Structure-adaptive sparse denoising for diffusion-tensor MRI
Graphical abstract
Highlights
► We propose a structure-adaptive sparse denoising method for diffusion tensor MRI. ► A similarity measure is defined to find similar patches to arrange into 3D arrays. ► A structure-adaptive window pursuit method is designed to achieve array sparsity. ► The proposed SASD performs well for images with sufficient structural redundancy. ► SASD facilitates subsequent tensor field and biological relevant metrics analysis.
Introduction
Magnetic resonance imaging (MRI) has benefited from many technological developments since its advent in the 1970s. However, the fundamental trade-offs between image resolution and signal-to-noise ratio (SNR) on the one hand, and between physiological and clinical constraints on acquisition speed on the other hand, often translate to spurious artifacts such as noise, partial volume, and bias field (Basser and Pajevic, 2000, Farrell et al., 2007). An eloquent example is diffusion-tensor MRI (DT-MRI), which has become quite popular over the last decade because of its potential for in vivo and non-invasive characterization of the three-dimensional (3D) fiber architecture of anatomical organs (Behrens et al., 2003, Bondiau et al., 2008, Clatz et al., 2005, Delingette et al., 2012, Deriche et al., 2009, Descoteaux et al., 2009, Durrleman et al., 2011, Fillard et al., 2011, Galanaud et al., 2010, Galban et al., 2005, Guevara et al., 2011, Le Bihan, 2003, Lenglet et al., 2009, Messe et al., 2011, Mori et al., 2009, Rohmer et al., 2007, Smith et al., 2006, Wakana et al., 2004, Wu and Tseng, 2006). The effects of Rician noise on DT-MRI are severe because of the inherent nature of the imaging process—the higher the tissue anisotropy, the lower the intensity in the diffusion-weighted images (DWIs), and hence the higher the sensitivity to noise (Awate and Whitaker, 2007).
Denoising magnetic resonance images is an important problem; the most popular approaches are Bayesian statistic approaches (Awate and Whitaker, 2007, Basser and Pajevic, 2003, Lu et al., 2006), PDE-based approaches (Chen and Edward, 2005, Fillard et al., 2007), wavelet-based methods (Pižurica et al., 2006, Yu et al., 2009), methods based on spatial correlation (Barash and Comaniciu, 2004, Kervrann and Boulanger, 2006, Manjón et al., 2008), and sparse representation denoising (Chatterjee and Milanfar, 2009, Donoho et al., 2006, Elad and Aharon, 2006, Varshney et al., 2008). No particular method shows good performance for all relevant aspects of MRI, including bias reduction, artifacts removal, structure- and edge-preservation, and good generality/reliability trade-off.
Buades et al. (2005) surveyed spatial correlation techniques and proposed the so-called non-local means (NL-means) method which is based on the assumption that natural images usually have structural redundancy. The underlying idea is that any pixel has similar pixels that are distributed not only in its local neighborhood but also in the whole spatial domain of the image, which allows to account for the global information associated with large structures, and hence to overcome the limitations of local search. The NL-means approach has been applied to denoising (Boulanger et al., 2010, Brox et al., 2008, Katkovnik et al., 2010, Katkovnik and Spokoiny, 2008), super-resolution restoration (Manjón et al., 2010, Rousseau, 2010), compressed sensing (Danielyan et al., 2010, Marim et al., 2010) and inpainting (Elad et al., 2005). NL-means detects pixel similarity by exploring patch similarity for better robustness to noise, and similar patches can be grouped together to achieve sparse representation in a suitable transform domain in which denoising reduces to a shrinkage operation. Therefore, non-local self-similarity and sparse representations are key elements of state-of-the-art filtering algorithms.
The similarity measure used in NL-means evaluates the correlation between patches globally via the standard Euclidean distance and neglects the local consistency between a patch and its center pixel. In particular, in the case where different patches are close in terms of Euclidean distance but have center pixels with quite different values, the standard similarity measure may cause a loss of fine structures and edge blurring – the same behavior is observed when the number of patches similar to the reference patch is too small. Alternatively, the block-matching algorithm (BM3D) proposed by Dabov et al. (2007), which exploits a specific non-local image model by grouping similar patches and by collaborative filtering, is another interesting denoising approach. The BM3D filter stacks similar two-dimensional (2D) image patches into 3D arrays which are processed in a 3D transform domain to attenuate the noise component. However, patches containing fine image details or singularities or sharp edges are examples where a non-adaptive transform is not able to produce good sparse representations (Foi et al., 2007, Hammond and Simoncelli, 2008, Sikora and Makai, 1995); for such patches, the filter may introduce artifacts around discontinuities where the visual attention is often mainly focused.
We propose a 3D structure-adaptive sparse denoising (SASD) algorithm for DT-MRI which we apply to human cardiac DWIs. Our method can be decomposed into four steps. First, we form groups of similar patches in the DWI to be denoised using a modified structure-similarity index. Second, each set of similar patches is arranged into a 3D array, and a structure-adaptive window pursuit method is used to adapt to the local image features. Third, the resulting structure-adaptive 3D arrays are denoised by Wiener shrinkage in the transform domain defined by 2D principal component analysis in the image domain and Haar transformation in the third dimension. Finally, the noise-free DWI estimate is obtained by weighted averaging of the denoised structure-adaptive patches.
The paper is organized as follows. Section 2 describes the SASD algorithm and its main components, including the definition of the proposed similarity measure, the design of the structure-adaptive window pursuit method, and the Wiener filtering operation in the transform domain. The experimental setup is discussed in Section 3, and our results are presented in Section 4, where our approach is compared to state-of-the-art denoising methods, namely, Bayesian least-squares Gaussian scale mixture (BLS-GSM) (Portilla et al., 2003) and Field of experts (FOE) (Roth and Black, 2005). Concluding remarks are given in Section 5.
Section snippets
Methodology
In a nutshell, the proposed approach is inspired by the NL-means method to filter sparse local representations of the image to be denoised. Therefore, we describe the standard NL-means method in Section 2.1 prior to our SASD algorithm in Section 2.2.
Simulated data
Since real noise-free cardiac DWIs cannot be obtained by real DT-MRI acquisitions, we simulated biologically informed DWIs based on physical measures from polarized light imaging (we used the simulation method proposed in Wang et al. (2012)). The simulated DWIs – an example of which is displayed in Fig. 5a – reveal the left and right ventricles of a human heart and contain various textures and fine structures. They form a set of realistic short-axis cardiac images (size 256 × 256, intensity range
Comparison of BLS-GSM, FOE and SASD
Fig. 5 shows the denoising results for the simulated DWI associated with the first diffusion direction. The image to be denoised is displayed in Fig. 5b; it was obtained by adding Rician noise with standard-deviation σ = 20 to the noise-free image shown in Fig. 5a (this noise component significantly affects the fine structures). The solution obtained by BLS-GSM is over-smoothed (Fig. 5c), and the output of FOE has chessboard artifacts pointed out by the arrows in Fig. 5d. By contrast, SASD shows
Conclusions
We proposed a new DWI denoising method, namely, structure-adaptive sparse denoising (SASD), that collects adaptive similar neighborhoods to increase redundancy and hence to facilitate the removal of the noise. To avoid the drawbacks of standard similarity measures, we have defined a new constrained similarity measure based on the local mean and the SSIM index. We also proposed to search for structure-adaptive neighborhoods in the reference patches, which allows local adaptation to image
Acknowledgments
This work was supported by the NNSF of China (61271092), the International S&T Cooperation Project of China (2007DFB30320), and the French ANR (ANR-09-BLAN-0372-01). The authors would like to thank L.H. Wang for supplying the simulated cardiac DWI data used in our experiments in Section 4.1.
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