Fan beam reconstruction

The Fan Beam Reconstruction. With a slit collimated fan beam of x-rays, a projection is formed by the illumination of a fixed line of detector cells. A common detector structure in this respect is the equally spaced collinear array. The projection data for this geometry is represented by the function P ), where )3 is the projection angle of a typical ray path SB and p is the distance along the detector line D,D2 to a point at B (Fig. 26.22). To simplify the algebra, the fan beam projection R ) is referred to the detector plane moved to DJ/Dj. The ray path integral along SB is now associated with the point A on this imaginary detector line aXp = OA (Fig. 26.23). 

The entire DSR system is conveniently divided into the DSR scanner section, the reconstruction section, and the data analysis section. The reconstruction system utilizes a filtered back-projection fan beam reconstruction system based on a floating point array processor (Robb et ai, 1980). Most important to the... 

The fan beam reconstruction algorithm given in Eq. (8) consists of three parts (1) weighting the projection data with J Pi), (2) convolution with g P ), and (3) weighted backprojection with weight l/V. ... 

In traditional Fan-Beam CT the radiation emitted from the X-ray tube is collimated to a planar fan, and so most of the intensity is wasted in the collimator blades (Fig. 2a). Cone-Beam CT, where the X-rays not only diverge in the horizontal, but also in the vertical direction, allows to use nearly the whole emitted beam-profile and so makes best use of the available LINAC photon flux (Fig. 2b). So fast scanning of the samples three-dimensional structure is possible. For Cone-Beam 3D-reconstruction special algorithms, taking in consideration the vertical beam divergence of the rays, were developed. 

Typical tomographic 2D-reconstruction, like the filtered backprojection teelinique in Fan-Beam geometry, are based on the Radon transform and the Fourier slice theorem [6]. 

A sinogram is a (a) hie containing individual annihilation events, (b) a matrix of parallel projection data, (c) a matrix of fan beam projection data, and (d) reconstructed image. 

Using the geometric relations (26.74), the reconstruction (26.76) can be expressed in terms of the fan beam projection R ), to give... 

To contribute to a voxel (r, s, z) for z 0 in the cone beam geometry, the fan beams must be tilted out of the r, s plane to intersect the particular voxel (r, s, z) from various x-ray source orientations. As a result, the location of the reconstruction point in the tilted system is now determined by a new coordinate system (r, s) (Fig. 26.25). Consequently, the fan beam geometry in these new coordinates will change. Specifically, the new source distance is defined by... 

Leng S, Zhuang T, Nett BE, Chen GH (2005) Exact fan-beam image reconstruction algorithm for truncated projection data acquired from an asymmetric half-size detector. Phys Med Biol 50 1805-1820... 

Early spiral-CT scanners (34) provided stacks of cross-sectional CT sections. However, the slices were separated in time. The DSR acquired all the slices simultaneously but cone-beam reconstructions required a prohibitive amount of computation time. Tomography slices in medical CT have, until recently, been generated from one-dimensional projections via fan-beam methods. 

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