Post-acquisition denoising methods ^1-4 have been shown to be a viable means of improving the signal-to-noise ratio (SNR) of diffusion MRI data without having to resort to repeat acquisitions. Among the plethora of methods that are dedicated to this purpose, the non-local means (NLM) ⁵ algorithm has been shown to offer particularly good performance in edge-preserving denoising. The key feature that distinguishes NLM from the other methods is its ability to increase significantly the information available for denoising by going beyond the local neighborhood and allowing non-local or spatially distant information to be involved in denoising. In this abstract, we will show that improved denoising performance can be attained by extending the NLM algorithm beyond the x-space (i.e., the spatial space) to include the q-space (i.e., the wave-vector space).

Our method utilizes neighborhood matching in both x-space and q-space for effective denoising. For each voxel at location $$$\mathbf{x}$$$, the measurement corresponding to wave-vector $$$\mathbf{q}$$$ is denoised by averaging over non-local measurements that have similar q-neighborhoods from non-local voxels that have similar x-neighborhoods.

For x-space neighborhood matching, we use a conventional approach that is similar to NLM. Let $$$\mathcal{N}(\mathbf{x}_{i})$$$ and $$$\mathcal{V}(\mathbf{x}_{i})$$$ be a 3D neighborhood block and a search volume centered at $$$\mathbf{x}_{i} \in \mathbb{R}^3$$$, respectively. The x-space weight between $$$\mathbf{x}_{i}$$$ and $$$\mathbf{x}_{j} \in \mathcal{V}(\mathbf{x}_{i})$$$ is computed as $$w_{X}(\mathbf{x}_{i},\mathbf{x}_{j})=\exp\left \{-\frac{\|\mathbf{u}(\mathcal{N}(\mathbf{x}_{i}))-\mathbf{u}(\mathcal{N}(\mathbf{x}_{j}))\|^{2}_{2}}{h_{X}} \right \},$$ where $$$\mathbf{u}(\mathcal{N}(\mathbf{x}_{i}))$$$ is a vector that represents the intensity values of all voxels within $$$\mathcal{N}(\mathbf{x}_{i})$$$ and $$$h_{X}$$$ is a variable controlling the attenuation of the exponential function.

Neighborhood matching in q-space is less straightforward since samples in the q-space are not typically acquired on a uniform grid. To overcome this problem, we compute for each q-space sample its neighborhood features in the form of moments for neighborhood similarity evaluation. If for the voxel at location $$$\mathbf{x}_{i}$$$, we have diffusion-weighted measurements $$$\{S(\mathbf{x}_{i},\mathbf{q}_{k})\}_{k=1}^{K}$$$, the central moments $$$\{\mu_{n}(\mathbf{x}_{i},\mathbf{q}_{k})\}_{n=1}^{N}$$$ are computed as$$\mu_{n}(\mathbf{x}_{i},\mathbf{q}_{k}) = \sum_{k'}{\left[S(\mathbf{x}_{i},\mathbf{q}_{k'}) - \bar{S}(\mathbf{x}_{i},\mathbf{q}_{k})\right]^{n}f\left[S(\mathbf{x}_{i},\mathbf{q}_{k'})\right]}$$where $$\bar{S}(\mathbf{x}_{i},\mathbf{q}_{k}) = \sum_{k'}{S(\mathbf{x}_{i},\mathbf{q}_{k'}) f\left[S(\mathbf{x}_{i},\mathbf{q}_{k'})\right]}$$and$$f\left[S(\mathbf{x}_{i},\mathbf{q}_{k'})\right] = \exp\left \{-\frac{1-(\hat{\mathbf{q}}_{k}^{\text{T}}\hat{\mathbf{q}}_{k'})^{2}}{h_{\text{angle}}} \right\} \exp\left\{- \frac{(\|\mathbf{q}_{k}\|-\|\mathbf{q}_{k'}\|)^{2}}{h_{\text{magnitude}}} \right \}.$$

The parameters $$$h_{\text{angle}}$$$ and $$$h_{\text{magnitude}}$$$ control the attenuation of the exponential functions and $$$\hat{\mathbf{q}}_{k}=\mathbf{q}_{k}/{\left\|\mathbf{q}_{k}\right\|}$$$.

If we let $$$\boldsymbol\mu(\mathbf{x}_{i},\mathbf{q}_{k}) = [\mu_{1}(\mathbf{x}_{i},\mathbf{q}_{k}),\dots,\mu_{N}(\mathbf{x}_{i},\mathbf{q}_{k})]^{\text{T}}$$$, the similarity between the neighborhood of $$$\mathbf{q}_{k}$$$ of the voxel at $$$\mathbf{x}_{i}$$$ and the neighborhood of $$$\mathbf{q}_{l}$$$ of the voxel at $$$\mathbf{x}_{j}$$$ is reflected by weight $$w(\mathbf{x}_{i},\mathbf{q}_{k};\mathbf{x}_{j},\mathbf{q}_{l})=w_{X}(\mathbf{x}_{i},\mathbf{x}_{j})\exp\left \{-\frac{\|\boldsymbol\mu(\mathbf{x}_{i}, \mathbf{q}_{k}) - \boldsymbol\mu(\mathbf{x}_{j}, \mathbf{q}_{l})\|^{2}_{2}}{h_{Q}} \right\}\exp\left\{ -\frac{ \left[ \bar{S}(\mathbf{x}_{i},\mathbf{q}_{k}) - \bar{S}(\mathbf{x}_{j},\mathbf{q}_{l})\right]^{2} }{h_{\text{mean}}} \right\},$$with tuning parameters $$$h_{Q}$$$ and $$$h_{\text{mean}}$$$.

Similar in spirit with NLM and by considering the Rician bias in MR magnitude signal, the denoised version of $$$S(\mathbf{x}_{i}, \mathbf{q}_{k})$$$ can be computed as$$\hat{S}(\mathbf{x}_{i}, \mathbf{q}_{k})=\sqrt{\frac{\sum_{j,l}w(\mathbf{x}_{i},\mathbf{q}_{k};\mathbf{x}_{j},\mathbf{q}_{l})S^2(\mathbf{x}_{j},\mathbf{q}_{l})} {\sum_{j,l}w(\mathbf{x}_{i},\mathbf{q}_{k};\mathbf{x}_{j},\mathbf{q}_{l})} - 2\sigma^2},$$where $$$\sigma$$$ is the noise standard variance, which can be estimated from the data.

For quantitative evaluation, we generated a multi-shell spiral synthetic dataset (with diffusion tensors oriented along the spiral, b=1000,2000,4000,6000 s/mm², and 81 gradient directions), added noise, and compared the denoised data with respect to the noiseless ground truth. For various percentage of noise, measured based on the maximum intensity value, the results shown in Figure 1 indicate that our method significantly improves the peak signal-to-noise ratio (PSNR). Compared with NLM, the largest improvement is 6.76 dB when the noise level is 3%. The root mean square error (RMSE) maps, shown in Figure 2, further indicate that our method produce results that are closer to the ground truth. We then evaluated the proposed method on a real dataset acquired using the same gradient directions and b-values as the synthetic dataset. Figure 3 shows that our method reduces the spurious peaks of fiber orientation distribution functions (fODFs) and yields fODFs that are more coherent than NLM.

The extension of NLM to include q-space affords the following advantages: (1) Non-local information can now be harnessed not only across the spatial space, but also across measurements in q-space, making available significantly more information for denoising; (2) In white matter regions with high curvature, q-space neighborhood matching corrects for such non-linearity so that information from structures oriented in different directions can be used more effectively for denoising without introducing artifacts.We have extended NLM beyond the x-space to include the q-space, allowing denoising using non-local information not only in the spatial domain but also in the wave-vector domain. Our method allows information from highly curve white matter structures to be used effectively for denoising.