A Reconstruction Approach for Imaging in 3D Cone Beam Vector Field Tomography

Algorithm 1

For cone beam VFT.

Given   : Measured data 𝐃₁𝐟(𝛼,𝜔) for 𝛼∈Γ, 𝜔∈𝑆²

Output    : Approximation 𝐟_𝛾 to 𝐟

Compute:

    (1) 𝐠(𝛼,𝜔)=𝐃₁𝐟(𝛼,𝜔)𝜔

    (2) (𝐟_𝛾)_𝑗(𝑥)=⟨𝐠,𝐕^𝑗_𝛾(𝑥)⟩_{𝐿2(Γ×𝑆2,ℝ3)}

           =∫_Γ∫_𝑆2⟨𝐠(𝛼,𝜔),𝐕^𝑗_𝛾(𝑥;𝛼,𝜔)⟩d𝑆(𝜔)d𝛼

   =∫_Γ∫_𝑆2𝐃₁𝐟(𝛼,𝜔)𝜔_𝑗𝑣_𝛾(𝑥;𝛼,𝜔)d𝑆(𝜔)d𝛼

                      for 𝑗∈{1,2,3}