Reconstruction from Line Integrals

Let E be a Euclidean space of dimension n and f be a function in E that decreases at infinity in such a way that any line integral of f converges absolutely. It is sufficient to assume that $$f\left( x \right)\, = O\,\left( {{{\left| x \right|}^{ - 1 - \varepsilon }}} \right)$$ at infinity for some $$\varepsilon \, > \,0.\,$$ We denote by g the line integral of f: 4.1$$g\left( {x,\theta } \right)\, = \,\int_{L\left( {x,\theta } \right)} {fd} s,$$ where L (x, θ) denotes the straight line (or the ray) through the point x ∈ E that is parallel to the unit vector θ. In the case n = 2, data of the line integrals defines the Radon transform of f as follows: R f (p, ω) = g (x, θ) where θ is orthogonal to the unit vector ω and x is any point such that $$\left\langle {\omega,x} \right\rangle \, = \,p.$$ In the case n > 2, the manifold A_l (E) of all straight lines has dimension 2n - 2 which implies that data of all line integrals are redundant. To avoid redundancy we state the reconstruction problem as follows: