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 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. A reproducing kernel Hilbert space (RKHS) forms the feature space where the smooth manifold is embedded in. Aiming at concise representations, KRIM identifies a small number of “landmark points” to define approximating “linear patches” that mimic tangent spaces to smooth manifolds. This geometric information is infused into the design through a novel bi-linear model which can be easily extended to accommodate multi-kernel contributions in the non-parametric approximations. 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.