Established techniques for three-dimensional radiographic reconstruction such as computed tomography (CT) or, more recently cone beam computed tomography (CBCT) require an extensive set of measurements/ projections from all around an object under study. The x-ray dose for the patient is rather high. Cutting down the number of projections drastically yields a mathematically challenging reconstruction problem. Few-view 3D reconstruction techniques commonly known as “tomosynthetic reconstructions” have gained increasing interest with recent advances in detector and information technology.
Radiographic Projection And 3D Reconstruction From Multiple Projections
The projection value measured by any x-ray sensitive receptor follows the well-known Lambert-Beer law: (1)
defining the intensity behind an absorber and I0
the input intensity, respectively. The parameters μ
denote the mass absorption coefficient and the thickness of the absorber. We aim to estimate μ
as shade of gray at discrete instances to obtain a reliable representation of the object.
In 1917, the Austrian mathematician Johann Radon discovered that the two-dimensional (2D) distribution of properties of an object may be obtained from an infinite number of line integrals sampled through the object. Mathematically, a function can be completely described by the complete number of straght line integrals through the support of, i.e. (2)
The famous method predominantly applied for image reconstructions in CT-scanners uses this formula in a process termed “Filtered Backprojection (FBP)”. Assuming parallel x-rays, the relationship between the projection data (P) and the object are given by: (3)
where Θ denotes the projection angle and t
the detector position in the beam (Fig. 1). Diracs delta function δ
is required to define the line interval. The CT image reconstruction problem is to compute
. Note, that the term “
” in equation (3)” represents a line equation (e.g. green “ray-line” in Fig. 1), (Beyerer & Leon, 2002) the sum of which the integrals are sampled. In other words, the measured data on the image receptor represent the integrals over a finite number of lines connecting the x-ray source with the detector cells.