Algebraic Reconstruction Technique Sample Clauses

Algebraic Reconstruction Technique. ‌ . ∫ Σ Rather than computing the linear attenuation coefficient µ analytically, we can also solve µ using iterative methods. If we use polar coordinates and discretize Equation (2.2) with respect to the j-th projection and the i-th detector pixel, we can obtain Ii = I0 exp − µ (˙r (t)) d t . (2.12) t∈li,j I0 To be convenient, we let bi = − log Ii and can obtain that ∫ bi = µ (˙r (t)) d t. (2.13) t∈li,j If we further decompose µ (˙r (t)) into an expansion Σ µ (˙r (t)) = µjφj (˙r (t)) , (2.14) where φj (˙r (t)) is a basis function of the image representation. The line integral of the basis function, ai,j, is the length of the x-xxx xxxx through the j-th pixel of the target image, incident onto the i-th element of detector pixels. So ai,j can be expressed as ai,j = ∫ t∈li,j φj (˙r (t)) dt. (2.15) The geometric meaning of ai,j is shown in Figure 2.4. In Figure 2.4, a x-ray intersects