Water-Fat Separation Using a Locally Low-Rank Enforcing Reconstruction

INTRODUCTION: Multi-contrast water-fat separation based on the Dixon method is steadily gaining importance in clinical routine. An accelerated examination using dedicated acquisition schemes in combination with an iterative reconstruction is therefore highly desirable. A direct reconstruction of water and fat images incorporating field inhomogeneities, relaxation effects as well as gradient delays and eddy current effects is, however, not straightforward as the optimization problem is rendered non-convex. Here we demonstrate that the reconstruction problem can be decoupled by first reconstructing the multiple echo images using a locally low-rank enforcing (LLR) regularization, followed by a conventional voxel-wise fit of the nonlinear parameters. The regularization enforces a local representation of the contrast images with as few chemical components as possible and assumes a low resolution for the phase evolutions. Both are common assumptions in water-fat separation. The approach also allows for varying sampling patterns across contrasts (blipping), adding further regularization terms, as well as the support of bipolar acquisitions and, in consequence, faster acquisitions. THEORY: For water-fat separation the contrast images at a given voxel and at echo times { } ,..., are modeled as , = ( ( ) + ( ) ( )) ( , ). Here ( ) is the water signal, ( ) the fat signal, ( ) the fat dephasing at echo time and ( , ) the phase at time . The phase evolution includes effects from field inhomogeneities, gradient delays and eddy currents in the case of bipolar acquisitions as well as the phase right after excitation for real-valued water and fat signals. The imaginary part addresses R2 * relaxation. A common assumption in water-fat separation is that the phase evolution is spatially very smooth. If we assume it to be spatially constant in sufficiently small patches { } ,..., , we directly see that the matrix ( ) = ( ) ( ) ⋮ ⋮ ( ) ( ) ( ) ⋯ ( ) ( ) ( ) ⋯ ( )