Kernel and Manifold Framework for Magnetic Resonance Imaging
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Magnetic resonance imaging (MRI) is one of the most advanced imaging technologies that encompasses both structural and functional imaging paradigm. It allows us to visualize not only qualitative structures but also quantitative and functional properties of cells, tissues and biological process in a body. However, MRI suffers from very slow data acquisition process due to various inherent physical and physiological constraints. Such slow signal acquisition consequences in several limitations and challenges in imaging such as low spatiotemporal resolution, noise and image artifacts, inefficiency in imaging fast temporal dynamics, and patient discomfort. Therefore, fast imaging has been of perennial interest since the emergence of MRI. One way to accelerate MRI is to reconstruct images from reduced signal acquisitions. Reduced acquisitions under-sample MR signals at sub-Nyquist rate and dedicated image reconstruction techniques are applied to reconstruct images. Current trends in image reconstruction from under-sampled signals rely solely on conventional signal reconstruction paradigms such as compressed sensing (CS) with sparsity constraints and low-rank modeling using singular value decomposition and principal component analysis. All these reconstruction methods depend on the signal priors that are based on linear correlation amongst signals. These methods overlook intrinsic nonlinear correlation inhibited by complex MR signals and hence reconstructed images are unable to characterize detail structural and functional properties often demanded in clinical settings. While many nonlinear manifold models have been studied for signal representation outside MR community and much theoretical advancement in signal reconstructions have been made, there is a clear gap between new theoretical developments in signal reconstruction and its application in medical imaging. The use of manifold learning models in MRI image reconstruction is challenging because of several inherent issues such as lack of sufficient training data, computational complexity, and unlike in image processing and other data classification applications, in MRI input and output signals are in different space i.e. the undersampled signal is acquired in k-space domain whereas the output is desired in the image domain. This necessitates dedicated reconstruction strategies for dMRI. Moreover, due to the undersampling of k-space in MRI, an optimization problem of image reconstruction from acquired k-space is highly ill-posed.