compared to the previous one, In real geometry, this factor is not critical factor for the image, because CT rotates 360 deg (average factor ~ 1) as well as itself is close to 1.

However, I think I omitted it for fast calculation, soon I will fix it for accuracy.

Thanks a lot.

ps) you can directly send e-mail to me, it is sometimes difficult to check comments here.

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11 Nov 2013 erick

erick (view profile)

Hi, I compared you work with the FDK equation in some papers, it seems that your FDK algorithm omited a factor. I added this factor in the "backprojection.m" of your algorithm, the code is shown as follows.

vol(:,:,iz) = Ratio.*interp2(uu,vv ,proj',pu,pv,'linear');

Do you think it is necessary to add this factor? I am waiting for your reply, Thank you.

28 Oct 2013 Kyung Sang Kim

Kyung Sang Kim (view profile)

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Hi Erick,

First Question:

If you have real projections ("RealProj"), then first calculate:

1. I0 = max(RealProj(:)); % if you don't know I0
2. Proj = -log(RealProj/I0);

Here, "I0" is the blank image (intensity of source) as mentioned in the text book.
If you think that the final image has a problem, check 1) "RealProj" has 0 values or 2) "I0" is too high than background values.

Solution of cases

1) step 3. Proj(isinf(Proj)) = 0;

2) step 1. Define I0 as a background (high intensity) average.
step 2. Proj = -log(min(RealProj/I0,1));

Then, you can execute FDK or MLEM as in Demo.m.

Second Question:

Parameter setting is very important. Image resolution is defined by detector size and distances of DSO, DSO. In your case, DSD is 700 and DSO is 560. so ratio is 560/700 = 0.8. Then detector size of a pixel is 50/1024, so maximum resolution of a voxel is 50/1024*0.8.

Now, voxel resolution "dx=dy=dz" is defined. then define the number of voxels that covers whole images (nx, ny, nz).
Thus, sx = nx*dx, sy=ny*dy, sz=nz*dz;

Last Question:
I modified just for fast calculation. But if you can see Jiang Hsieh's book (second edition,page 96), it is easy to understand.


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28 Oct 2013 erick

erick (view profile)

Thank you for your answers. I have a few questions left.
If I have already got real projections from x-ray imaging system, for example, I got the projections of a mouse, do I need to do the "CTprojection"? Under this circumstance, if I only need to do "CTbackprojection", how to set following params about the reconstructed object:nx, ny, nz, sx, sy, and sz.
The params of the detector are setted as follows:;%mm;%mm;

Last question, would you please give some reference about FDK algorithm not using iso-center(DSO) domain.

Thank you.

27 Oct 2013 Kyungsang

Hi Erick,

1. w = cos(theta), to calculate w, my code considers distances in detectors, as your comment, many books calculate in iso-center (DSO) domain. In this case, uu and vv should be differently calculated by considering distance ratio from source to center and source to detector. Please think about angle between iso-center line and extension line source-voxel-detector.

2. In my code, pi/2 rotate shift is for calculating exactly your comment. "interp2" function results in 90 degree rotated image. So I did "angle" -> "angle-pi/2", and I think DSO+ry is right.

3. If you want to get as real projection data, first set blank scan intensity "b", this is related to "dose", then measurement M = b*exp(-projection(obj)).

Thanks your comment.

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27 Oct 2013 erick

erick (view profile)

I am wondering if I have real projected images, do I need to use the function "CTprojection"?

27 Oct 2013 erick

erick (view profile)

Good work. I have some questions.
Firstly, in the file of "backprojection.m", you set "w = (param.DSD)./sqrt((param.DSD)^2+uu.^2 + vv.^2);", but in many books, they said that w should be setted as "w = (param.DSO)./sqrt((param.DSO)^2+uu.^2 + vv.^2);". Can you give me some reason?
Secondly, also in the "backprojection.m", I think the description of rx, ry, pu and pv should be:

Would you please give me some suggestion, thank you.

04 Apr 2013 Kyung Sang Kim

Kyung Sang Kim (view profile)

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Hi, Masoud Hashemi,
This program is for flat detectors,
I think you should modify for the arc detector geometry,
Thanks for using my software.

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30 Mar 2013 Masoud Hashemi

I am wondering if the projection function is for Arc detectors (3rd Generation) or Flat detectors?

02 Mar 2013 Tai Chieh


16 Dec 2012 ljpzgx

ljpzgx (view profile)

The codes can run but the result is not very good,can you upload the referrences you refer for the codes,so tha we can understand clearly. Thank you!

13 Mar 2012 Kyung Sang Kim

Kyung Sang Kim (view profile)

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This program is similar to other programs but
much faster.

To accelerate speed,
(1) define the reconstruction object and projection images as a "single" precision,
and then (2) use "matlabpool".

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12 Mar 2012 Kyung Sang Kim

Kyung Sang Kim (view profile)

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You can use "matlabpool",


for i=1:n (angle)
projection or backprojection


parfor i=1:n (angle)
projection or backprojection
!!please delete figure plot functions

This increases the performance more than twice in quad cores.

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12 Mar 2012 Kyung Sang Kim

Kyung Sang Kim (view profile)

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Please change the function code "projection_pixel.m":


dist = (SAD+SDD)./sqrt((SAD+SDD)^2 + uu.^2 + vv.^2)*abs(ys(2)-ys(1));


dist = sqrt((SAD+SDD)^2 + uu.^2 + vv.^2)./(SAD+SDD)*abs(ys(2)-ys(1));

Thank you, :)

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20 Mar 2012

I optimized the source codes

16 Nov 2012

Fixed a bug

13 Dec 2012

I changed title, tag

14 Jan 2013

Change description

24 Apr 2013

I modified codes simply.

17 Dec 2013

Update code and debug

28 Feb 2014

Debug and add some descriptions

25 Oct 2014

Fixed filtering function

18 Feb 2015

Examples updated, Matlab built-in function based GPU code updated, Parker weight updated.

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