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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

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This program is about 3D cone-beam CT for flat detector.
This is made for students who start to learn the CT medical imaging.
GPU option:
- I provide the projection, back-projection codes using built-in GPU functions (semi-GPU codes)
- Above version Matlab 2013b with parallel computing toolbox are needed.
- 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.

- FDK ( known as filtered backprojection)
- MLEM (Maximum Likelihood Expectation Maximization),
- SART (Simultaneous Algebraic Reconstruction Technique)
- SQS (Separable Quadratic Surrogates).

You can run:
0. Please check "ParamSetting.m" carefully.
1. Make a measurement: MeasurementGen.m: F5
    1.1 Poisson Noise can be imposed.
2. Reconstruction examples
    2.1 FDK
    2.2 MLEM
    2.3 SART
    2.4 SQS

Please see this:!4249&authkey=!AJPJM8_qtvrhuGc&ithint=file%2cpdf

Thank you

Comments and Ratings (47)


121 (view profile)

To use this package,
1. Extract the ZIP file to a folder and add that folder to your path.
2. Run MeasurementGen(). This creates the proj.mat file which is missing when you try to run the other files.

Hany Kashani

Hi all I dont seem to find the main Demo.m code , could someone please advise?

Carla Romano

I put in input a matrix 138*157*101. The only way to run MeasurementGen.m is to impose in param setting x=157 and y=138. Is it normal? After that the code back projection gave me this error
Error in backprojection (line 25)
        vol(:,:,iz) = (Ratio.*interp2(proj',pu,pv,param.interptype));
Could someone help me?

Ka Mirul

Hi Aubrey, you should convert your projection image to double or single data format. When you have image input yo can use imread and then stack it into 3D data.

Hello All,
What format should i insert 'img' as?
I have the series of 2D image projections as TIF

Matheus Silva

Hello, Kyungsang Kim!

Could you refer me to the paper where you studied to implement the MLEM?

Thanks in advance.

hyt 1220

Who knows which paper this file refered to?


thanks you for the codes!
it`s helpful for me!


dect (view profile)

nice program. just wonder the following line in filtering.m

proj(:,:,i) = fproj(end/,:)/2/param.du*(2*pi/param.nProj)/2*(param.DSD/param.DSO);

can you explain why we need the last several terms instead of just fproj(end/,:)?



sigrid (view profile)


Thank you for the code.
Is there beam hardening correction algorithm?

bo zhou

The projection x-ray is mono-energitic or poly-energitic spectrum and what with kVp and mAs?

Thank you so much

Huang YanYan

good job!besides I wonder if you can tell me which paper did your work refer to ?

Mu Sun

Mu Sun (view profile)

good code.

Ander Biguri

Ander Biguri (view profile)

Note that the projection implementation has an asymptotically increasing error when it reaches to vertical edges at 45 degrees. This is due to the fact that the way to project and back project is to "stretch" the image to transform the cone to a parallel, thus making the closest single voxels (at 45) incredibly wide. You can test this by projecting a single image with a full white "cube image" at 45 degrees. Strange artifacts arise. This is less relevant the closer to the centre of the image one is. The toolbox is still great to learn about medical imaging, but this error makes it (unfortunately) not a good toolbox for research purposes or real image reconstruction.

Nick Tsui

You need to run MeasurementGen.m first to generate the proj.mat. Could have been a lot less confusing if proj.mat is pre-generated in the folder. Thanks though.

Nick Tsui

Where is your proj.mat file? cannot find it?

Farid Z

Dear Kyungsang, may I ask two questions?
1- In comments you mentioned page 96 Jiang Hsieh 2nd edition; however, I cannot follow the code in projection.m, i.e., how it is representing the formula 3.59. May I ask for your hints on it?
2- How can we reach from a CT images of a patient to the sample file which you have provided?
Thank you.


yw628 (view profile)

Sorry my mistake, I found proj.mat


yw628 (view profile)


Where is proj.mat file?

Diego Garcia

Great contribution, thank you very much for the code.

Great contribution for others!
Thank you Kyungsang Kim!


Anton (view profile)

Really cool! Thank you for this code! Especially I like to see a real phantom and animated reconstructions.


Seta (view profile)

Thank you Kyung :-)

Kyungsang Kim

Kyungsang 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.


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.


mat (view profile)


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!

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

Kyungsang Kim

Kyungsang 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.


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.

Kyungsang Kim

Kyungsang 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.



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.


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.



erick (view profile)

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


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.

Kyungsang Kim

Kyungsang 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.

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

Tai Chieh



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!

Kyungsang Kim

Kyungsang 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".

Kyungsang Kim

Kyungsang 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.

Kyungsang Kim

Kyungsang 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, :)



Add descriptions


Update descriptions


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


Fixed filtering function


Debug and add some descriptions


Update code and debug


I modified codes simply.


Change description


I changed title, tag


Fixed a bug


I optimized the source codes

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