Improving Diffusion Tensor Estimation using Adaptive ...

Improving Diffusion Tensor Estimation using Adaptive and Optimized Filtering based on Local Similarity ´ ´ ´ Orozco Andr´es F. L´ opez-Lopera, Mauricio A. Alvarez and Alvaro A. Universidad Tecnol´ ogica de Pereira, Electrical Engineering Program, Pereira, Colombia {anfelopera,malvarez,aaog}@utp.edu.co

Abstract. The diffusion-weighted magnetic resonance imaging (DWMRI) has been used to diagnose anomalies in human brain by describing the magnitude and directionality of water diffusion per voxel. Such information can be represented alternatively in diffusion tensor imaging (DT-MRI), yielding images of normal and abnormal white matter fiber structures, and maps of brain connectivity through fiber tracking. A DW-MRI study is usually characterized by a low signal to noise ratio, which may reflect in the poor estimation of DT-MRI. Filters based on local similarity have been receiving increasing attention, but they have been barely studied for DT-MRI. In this proposal we introduce adaptive and optimized filtering techniques based on local similarity for MRI to remove the biasing in both DW-MRI filtering and DT-MRI estimation, evidencing a better performance respect to classical filters and robust DT estimation algorithms. We estimate the DT-MRI extracting metrics computed from the DT to evaluate the filtering performance. Keywords: adaptive and optimized filtering, diffusion tensor imaging, diffusion-weighted magnetic resonance imaging

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Introduction

The diffusion-weighted magnetic resonance imaging (DW-MRI or DWI) studies 4D diffusion images, and widely used to diagnose several pathological anomalies in human brain [1]. The DW-MRI describes the neuronal pathways which are used for communication among several centers of brain activity. In the last decades, different clinical communities have focused on DW-MRI for studying the microscopic water particles’ motion within brain tissue for the diagnosis of pathologies. This information can be represented alternatively in diffusion tensor imaging (DT-MRI or DTI), where the diffusion tensor (DT) describes the magnitude and the directionality of water diffusion in a specific voxel, producing images of normal and abnormal white matter fiber structure. Also, DT-MRI yields maps of brain connectivity through fiber tracking [2]. Therefore, the studies of DWMRI and DT-MRI are promising in pre-operative planning of neurodegenerative diseases [3].

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The DW-MRI has an inherent low signal to noise ratio (SNR) due to a low signal amplitude and a pronounced thermal noise, complicating and biasing the estimation of diffusion tensors [1]. A common practice for increasing the SNR in DW-MRI is averaging several DW-MRI studies in order to reduce noise variance [2]. This practice does not remove noise bias and consume a long acquisition time, therefore it is not adequate for typical clinical settings [1]. A potential alternative to the averaging of several DW-MRI studies, is to include an additional denoising stage in DWI that helps to improve the DTI estimation. Different filtering methods have been applied in the preprocessing stages, aiming to achieve denoised images that help to improve the DT-MRI estimation [1]. Classic denoising techniques in DW-MRI filtering such as Gaussian filters [4] and the algorithm proposed by Perona and Malik in [5], assume an equal noise distribution across the image, modifying important information (e.g. edges, structures and type of matter), leading to suboptimal filtering results. Methods based on local similarity have been receiving increasing attention due their local filtering adaptation, having the potential to improve qualitatively and quantitatively the estimation of diffusion information with respect to another filtering techniques [1, 2, 6]. Moreover, adaptive and optimized filters based on local similarity can mitigate the drawbacks of having to deal with tuning the filter parameters, but they have been barely studied for DW-MRI data. In this proposal, we introduce some adaptive and optimized methods grounded in non-local means (NLM) similarity, and principal component analysis decomposition (PCA) in order to achieve better results in both denoising DW-MRI and DT-MRI estimation. We experimentally show that by using filters that take into account local similarity in the DWI, it is possible to outperform classical filters previously used in this literature. In this paper, we compare the DTI estimation results using different adaptive and optimized filtering techniques based on local similarity to remove Rician noise, smoothing images and enhancing the edges of the brain structures. We use several metrics computed from the diffusion tensors to evaluate the performance for each filter. This paper is organized as follows. The materials and methods are described in section 2. In section 3, we discuss and compare the final results obtained by using the different filtering techniques, evaluating the performance for each one using the DT estimated. Finally, conclusions are given in section 4.

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Materials and Methods

In order to introduce the solution for a better DW-MRI filtering, ensuring a reliable diffusion information, in this section we will describe the advantages and disadvantages for the different filters that were used in this paper. We will describe three classic filters used to denoising DW-MRI, evidencing the tuning problem of the filtering parameters. Second, we introduce a series of adaptive and optimized filtering techniques based on local similarity used in brain imaging preprocessing, aiming to achieve a better DW-MRI filtering. Third, we introduce the concept of DT estimation to evaluate the performance of the above filters. Finally, we describe the procedure and the experimental background used to obtain the results in section 3.

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2.1

Gaussian Filter (GF)

GF is the first of the most common filters used in DW-MRI. This filter can be applied in brain image preprocessing, aiming to achieve an isotropic or anisotropic smoothing by the convolution between a multivariate Gaussian kernel with the corrupted image. This filter can remove noise from images, but a poor tuning of the scale parameter tends to blur the edges from brain structures [4]. 2.2

Perona-Malik Algorithm

Perona and Malik in [5], proved that the problem of denoising DW-MRI can be based on the diffusion equation expressed in (1), if we assume that the images are both spatial and temporal functions. The diffusion equation over the intensities u(x) per voxel x is given by ∂u(x, t) = div {g(k∇u(x, t)k)∇u(x, t)} , ∂t

(1)

where div{·} is the divergence operator, g(·) is a monotonically decreasing function, and the initial condition from (1) is the noisy image. The performance of this filter depends of the type and monotonic trend of g(·) [5]. 2.3

Non-Local Means filter (NLM)

Filters based on NLM, originally proposed by Baudes et al. in [7], have become an attractive way to remove noise preserving original structures. NLM considers the high level of pattern redundancy from images, aiming to achieve high-quality image denoising by averaging similar realizations of the noisy signals [6]. In a 2D ˆ (xi ) of the position or voxel set of parallel images u, the restored intensities u xi , is a weighted average of the surrounding voxels intensities u(xi ) in a sliding volume Vi [1], given by Equation (2) X ˆ (xi ) = u w(xi , xj )u(xj ), (2) xj ∈Vi

where w(xi , xj ) ∈ [0, 1] is the weight assigned to u(xj ) to restore the intensity u(xi ), based on the square of the Euclidean distance given as   1 ku(xi ) − u(xj )k22 w(xi , xj ) = exp , (3) Zi h2 with Zi a normalization constant and h controls the contributed information of the surrounding voxels in Vi . Once again, the performance of this filter depends on the parameters h and Vi . The performance of the above filters heavily depends on how well we tune the parameter for each one. To avoid this problem, several researches have proposed a series of adaptive and optimized filter techniques based on local similarity for brain imaging preprocessing (e.g. NLM, DCT and PCA). Next, we briefly review some optimized variants of filters based on NLM, DCT and PCA decomposition.

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2.4

Adaptive and Optimized NLM filter (AONLM)

The AONLM filter was proposed by Maj´on et al. in [1], designed for spatially varying noise typically presented in MRI. This filter includes a noise estimation module (e.g. Gaussian or Rician noise), making it an automatic and robust to outliers. This method deals with the non stationary noise with the introduction of the local noise estimation proposed by Wiest-Daessl [8], removing the bias intensity using the properties of the second-order moment of the Rician law. In practice, AONLM is more robust than an optimized NLM but it does not achieve the intensity bias correction. Results in [1], have showed a better performance with respect to NLM for different image types and noise levels. 2.5

Oracle-based 3D Discrete Cosine Transform filter (ODCT)

In [6], the authors propose an extension of the original method proposed by Guleryuz in [9]. ODCT is based on a discrete cosine transform (DCT) that takes advantages of the sparseness information from images. In contrast to DCT, ODCT assumes that the previous well known null coefficients of the denoised image can be applied to improve the denoising. This method minimizes the mean squared error with respect to a pre-filtered image obtained by DCT. In [10], the authors propose the Rician median absolute deviation method (RMAD) in order to estimate a slanted the noise variance making the filter to be fully automatic. Results in [6] evidence a successful filtering of MRI in a few time [6]. 2.6

Prefiltered Rotationally Invariant NLM filter (PRINLM)

Coup´e et al. propose an approach that uses the extended ODCT preprocessing results for a new filtering stage based on a rotationally invariant version of the NLM filtering to compute patch similarities. It is known as PRINLM [6]. This filter takes advantage of both sparseness and self-similarity properties between voxel intensities and the corresponding local patch mean using a Gaussian kernel. The authors propose the RMAD method in order to estimate the noise variance. Whereas ODCT tends to slightly oversmooth edges and some fine details, the PRINLM seems to retain more details in the denoised image because of its voxelwise processing. Experimental works in [6] have shown a high compressibility of MRI data, allowing a more efficient noise reduction and benefiting visual diagnostics in MRI brain tissue segmentation. 2.7

Local PCA filter (LPCA)

Recently, Maj´ on et al. have proposed an efficient denoising filter that takes into account the nature of 4D images structures (e.g. DWI) in [2], integrating a noise estimation module making it an automatic and a robust filter. This filter computes the PCA decomposition over a 4D sliding window at each image position in order to locally find a reduced representation of diffusion information, promoting sparser representation in contrast to related PCA methods [11]. In [2], the authors have proposed an automatic method for estimate and provide the

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local noise variance prior for the LPCA filter. In DW-MRI preprocessing, the LPCA filter takes advantages of the locality to reduces the bias induced by the Rician noise in DWI, producing a better diffusion parameter estimations that reflect the real characteristics of tissues, rather than noise-biased measurements. Results in clinical datasets [2], evidence high degree of confidence, potentially improving any quantitative measure derived from them. 2.8

Diffusion Tensors (DT)

As previously stated, the DWI contains information of the neuronal pathways used for communication in the brain [3], and it is possible to embed this information in symmetric 3 × 3 matrices known as diffusion tensors (DT) per voxel [12]. Stejskal-Tanner in [13], relates the DWI and DTI fields by the Equation (4) n o Sk = S0 exp −bˆ g> gk , k Dˆ

(4)

where Sk is the DW-MRI associated to the k-th diffusion direction with k = ˆ is 1, 2 · · · , K and K is the number of directions. S0 is the baseline image, g a normalized gradient vector, b is a diffusion parameter and D is the tensor matrix per voxel. Also it is possible to visualize the diffusion information from the DT calculating metrics based on diffusion information such as the fractional anisotropy (FA) and the mean diffusivity (MD). Several researches have proposed different methods for DTI estimation such as linear systems with non-negative constraints [14] and methods based on least-squares criterion (e.g. RESTORE)1 [15]. Recently, clinical studies have proved that there is a relationship between some diseases and the changes in the connectivity of certain brain areas. Therefore, the DW-MRI potential and the use of DT-MRI are promising in terms of diagnosis and pre-operative planning of neurodegenerative diseases [3]. 2.9

Procedure

Experimental Background: In this proposal, we worked with a complete set of DWI images in the Nifti format2 (dti30.nii), with both the direction and header information (dti30.bvec, dti30.bval). This dataset contains 60 slices (102 × 102 each one) and 62 gradient directions per slice. We applied the LPCA filter over the dataset in order to remove the intrinsic noise. We refer to this pre-processed dataset as the DTI30-DWI.nii dataset. We use the new dataset as the goal standard to compare the preprocessing methods listed above in section 2. Subsequently, we introduce a Rician noise with σ = 25 for evaluating the filtering performance. 1

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The robust estimation of tensors by outlier rejection (RESTORE), uses iteratively reweighted least-squares regression to identify potential outliers and exclude them. Available on http://nipy.org/dipy/examples\_built/restore\_dti.html. Dataset used by Leigh Morrow et al. from a 3-Tesla Siemens Trio. Available at http://www.cabiatl.com/CABI/resources/dti-analysis/.

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(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

(m)

(n)

(o)

(p)

Fig. 1. DWI and DTI results for the proposed filtering methods using the 38th direction and the 30th slide from the DTI30-DWI.nii. In (a) and (i) both DWI and DTI estimated from DTI30-DWI are showed, respectively. In (b) and (j) we show the corrupted DWI with Rician noise and its estimated DTI. In (c), (d), (e), (f), (g) and (h) we show the filtering results using GF, PM, AONLM, ODCT, PRINLM and LPCA, respectively. Finally in (k), (l) (m), (n), (o) and (p) we show the estimated DTI from the yellow square window drawn in (a) in the same order of the filtering techniques listed above.

Filtering techniques and DT estimation method: The parameters for GF and PM were tuned by cross-validation, aiming to achieve the best performance. For PM implementation, we chose a monotonic function privileging wide regions over smaller ones proposed in [5]. For the optimized filters described in section 2, we used the package dedicated for 4D MRI proposed by Coup´e3 . Finally, we rely on the DT estimation method proposed by Barmpoutis in [14]. Also, we use RESTORE algorithm to evaluate the filtering performance.

3

Results and discussions

In order to visualize the filtering performance for each preprocessing technique, in Figure 1 we show the 38th direction and the 30th slide from the DTI30-DWI.nii dataset with their both respective noisy and filtered images using the above filters mentioned in section 2. Also, we show the estimated DTI for each previous DWMRI cases. We can observe a bias in the orientation of the DTI estimated by 3

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VII Table 1. Frobenius norm from the difference of metrics computed from the DTI between the case under study and DTI30-DWI.nii. MD: mean diffusivity error. RA: relative anisotropy error. FA: fractional anisotropy error. Frob: Frobenius error norm. The mean performance and standard deviation are drawn using all the slices (µ ± σ). Scheme

MD (×10−4 )

RA (×10−3 )

FA

Frob (×10−3 )

Noisy GF PM AONLM ODCT PRINLM LPCA RESTORE

0.1234 ± 0.0131 0.0679 ± 0.0078 0.1234 ± 0.0131 0.0145 ± 0.0014 0.0112 ± 0.0017 0.0090 ± 0.0019 0.0104 ± 0.0017 0.0508 ± 0.0046

0.1912 ± 0.0190 0.1366 ± 0.0149 0.1912 ± 0.0190 0.0558 ± 0.0070 0.0448 ± 0.0059 0.0300 ± 0.0040 0.0437 ± 0.0057 0.0857 ± 0.0072

0.0061 ± 0.0006 0.0060 ± 0.0006 0.0061 ± 0.0006 0.0058 ± 0.0007 0.0059 ± 0.0006 0.0059 ± 0.0006 0.0058 ± 0.0006 0.0044 ± 0.0005

0.9274 ± 0.0974 0.5371 ± 0.0449 0.9274 ± 0.0974 0.1960 ± 0.0237 0.1453 ± 0.0198 0.1148 ± 0.0239 0.1296 ± 0.0161 0.4130 ± 0.0077

the introduction of Rician noise over the DWI data. All the filtering techniques tend to remove this bias, but the optimized filters denoise the image in a better way with respect to the classical filters, concluding that the methods based on local similarity have a better performance in both DWI filtering and DTI estimating. Finally, in Table 1 we evaluate quantitatively the filters computing the Frobenius error norm from the difference of metrics computed from the DTI between the case under study and the goal standard DTI30-DWI. We draw the mean performance and standard deviation using all the slices from each dataset. Also, we extract this error metrics for the results using RESTORE. This robust algorithm obtains a better FA performance due it was designed to recover the trace and the FA from the DT by using Monte Carlo simulations [15]. However, the optimized filters have better behaviour among the other metric errors. Finally, we can conclude that PRINLM and LPCA filters show the best performances among the optimized filters including the RESTORE algorithm.

4

Conclusions

The DW-MRI study has an inherently low signal to noise ratio, biasing the estimation of diffusion parameters. Classic denoising techniques that assume an equal noise distribution across the image modifying important information, produce suboptimal filtering results. Moreover, methods based on local similarity improves the performance of diffusion information. However, the performance of the above filters depend on how well we tune the filtering parameters. Adaptive and optimized filter techniques based on local similarity for brain images, tend to remove the bias in both DWI filtering and DTI estimation by local information, showing a better performance respect to the classical filters and the robust RESTORE algorithm. According to our results, the PRINLM and LPCA filters show the best performances among the optimized filters, making them the most suitable for DWI.

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Acknowledgment This work was funded by COLCIENCIAS under the project 1110-569-34461. Authors were also supported by the 617 agreement, “J´ovenes Investigadores e Innovadores”, funded by COLCIENCIAS. Finally, the authors are thankful to the research group in Autom´atica ascribed to the engineering program at the Universidad Tecnol´ ogica de Pereira, and M.Sc. H.F. Garc´ıa for technical support.

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