Bi-Linear Modeling of Manifold-Data Geometry for Dynamic-MRI Recovery

This paper establishes a modeling framework for data located onto or close to (unknown) smooth manifolds, embedded in Euclidean spaces, and considers its application to dynamic magnetic resonance imaging (dMRI). The framework comprises several modules: First, a set of landmark points is identified to describe concisely a data cloud formed by highly under-sampled dMRI data, and second, low-dimensional renditions of the landmark points are computed. Searching for the linear operator that decompresses low-dimensional data to high-dimensional ones, and for those combinations of landmark points which approximate the manifold data by affine patches, leads to a bi-linear model of the dMRI data, cognizant of the intrinsic data geometry. Preliminary numerical tests on synthetically generated dMRI phantoms, and comparisons with state-of-the-art reconstruction techniques, underline the rich potential of the proposed method for the recovery of highly under-sampled dMRI data.