High-resolution diffusion weighted imaging (DWI) with reduced susceptibility artifacts^1,2 can be acquired using readout-segmented echo planar imaging (rs-EPI). The clinical applicability of this technique is limited by the poor SNR thus increasing the awareness for denoising strategies. Many methods have been developed and optimized for DWI to improve SNR under the constraint of edge preservation. However, most filter applications require a-priory information about the noise characteristics. The method described in ref. (3) allows the robust noise estimation also for complex setups by incorporating the correlation of multiple coils and parallel imaging. We took advantage of this method and incorporated the estimated noise in a regularization algorithm based on total generalized variation (TGV)⁴ in order to improve the SNR of DWI data. Compared to total variation (TV) TGV uses a less restrictive assumption of a piecewise constant signal avoiding staircasing artifacts often observed in TV regularization. TGV regularization along with noise estimation allows for user independent denoising of DWI data with excellent preservation of small details specifically for data with low SNR. The feasibility of the proposed method was tested on synthetic DWI data at different noise levels allowing for the comparison with a noise-free gold standard. Furthermore, results from TGV denoising were compared with non-local-means filtering (NLM)⁵, incorporating the same noise estimation, with respect to structure similarity index measure (SSIM), peak signal-to-noise-ration (PSNR) and root mean square error (RMSE).

High resolution DWI data were constructed using the IIT Human Brain Template⁶. Based on the available diffusion tensor DWI data were emulated with 12 diffusion sensitizing gradient directions by multiplying the tensor with appropriate b-vectors. To simulate realistic noise conditions, a 32 channel head-coil was assumed with Gaussian noise overlain the real and imaginary part for each individual coil element. Image reconstruction was performed using sum-of-squares without parallel imaging. Three-dimensional TGV regularization was carried out for every volume of the DWI dataset by minimizing the cost function:

$$\min_{u(w):Ω\rightarrow{Sym}^{2(3)}}\frac{1}{2}\|\lambda^{-1}\left(u-f\right)\|_2^2+\alpha_1\|\epsilon u-w\|_1+\alpha_0\|\epsilon w\|_1$$

where Ω is the 3D domain of computation, u is the denoised DWI volume, λ a spatially varying 3D regularization parameter that is updated during optimization , w a third-order tensor field arising from tensor TGV2 regularization⁷, ε denotes the symmetrized derivative and α₀, α₁ are fixed parameters. Solutions were found numerically using an iterative primal-dual first-order optimization algorithm⁸. The initial 3D regularization parameter λ is the spatial dependent 3D noise estimate σ and is updated during the iteration using the following rule:

$$\lambda\leftarrow\lambda\frac{\|u-f\|_2}{\|\sigma\|_2}$$

The NLM algorithm according to ref. (5) and the TGV algorithm were implemented in Matlab (The MathWorks, Inc., MA, USA). All DWI data (noise free, noisy, NLM denoised, TGV denoised) were processed with FSL 4.1 to calculate fractional anisotropy (FA) maps. Comparisons of SSIM, PSNR and RMSE for DWI data and for FA were also performed using Matlab software.

Figure 1 shows a slice of the simulated DWI dataset with diffusion sensitizing gradients at two different noise levels. The noise-free gold standard, the noisy image the denoised image using the NLM method and the proposed TGV method are shown. NLM denoising provided more inhomogeneous results due to the search window (7x7x7 voxel) used for the non-local means operation. In contrast, TGV results appeared smoother while edges were still preserved to high extent. SSIM, PSNR and RMSE showed a reduced dependence on different noise levels (σ² varying from 500 to 3500) compared to NLM (figure 2). Less inhomogeneity was observed in FA-maps (figure 3). This could be explained by the fact that FA is obtained from the diffusion tensor, which is a projection from a 12 dimensional space into a 6 dimensional subspace. Differences in SSIM, PSNR and RMSE for FA values are displayed in figure 4.In this work we introduce a novel user independent algorithm for denoising DWI data utilizing TGV regularization under consideration of the spatial dependent noise distribution. It was demonstrated, that DWI data, denoised with TGV, are less dependent from the noise level compared to the widely used NLM filter method with respect to the parameters SSIM, PSNR and RMSE. Overall the denoising abilities of TGV were better compared with NLM when the same noise estimate was used for both methods. This behavior has also been found for FA but to a smaller extent. The quality of denoising using TGV, similar to other methods, depends on the accuracy of the noise estimation. Considering that the regularization parameter that is updated in every iteration step depends on the norm of the noise estimate, the role of noise estimation is crucial. However, any appropriate method for noise estimation can be used with our proposed algorithm providing a fast and robust denoising method.