Denoise diffusion-weighted images using higher-order singular value decomposition
Introduction
Diffusion-weighted (DW) magnetic resonance imaging (MRI) provides a non-invasive technique to map the molecular diffusion of in vivo biological tissues. The random molecular Brownian motion in tissues is restricted by structures such as cell membranes, myelin sheaths, macromolecules, etc (Feng et al., 2014). Thus, the diffusion anisotropy information extracted from multidirectional DW images reflects the microscopic characteristic of tissue structures. Quantitative maps and fiber tracking from DW magnetic resonance (MR) data are widely used in the clinical diagnosis of patients with acute ischemic stroke (Kloska et al., 2010, Sotak, 2002), multiple sclerosis (Tievsky et al., 1999, Werring et al., 1999), epilepsy (Eriksson et al., 2001, Hugg et al., 1999) and brain tumors (Bode et al., 2006, Krabbe et al., 1997), as well as in the scientific research of brain anatomical connectivity (Jones et al., 1999, Poupon et al., 2001).
The quantitative analysis of DW data is usually challenged by noise (Bastin et al., 1998, Kristoffersen, 2012), especially in high-spatial-resolution and/or high-b-value diffusion imaging where signal-to-noise ratio (SNR) is usually low. In regions where white matter fiber bundles intersect or split, high-spatial-resolution DW imaging is required for the accurate estimation of the orientation of fiber bundles (Uğurbil et al., 2013). However, the reduced voxel size in high-spatial-resolution DW imaging will decrease SNR and challenge the quantification of diffusion-related parameters. In addition, to model the non-Gaussian diffusion characteristics in biological tissues, high-b-value DW images are usually acquired in multiple-b-value diffusion MRI (Grinberg et al., 2011, Shinmoto et al., 2009). Because the signal magnitude decreases rapidly with b values, the low SNR in high-b-value DW images will degrade the accuracy and reliability of the subsequent quantitative analysis (Jones and Basser, 2004, Wu and Cheung, 2010).
The SNR can be improved by averaging repeated acquisitions at the expense of prolonging scan time and increasing the susceptibility to subject motion. Alternatively, the noise in the DW images and its effect on parameter quantification can be reduced by post-processing techniques that do not increase acquisition time. A number of filtering methods in image processing domain have been applied to denoise DW images, and these methods can be mainly classified into two categories. One category is to impose parameter smoothing constraint during quantification (Liu et al., 2013, Tschumperlé and Deriche, 2003, Wang et al., 2003, Wang et al., 2004) or to filter quantified parameter maps (Castaño-Moraga et al., 2007, Hamarneh and Hradsky, 2007). The other category is to first denoise DW images and then estimate diffusion parameters. The latter approach avoids the bias introduced by diffusion models, and can be integrated with various models for the subsequent analysis. Thus, the latter approach has been widely adopted to reduce the effect of noise on the diffusion parameter mapping in the past few decades. The current DW image denoising methods include algorithms based on wavelet transforms (Wirestam et al., 2006), partial differential equations (Ding et al., 2005, Parker et al., 2000), propagation-separation approaches (Becker et al., 2014, Becker et al., 2012), nonlocal means (Descoteaux et al., 2008, Wiest-Daessle et al., 2007), principal component analysis (Arfanakis et al., 2002, Bao et al., 2013, Manjon et al., 2013), low-rank approximations (Lam et al., 2014, Lam et al., 2016), and sparseness among grouped similar patches (Bao et al., 2013).
Recently, we developed an MR image denoising algorithm based on the higher-order singular value decomposition (HOSVD) transform (Zhang et al., 2015a), which exhibits state-of-the-art performance in denoisingT1-, T2- and proton density-weighted MR images. The HOSVD denoising algorithm first groups similar small local cubes (3D patches) into a 4D tensor, then performs the HOSVD transform of the tensor, thresholds the transformed coefficients, and performs the invert HOSVD transform, and finally aggregates multiple estimates of each pixel by a weighted average. The HOSVD transform offers a simple, adaptive and natural way to exploit sparsity along all dimensions of multidimensional data. Compared with conventional MR images, DW images not only contain redundant information across the spatial domain but are also highly correlated along different diffusion directions. The sparse representation of multidimensional structures in DW images can be effectively exploited for noise reduction by using the HOSVD transform.
In this paper, we aim to investigate the application of the HOSVD transform to denoising DW images. In the case of low SNR, we found by experiment that the patch-based HOSVD algorithm introduced stripe-like artifacts in the homogeneous regions of the denoised image. Similar phenomenon was also observed in previous studies on transform-based image denoising (Knaus and Zwicker, 2013, Knaus and Zwicker, 2014; Pierazzo et al., 2014). We showed by experiment that these artifacts are caused by the noise-deteriorated HOSVD bases learned from the patches with severe noise. To mitigate the artifacts, an intuitive approach is to transform the grouped noisy patches using the HOSVD bases learned from the corresponding noise-free or noise-less patches. Thus, we introduce a global HOSVD prefiltering stage to guide the patch-based local HOSVD method for the improved denoising of DW images. The proposed algorithm is referred as GL-HOSVD for brevity, and is detailed in the Methods section. The performance of GL-HOSVD was evaluated on simulation as well as in vivo DW data contaminated by the spatially-invariant Rician or noncentral Chi noise.
Section snippets
The HOSVD-based denoising strategy
The HOSVD is a multilinear generalization of the matrix SVD to high-order tensors, and it can achieve the sparse representation of a tensor with adaptive bases (De Lathauwer et al., 2000, Kolda and Bader, 2009). Similar to other transform-based denoising methods, HOSVD-based denoising consists of three main steps. First, the HOSVD transform of the noisy tensor is performed. Then, the transformed coefficients are manipulated by thresholding based on the assumption that small transform
Simulated data
To quantitatively evaluate the performance of denoising algorithms, DW images were synthesized by using the tensor model , where S0 is the non-diffusion signal, g a unit gradient vector, and D the diffusion tensor. An adult mouse diffusion tensor atlas from the Biomedical Informatics Research Network Data Repository as in (Lam et al., 2014) was used for simulation. Forty-five DW images with size of 256×256 were generated including one with b=0 and forty-four with b=2000 s/mm2 along
Discussion and conclusion
This paper investigates the feasibility of denoising DW images by using the HOSVD transform. We developed a novel approach which employs a global HOSVD prefiltering stage to guide the conventional local HOSVD denoising algorithm. The proposed GL-HOSVD algorithm can successfully mitigate stripe artifacts produced by the local HOSVD denoising algorithm. GL-HOSVD can simultaneously exploiting information redundancy across spatial domain and along diffusion directions using the HOSVD transform. The
Acknowledgements
We wish to thank Fan Lam and Zhipei Liang for the source code of their denoising algorithm and constructive discussions. This study was supported by the National Natural Science Foundation of China (81601564, 61671228, 61471187 and U1501256), the China Postdoctoral Science Foundation Projects (2016M602490), the Province Natural Science Fund of Guangdong (2016A030310380), the National Key Technology R&D Program (2015BAI01B03) and a grant from Hong Kong Research Grant Council (RGC C7048-16G). The
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