POAS4SPM: A Toolbox for SPM to Denoise Diffusion MRI Data
Abstract
We present an implementation of a recently developed noise reduction algorithm for dMRI data, called multishell position orientation adaptive smoothing (msPOAS), as a toolbox for SPM. The method intrinsically adapts to the structures of different size and shape in dMRI and hence avoids blurring typically observed in nonadaptive smoothing. We give examples for the usage of the toolbox and explain the determination of experimentdependent parameters for an optimal performance of msPOAS.
Keywords
dMRI Noise reduction POAS msPOAS SPM (RRID:nif000000343)Introduction
Diffusionweighted magnetic resonance imaging (dMRI) has developed into an extremely versatile tool for the invivo structural analysis of tissue, for example in the human brain (JohansenBerg and Behrens 2009). One reason is that the diffusion signal obtained with the pulsed gradient spin sequence echo (PGSE, Stejskal and Tanner 1965) directly relates, via threedimensional Fourier transform, to the diffusion propagator which is the probability density of the underlying Random Walk the spin particles experience (Mitra and Sen 1992). Therefore, if we measured the diffusion signal for all possible diffusion gradient directions, times and strengths, i.e. cover the whole qspace, we would know the full propagator. Its spatial and directional dependence would allow us to infer on boundaries for the diffusing particles and hence the underlying structure. However, in practice, only a limited coverage of the qspace is feasible. Therefore, a number of models have been developed in the past, which reveal at least partial information contained in the diffusion propagator. Most require dMRI data measured on at least one qshell, that is characterized by a single bvalue (Basser et al. 1994b) subsuming diffusion gradient strength and diffusion time.
The most prominent example of a diffusion model gives rise to Diffusion Tensor Imaging (DTI, Basser et al. 1994a, b). Surprisingly, although this model actually describes free diffusion in anisotropic media it has proven to relate well to the underlying tissue geometry in the brain in general, and to the main fiber directions in the white matter in particular (JohansenBerg and Behrens 2009). More sophisticated descriptions of the diffusion signal have been examined to infer on more complicated subvoxel structure like multiple fiber directions. These include HARDI (Frank 2001; Tuch et al. 2002), tensor mixture models (Behrens et al. 2003; Assaf and Basser 2005; Tabelow et al. 2012), higher order tensor approximations (Özarslan and Mareci 2003; Liu et al. 2003; Jensen et al. 2005) and methods to determine the full diffusion propagator (Özarslan et al. 2006; Cheng et al. 2010), see Assemlal et al. (2011) for a recent review.
In any case the measures of interest, like fiber directions or quantitative measures like the fractional anisotropy (FA) in DTI, are estimated based on the raw diffusion images. Hence, the accuracy of the estimates depends on the data quality, which in turn typically requires retrospective correction of artifacts due to eddycurrents, motion (Mohammadi et al. 2010), susceptibilityrelated distortions (Ruthotto et al. 2012, 2013), instrumental (Mohammadi et al. 2012a, b) or physiological noise (Mohammadi et al. 2013a, b). Signaltonoise ratio (SNR) is especially low in dMRI because of the additional exponential dependence in the diffusionweighted signal, see, e.g., Stejskal and Tanner (1965). As a result the SNR in dMRI decreases with increased diffusion weighting, i.e. higher qshells have lower SNR. Furthermore, beyondtensor models with a larger number of parameters are more sensitive to data noise, making sophisticated denoising strategies mandatory for cuttingedge dMRI. In order to reduce noise in dMRI data a number of different approaches have been developed in recent years starting from Gaussian filtering (Westin et al. 1999), smoothing procedures in tensor space for DTI (Fletcher 2004), to denoising algorithms based on partial differential equations (Ding et al. 2005; Parker et al. 2000; Duits and Franken 2011) to name only a few.
Recently, we developed a positionorientation adaptive smoothing algorithm (POAS, Becker et al. 2012) based on the propagationseparation (PS) approach (Polzehl and Spokoiny 2006; Becker and Mathé 2013). The method is edgepreserving and avoids blurring of the fine anisotropic structures observed in dMRI. The method has been extended to be applicable to dMRI measured on multiple shells and named multishell POAS (msPOAS, Becker et al. 2014). It capitalizes on the additional information on the different shells. Furthermore, several improvements make msPOAS feasible from a computational point of view. Finally, msPOAS can also be applied to singleshell dMRI data. Thus, in this paper we will only consider the more general msPOAS approach.
Compared to previous diffusionmodelbased adaptive smoothing methods, e.g., Tabelow et al. (2008), (ms)POAS has the advantage that it directly denoises dMRI data without requiring any diffusionmodel assumptions. Thus, it avoids a bias towards any diffusion model such as DTI, HARDI, or tensor mixture models (see Becker et al. (2012) for details).
(Ms)POAS has been originally implemented in the R language and environment for statistical computing and graphics (R Development Core Team 2014). While the corresponding package dti (Tabelow and Polzehl 2014) is easy to install (Polzehl and Tabelow 2011) and allows for applying msPOAS to dMRI data using very few code lines, the use of R in the neuroimaging community is still limited. We therefore implemented msPOAS as a toolbox for Statistical Parametric Mapping (SPM) (Friston et al. 2006), the most widely used neuroimaging analysis package, to make the method available to a broader audience, see www.diffusiontools.com.
In this paper we shortly review the method in a simplified way, present the new toolbox for SPM, describe the usage of the toolbox, and suggest methods to determine experimentdependent measures and to set method parameters. We present some worked examples with single and multishell data. For a more indepth review of the theory of POAS and msPOAS we refer the reader to the original work in Becker et al. (2012, 2014).
Methods
MsPOAS is a noise reduction method for dMRI data that is measured on at least one qshell, i.e., for at least one bvalue and a sufficiently large number of diffusion gradient directions. Although msPOAS also works for very few diffusion gradient directions, it particularly benefits from the information from more gradient directions (Becker et al. 2014). MsPOAS is also suitable for singleshell dMRI data.
Review: MultiShell PositionOrientation Adaptive Smoothing (msPOAS)
The design space in dMRI for a single qshell forms an \( {\mathbb{R}}^3\times {\mathbb{S}}^2 \) space that contains a collection of points described by their voxel positions in ℝ ^{3} and gradient directions in \( {\mathbb{S}}^2 \) (Duits and Franken 2011). Let S _{ b }(m) denote the observed signal for the bvalue and point m defined as \( m=\left(\mathrm{v}\in {\mathbb{R}}^3,\mathrm{g}\in {\mathbb{S}}^2\right) \), where v is its voxel location and g is its gradient direction. The nondiffusion weighted b _{0}image is denoted by S _{0}(v, 0) = S _{0}(v) and does of course not depend on g. In case of several b _{0}images we consider their mean. Interpolation is used to account for discrepancies in gradient directions across shells, see Becker et al. (2014).
The measured random values S _{ b }(m) are distributed according to some probability distribution that is parametrized by a “true” intensity parameter θ, the noise standard deviation σ and a number of degrees of freedom 2L where L denote the effective number of coils for parallel imaging (AjaFernández et al. 2009). Then S _{ b }(m)/σ is assumed to be noncentral χdistributed with 2L degrees of freedom.
In msPOAS we assume that similar signal values in \( {\mathbb{R}}^3\times {\mathbb{S}}^2 \) extend over sets of neighboring points n = (v_{ n }, g_{ n }). This can be used to obtain an improved estimate for the image value at any point \( m\in {\mathbb{R}}^3\times {\mathbb{S}}^2 \). If the definition of the neighborhood is specific for the point (and based on the data) we call the method adaptive, or nonadaptive otherwise. The notion of neighborhood typically requires the definition of a metric in the considered design space, here \( {\mathbb{R}}^3\times {\mathbb{S}}^2 \).
(with (x)_{+} denoting the maximum of x and 0), due to its higher efficiency and computational simplicity, see Fan and Gijbels (1996).
 1.
Redefine the weighting schemes given in Eq. (2) by an additional term that evaluates the distance of the signal in two measurement points m and n making the weights adaptive, see Eq. (4), cf. the construction of bilateral filters.
 2.
Repeat the estimation step in Eq. (1) using adaptive weights (4) for an increasing (typically geometric) sequence of bandwidths h _{ k } for k = 1, …, k ^{⋆} instead of a single bandwidth h, see Eq. (2), in an iterative procedure, see (Becker et al. 2014). This approach is, therefore, a scalespace method. The specific choice of the sequence of bandwidths is described in detail in Appendix B, Becker et al. (2014).
The summation in Eq. (3) is over all signal from the same qshell. For b _{0}images averaging in Eq. (3) is reduced to voxel space, see Becker et al. (2014) for details.
Toolbox Implementation and Installation
The toolbox POAS4SPM for the neuroimaging software SPM has been implemented using C and Matlab. It comes as opensource software with GPL2.
The toolbox is part of the ACID toolbox for “Artefact correction in diffusion MRI” and can be downloaded from its homepage at http://www.diffusiontools.com/. It is listed on the SPM extension homepage http://www.fil.ion.ucl.ac.uk/spm/ext/, too. Installation is done by extracting the toolbox into the toolbox folder of SPM, and compiling all mexfiles in the Preprocessing/POAS subfolder. Running the make_ACID.m utility in the cfiles folder of the ACID toolbox will automatically compile all necessary cfiles.
Usage of the Toolbox

Diffusion weighted images: Choose the N images including N _{ g } diffusion weighted and N _{0} nondiffusion weighted data files. Data should be given in separate 3D volumes.

Diffusion directions: Add a 3 × N–array consisting of the diffusion gradient directions with normalized vectors that appear in the same order as the DTI images were entered. Choose a vector with three zeros for each b _{0}image. If a name of a variable is entered here, e.g., with gradient direction data read from a file, its value is automatically evaluated.

bvalues: Add a 1 × N–array with bvalues. They should appear in the same order as the DTI images were entered. The bvalue is given in units of s/mm^{2}. b _{0}images should have b = 0. If the data contains images with a small bvalue (b < 100s/mm^{2}), which serve as reference image without directional information, mark them by using b = 0 as well. The diffusionweighted images corresponding to different shells will be identified by their bvalue. Also here, a variable name can be entered.

k star: This is the parameter k ^{⋆} of msPOAS that defines the number of iterations and thus the maximal location bandwidth \( {h}_{k^{\star }} \).

kappa: This is the parameter κ _{0} of msPOAS that defines the initial ratio of the spatial and spherical distance in Eq. (5).

lambda: This is the adaptation parameter λ of msPOAS, see Becker et al. (2012, 2014) for more details.

sigma: The value σ is the noise level in the data and must be obtained from the data, see option Estimate sigma in the toolbox. Although the value of σ may vary spatially due to effects of parallel imaging, the current implementation of msPOAS assumes a homogeneous variance. The effect of a misspecified σ can be partly compensated by the choice of λ (Becker et al. 2014).

ncoils: This parameter specifies the parallel imaging factor L, i.e., the number of different receiver coils that contributed to the measured signal value. It may also vary with spatial location, but the current implementation of msPOAS assumes a global value for L. msPOAS has been shown to be relatively robust against misspecifications of L (Becker et al. 2014).
After running the batch script, the smoothed diffusion weighted volumes are written to disk using “poas” as a prefix. Only one b0image, obtained as smoothed average of all original b0images, is written to disk, see Becker et al. (2012). For further processing the corresponding gradient orientations and bvalues are written to disk as a .matfile. The input directory is used as the target directory for the script’s output.
Choice of the Method Parameters k ^{⋆}, κ _{0}, λ
λ is the adaptation bandwidth of the procedure that steers the amount of adaptation of msPOAS. For λ = 0 the adaptive weights in Eq. (4) vanish for all m ≠ n. As a consequence, all estimates at any iteration step will coincide with the original data. In contrast, for λ = ∞ the adaptive weights coincide with the nonadaptive weights, i.e. \( {\tilde{w}}_{mn}^{(k)}={\overline{w}}_{mn} \) for all m, n, see Eq. (4), and msPOAS finally generates a nonadaptive kernel estimate in the space \( {\mathbb{R}}^3\times {\mathbb{S}}^2 \) with kernel K _{loc}, bandwidth \( {h}_{k^{\star }} \) and \( {\kappa}_{k^{\star }}={\kappa}_0/{h}_{k^{\star }} \).
Basically, λ can be chosen to satisfy a propagation condition using simulation independent from the processed data (Becker et al. 2012) but for the specific noise distribution. In case of dMRI data, the distribution can be assumed to be a noncentral χdistribution with 2L degrees of freedom and noncentrality parameter θ. We suggest a value of λ = 12 for all datasets. Depending on the quality of the estimates for the datadependent parameters σ, L slight adjustments may be required, as discussed below. The propagation condition ensures with high probability that in homogenous image regions the msPOAS estimator basically coincides with the nonadaptive in Eq. (1).
How to Estimate σ and L?
MsPOAS requires two datadependent input parameters σ and L, that fix the properties of the noise distribution and enter the definition of the statistical penalty s _{ mn } ^{(k)} , see Eq. (6).
A suitable estimate for the standard deviation σ of the noise can be obtained by any method available to the user, see (AjaFernández et al. 2009) for a review. The methods typically determine σ from the Rayleigh L = 1 or central χdistribution L > 1 in the image background.
We implemented the method described in (Constantinides et al. 1997) in the toolbox. It can be accessed in the batch editor via SPM > ACID Toolbox > PreProcessing > Choose POAS options > Sigma estimation and requires as input the diffusion weighted images from the data, a binary mask file defining background voxel only and the specification of L, see below. Running the estimation returns a mean value over all diffusion weighted images. It also writes a .txtfile into the data directory with individual values for each diffusion weighted volume. b _{0}image typically lead to larger estimates for σ, than the diffusion weighted volumes. Note, that the method of Constantinides et al. (1997) does not account for noise correlation due to multichannel receiver coils (Hutton et al. 2012). Furthermore, nonbackground structure like ghosts within the defined background mask also leads to overestimates of σ. For msPOAS it is advisable to use a conservative small estimate for σ and potentially correct a misspecification by adjusting λ.
The parameter L depends on the reconstruction algorithm, see, e.g., AjaFernández et al. (2011). It is very difficult to estimate from the data. For some reconstruction methods it can be shown that L = 1 (Sotiropoulos et al. 2013). Generally L equals the total number of receiver coils for a SumofSquares reconstruction and is spatially varying for GRAPPA (AjaFernández et al. 2011). Fortunately, msPOAS has been shown to be relatively robust against misspecification of L (Becker et al. 2014). We therefore suggest to use a value of L = 1 consistently in the estimation of σ and for msPOAS, if no other estimate is available.
Experimental Data
Two healthy volunteers (male) participated in the study approved by the local ethics committee after giving written informed consent. The example data used in this paper has been acquired as follows: Experiments were performed on a MAGNETOM Trio, a Tim System 3T scanner (Siemens AG, Healthcare Sector, Erlangen, Germany). Two highresolution diffusion magnetic resonance imaging (dMRI) data sets were acquired using a reduced fieldofview (FoV) technique (Heidemann et al. 2010), one multishell, and one singleshell data set.
For the multishell data, the 161 × 58mm FoV was centered on the motor cortex. It had 1.2mm isotropic resolution, and 10 % slice gap, resulting in an effective slice thickness of 1.3mm. The images were acquired at 3 different bvalues: 21 at b = 20s/mm^{2}, 100 at b = 800s/mm^{2}, and 100 at b = 2000s/mm^{2} using the directions suggested by Caruyer et al. (2011). The total scan time was about 22 min. This dataset was also used in Becker et al. (2014).
The 156 × 56mm FoV of the singleshell data set was centered on the thalamus with 1mm isotropic resolution, and 10 % slice gap, resulting in an effective slice thickness of 1.1mm. The images were acquired at 2 different bvalues: 5 at b = 0s/mm^{2} and 64 at b = 1000s/mm^{2} using the directions provided by Siemens. This dataset was acquired three times, giving a total scan time of about 20 min.
Prior to POAS the data were corrected for motion and eddy current artifacts using the method detailed in Mohammadi et al. (2010), which is implemented as part of the ACID toolbox pipeline. For the analysis in this paper we then estimated the diffusion tensor and FA.
Parameter Choices for the MultiShell Dataset
We repeatedly defined an arbitrary region within the background of the data and used the method implemented in the toolbox and described in the “Methods” section to estimate the noise standard deviation σ. We consistently found a value of σ = 30. We used L = 1 for all calculations as in (Becker et al. 2014).
We used msPOAS parameter values κ _{0} = 0.3 and λ = 12 to match the choice in Becker et al. (2014). The number of iteration steps was fixed at k ^{⋆} = 12 that provided a good balance between computational costs and achieved noise reduction.
Parameter Choices for the SingleShell Dataset
Estimation of the noise level in the image background for the singleshell dataset consistently provided a value of σ = 45. We used L = 1, k ^{⋆} = 12, and λ = 12 for all calculations. This dataset was measured with three repetitions. First, we smoothed the data for a onerepetition dataset (using the first repetition), here the κ _{0} value was used to be κ _{0} = 0.5. Then, we smoothed the data for all three repetitions, with κ _{0} = 0.3.
Finally, to demonstrate the dependence of the msPOAS outcome on the different method parameters, we consecutively changed one of the three parameters κ _{0}, λ, and k ^{⋆}, while leaving the others constant. To this end, we used the onerepetition dataset and varied κ _{0} = 0.3, 0.5, 0.8, λ =1, 5, 10, 12, 50, 100, 500, ∞, k ^{⋆} = 4, 8, 12, 16, 20, 24, 28.
Hardware
We performed the example analysis on a HP Workstation XW4600 with Intel^{®} Core™2 Duo CPU E6850@3.00GHz and 8GB RAM running with OpenSuSE 12.3 and Matlab 2012b with SPM8.
Results
The calculations on the described hardware using the optimal parameters given above took approximately 1200 sec = 20 min for the multishell dataset, 700 sec ≈ 12 min for the onerepetition singleshell dataset, and 1900 sec ≈ 30 min for the three repetition singleshell dataset.
Discussion
We implemented a method for adaptive denoising diffusion weighted MRI called msPOAS as a toolbox for SPM. The program is part of a general toolbox for artefact correction in diffusion MRI data named ACID.
Dependence of the Results on the Method Parameters and its Interaction
MsPOAS requires the specification of datadependent quantities σ and L and method parameters h ^{⋆}, λ, and κ _{0}. In this paper, we analyzed the dependence of the outcome of msPOAS on the choice of the method parameters.
As shown in Fig. 6 the effect of noise reduction increases with the number of iteration steps k ^{⋆}. In homogeneous compartments of the image, the smoothness increases with k ^{⋆}. On the other hand, the adaptivity of the procedure obviously avoids blurring at the borders. Thus in principle increasing k ^{⋆} improves the noise reduction effect of msPOAS without blurring structural border. This can be interpreted as an intrinsic local stopping criterion for the method. However, the computational cost increases exponentially with k ^{⋆}, see Fig. 1. For very large k ^{⋆} deviations in the diffusion weighted images from the structural assumption of local constant image intensities, e.g., by smooth and gradual changes, lead to a stepfunction approximation of the image intensities, see Becker and Mathé (2013), Becker et al. (2012). Best results are achieved for intermediate k ^{⋆}, such that a suitable compromise between computation time, variancecontrolled bias and required noise reduction has to be made, e.g. k ^{⋆} = 12.
The adaptation bandwidth λ controls the adaptivity of the method, ranging from complete adaptation at λ = 0, where the original is not changed at all by msPOAS, to a nonadaptive estimate at λ = ∞, where the adaptivity of msPOAS is turned off, see Fig. 7. Best results can be achieved at λ = 12.
Different choices of λ have a similar effect on the results of msPOAS as the estimates for σ. Generally, if σ is underestimated, msPOAS will be to restrictive and only little smoothing effect will appear in the result. If σ is overestimated, this may lead to blurring in the result. This also means that the effect a misspecification of σ can be compensated (to some extent) by adjusting λ accordingly.
The choice of κ _{0} influences the amount of smoothing on the sphere. For relatively good SNR κ _{0} can be chosen smaller then the recommended value to reduce an estimation bias due to the violation of the a local constant signal function on the sphere. For lower SNR the initial estimates benefit from larger values of κ _{0} through stabilization. Our construction of the sequences of bandwidths h _{ k } and κ _{ k }, see Eq. (5) and its discussion, automatically lead to increased noise reduction by larger values of κ _{0} (with higher computational costs).
The choice of L has only a minor effect on the result of msPOAS, see Becker et al. (2014). Nevertheless, msPOAS may benefit from precise specification of L, if available. If L is unknown, we recommend to use L = 1.
Suggestions for Parameter Choices
 1.
L = 1, if L is unknown.
 2.
Determine σ by some suitable method (e.g. the method given in the toolbox) using L as previously chosen.
 3.
λ = 12.
 4.
Choose κ _{0} such that \( 5\le \widehat{N}\left(1 cos\left({\kappa}_0\right)\right)\le 10 \), depending on the mean number \( \widehat{N} \) of diffusion gradients per shell, see Fig. 2 for a graphical tool for this choice.
 5.
k ^{⋆} = 12.
 6.
Run msPOAS.
 7.
Adjust parameters: slightly decrease λ if oversmoothing at borders occurs, which looks like Fig. 7d or e. Slightly increase λ if the noise reduction in homogeneous regions is less then expected for the utilized k ^{⋆}: In this case increasing k ^{⋆} does not increase noise reduction in homogeneous regions.
 8.
Adjust k ^{⋆} if more or less noise reduction for homogeneous regions is required.
 9.
Rerun msPOAS if adjustments are necessary.
The evaluation of the msPOAS result can be done at the level of diffusion weighted images or for diffusion model parameters, like FA maps.
Conclusion
MsPOAS is a powerful method for adaptive noise reduction in diffusion MRI data that is now available as a toolbox for SPM. We demonstrated and discussed the effect of different method parameters and datadependent quantities on the results of msPOAS and gave recommendations for their choice and determination, respectively.
Information Sharing Statement
POAS4SPM is part of the ACIDtoolbox available at http://www.diffusiontools.com.
SPM is a MATLAB toolbox that is freely available from http://www.fil.ion.ucl.ac.uk/spm/software/spm8/
Notes
Acknowledgments
This work has been supported by the DFG research center Matheon. The Wellcome Trust Centre for Neuroimaging is supported by the Wellcome Trust. S.M. was supported by the Deutsche Forschungsgemeinschaft (DFG, MO 2397/11).
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