On a bilinear optimization problem in parallel magnetic resonance imaging

This work is concerned with the structure of bilinear minimization problems arising in recovering subsampled and modulated images in parallel magnetic resonance imaging. By considering a physically reasonable simplified model exhibiting the same fundamental mathematical difficulties, it is shown that such problems suffer from poor gradient scaling and non-convexity, which causes standard optimization methods to perform inadequately. A globalized quasi-Newton method is proposed which is able to reconstruct both image and the unknown modulations without additional a priori information. Thus the present paper serves as a first contribution toward understanding and solving such bilinear optimization problems.