Accelerating Dynamic Cardiac MR Imaging Using Structured Sparse Representation

Algorithm 1

NCSR-based dynamic cardiac cine MR imaging.

(1) Initialization:

  Set the initial estimate 𝐗⁽¹⁾=𝐘,𝐗⁽⁰⁾=𝟎 and the regularization parameter 𝜆;

(2) Outer loop (dictionary learning and clustering): iterate on 𝑙=1,2,…,𝐿

  (a) Update the dictionaries Φ_𝑘 via 𝑘-means and PCA;

  (b) Inner loop (clustering): iterate on 𝑗=1,2,…,𝐽

  (b.1) 𝑎𝑗=1+𝑗−1/𝑗+2, 𝑏𝑗=−𝑗−1/𝑗+2

  (b.2) Compute 𝑔_𝑠^(𝑗)=𝑎𝑗𝑥_𝑠^(𝑗)+𝑏𝑗𝑥_𝑠^𝑗−1 and

     𝑔_𝑠^(𝑗+1/2)=𝑔_𝑠^(𝑗) + 𝜆𝐅^𝑇_𝐮𝑠𝑦_𝑠−𝐅_𝐮𝑠𝑔_𝑠^(𝑗) for 𝑠=1,2,…,𝑆;

  (b.3) Compute 𝑣^(𝑗)=Φ^𝑇_𝑘1𝐱^′₁^(𝑗+1/2),…,Φ^𝑇_𝑘𝑁𝐱^′_𝑁^(𝑗+1/2), where Φ_𝑘𝑛 is the dictionary assigned to patch;

  (b.4) Compute the 𝛼_𝑖^(𝑗+1) using the shrinkage operator given in (8);

  (b.5) Update the estimate 𝛽_𝑖;

  (b.6) Image matrix estimate update: 𝐗^(𝑗+1)=Φ𝛼_𝑦^(𝑗+1).