Tomographic reconstruction

Figure 1: Parallel beam geometry utilized in tomography and tomographic reconstruction. Each projection, resulting from tomography under a specific angle, is made up of the set of line integrals through the object.

Resulting tomographic image from a plastic skull phantom. Projected X-rays are clearly visible on this slice taken with a CT-scan as image artifacts, due to limited amount of projection slices over angles.

The projection of an object, resulting from the tomographic measurement process at a given angle ${\displaystyle \theta }$, is made up of a set of line integrals (see Fig. 1). A set of many such projections under different angles organized in 2D is called sinogram (see Fig. 3). In X-ray CT, the line integral represents the total attenuation of the beam of x-rays as it travels in a straight line through the object. As mentioned above, the resulting image is a 2D (or 3D) model of the attenuation coefficient. That is, we wish to find the image ${\displaystyle \mu (x,y)}$. The simplest and easiest way to visualise the method of scanning is the system of parallel projection, as used in the first scanners. For this discussion we consider the data to be collected as a series of parallel rays, at position ${\displaystyle r}$, across a projection at angle ${\displaystyle \theta }$. This is repeated for various angles. Attenuation occurs exponentially in tissue:

${\displaystyle I=I_{0}\exp \left({-\int \mu (x,y)\,ds}\right)}$

where ${\displaystyle \mu (x,y)}$ is the attenuation coefficient as a function of position. Therefore, generally the total attenuation ${\displaystyle p}$ of a ray at position ${\displaystyle r}$, on the projection at angle ${\displaystyle \theta }$, is given by the line integral:

${\displaystyle p_{\theta }(r)=\ln \left({\frac {I}{I_{0}}}\right)=-\int \mu (x,y)\,ds}$

Using the coordinate system of Figure 1, the value of ${\displaystyle r}$ onto which the point ${\displaystyle (x,y)}$ will be projected at angle ${\displaystyle \theta }$ is given by:

${\displaystyle x\cos \theta +y\sin \theta =r\ }$

So the equation above can be rewritten as

${\displaystyle p_{\theta }(r)=\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }f(x,y)\delta (x\cos \theta +y\sin \theta -r)\,dx\,dy}$

where ${\displaystyle f(x,y)}$ represents ${\displaystyle \mu (x,y)}$ and ${\displaystyle \delta ()}$ is the Dirac delta function. This function is known as the Radon transform (or sinogram) of the 2D object.

The Fourier Transform of the projection can be written as

${\displaystyle P_{\theta }(\omega )=\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }f(x,y)\exp[-j\omega (x\cos \theta +y\sin \theta )]\,dx\,dy=F(\Omega _{1},\Omega _{2})}$ where ${\displaystyle \Omega _{1}=\omega \cos \theta ,\Omega _{2}=\omega \sin \theta }$[2]

${\displaystyle P_{\theta }(\omega )}$ represents a slice of the 2D Fourier transform of ${\displaystyle f(x,y)}$ at angle ${\displaystyle \theta }$. Using the inverse Fourier transform, the inverse Radon transform formula can be easily derived.

${\displaystyle f(x,y)={\frac {1}{2\pi }}\int \limits _{0}^{\pi }g_{\theta }(x\cos \theta +y\sin \theta )d\theta }$

where ${\displaystyle g_{\theta }(x\cos \theta +y\sin \theta )}$ is the derivative of the Hilbert transform of ${\displaystyle p_{\theta }(r)}$

In theory, the inverse Radon transformation would yield the original image. The projection-slice theorem tells us that if we had an infinite number of one-dimensional projections of an object taken at an infinite number of angles, we could perfectly reconstruct the original object, ${\displaystyle f(x,y)}$. However, there will only be a finite number of projections available in practice.

Assuming ${\displaystyle f(x,y)}$ has effective diameter ${\displaystyle d}$ and desired resolution is ${\displaystyle R_{s}}$, rule of thumb number of projections needed for reconstruction is ${\displaystyle N>\pi d/R_{s}}$[2]

Practical reconstruction algorithms have been developed to implement the process of reconstruction of a three-dimensional object from its projections.[3][2] These algorithms are designed largely based on the mathematics of the Radon transform, statistical knowledge of the data acquisition process and geometry of the data imaging system.

Iterative reconstruction algorithm

Main article: Iterative reconstruction

The iterative algorithm is computationally intensive but it allows the inclusion of a priori information about the system ${\displaystyle f(x,y)}$.[2]

Let ${\displaystyle N}$ be the number of projections and ${\displaystyle D_{i}}$ be the distortion operator for the ${\displaystyle i}$th projection taken at an angle ${\displaystyle \theta _{i}}$. ${\displaystyle \{\lambda _{i}\}}$ are a set of parameters to optimize the conversion of iterations.

${\displaystyle f_{0}(x,y)=\sum _{i=1}^{N}\lambda _{i}p_{\theta _{i}}(r)}$

A fan-beam reconstruction of Shepp-Logan Phantom with different sensor spacing. Smaller spacing between the sensors allow finer reconstruction. The figure was generated by using MATLAB.

${\displaystyle f_{k}(x,y)=f_{k-1}(x,y)+\sum _{i=1}^{N}\lambda _{i}[p_{\theta _{i}}(r)-D_{i}f_{k-1}(x,y)]}$

An alternative family of recursive tomographic reconstruction algorithms are the algebraic reconstruction techniques and iterative sparse asymptotic minimum variance.

Deep learning reconstruction

The influence of Poisson noise in deep learning reconstruction where Poisson noise causes the U-Net fail to reconstruct an existing high contrast lesion-like object.

Deep learning methods are widely applied to image reconstruction nowadays and have achieved impressive results in various image reconstruction tasks, including low-dose denoising, sparse-view reconstruction, limited angle tomography and metal artifact reduction. An excellent overview can be found in the special issue [5] of IEEE Transaction on Medical Imaging. One group of deep learning reconstruction algorithms apply post-processing neural networks to achieve image-to-image reconstruction, where input images are reconstructed by conventional reconstruction methods. Artifact reduction using the U-Net in limited angle tomography is such an example application.[6] However, incorrect structures may occur in an image reconstructed by such a completely data-driven method,[7] as displayed in the figure. Therefore, integration of known operators into the architecture design of neural networks appears beneficial, as described in the concept of precision learning.[8] For example, direct image reconstruction from projection data can be learnt from the framework of filtered back-projection.[9] Another example is to build neural networks by unrolling iterative reconstruction algorithms.[10] Except for precision learning, using conventional reconstruction methods with deep learning reconstruction prior [11] is also an alternative approach to improve the image quality of deep learning reconstruction.

For flexible tomographic reconstruction, open source toolboxes are available, such as PYRO-NN,[12] TomoPy,[13] CONRAD,[14] ODL, the ASTRA toolbox,[15][16] and TIGRE.[17] TomoPy is an open-source Python toolbox to perform tomographic data processing and image reconstruction tasks at the Advanced Photon Source at Argonne National Laboratory. TomoPy toolbox is specifically designed to be easy to use and deploy at a synchrotron facility beamline. It supports reading many common synchrotron data formats from disk through Scientific Data Exchange,[18] and includes several other processing algorithms commonly used for synchrotron data. TomoPy also includes several reconstruction algorithms, which can be run on multi-core workstations and large-scale computing facilities.[19] The ASTRA Toolbox is a MATLAB and Python toolbox of high-performance GPU primitives for 2D and 3D tomography, from 2009 to 2014 developed by iMinds-Vision Lab, University of Antwerp and since 2014 jointly developed by iMinds-VisionLab (now imec-VisionLab), UAntwerpen and CWI, Amsterdam. The toolbox supports parallel, fan, and cone beam, with highly flexible source/detector positioning. A large number of reconstruction algorithms are available through TomoPy and the ASTRA toolkit, including FBP, Gridrec, ART, SIRT, SART, BART, CGLS, PML, MLEM and OSEM. In 2016, the ASTRA toolbox has been integrated in the TomoPy framework.[20] By integrating the ASTRA toolbox in the TomoPy framework, the optimized GPU-based reconstruction methods become easily available for synchrotron beamline users, and users of the ASTRA toolbox can more easily read data and use TomoPy's other functionality for data filtering and artifact correction.

Shown in the gallery is the complete process for a simple object tomography and the following tomographic reconstruction based on ART.

• 
  Fig. 2: Phantom object, two kitty-corner squares.
• 
  Fig. 3: Sinogram of the phantom object (Fig.2) resulting from tomography. 50 projection slices were taken over 180 degree angle, equidistantly sampled (only by coincidence the x-axis marks displacement at -50/50 units).
• 
  Fig.4: ART based tomographic reconstruction of the sinogram of Fig.3, presented as animation over the iterative reconstruction process. The original object could be approximatively reconstructed, as the resulting image has some visual artifacts.

• Operation of computed tomography#Tomographic reconstruction
• Cone beam reconstruction
• Industrial CT scanning
• Industrial Tomography Systems

• Avinash Kak & Malcolm Slaney (1988), Principles of Computerized Tomographic Imaging, IEEE Press, ISBN 0-87942-198-3.
• Bruyant, P.P. "Analytic and iterative reconstruction algorithms in SPECT" Journal of Nuclear Medicine 43(10):1343-1358, 2002

• Slaney, A. C. Kak and Malcolm. "Principles of Computerized Tomographic Imaging". Slaney.org. Retrieved 7 September 2018.
• Insight ToolKit; open source tomographic support software
• "TomoPy — TomoPy 1.1.3 documentation". Tomopy.readthedocs.org. Retrieved 7 September 2018.
• ASTRA (All Scales Tomographic Reconstruction Antwerp) toolbox; very flexible, fast and open source software for computed tomographic reconstruction
• NiftyRec; comprehensive open source tomographic reconstruction software; Matlab and Python scriptable
• Open-source tomographic reconstruction and visualization tool
• "ITS plc - Electrical Process Tomography For Industrial Visualization". Itoms.com. Retrieved 7 September 2018.