Denoising of diffusion MRI using random matrix theory

Highlights

• Denoising enhances the image quality for improved visual, quantitative, and statistical interpretation.

• Random matrix theory enables data-driven threshold for PCA denoising.

• The Marchenko-Pastur distribution is a universal signature of noise.

• The technique suppresses signal fluctuations that solely originate in thermal noise.

• Precision of diffusion parameter estimators increases without lowering accuracy.

Abstract

We introduce and evaluate a post-processing technique for fast denoising of diffusion-weighted MR images. By exploiting the intrinsic redundancy in diffusion MRI using universal properties of the eigenspectrum of random covariance matrices, we remove noise-only principal components, thereby enabling signal-to-noise ratio enhancements. This yields parameter maps of improved quality for visual, quantitative, and statistical interpretation. By studying statistics of residuals, we demonstrate that the technique suppresses local signal fluctuations that solely originate from thermal noise rather than from other sources such as anatomical detail. Furthermore, we achieve improved precision in the estimation of diffusion parameters and fiber orientations in the human brain without compromising the accuracy and spatial resolution.

Introduction

In vivo exploration of microstructure of biological tissues has been made possible by the development of diffusion Magnetic Resonance Imaging (dMRI) (Jones, 2010a). The dMRI signal is sensitized to the stochastic thermal motion of water molecules and their interaction with surrounding microstructure by applying diffusion-encoding gradients. Unfortunately, due to the signal-attenuation induced by diffusion-sensitization and T₂ relaxation resulting from the long echo time necessary to accommodate gradient pulses, the signal-to-noise ratio (SNR) of the diffusion-weighted (DW) MR signals is inherently low (Jones, 2010b).

Thermal noise that corrupts dMRI measurements propagates to the diffusion parameters of interest and, as such, hampers visual inspection and quantitative interpretation of the underlying diffusion process. Although attempts have been made to minimize the noise propagation by optimizing diffusion encoding settings (Hansen et al., 2013, Hansen et al., 2015, Jones et al., 1999, Poot et al., 2010), scan time limitations put a bar on what is to gain with protocol optimization in terms of precision. Therefore, image denoising, i.e. minimizing the variance of the dMRI signals in a post-processing step, is essential to raise that bar.

Denoising has been an important and long-standing problem in image processing. Some of the proposed techniques were adopted by the dMRI community in the last few years because of their ability to deal with the spatially varying and non-Gaussian nature of noise in magnitude MR data. Overall, many of these methods share an underlying similarity in terms of their structure, which is based on weighted averages of voxels, where the voxels (and weights) are selected by metric similarity of patches (Buades et al., 2005, Coupé et al., 2006, Foi, 2011, Manjón et al., 2008, Manjón et al., 2010, Orchard et al., 2008, Rajan et al., 2011, Rajan et al., 2012). Limitations are typically loss of spatial resolution of the image (blur) and introduction of additional partial volume effects that lead to complications in further quantitative analyses or to biases in diffusion modeling. An alternative approach was pioneered by Rudin et al. (1992), who proposed total variation (TV) minimization, which is a method based on the principle that local signal fluctuations increase the L₁ norm of image gradient. The main benefit of such TV-based noise removal techniques is that they are well-suited to remove local noise variations while preserving the edges in the images. Limitations are the dependency on a regularization term, introduction of reconstruction artifacts, and the fact that thermal noise is not the sole source of local variations. Indeed, fine anatomical details might be removed as well by this non-selective technique (Block et al., 2008, Knoll et al., 2011, Perrone et al., 2015, Veraart et al., 2015).

In 1933, Hotelling seeded the idea of noise removal by means of transforming a redundant dataset into a principal component basis and preserving only the signal-carrying principal components by suggesting “perhaps neglecting those whose contributions to the total variance are small.” (Hotelling, 1933). Indeed, the principal component analysis (PCA) of redundant data shows that most of the signal-related variance is contained in a few components, whereas the noise is spread over all components. Redundancy in data is commonly pursued by a local or non-local selection of image patches, a non-trivial and time-consuming approach, especially in case of spatially varying noise (Deledalle et al., 2011, Manjón et al., 2015). Fortunately, it has been shown that typical dMRI data exhibit sufficient redundancy due to common practice of oversampling the q-space (Veraart et al., 2016).

The number of signal-carrying principal components, i.e. the number of components that significantly contribute to the description of the underlying diffusion process, is unknown and is expected to depend on imaging factors such as resolution, b-value and SNR. Hence, an objective criterion to discriminate between the signal-carrying and noise-only components has been missing. In other words, it had remained unclear what “small contributions to the total variance” (Hotelling, 1933) actually means. Commonly used criteria include thresholding of the eigenvalues associated with the principal components by an empirically set value (Manjón et al., 2013).

In this work, we will objectify the above-mentioned threshold for PCA denoising by exploiting the fact that noise-only eigenvalues are expected to obey the universal Marchenko-Pastur law, a result of the random matrix theory for noisy covariance matrices (Marchenko and Pastur, 1967). This article is an extension of our previous work that was concerned with the estimation of spatially varying noise maps using random matrix theory (Veraart et al., 2016). Whereas we previously focused on the noise level estimation, here we will demonstrate that an objective threshold on the eigenvalues for PCA denoising can be derived from the noise level. We will show that the proposed technique preserves the underlying signal better than other state-of-the-art techniques at the level of the diffusion sensitized images and diffusion MR parameters of general interest. Indeed, we propose here a denoising technique that preserves local signal fluctuations of any origin different than thermal noise, including fine anatomical detail. The noise level, being an additional product of the method (Veraart et al., 2016), offers the opportunity to correct the denoised signal for Rician or noncentral-χ distributed noise bias (Aja-Fernández et al., 2011, Gudbjartsson and Patz, 1995) using the method of moments (Koay and Basser, 2006).