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DT-MRI (diffusion tensor magnetic resonance imaging) tractography is a method to determine the architecture of axonal fibers in the central nervous system by computing the direction of the principal eigenvectors obtained from tensor matrix, which is different from the conventional isotropic MRI. Tractography based on DT-MRI is known to need many computations and is highly sensitive to noise. Hence, adequate regularization methods, such as image processing techniques, are in demand. Among many regularization methods we are interested in the median filtering method. In this paper, we extended two-dimensional median filters already developed to three-dimensional median filters. We compared four median filtering methods which are two-dimensional simple median method (SM2D), two-dimensional successive Fermat method (SF2D), three-dimensional simple median method (SM3D), and three-dimensional successive Fermat method (SF3D). Three kinds of synthetic data with different altitude angles from axial slices and one kind of human data from MR scanner are considered for numerical implementation by the four filtering methods.

DT-MRI tractography is a method of noninvasively tracing neuronal fiber bundles. DT-MRI measures anisotropy per pixel and provides the directional information of eigenvectors relevant for fiber tractography. Since DT-MRI data which eigenvectors are computed from usually contain noise, the calculated principal eigenvector direction assumed to be the fiber direction may be different from the real direction in the voxel. As the propagation becomes longer, these differences in the voxels, how small it is, make the whole computed fiber direction deviate far away from the real fiber direction [

Many approaches have been attempted to stabilize the noise problem. The approaches are to stabilize six or more diffusion weighted tensors. Since a diffusion-weighted image is a scalar image, there are many conventional image processing techniques [

Among many regularization techniques we are interested in the median filtering of diffusion tensor data [

This paper is organized as follows. Section

In this section, we describe general median filtering and explain briefly SM2D and SF2D. We also extend the two-dimensional methods to the three-dimensional methods: SM3D and SF3D. In this paper, we use the following Frobenius norm as a matrix norm of a matrix

If all matrices in

If

Computations of (a) SM2D and (b) SM3D medians.

The SM2D method is to find a simple approximation

Fermat median filtering algorithm proposed by Kwon et al. [

The above problem is the same as the minimizing solution of (

If one of the three angles in the triangle is greater than or equal to 120°, the Fermat point is the vertex at that angle (Figure

If all the three angles in the triangle are less than 120°, the Fermat point is the intersection of the two straight lines joining any vertex of the triangle and its symmetrical point. The symmetrical point of

Computing Fermat point

We call the algorithm to find Fermat point for three tensors using the above properties, Fermat algorithm. Then, SF2D is a method to find the approximation of

Computations of (a) SF2D and (b) SF3D medians.

To define a three-dimensional SF3D, we divided 27 points in

To evaluate the four filtering methods, we prepared three kinds of synthetic data depending on the angle deviating from each axial slice and human data from the MR scanner. These data are three-dimensional.

The flowchart of our numerical experiment is given in Figure

Flowchart of the fiber tractography loaded from synthetic data (Section

The fiber tracking was based on the fiber assignment continuous tracking (FACT) algorithm and a brute-force reconstruction approach [

Error measures used in the analysis of numerical examples are also given in this section. Let us define the followings:

: The number of voxels

The tensor data for each voxel was constructed using the following diagonalization with a diagonal matrix having three eigenvalues

The number of voxels for the synthesized data was chosen

We have considered three kinds of synthetic data, with different altitude angles

The synthesized data and their projection to

Synthesized data

The whole procedure obtaining fiber tracts for DT-MR images from the MR scanner data is as follows. The single-shot spin echo is used in the image acquisition and data preprocessing from MR scanner. Echo planar imaging (SE-EPI) pulse sequence with two diffusion sensitizing gradients placed on both sides of the 180° refocusing pulse. Fifty contiguous DT-MR images were obtained at 1.5 T Philips Gyroscan MR scanner with the following imaging parameters: field of view = 224 ^{2}, slice thickness = 3 mm, acquisition matrix = 96, reconstruction matrix = 128, TR = 10,000 ms, TE = 76 ms,

To correct subject head motion and the image distortion due to eddy current, every DTI 3D volume image was realigned to

MRI data for the pons in the (a) sagittal and (b) axial planes. The rectangle in (b) was chosen as ROI to select the corticospinal tract (CST).

The numerical results about the three synthetic data in Section

Errors with respect to

Filtering of synthetic data

Errors with respect to

In Section

In Figures

Errors when we recover human data from MR scanner corrupted with Gaussian noise having zero mean and

CSTs passing through the pons of a human subject: corticospinal tracts obtained from the original data with (a) no filter, (b) SM3D, and (c) SF3D and the CSTs obtained from the disturbed data (

Computation time for the four methods to reconstruct the DT-MR data: the blue solid bar and small vertical bar at the top end of the blue bar represent the average and the standard deviation of computational times for the four methods, respectively, after testing the four median filters five times for each.

The calculation time average and standard deviation for the four methods are graphed in Figure

In this study, we developed three-dimensional median filters SM3D and SF3D, extending previously developed SM2D and SF2D in [

The authors declare that there is no conflict of interests regarding the publication of this paper.

This research was supported by the Leading Foreign Research Institute Recruitment Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT & Future Planning (2010-00757), and by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2013R1A1A2010624).