ALTERNATING DIRECTION OF MULTIPLIERS METHOD FOR PARALLEL MRI RECONSTRUCTION

摘要

A method for reconstructing parallel magnetic resonance images includes providing a set of acquired k-space MR image data y, and finding a target MR image x that minimizes ½∥Fv−y∥₂²+λ∥z∥₁ where v=Sx and z=Wx where S is a diagonal matrix containing sensitivity maps of coil elements in an MR receiver array, F is an FFT matrix, W is a redundant Haar wavelet matrix, and λ≧0 is a regularization parameter, by updating; $$ x^{k+1}={\left({\mu }_1I+{\mu }_3S^HS\right)}^{-1}\left[{\mu }_1W^H\left(z^k-b_z^k\right)+{\mu }_3S^H\left(v^k-b_v^k\right)\right],z^{k+1}=\mathrm{soft}\left({\mathrm{Wx}}^{k+1}b_z^k,\frac{1}{{\mu }_1}\right)\mathrm{where}$$ $$ \mathrm{soft}\left(x,T\right)=\{\begin{array}{cc}x+T& \mathrm{if}x\le -T,\\ 0& \mathrm{if}x\le T,\\ x-T& \mathrm{if}x\ge T,\end{array}\mathrm{and}v^{k+1}={\left(F^HF+{\mu }_3I\right)}^{-1}\left[F^Hy+{\mu }_3\left({\mathrm{Sx}}^{k+1}+b_v^k\right)\right],$$ ;where k is an iteration counter, μ₁ and μ₃ are parameters of an augmented Lagrangian function, and b_z and b_v are dual variables of the augmented Lagrangian.