Kernel Regression Imputation in Manifolds via Bi-Linear Modeling: The Dynamic-MRI Case
preprintposted on 21.06.2021, 16:41 by Konstantinos Slavakis, Gaurav Shetty, Loris Cannelli, Gesualdo Scutari, Ukash Nakarmi, Leslie Ying
This paper introduces a non-parametric kernel-based modeling framework for imputation by regression on data that are assumed to lie close to an unknown-to-the-user smooth manifold in a Euclidean space. The proposed framework, coined kernel regression imputation in manifolds (KRIM), needs no training data to operate. Aiming at computationally efficient solutions, KRIM utilizes a small number of ``landmark'' data-points to extract geometric information from the measured data via parsimonious affine combinations (``linear patches''), which mimic the concept of tangent spaces to smooth manifolds and take place in functional approximation spaces, namely reproducing kernel Hilbert spaces (RKHSs). Multiple complex RKHSs are combined in a data-driven way to surmount the obstacle of pin-pointing the ``optimal'' parameters of a single kernel through cross-validation. The extracted geometric information is incorporated into the design via a novel bi-linear data-approximation model, and the imputation-by-regression task takes the form of an inverse problem which is solved by an iterative algorithm with guaranteed convergence to a stationary point of the non-convex loss function. To showcase the modular character and wide applicability of KRIM, this paper highlights the application of KRIM to dynamic magnetic resonance imaging (dMRI), where reconstruction of high-resolution 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.