Introduction

Ultra-high resolution diffusion MRI holds the promise of being an essential tool to explore human brain at an unprecedented level of detail. However, at that resolution, severe noise often obscures fine anatomical details and hampers further dMRI analysis. We introduce kernel PCA (principal component analysis), a non-linear generalization of the traditional linear PCA to Hilbert spaces, to suppress noise while preserving the fine details in the diffusion-weighted images. The proposed method holds promise for denoising at very low SNR levels outperforming standard PCA denoising methods.

Methods

Kernel PCA
Kernel PCA (K-PCA) is a nonlinear extension of the traditional linear PCA where principal component analysis is performed not in the native space, but in a high dimensional reproducing kernel Hilbert space (RKHS) where the data lies on a flat manifold¹. Let the input data set be $$$\{ {\bf{x}}_k \in \mathbb{R}^D, k=1,...,N\}$$$ and the mapped data be $$$\psi({\bf{x}}_k ) \in F$$$, where $$$\psi(\cdot)$$$ is a non-linear mapping from the original space to the RKHS^1,2. As in PCA, the directions of maximum variance in the RKHS feature space can be obtained using singular value decomposition in the feature space $$$F$$$^1,2. The power of K-PCA lies in its ability to capture non-linear dependencies/redundancies that may exist in the data, i.e., diffusion signal, and that are not possible to model with linear PCA. In terms of denoising, given an input vector (dMRI signal), $$$ \bf x $$$, a denoised version $$$\hat{\bf x }$$$ is obtained as a pre-image of the projection $$$P\psi( \bf x )$$$ of $$$ \bf x $$$ in the feature space. To accomplish the whole process, there is no need to know the mapping $$$\psi(\cdot)$$$ explicitly but just to define a kernel function, $$$K( {\bf{x}}_k , {\bf{x}}_{k'})$$$, which measures the similarity between two vectors $$${\bf{x}}_k$$$ and $$${\bf{x}}_{k'}$$$,³ allowing to define a metric on $$$F$$$^1,2.

Experiments
We demonstrate that K-PCA denoising outperforms state-of-the-art linear PCA based denoising methods, e.g. Marchenko-Pastur PCA, MP-PCA^4,5, in removing noise while preserving the fine structure in ultra-high resolution dMRI data. We implement K-PCA in a voxel-wise fashion with a patch size of [5x5x5]. A Gaussian kernel is a typical choice for KPCA with bandwith $$$\sigma$$$ defined automatically and adaptively in a data-driven manner as given in^6-8. To obtain the denoised signal $$$\hat{\bf x }$$$ in closed-form, we used the method presented in⁸. We note that no truncation of the eigenvalues is required in this method, unlike MP-PCA. Finally, MP-PCA was implemented in a similar voxel-wise fashion with the same patch size for comparative analysis.
Dataset:
Whole-brain gSlider-SMS dMRI data (D = 64 directions, $$$b = 1500 s/{mm}^2$$$, 7b0 volumes) was acquired (sagittally), with three repetitions, and reconstructed with conventional gSlider processing⁹ to create 660 $$${\mu}m$$$ isotropic resolution data( gslider-1av). K-PCA and MP-PCA were then used to denoise the 64 images of one of the three DWI datasets (gslider-1av) and compared to a reference: gSlider reconstruction averaged three times(gSlider-3av).
Validation:
We used the normalized mean square error (NMSE) of the DWI signals and the relative absolute error of the fractional anisotropy (FA), and mean diffusivity (MD) estimated after applying DTI (dtifit FSL) to compare the denoised DWI datasets. Reference FA and MD values were those derived from gSlider-3av: $$$FA_{3AV}$$$, $$$MD_{3AV}$$$ (estimated with the same DTI algorithm).

Results

An axial slice of the denoised DWI image with MP-PCA and K-PCA is shown in Fig.1c-d, respectively, whereas the original (noisy) DWI set (gSlider-1av) is shown in Fig.1b. Clearly, K-PCA achieves substantially more noise reduction than MP-PCA. With three-times shorter acquisition time, visual quality seems superior to the gSlider-3av case (Fig.1a). This is also evident quantitatively from the NMSE results (Fig.2). Further, K-PCA has much lower NMSE in several regions compared to MP-MPCA. Residual map of K-PCA in Fig.3 does not present any structure, demonstrating that no signal information is removed. Superior noise removal can also be observed in the color-encoded FA maps that are presented in Fig 4. Fine details (see cerebellum area) remain obscured by noise in gSlider-1av and in MP-PCA. However, much better delineation of cerebellum structures, due to noise suppression, are seen with K-PCA (Fig.4d). Finally, we report (Fig.5) whole-brain errors in FA and MD. While comparable for MD, K-PCA achieves significant reduction in NMSE as well as in relative error in FA.

Discussion

K-PCA can achieve a higher noise suppression than MP-PCA while preserving intricate anatomical details. An important benefit of K-PCA is that it does not require a large number of samples to perform at its best, as MP-PCA does for the MP assumption to hold. K-PCA does not need eigenvalue truncation and can work with any dMRI protocol.

Conclusion

We have introduced K-PCA, a powerful and nonlinear generalization of linear PCA, that can reliably reduce noise in dMRI substantially more than state-of-the art PCA methods (MP-PCA) while preserving the underlying true signal. Showcased with 660 isotropic micrometer resolution gSlider dMRI data, we demonstrate that fine details that are obscured by noise with MP-PCA, are visible by K-PCA due to its superior noise removal performance. We believe K-PCA denoising can be an essential tool in any dMRI analysis pipeline, especially for highly noisy datasets.

Acknowledgements

R01MH116173 (PIs: Setsompop, Rathi)