Introduction

Diffusion in Magnetic Resonance Imaging (d-MRI), in forms of the diffusion tensor imaging [ 1 ] and high angular resolution diffusion imaging, such as q-ball [ 2 ] diffusion spectrum imaging [ 3 ] spherical de-convolution [ 4 ], and other novelty methods, is a technique based on movements of water molecules inside the tissues, including brain tissues [ 5 ]. The Difference in the diffusion of water molecules in the brain tissues reveals anatomical findings [ 6 ], and tractography detects the configurations of microstructures in the brain [ 7 ]. Resolution in diffusion data is high when diffusivity is high, which is desirable for clinical applications. D-MRI with higher spatial resolution can result in improving detection structures of the brain [ 8 - 12 ]. However, a major challenge for d-MRI is the intrinsically low signal-to-noise ratio (SNR), and increasing the SNR is caused due to decreasing b-value parameter, angular resolution declined related to the detection of microstructures. Sotiropoulos et al. [ 10 ] proposed a model of data based on combining high-spatial-resolution data with data of higher angular resolution and contrast to achieve benefits both.

Statistical model of signal and noise has an important effect on medical image analysis, and many applications in MRI are more sensitive when the data is well-defined based on a statistical model. In some types of researche, these methods are described as follows: 1) noise removal and signal estimation [ 13 ], 2) signal and noise maximum likelihood estimation [ 14 ], 3) linear minimum mean square error (LMMSE) filtering based schemes [ 15 ], or unbiased nonlocal mean filters [ 16 ].

The distribution of noise may also affect the results of image processing in MRI, and the effects of Rician noise on a coil are important for estimation of SNR and filtration of images especially in diffusion tensor estimation [ 15 ].

Averaging is another method to promote the SNR; for unreal data and less than threshold SNR, the prepared model cannot fit the signal by calculating the average of data mixed with correlated Gaussian noise [ 17 ].

The final goal in the acquisition of diffusion images is to determine the configurations of neurons in the structures of the brain with the tractography algorithm to predict diseases, such as Alzheimer’s and epilepsy. Many researchers focused on the effectiveness of the tractography methods to predict the numbers and shapes of fibers passing through a voxel, also studied structural tractography and arrangement of nerve fibers based on internal structures obtained from data [ 1 - 6 , 18 ]. Additionally, the target could be a phantom or white matter of the human brain; the relationship between lines obtained from outputs of the tractography algorithms with actual fibers can be a question. Since the tractography algorithms are affected by parameters, such as noise, masks, and region of interest (ROI), selected by users according to the type of cases, small changes lead to incorrect findings. The connection index in the most possible tractography algorithms is a simple voxel count, which is a simple count [ 18 ]. Accordingly, these limitations lead to paths for reducing errors in the tractography algorithms.

The local structures of diffusion images considered a solution in all directions can lead to a more accurate estimation of the fibers. This paper aimed to promote tractography outputs in presence of noise.

In this paper, the pixels are addressed based on their property of the local frequency in all directions of diffusion signals. First, a digital phantom was selected based on the configurations of nerve fibers [ 19 ], and the images were in 60 directions using the local phase algorithm; the local frequency feature for all of the images was calculated. In the following, the value of each pixel was compared with the average of the total pixels of each image, and all pixels with smaller values than a threshold (mean of total pixels) were replaced with zero to design the desired mask. However, the address of non-zero pixels in the data is the same as their address in a mask, values of those pixels are just selected from data as the input of the tractography algorithm. Finally, the overall tracts of the fibers in the phantom were detected by the Hough transform [ 20 ].

Material and Methods

This simulation and analytical study aimed to examine the impact of the local phase features on the tractography algorithm in noisy data. Some voxels may alter the tractography results due to unintentional alteration of noises and other unwanted factors. Further, in the proposed method, the aim was to improve the output of the tractography algorithm in noisy images based on the local structures to select appropriate pixels and remove other pixels considered wrong selections of nerve fibers. This proposed technique employs the local structures of images along anatomical structures to investigate the brain fibers. Accordingly, the monogenic algorithm proposed by Felsberg et al. was used with the implementation of the local phase feature [ 21 ].

Local frequency as a feature of local phase

Implementation of Local frequency

A difference of Poisson filter (DOP) was used with lambda = 10; the Poisson distribution is specified with one parameter: lambda (λ) that equals the mean and variance. Increasing lambda to sufficiently large values can lead to approximate the Poisson distribution by the normal distribution (λ) that λ was considered 10 since the distribution function of the designed filter has the least attenuation at intermediate frequencies (Figure 1).

Figure 1. The distribution function of Poisson based on lambda parameter

The Fourier transform (in frequency domain) of the diffusion images was used in different directions and slices for given results as an input of Spherical Quadrature Filter. In the spatial domain, the procedure is as follows [ 22 ]:

$p=\mathrm{(diffusion\; images\; )\; \otimes \; DOP)}$ (1)

$q=\mathrm{(diffusion\; images\; )\otimes [(icos\varnothing \; ,\; isin\varnothing )DOP]}$ (2)

The symbol ⨂ is the convolution in the spatial domain and DOP is a difference of Poisson filter. The local phase model is as follow, including the local frequency f_x:

$r=\stackrel{\_\_}{\varnothing }{\left(\mathrm{cos\theta \; ,\; sin\theta }\right)}^T=\frac{q}{\left|q\right|}arg(p+i|q\left|\right)$ (3)

In Equation 3, r is obtained from a combination of directions and the local Phase. By using the derivative direction in the direction n_x to $y(\nabla =({\mathrm{\partial y}}_1,{\mathrm{\partial y}}_2\left)\right)$, we have:

$f\triangleq {\mathrm{(\nabla \varnothing )}}^T.n=\frac{{\mathrm{p\nabla }}^{T}q-q^T\mathrm{\nabla p}}{p^2+{\left|q\right|}^2},n={\left[\mathrm{cos(\theta )\; ,\; sin(\theta )}\right]}^T$ (4)

It was shown that the local frequency can be directly obtained by using the response of the four proposed filters.

Creating a mask using local frequency information

According to the previous section, a mask was designed with pixels introducing information about the local structure of the original image. This mask is multiplied by diffusion images to eliminate the values of some pixels in the tractography algorithm. In the case of Gaussian noise, we have:

${\mathrm{Mask}}_{\mathrm{d,s}}\mathrm{(x,y)}=0\mathrm{,\; if}{\mathrm{(local\; f\; Matrix)}}_{\mathrm{d,s}}\mathrm{(x,y)}<(L_{\max }-\sigma )\mathrm{,if\; not,\; Mask}=1$

$L_{\max }=\max \left(\max \right({\mathrm{(local\; f\; Matrix)}}_{\mathrm{d,s}}\mathrm{(:,:)}\left)\right)$

$\sigma =\left\{\begin{array}{c}0.1<\sigma <\mathrm{0.3\; if\; for\; low\; SNR}\\ 0.3<\sigma <\mathrm{0.7\; if\; SNR\; \gg \; 1}\end{array}\right\}$ (5)

In the case of random noise $\sigma =0$.

Here, the coordination of (x,y) indicates each pixel; ${\mathrm{(local\; f\; Matrix)}}_{\mathrm{d,s}}\mathrm{(x,y)}$ is a value of each pixel based on a local frequency feature of a pixel. ${\mathrm{(local\; f\; Matrix)}}_{\mathrm{d,s}}\mathrm{(:,:)}$ is the local frequency matrix of a diffusion image in each slice and all directions; d and s indicate the direction and the slice number of each image, respectively. σ depends on SNR (Signal to Noise Ratio) and a parameter determining the lower band of the local structures in each image. Based on Equation 5, if the value under the Mask in each pixel is equal to one, this pixel is a candidate for the main structure of the image. Finally, the diffusion images are multiplied pixel by pixel with the local mask; therefore, only the pixels that represent the local changes are selected as the inputs of the tractography algorithm, and fiber bundles can be detected as the local structures in diffusion images.

$I_{\mathrm{d,s}}\mathrm{(x,y)}=I_{\mathrm{d,s}}\mathrm{(x,y)}<•*>{\mathrm{Mask}}_{\mathrm{d,s}}\mathrm{(x,y)},\mathrm{\forall \; x},x,y,d,s$ (6)

The Symbol $<•*>$ is the pixel by pixel multiplication.

Hough Tractography

The Hough transform was used by Aganj as a tractography algorithm [ 20 ] that the brain fibers are modeled as 3D curves and parameterized with an arc length S. The aim is to obtain the value of each point x of the curve. First, it is assumed that the starting point of curve s has two-sides from each seed point, then the parametric line fits the curve as follow:

$\overrightarrow{x_s}=\overrightarrow{x_0}+{\int }_s^0{t}_s\mathrm{ds}$ (7)

which $x_0$ represents the starting point or the same as a seed point, and $t_s$ represents the unit tangent vector at any point ${\mathrm{(x,y,z)}}_s$ of curve S. Now, the coordinates of $t_s$ is written in a spherical space.

$t_s={\left(\begin{array}{c}x\\ y\\ z\end{array}\right)}_s\to {\left(\begin{array}{c}\mathrm{sin\theta \times cos\varphi }\\ \mathrm{sin\theta \times sin\varphi }\\ \mathrm{cos\theta }\end{array}\right)}_s$ (8)

The θ and ϕ are fitted with a polynomial in order of N. Using the Hough transform, the coefficients of each polynomial are calculated based on available diffusion data. The desired fibers are selected to log-probability of the existence of fiber. The expression of probability is directly related to Generalized Fractional Anisotropy (GFA) inside the region of interest. In the current study, GFA was determined by the local phase mask, i.e. local mask represents the probability for the point $\overrightarrow{x_s}$ located inside fiber bundles passing in each direction.

Noise Distributions

In this paper, the noise distributions added to d-MRI images are Gaussian and random as follow:

$\mathrm{Gaussian\; noise}=A\sqrt{v}\times \mu \sqrt{K}+M$ (9)

$\mathrm{Random\; noise}=A\times \mu \sqrt{K};K=\mathrm{size(data)}$ (10)

In Equation 9 and 10, the amplitude of the noise is A, v is the variance of the noise, M is mean of data that correspond to mean of data, and μ(K) is a random function based on the size of data. In the case of Gaussian, two approaches are considered: the distribution of noise with zero mean and a certain mean as similar as the mean of data.

Results

Select the data type

The data was obtained from a digital phantom made by Leemans [ 23 ], and the image parameters are as follows:

6 non-DWIs (b = 0 s/mm²), 60 DWIs (b = 1200 s/mm²), voxel size = 1×1×1 mm³ and SNR = INF (no noise added). Figure 2a and b show the phantom images and the results of the local phase model on the phantom images, respectively.

Figure 2. (a) A digital phantom (b) applying the Local phase algorithm on a digital phantom

Results for a Gaussian noise

The Gaussian noise is firstly generated by the variance and mean of the data. The amplitude of the noise changed from upward to downward and added the modified noise to the diffusion images to reach multi SNR ranges. In Figure 3, the left column is the orientation distribution function (ODF) of the digital phantom when the local phase mask was applied and the middle column showed the tractography results. To compare the results of the effects of noise on the tractography algorithm, two steps are implemented as follows: 1) the data was mixed by Gaussian noise with a mean that correspond to the mean of the data (Figure 3 (a1 to c3)), and 2) distribution of noise is considered with zero means (Figure 3 (a4 to c4)).

Figure 3. Gaussian noise added to the digital phantom. The Orientation distribution function (ODF) of digital phantom (left), Local phase tractography (middle), and Ordinary tractography (right)

Results for a Random noise

In Figure 4, the left column is the local phase tractography for mixed data by a random noise; the ordinary tractography results are shown in the right column.

Figure 4. Random noise added to the digital phantom. Local phase tractography (left) and Ordinary tractography (right)

Discussion

In the current study, the innovative idea is to consider the local image structures to find the main directions of the tractography algorithm without any post-process results of the tractography since only the pixels are selected as the input of the tractography algorithm that contains malicious information about the positions of the nerve fibers. Other methods used some manual techniques [ 13 - 16 ], such as determining a specific area on results of tractography to eliminate the area of interest and decreasing the false negative and false positive cases that the ROI was changed case by case whereas the proposed method automatically omits the pixels that are not a candidate to the tractography.

However, for diffusion signals with low intrinsic SNR, and in some cases, the noisy diffusion images, de-noising methods has a key role, in our proposed method, de-noising is not needed [ 24 , 25 ] since the proposed innovative algorithm is capable of separating noise from the main structures of the images, resulting in reducing the time of implementation of the algorithm.

Furthermore, the present study aimed to reduce the number of directions of gradients for image reconstruction for the first time in studies. The local phase features indicate changes in the gray level of the images, pixels, including more complex structures of images can reduce the number of ODF’s. For an imaging sequence with the minimum gradients and calculated the main structures of the diffusion images, we used a better imaging sequence by using the local phase models.

One of the limitations of tractography algorithms of brain fibers is their unpredictability [ 6 ], related to the micrometers dimensions of fibers while the minimum dimension of an imaging voxel is not less than 0.5 mm. Based on the evidence, the proposed tractography algorithm should be more sensitive to the pixels of all directions to reduce the number of false-positive and false-negative cases [ 18 ]. Therefore, it is important to avoid getting stuck in pixels whose values may change due to noise [ 18 ]. The local phase features are good candidates to reduce misalignment tracts and tractography errors.

Conclusion

The local phase model of the diffusion images is presented based on designing a local mask; at the heart of our mask, a quadrature filter is embedded and the output of the mask is multiplied by the Fourier of diffusion images in all directions of the digital phantom. All directions of images are uniform with no distortions of noise in the data. Two types of noise were separately added to investigate the role of the local phase features on the tractography algorithm. The noise is considered with a distribution of Gaussian and random noise; the diffusion images were separately mixed by a distribution of Gaussian noise in two forms: a zero-mean noise and a noise with a mean of data. Therefore, the tractography algorithm shows that the local phase features promote the efficiency of findings the configurations of fibers when the diffusion images are mixed with noise.

Acknowledgment

This research has been supported by the Tehran University of Medical Sciences & Health Services grant No. 38097.

Authors’ Contribution

All authors contributed to the study’s conception and design. AR. Shirazinodeh performed data collection and analysis. The first draft of the manuscript was written by AR. Shirazinodeh, H. Faraji, S. Sharifzadeh Javidi, AH. Jafari, MR. Nazemzadeh and HR. Salighehrad commented on previous versions of the manuscript. All the authors read, modified, and approved the final version of the manuscript.

Ethical Approval

The research was approved by the Institutional Ethical Committee of Tehran University of Medical Sciences (Approval code: (IR.TUMS.MEDICINE.REC.1398.771).

Informed consent

All experimental procedures were conducted according to the declaration of Helsinki; written informed consent was obtained from participants.

Funding

This project was supported by Tehran University of Medical Sciences Grant No. 38097.

Conflict of Interest

None

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