chapter  2
22 Pages

Ray and Line Integral Transforms

The ray transform of a function f with compact support in a vector space V is defined by

Xf (y, v) =

f (y + tv) dt, v ∈ V \0,

where y is called the vertex or source point of the integration ray. The ray integral Xf (y, v) is a homogeneous function of v of degree -1. The line transform of a function f is defined in the similar way:

Lf (y, v) =

f (y + tv) dt = Xf (y, v) + Xf (y,−v) . (2.1)

Reconstruction problem is to find a reconstruction formula Lf |Σ 7→ f for functions f supported in a compact set K ⊂ V , where Σ is a family of lines in V. We call the family Σ non redundant if dim Σ = n. In particular the manifold L (V ) of all lines in V has dimension 2n− 2. It is redundant for n > 2 and non redundant for n = 2.