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Register a new userSNR-enhanced diffusion MRI with structure-preserving low-rank denoising in reproducing kernel Hilbert spaces
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Purpose: To introduce, develop, and evaluate a novel denoising technique for diffusion MRI that leverages non-linear redundancy in the data to boost the SNR while preserving signal information. Methods: We exploit non-linear redundancy of the dMRI data by means of Kernel Principal Component Analysis (KPCA), a non-linear generalization of PCAto reproducing kernel Hilbert spaces. By mapping the signal to a high-dimensional space, better redundancy is achieved despite nonlinearities in the data thereby enabling better denoising than linear PCA. We implement KPCA with a Gaussian kernel, with parameters automatically selected from knowledge of the noise statistics, and validate it on realistic Monte-Carlo simulations as well as with in-vivo human brain submillimeter resolution dMRI data. We demonstrate KPCA denoising using multi-coil dMRI data also. Results: SNR improvements up to 2.7 X were obtained in real in-vivo datasets denoised with KPCA, in comparison to SNR gains of up to 1.8 X when using state-of-the-art PCA denoising, e.g., Marchenko- Pastur PCA (MPPCA). Compared to gold-standard dataset references created from averaged data, we showed that lower normalized root mean squared error (NRMSE) was achieved with KPCA compared to MPPCA. Statistical analysis of residuals shows that only noise is removed. Improvements in the estimation of diffusion model parameters such as fractional anisotropy, mean diffusivity, and fiber orientation distribution functions (fODFs)were demonstrated. Conclusion:Non-linear redundancy of the dMRI signal can be exploited with KPCA, which allows superior noise reduction/ SNR improvements than state-of-the-art PCA methods, without loss of signal information.
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The geometry of spaces with indefinite inner product, known also as Krein spaces, is a basic tool for developing Operator Theory therein. In the present paper we establish a link between this geometry and the algebraic theory of *-semigroups. It goes via the positive definite functions and related to them reproducing kernel Hilbert spaces. Our concern is in describing properties of elements of the semigroup which determine shift operators which serve as Pontryagin fundamental symmetries
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