Improved Compressed Sensing-Based Algorithm for Sparse-View CT Image Reconstruction

Algorithm 1

Iterative algorithm for ℓ₁ minimization.

INPUTS:

𝑦: undersampled projection data

𝐴: system matrix associated with the measurements

Φ: wavelet transform operator

𝜆₁,𝜆₂: tuning constants

OPTIONAL PARAMETERS:

Tol: stopping criteria by gradient magnitude (default 10⁻⁴)

Iter: stopping criteria by number of iterations (default 100)

𝛼,𝜍: line search parameters (defaults 𝛼=0.01, 𝜍=0.6)

OUTPUTS:

𝜇: the numerical approximation to (18)

% Initialization

𝑘=0;𝜇₀=𝜇_FBP;𝑔_𝑜=∇𝐽(𝜇₀);Δ𝜇₀=−𝑔_𝑜

% Iterations

while (𝑔_𝑘2<Tol)

{

% Backtracking line-search

𝑡=5; while (𝐽(𝜇_𝑘+𝑡Δ𝜇_𝑘)>𝐽(𝜇_𝑘)+𝛼𝑡⋅𝑅𝑒𝑎𝑙(𝑔_𝑘∗Δ𝜇_𝑘)

and 𝑘<Iter)

{𝑡=𝜍𝑡}

𝜇_𝑘+1=𝜇_𝑘+𝑡Δ𝜇_𝑘

𝑔_𝑘+1=∇𝐽(𝜇_𝑘+1)

𝜂_𝑘=𝑔_𝑘+1−𝑔_𝑘

𝛽_𝑘+1𝑔=max0,min^𝑇𝑘+1𝜂_𝑘Δ𝜇^𝑇_𝑘𝜂_𝑘,𝑔_𝑘+1²Δ𝜇^𝑇_𝑘𝜂_𝑘

Δ𝜇_𝑘+1=−𝑔_𝑘+1+𝛽_𝑘+1Δ𝜇_𝑘

𝑘=𝑘+1}