Tomographic reconstruction

Tomographic reconstruction is a type of multidimensional inverse problem where the challenge is to yield an estimate of a specific system from a finite number of projections. The mathematical basis for tomographic imaging was laid down by Johann Radon. A notable example of applications is the reconstruction of computed tomography (CT) where cross-sectional images of patients are obtained in non-invasive manner. Recent developments have seen the Radon transform and its inverse used for tasks related to realistic object insertion required for testing and evaluating computed tomography use in airport security.

This article applies in general to reconstruction methods for all kinds of tomography, but some of the terms and physical descriptions refer directly to the reconstruction of X-ray computed tomography.

Introducing formula

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 $\theta$ , 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 $\mu (x,y)$ . 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 $r$ , across a projection at angle $\theta$ . This is repeated for various angles. Attenuation occurs exponentially in tissue:

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

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

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

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

$x\cos \theta +y\sin \theta =r\$

So the equation above can be rewritten as

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

where $f(x,y)$  represents $\mu (x,y)$  and $\delta ()$  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

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

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

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

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

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, $f(x,y)$ . However, there will only be a finite number of projections available in practice.

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

Reconstruction algorithms

Practical reconstruction algorithms have been developed to implement the process of reconstruction of a 3-dimensional object from its projections. 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.

Fourier-Domain Reconstruction Algorithm

Reconstruction can be made using interpolation. Assume $N$ -projections of $f(x,y)$  are generated at equally spaced angles, each sampled at same rate. The Discrete Fourier transform on each projection will yield sampling in the frequency domain. Combining all the frequency-sampled projections would generate a polar raster in the frequency domain. The polar raster will be sparse so interpolation is used to fill the unknown DFT points and reconstruction can be done through inverse Discrete Fourier transform. Reconstruction performance may improve by designing methods to change the sparsity of the polar raster, facilitating the effectiveness of interpolation.

For instance, a concentric square raster in the frequency domain can be obtained by changing the angle between each projection as follow:

$\theta '={\frac {R_{0}}{\max\{|\cos \theta |,|\sin \theta |\}}}$

where $R_{0}$  is highest frequency to be evaluated.

The concentric square raster improves computational efficiency by allowing all the interpolation positions to be on rectangular DFT lattice. Furthermore, it reduces the interpolation error. Yet, the Fourier-Transform algorithm has a disadvantage of producing inherently noisy output.

Back Projection Algorithm

In practice of tomographic image reconstruction, often a stabilized and discretized version of the inverse Radon transform is used, known as the filtered back projection algorithm.

With a sampled discrete system, the inverse Radon Transform is

$f(x,y)={\frac {1}{2\pi }}\sum _{i=0}^{N-1}\Delta \theta _{i}g_{\theta _{i}}(x\cos \theta _{i}+y\sin \theta _{i})$

$g_{\theta }(t)=p_{\theta }(t)\cdot k(t)$

where $\Delta \theta$  is the angular spacing between the projections and $k(t)$  is radon kernel with frequency response $|\omega |$ .

The name back-projection comes from the fact that 1D projection needs to be filtered by 1D Radon kernel (back-projected) in order to obtain a 2D signal. The filter used does not contain DC gain, thus adding DC bias may be desirable. Reconstruction using back-projection allows better resolution than interpolation method described above. However, it induces greater noise because the filter is prone to amplify high-frequency content.

Iterative Reconstruction Algorithm

Iterative algorithm is computationally intensive but it allows to include a priori information about the system $f(x,y)$ .

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

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

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.

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

An alternative family of recursive tomographic reconstruction algorithms are the Algebraic Reconstruction Techniques and iterative Sparse Asymptotic Minimum Variance.

Fan-Beam Reconstruction

Use of a noncollimated fan beam is common since a collimated beam of radiation is difficult to obtain. Fan beams will generate series of line integrals, not parallel to each other, as projections. The fan-beam system will require 360 degrees range of angles which impose mechanical constraint, however, it allows faster signal acquisition time which may be advantageous in certain settings such as in the field of medicine. Back projection follows a similar 2 step procedure that yields reconstruction by computing weighted sum back-projections obtained from filtered projections.

Tomographic reconstruction software

For flexible tomographic reconstruction, open source toolboxes are available, such as TomoPy, ODL, the ASTRA toolbox, and TIGRE. 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, 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. The ASTRA Toolbox is a MATLAB and Python toolbox of high-performance GPU primitives for 2D and 3D tomography, from 2009–2014 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. Recently, the ASTRA toolbox has been integrated in the TomoPy framework. 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.

Gallery

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