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.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. 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. The projection of an object, resulting from the tomographic measurement process at a given angle θ {displaystyle heta } , 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 μ ( x , y ) {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 r {displaystyle r} , across a projection at angle θ {displaystyle heta } . This is repeated for various angles. Attenuation occurs exponentially in tissue: where μ ( x , y ) {displaystyle mu (x,y)} is the attenuation coefficient as a function of position. Therefore, generally the total attenuation p {displaystyle p} of a ray at position r {displaystyle r} , on the projection at angle θ {displaystyle heta } , is given by the line integral: Using the coordinate system of Figure 1, the value of r {displaystyle r} onto which the point ( x , y ) {displaystyle (x,y)} will be projected at angle θ {displaystyle heta } is given by: