Multi-Linear Kernel Regression and Imputation in Data Manifolds
This work addresses data regression and imputation challenges in dynamic MRI, offering a novel approach that is incremental in its application to this domain.
The paper tackles the problem of efficient regression and imputation for data on manifolds, specifically in dynamic MRI, by introducing a multi-linear kernel-based framework that uses landmark points and linear patches to reduce dimensionality and extract geometry without training data. It demonstrates remarkable improvements in efficiency and accuracy over existing methods in numerical tests on severely under-sampled dMRI data.
This paper introduces an efficient multi-linear nonparametric (kernel-based) approximation framework for data regression and imputation, and its application to dynamic magnetic-resonance imaging (dMRI). Data features are assumed to reside in or close to a smooth manifold embedded in a reproducing kernel Hilbert space. Landmark points are identified to describe concisely the point cloud of features by linear approximating patches which mimic the concept of tangent spaces to smooth manifolds. The multi-linear model effects dimensionality reduction, enables efficient computations, and extracts data patterns and their geometry without any training data or additional information. Numerical tests on dMRI data under severe under-sampling demonstrate remarkable improvements in efficiency and accuracy of the proposed approach over its predecessors, popular data modeling methods, as well as recent tensor-based and deep-image-prior schemes.