Total Variation-Stokes Strategy for Sparse-View X-ray CT Image Reconstruction

. Author manuscript; available in PMC: 2014 Apr 1.

Published in final edited form as: IEEE Trans Med Imaging. 2014 Mar;33(3):749–763. doi: 10.1109/TMI.2013.2295738

initial: µ⁽⁰⁾

initial: Δ t₁, Δ t₂, ε₁, ε₂ and ε₃;

caculate the initial vectors u and v from µ⁽⁰⁾;

while stop criterion is not met;

for n =1,2,…, N; (TFS)

calculate τⁿ,

T_{1}^{n}, T_{2}^{n}

v^{n + 1} = v^{n} + Δ t_{1} (D_{x}^{-} (\frac{D_{x}^{+} v^{n}}{T_{1}^{n}}) + D_{y}^{-} (\frac{D_{y}^{+} v^{n}}{T_{2}^{n}}) + D_{x}^{-} (λ^{n} + D i v (τ^{n}))),

u^{n + 1} = u^{n} + Δ t_{1} (D_{x}^{-} (\frac{D_{x}^{+} u^{n}}{T_{2}^{n}}) + D_{y}^{-} (\frac{D_{y}^{+} u^{n}}{T_{1}^{n}}) + D_{y}^{-} (λ^{n} + D i v (τ^{n}))),

λ^{n + 1} = λ^{n} + Δ t_{1} (D_{x}^{+} v^{n} + D_{y}^{+} u^{n}),

end for;

for j 1, 2,…, J; (POCS)

if j==1;

10:

µ^(j):=SART(µ⁽⁰⁾,ω);

11:

else µ^(j) :=SART(µ^{( j-1)},ω);

12:

end if;

13:

end for;

14:

μ_{x, y}^{(J)} > 0

, then

μ_{x, y}^{(J)} = μ_{x, y}^{(J)}; x = 1, 2, \dots, X, y = 1, 2, \dots, Y;

15:

else

μ_{x, y}^{(J)} ≔ 0; x = 1, 2, \dots, X, y = 1, 2, \dots, Y;

16:

end if;

17:

for i = 1, 2,…, I; (Image Fitting)

18:

calculate n₁, n₂,

T_{3}^{i}, T_{4}^{i};

19:

μ^{i + 1} = μ^{i} + Δ t_{2} (D_{x}^{-} (\frac{D_{x}^{+} μ^{i}}{T_{3}^{i}} - n_{1}) + D_{y}^{-} (\frac{D_{y}^{+} μ^{i}}{T_{4}^{i}} - n_{2})), x = 1, 2, \dots, X, y = 1, 2, \dots, Y;

20:

end for;

21:

end if stop criterion is satisfy.