Xiaoli Yang, Halil Pasic, Danah Tonne

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In the ultrafast tomography imaging, fewer projections are actually obtained because of the high rotation speed of the sample. This may cause a decline in reconstruction image quality if the established reconstruction method, filtered back-projection (FBP), is applied. As the FBP method can only be applied when the object is densely sampled. Otherwise, severe aliasing artifacts such as sharp streaks could appear. Thus, we resort to another entirely different tomographic reconstruction method from FBP called algebraic reconstruction technique. It is easier to integrate the prior knowledge obtained from the data acquisition procedure and sample in the experiment.

The Radon transform is the mathematical description of signal formation in computed tomography. A ray in the forward model is generally a straight line going through the object, and projection measured with this ray is obtained from the integral of a function over the straight line. However, this approximations do not hold for typical CT experiments since the sensors in the camera are of some area and separated by gaps between each other. Taking the sensor geometry into account, it is more appropriate to describe the rays as having a finite width and a given distance. Therefore, a new ray model in ART possesses two features: 1) a ray is of a certain width $\tau$; 2) a distance exists between the neighbor rays. To illustrate this, shaded rays are showed in the following figure.

Generally, ART assumes that the cross section of the object consists of an array of  unknowns representing the X-ray absorption coefficients of the object, denoted by $s_j, (j=1,2,\dots,N)$. A sum version of the line integral is used here to calculate these projections. The intersection area represents the contribution of image cell $s_j$ to the $i^{\mbox{\footnotesize{th}}}$ ray projection. It is expressed as a weighting factor $a_{ij}$, defined as
$a_{ij} = \frac{\mbox{intersection--area}}{\mbox{cell--area}}$.

Let $p_i$ be the projection measured with the $i^{\mbox{\footnotesize{th}}}$ ray which equals
$p_i = \sum_{j=1}^{N}{a_{ij}s_j}, \quad i=1,2,\dots,M$,
where $M$ is the total number of rays. These $M$ projection equations build up a equation system,
which can be simply written as a matrix form
$p=\mathbf{A}s$,
where $s =\{ s_1, s_2, \dots, s_N\}$ is the image vector; $p =\{p_1, p_2, \dots, p_M\}$ is the set of all measured projections; $\mathbf{A}$ is viewed as the system transform matrix of the object image to the measurement space, and determined by the forward model. The size of $\mathbf{A}$ is $M\times N$.
$\mathbf{A}=\{a_{ij}\}, (i=1,\dots,M; j=1\dots,N)$.

The reconstruction algorithm aims to obtain the image vector by solving the equation system in terms of the system transform matrix and practical measurements.

Copyright by SWM, KIT – Universität des Landes Baden-Württemberg und nationales Forschungszentrum in der Helmholtz-Gemeinschaft