Kernel Regression Imputation in Manifolds via Bi-Linear Modeling: The Dynamic-MRI Case

This paper introduces a non-parametric approximation framework for imputation-by-regression on data with missing entries. The proposed framework, coined kernel regression imputation in manifolds (KRIM), is built on the hypothesis that features, generated by the measured data, lie close to an unknown-to-the-user smooth manifold. The feature space, where the smooth manifold is embedded in, takes the form of a reproducing kernel Hilbert space (RKHS). Aiming at concise data descriptions, KRIM identifies a small number of ``landmark points'' to define approximating ``linear patches'' in the feature space which mimic tangent spaces to smooth manifolds. This geometric information is infused into the design through a novel bi-linear model that allows for multiple approximating RKHSs. To effect imputation-by-regression, a bi-linear inverse problem is solved by an iterative algorithm with guaranteed convergence to a stationary point of a non-convex loss function. To showcase KRIM's modularity, the application of KRIM to dynamic magnetic resonance imaging (dMRI) is detailed, where reconstruction of images from severely under-sampled dMRI data is desired. Extensive numerical tests on synthetic and real dMRI data demonstrate the superior performance of KRIM over state-of-the-art approaches under several metrics and with a small computational footprint.