ABSTRACT
In a 2D case, the spatial distribution of an object’s attenuation coecient can be dened as m : . 2 ® We dene W Ì 2 as the spatial support of the object. e attenuation coecient outside of the object, that is, in the set 2\Ω, is dened as zero. A projection dataset is also dened as function l : G ® , where Γ = [ρmin, ρmax] × [0, π] is the sinogram space. e position along the detector array is described by variable ρ ∈ [ρmin, ρmax] while the angle of the x-ray beam is given by variable θ ∈ [0, π]. e spatial position is described by variables r1 and r2. If one assumes that the projection that includes point (r1, r2) = (0, 0) is located at ρ = 0 for all θ, we can write:
l r q m d r q q( , ) ( , ) ( cos( ) sin( ))= - - -¥
òò dr dr r r r r1 2 1 2 1 2
(14.4)
where the Dirac δ distribution is used according to its customary denition. In essence, this relationship denes nothing but a set of line integrals over the object. It is known widely as the Radon transform.
