Compressed Sensing-Based MRI Reconstruction Using Complex Double-Density Dual-Tree DWT

Algorithm 1

INPUTS:

𝑦: undersampled k-space data

𝐹_𝑢: system matrix associated with the measurements

Φ: complex double-density dual-tree wavelet 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:

𝑥: the numerical approximation to (6)

% Initialization

𝑘=0; 𝑥₀; 𝑔_𝑜=∇𝐽(𝑥₀); Δ𝑥₀=−𝑔_𝑜

% Iterations

while (𝑔_𝑘2>Tol)

{

% Backtracking line-search

𝑡=5; while (𝐽(𝑥_𝑘+𝑡Δ𝑥_𝑘)>𝐽(𝑥_𝑘)+𝛼𝑡⋅𝑅𝑒𝑎𝑙(𝑔^∗_𝑘Δ𝑥_𝑘) and k < Iter)

{𝑡=𝜍𝑡}

𝑥_𝑘+1=𝑥_𝑘+𝑡Δ𝑥_𝑘

𝑔_𝑘+1=∇𝐽(𝑥_𝑘+1)

𝜂_𝑘=𝑔_𝑘+1−𝑔_𝑘

𝛽_𝑘+1𝑔=max0,min^𝑇𝑘+1𝜂_𝑘Δ𝑥^𝑇_𝑘𝜂_𝑘,𝑔_𝑘+1²Δ𝑥^𝑇_𝑘𝜂_𝑘

Δ𝑥_𝑘+1=−𝑔_𝑘+1+𝛽_𝑘+1Δ𝑥_𝑘

𝑘=𝑘+1

}