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