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. The projection of an object, resulting from the tomographic measurement process at a given angle formula_1, 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 formula_2. 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 formula_3, across a projection at angle formula_1. This is repeated for various angles. Attenuation occurs exponentially in tissue: where formula_2 is the attenuation coefficient as a function of position. Therefore, generally the total attenuation  formula_7 of a ray at position formula_3, on the projection at angle formula_1, is given by the line integral: Using the coordinate system of Figure 1, the value of formula_3 onto which the point formula_12 will be projected at angle formula_1 is given by: So the equation above can be rewritten as where formula_16 represents formula_2 and formula_18 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 formula_19 where formula_20 formula_21 represents a slice of the 2D Fourier transform of  formula_16 at angle formula_1. Using the inverse Fourier transform, the inverse Radon transform formula can be easily derived. formula_24 where formula_25 is the derivative of the Hilbert transform of formula_26 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, formula_16. However, there will only be a finite number of projections available in practice. Assuming formula_16 has effective diameter formula_29 and desired resolution is formula_30 , rule of thumb number of projections needed for reconstruction is formula_31 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 formula_32-projections of formula_16 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: formula_34 where formula_35 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 formula_36 formula_37 where formula_38 is the angular spacing between the projections and formula_39 is radon kernel with frequency response formula_40. 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 formula_16. Let formula_32 be the number of projections, formula_43 be the distortion operator for formula_44th projection taken at an angle formula_45. formula_46  are set of parameters to optimize the conversion of iterations. formula_47 formula_48 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. Deep learning reconstruction. 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 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. However, incorrect structures may occur in an image reconstructed by such a completely data-driven method, 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. For example, direct image reconstruction from projection data can be learnt from the framework of filtered back-projection. Another example is to build neural networks by unrolling iterative reconstruction algorithms. Except for precision learning, using conventional reconstruction methods  with deep learning reconstruction prior is also an alternative approach to improve the image quality of deep learning reconstruction. Tomographic reconstruction software. For flexible tomographic reconstruction, open source toolboxes are available, such as PYRO-NN, TomoPy, CONRAD, 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 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. 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.