Method of reconstructing images for spiral and non-spiral computer tomography

A method of computing exact images derived from computer tomography with detectors, comprising the steps of:

• (a) collecting cone beam data obtained by a two dimensional detector in the process of scanning of an object;
• (b) identifying families of lines on a plane DP(s0), where s0 is a parameter value along the scan path and DP(s0) is any plane intersecting the cone beam;
• (c) preprocessing of the said data at selected points on said families of lines;
• (d) convolution-based filtering the data on said families of lines;
• and
• (e) back projecting said filtered data to update an intermediate version of the said exact image;
• and
• (f) repeating steps (a), (b), (c), (d) and (e) until an exact image of the object is completed.

A method of computing images derived from computer tomography detectors, comprising the steps of:

• (a) collecting cone beam data from a detector during a scan of an object;
• (b) preprocessing said data;
• (c) identifying lines on a plane DP(s0), where s0 is the parameter along the scan path and DP(s0) is any plane intersecting the cone beam, wherein the step (c) of identifying lines includes the steps of
• (ci) Taking the weight function n0(s,x,α), where
• α is a unit vector,
• x is a reconstruction point,
• s=s0 for simplicity of notation,
• with the properties
• nΣ(x,α)>0 and n(s,x,α) is piece-wise constant,
• where n ∑ ⁡ ( x , α ) = ∑ j ⁢ n 0 ⁡ ( s j , x , α ) , n ⁡ ( s , x , α ) = n 0 ⁡ ( s , x , α ) n ∑ ⁡ ( x , α ) ,
• sj=sj(α,α·x) are values of the parameter along the scan path that correspond to points of intersection of the plane Π(x,α) with the scan path,
• Π(x,α) is the plane through x and perpendicular to α,
• α is a non-critical direction, ∑ j
• is the summation over all sj such that y(sj)εC∩Π(x,α);
• (cii) Computing the function φ(s,x,θ):=sgn(α·{dot over (y)}(s))n(s,x,α), where
• {dot over (y)}(s)=dy(s)/ds,
• y(s) is the source position,
• θ is the polar angle in the plane perpendicular to β(s,x), which parameterizes angles α(s,θ) belonging to that plane, θ is chosen in such a way that α ⊥ ⁡ ( s , θ ) = ∂ ∂ θ ⁢ α ⁡ ( s , θ ) ⁢ ⁢ and ⁢ ⁢ α ⊥ ⁡ ( s , θ ) = ∂ ∂ θ ⁢ α ⁡ ( s , θ ) ,
• α=α(s,θ) is a unit vector in the plane perpendicular to β(s,x),
• β(s,x) is the unit vector pointing from y(s) towards x;
• (ciii) Finding values of the polar angle θm and the corresponding directions α⊥(s,θm) where φ(s,x,θ) is discontinuous;
• (civ) Finding lines obtained by intersecting the planes through y(s) and parallel to α⊥(s,θm) and β(s,x) with the plane DP(s);
• (d) Computing backprojection coefficients cm(s,x) by the formula c m ⁡ ( s , x ) := lim ϵ → 0 + ⁢ ( φ ⁡ ( s , x , θ m + ϵ ) - φ ⁡ ( s , x , θ m - ϵ ) ) ,
• where θm are values of the polar angle where φ(s,x,θ) is discontinuous;
• (e) shift invariant filtering said preprocessed data along said lines;
• (f) back projecting said filtered data using the said back-projection coefficients to form a precursor of said image;
• and
• (g) repeating steps a, b, c, d, e, and f until an image of the object is completed.