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I have some CT images for some rock, and I want to build a 3D image out of these images. This is the first time I work on image processing, and constructing 3D image out of CT 2D images sounds complicated for me. I've searched a lot, after searching I've ended up with these thoughts,

  1. To construct a 3D image from 2D one you need to find depth map which indicates how far or near an object is to the camera. Then this map will help constructing the 3D image (I don't know how, and I didn't find a clear resource about that).

  2. There exist multiple algorithms for such a reconstruction, like analytical, iterative, and statistical. After searching it seems to me that the last two types are better than the first one.

I've read too many resources like research papers and survey papers, but they were very technical and I couldn't connect the information mentioned within them.

Here are few results of my searching process,

Getting 3D image from 2D image

https://www.aapm.org/meetings/02AM/pdf/8372-23331.pdf

https://en.wikipedia.org/wiki/Iterative_reconstruction

None of the above resources explains the process in a simple way.

I know that there are some software that can do this job, but before using any, I want to understand the algorithm mathematically itself.

Can anyone please explain in steps how the algorithm of 3D image reconstruction from 2D CT images work, specially for the iterative and statistical methods?

Community
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Dania
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    If these are 2D slices through a 3D volume, you need to google for "volume rendering" as this is a different technique and has plenty of resources available. For example, [look at VTK](http://www.vtk.org/). – Roger Rowland Jul 28 '15 at 08:46
  • Thanks Roger Rowland, by 2D slices do you mean that the CT images should be taken every fixed distance like 1mm? – Dania Jul 28 '15 at 08:51
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    How dense are the slices compared to the 2D resolution ? Can you show us a few of them ? –  Jul 28 '15 at 09:00
  • Thanks Yves, the images are not my own, they are confidential, so unfortunately I can't show them. I just want an explanation of one of the well know algorithm so that I can understand the process more. – Dania Jul 28 '15 at 09:03

2 Answers2

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From your description I deduce that you have already processed data as 2D images(slices of 3D object). Reconstruction of CT images is performed by CT device manufacturers software and this is a process of transforming raw data gathered by detectors into 3D matrix of voxels using algorithms like for example backprojection.

Images that you have are 2D slices of that 3D voxels matrix. To reproduce 3D image from that 2D data you only need distance between slices(slices thickness) and size of 2D image pixel. If You have DICOM image there is a tag: Slice Thickness representing slice thickness in mm.

If you want to understand reconstruction process I suggest to start with Iterative reconstruction methods. Subject is to broad to elaborate in single post so I provide some links where you could start:

http://rsta.royalsocietypublishing.org/content/373/2043/20140399

http://www.dspguide.com/ch25/5.htm

Springfield762
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    Seems helpful, thanks a lot. But, before a use any software I want to understand the process from the aspect of mathematics and algorithms. Can you please elaborate more on one of the algorithms? For example how the raw data is converted into voxels? Thanks. – Dania Jul 28 '15 at 09:01
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    CT images are not slices of the whole volume, they are projections of it onto a plane which is what makes the reconstruction complicated. – ltjax Jul 28 '15 at 09:23
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    Subject is to broad to elaborate in single post I've edited my answer. I added some links where you could start. – Springfield762 Jul 28 '15 at 09:23
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From your description, it looks like you search for Computed Tomography or simply CT Scan. In this approach an object of interest is rotated inside the scanner (or the scanner's source and detector are rotated around the object). At every small angle increment a 2D x-ray image is taken (also known as radiograph or x-ray projection). At the end of scanning we have a set of 2D images, which must be converted in 3D volumetric image.

The most popular algorithm for solving this task is Filtered Back Projection or simply FBP.

In short, FBP algorithm consists of two major steps:

  1. Filtering. Every row of the image is filtered to amplify high frequencies.
  2. Back projection. Every filtered image is smeared back on the volume following the trajectories of x-rays.

You can find more details about FBP in the book Principles of Computerized Tomographic Imaging, see chapter 3.

As to existing implementations, there are many. Take for example the Reconstruction Toolkit RTK - an open-source and cross-platform software for fast circular cone-beam CT reconstruction.

Fedor
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