1 Introduction

Magnetic resonance imaging (MRI) is a significant powerful imaging techniques and effective diagnostic tool developed to analyze the anatomical and physiological information of internal body parts [38]. MRI images are affected by thermal noise during acquisition and decrease the quality of the acquired images [11, 16, 21, 26]. The restoration of noisy image without degrading the fine structural features has been a crucial issue in MRI analysis [10]. Need of noise reduction technique for medical image is a vital issue in medical field. Different kinds of noises like salt & pepper, speckle, gaussian, rician, rayleigh, gamma, uniform, poisson etc. are present in medical image [1, 2]. The most frequently occurring noise in MRI images is the rician noise. The complexity in restoring high resolution MRI increases due to rician noise. The low signal to noise ratio (SNR) creates random variation and the signals dependent on bias which reduces the brightness and contrast of the image [28, 29]. This work is to improve energy consumption, reliability and better lifetime. It provides the hybrid solution. The watermarking has to be done for the restoration for achievement of good quality in the images [4, 6, 22].

Many researchers developed diverse techniques to restore the MRI images affected by rician noise. The estimation of magnitude MRI from the degraded image has been proposed but limits to low noise variation [13]. Gaussian filters have been extensively used in MRI pre-processing during restoration, results in blurred edges at high-frequency signals [8]. Various edge-preserving methods have been projected for medical image denoising to overcome the blurring effects and they do not degrade the morphological edges during smoothing [17]. The anisotropic diffusion filters removes noise using gradient information at low noise level rather than high noise level [12]. Homogeneity mean difference (HMD) method is sensitive to noise and insensitive to contrast during image restoration [37]. In [23] noises in the brain images were detected and removed using directional filtering algorithm but it takes more complexity. Wavelet based bilateral filter with neigh shrink preserves the structural features, but the strong noisy pixels in homogeneous regions were not reduced [7]. MRI enhancement using genetic programming for rician noise removal preserves the structural details, but complexity increases in setting the parameters for genetic programming and requires more time to converge [20].

Rough set based bilateral filter approach derives pixel level edge map and class labels to achieve better restoration but the restoration potential needs to be improved [31]. Non-local means (NLM) algorithms were efficiently used in MRI denoising, but limited due to high computational complexity. The random sampling based NLM algorithm decreases the computational burden, but the restoration potential needs to be improved [15]. Linear minimum mean square error (LMMSE) filtering removes the rician noise with less restoration efficiency [36]. The choice of bilateral filter parameters affects the performance of denoising. In order to get the optimised parameters, genetic algorithms have been applied to the noisy images in searching regions of different window size and the efficiency of the filter mainly depends upon the optimal parameter selection [3]. Iterative bilateral filter improves the denoising efficiency, preserves the fine structures and reduces the bias due to rician noise but the complexity increases due to increase in number of iterations [32].

To overcome these challenges, machine learning techniques such as fuzzy logic based system are widely used for solving problems in different domain of engineering [5, 9] and in medical application it is based on image-based diagnosis, disease detection and disease prognosis to reduce the operator dependency and get better diagnostic accuracy [19]. Trapezoidal fuzzy based hybrid filter preserves edges but does not give a suitable degree of membership to the filters to restore the MR image [35]. In fuzzy similarity based NLM filter for rician noise removal, the structural information has not been retrieved properly from noisy image [34]. The adaptive hexagonal fuzzy hybrid filter for rician noise removal in MR images restores the image at low and high densities of rician noise [18]. However, the problem of image restoration still remains open because it is an ill-posed inverse problem; it requires necessary information about the degraded image to reconstruct the original image. Therefore, the need for more effective methods keeps increasing. The proposed adaptive fuzzy hexagonal bilateral filter removes the rician noise by preserving the structural information. The proposed filter combines the median filter and bilateral filter adaptively by constructing the fuzzy hexagonal membership function using local and global statistical parameters.

The rest of the paper is organised as follows. Section 2 briefly discusses the proposed adaptive fuzzy hexagonal bilateral filter for brain MRI restoration. Section 3 provides the information on experimental setup. Section 4 discusses the analysis of the proposed adaptive fuzzy hexagonal bilateral filter effect for synthetic and clinical MRI data. Finally Section 5 concludes the paper.

2 Adaptive fuzzy hexagonal bilateral filter

The proposed adaptive fuzzy hexagonal bilateral filter along with local order filter removes the rician noise in MRI under low and high noise levels. The acquired MRI data are complex and represented in frequency domain as k-space [35]. The raw data has been corrupted by thermal noise during acquisition [36]. After inverse fourier transformation of the k-space data, the resultant data are complex and still corrupted by noise. The magnitude computation is a non-linear operation, changes to rician by realizing the probability distribution function (PDF) of noise in the image [35]. The PDF of MR signal (M) magnitude data is given as

$$ \mathrm{p}\left(\mathrm{M}\mid \mathrm{A},\upsigma \right)=\frac{\mathrm{M}}{\upsigma^2}\exp \left(-\frac{{\mathrm{M}}^2+{\mathrm{A}}^2}{2{\upsigma}^2}\right){\mathrm{I}}_{\mathrm{o}}\left(\frac{\mathrm{A}\mathrm{M}}{\upsigma^2}\right)\mathrm{u}\left(\mathrm{M}\right) $$
(1)

where, A represents the amplitude of noise free signal, σ2 refers to variance of white Gaussian noise, Io denotes the modified Bessel Function in zero order and u (M) represents unit step Heaviside function that indicates the PDF of M is valid for non negative values of M [36].

The proposed adaptive fuzzy hexagonal bilateral filter removes the rician noise in MRI is shown in Fig. 1. The method is based on fuzzy logic that combines local order filter and bilateral filter to restore the MRI. The proposed method determines the statistical features such as local mean, global mean and standard deviation of the noisy image. The fuzzy weights are computed adaptively using hexagonal membership function at both low and high noise levels to restore the MR image as shown in Fig. 1 where Wbilateral and Wlocal are the fuzzy weights for the noisy MRI image.

Fig. 1
figure 1

Block diagram of the proposed adaptive fuzzy hexagonal bilateral filter

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2.1 Statistical features

The statistical features such as standard deviation, local mean, and global mean are computed using (2) and (3) [35]. To differentiate background and foreground regions of the image, local mean (μi) of a local neighbourhood and global mean (μg) of a noisy image are considered to construct fuzzy membership function. In magnitude MR data the standard deviation of the rician noise is computed using (2)

$$ {\upsigma}_{\mathrm{g}}=\sqrt{\frac{\upmu_{\mathrm{b}}}{2}} $$
(2)

where μb is the mean value of the background region of MR image.

$$ {\upmu}_{\mathrm{b}}=\frac{1}{\mathrm{m}\times \mathrm{n}}\ {\sum}_{\mathrm{i},\mathrm{j}=1}^{\mathrm{m},\mathrm{n}}\mathrm{f}\left(\mathrm{i},\mathrm{j}\right) $$
(3)

f(i, j) is the input MRI image of size m × n. Background is extracted using Otsu threshold method [35].

2.2 MRI denoising using local order and bilateral filter

The local order and bilateral filter are applied to the noisy MR image. The local order filter is the high pass filter works well at low level noise in MRI image by retaining the edges. The bilateral filter is a non linear filter, preserves the edges and removes the noise effectively. The local order and bilateral filter are applied along with the corresponding fuzzy weights determined by using hexagonal membership function suppress the rician noise.

2.2.1 Local order filter

The local order filter is used to remove the highly corrupted pixels. It uses the search window size (2 × Mlocal + 1) with a mask of size (2 × Rlocal + 1) to perform convolution over the complete noisy MR image. The Mlocal and Rlocal are set to one and compute the median value for each pixel in the image [35]. The restored image for local-order statistical filter is given by

$$ {\mathrm{L}}_{\mathrm{local}}=\mathrm{LocalFilter}\ \left(\mathrm{L},{\mathrm{R}}_{\mathrm{local}}\right) $$
(4)

where, Llocal is the local order filter image of the noisy MR image L and Rlocal is the radius of squared neighbourhood pixel.

2.2.2 Bilateral filter

The bilateral filter uses the combination of the weight obtained in range filter and domain filter for removing the noise in MR image. The bilateral filter smoothens and preserves edges of the noisy MR image. The bilateral filter for image restoration is given in (5). The weights of the domain and range filter are given in (6) & (7) respectively. The weight of the domain filter is computed as spatial distance between the pixels and the range filter is the intensity difference between the pixels.

$$ {B}_{bilateral}=\frac{1\ }{w_p}\ \sum \limits_{x\in \Omega}{w}_d\kern0.5em {w}_r\kern0.5em f(x) $$
(5)

where,

$$ {\mathrm{w}}_{\mathrm{d}}={\mathrm{e}}^{-\frac{1}{2}{\left(\frac{\mathrm{d}\left(\mathrm{p}-\mathrm{s}\right)}{\upsigma_{\mathrm{d}}}\right)}^2} $$
(6)
$$ {\mathrm{w}}_{\mathrm{r}}={\mathrm{e}}^{-\frac{1}{2}{\left(\frac{\updelta \left({\mathrm{I}}_{\mathrm{p}}-{\mathrm{I}}_{\mathrm{s}}\right)}{\upsigma_{\mathrm{r}}}\right)}^2} $$
(7)

and

$$ {\mathrm{w}}_{\mathrm{p}}=\kern0.75em \sum \limits_{\mathrm{x}\in \Omega}{\mathrm{w}}_{\mathrm{d}}\kern0.5em {\mathrm{w}}_{\mathrm{r}} $$

f(x) is the input MR image, wd, wr and wp are the weights of the domain filter, range filter and normalization parameter respectively. Ω is the neighbourhood of a centre pixel. σd and σr controls the decay of two weight factors, d (p-s) is the Euclidean distance between current pixel (p) and neighbour pixel (s), δ(Ip - Is) is the difference between two intensity values [32].

2.3 Adaptive fuzzy hexagonal membership function

An adaptive fuzzy hexagonal membership function restores the noisy MRI using the statistical features. Membership function (MF) is a hexagonal curve that defines the mapping of each pixel in the input MR image to the corresponding membership value between 0 and 1 as given in (9). Fuzzy hexagonal membership function has been constructed adaptively by using statistical features for better restoration. The hexagonal MF shown in Fig. 2 has been denoted using (9).

$$ f\left(x;a,b,c,d,e,f\right)=\Big\{{\displaystyle \begin{array}{ll}0& for\ x<a\\ {}\frac{1}{2}\left(\frac{x-a}{b-a}\right)& for\ a\le x\le b\\ {}\frac{1}{2}+\frac{1}{2}\left(\frac{x-b}{c-b}\right)& for\ b\le x\le c\\ {}1& for\ c\le x\le d\\ {}1-\frac{1}{2}\left(\frac{x-d}{e-d}\ \right)& for\ d\le x\le e\\ {}\ \frac{1}{2}\left(\frac{f-x}{f-e}\right)& for\ e\le x\le f\\ {}0& for\ x>f\end{array}} $$
(9)

where x is the input value for the hexagonal function and a, b, c, d, e and f are scalar parameters, computed using (10).

$$ {\displaystyle \begin{array}{l}\mathrm{a}={\mathrm{k}}_1\times \min\ \left({\upmu}_{\mathrm{i},}{\upmu}_{\mathrm{g}}\right)\\ {}\mathrm{b}={\mathrm{k}}_2\times \max\ \left({\upmu}_{\mathrm{i},}{\upmu}_{\mathrm{g}}\right)\\ {}\mathrm{c}={\mathrm{k}}_3\times \mathrm{b}\\ {}\mathrm{d}={\mathrm{k}}_4\times \mathrm{c}\\ {}\mathrm{e}={\mathrm{k}}_5\times \mathrm{d}\\ {}\mathrm{f}={\mathrm{k}}_6\times \mathrm{e}\end{array}} $$
(10)

where k1, k2, k3, k4, k5, and k6 are adjusting parameters and it depends on noise level σg. μi is the mean of a local neighbourhood centred around a pixel i and μg is the mean of the complete image. The adjusting parameters k1 and k2 have been considered as given in (11) [35]. The k3, k4, k5, and k6 are obtained by conducting the experiment based on trial and error method.

$$ {\displaystyle \begin{array}{l}{\mathrm{k}}_1=3.1\times {\upsigma}_{\mathrm{g}}\\ {}{\mathrm{k}}_2=0.98+0.8\times {\upsigma}_{\mathrm{g}}\\ {}{\mathrm{k}}_3=4.1\\ {}{\mathrm{k}}_4=3.1\\ {}{\mathrm{k}}_5=2.1\\ {}{\mathrm{k}}_6=1.1\end{array}} $$
(11)
Fig. 2
figure 2

Hexagonal Fuzzy Membership Function

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After the construction of fuzzy hexagonal membership function, weight of bilateral and local order are computed as given in (12),

$$ {\displaystyle \begin{array}{l}{\mathrm{W}}_{\mathrm{bilateral}}=\mathrm{f}\left({\upmu}_{\mathrm{i}};\mathrm{a},\mathrm{b},\mathrm{c},\mathrm{d},\mathrm{e},\mathrm{f}\right)\\ {}{\mathrm{W}}_{\mathrm{local}}=1\hbox{--} {\mathrm{W}}_{\mathrm{bilateral}}\end{array}} $$
(12)

where Wbilateral and Wlocal are the near optimal contributions of bilateral and local filters respectively.

The restored image is obtained using (13),

$$ \mathrm{f}\ \left(\mathrm{x},\mathrm{y}\right)={\mathrm{W}}_{\mathrm{bilateral}}\times \kern0.5em {\mathrm{B}}_{\mathrm{bilateral}}+\kern0.5em {\mathrm{W}}_{\mathrm{local}}\times \kern0.5em {\mathrm{L}}_{\mathrm{local}} $$
(13)

PSEUDOCODE

figure c

3 Experimental setup

Quantitative and Qualitative analysis are performed for the synthetic and clinical MRI using the proposed method and existing methods. Quantitative analysis is performed by the performance metrics peak signal to noise ratio (PSNR), root mean squared error (RMSE) and the structural similarity index measure (SSIM). Qualitative analysis is done by the visual assessment of the restored MRI image.

3.1 Synthetic MRI data set

The synthetic data sets of the normal brain MR images are taken from the BrainWeb for analysis. The three co-registered modalities named: T1- weighted, T2-weighted, and PD-weighted are considered with the size 181× 128× 181. The voxel resolution of the datasets is 1mm3. Different slices are taken for the analysis.

3.2 Clinical MRI data set

The clinical MR data sets of the brain MR images are obtained for analysis from the Medall Diagnostics at Tirunelveli for co-registered three modalities: T1-weighted, T2-weighted, and PD-weighted. Conventional T1, T2, and PD weighted MR image are measured with angles 700, 2200 and 2200 respectively, of the same spin echo sequence are considered for analysis with 30 number of 2-D slices in each volume. The public MR data sets of the brain MR images [14] for T1-weighted, T2-weighted, and PD-weighted have been analyzed for different slices.

3.3 Performance metric and parameter setup

The restoration of noisy MR images has been analysed using synthetic MRI data sets from Brain Web and clinical data using the proposed adaptive hexagonal fuzzy bilateral filter. The quality of the MR image has been evaluated using PSNR, RMSE and SSIM.

The proposed adaptive hexagonal fuzzy bilateral filter and the existing methods are carried out using MATLAB 2013. The proposed adaptive hexagonal fuzzy bilateral filter has been compared with the existing techniques such as Median filter [24], NLM filter [27], Adaptive filter [12, 30, 33], Wiener filter [25], Bilateral filter [32], Fuzzy trapezoidal with NLM filter [35] and Fuzzy hexagonal with NLM filter [18].

4 Results and discussion

This section represents extensive experimental validation of the proposed adaptive hexagonal fuzzy bilateral filter using the synthetic MR image and clinical MR image as discussed in Section 3.1 and 3.2 respectively. The noise in MRI has been removed by restoration process using the proposed adaptive fuzzy hexagonal bilateral filter and its performance were compared with the existing methods such as Median filter, NLM filter, Adaptive filter, Bilateral filter, Wiener filter, Fuzzy trapezoidal with NLM filter and Fuzzy hexagonal with NLM filter. The efficiency of the proposed adaptive fuzzy hexagonal bilateral filter were analysed using the performance metrics PSNR, RMSE and SSIM.

4.1 Validation on synthetic MRI data

The synthetic MRI data with T1 weighed, T2 weighed and PD weighed were exploited at different noise levels. Table 1 shows the performance measures from low to high level noise for synthetic MRI image with T1 weighed, T2 weighed and PD weighed. The PSNR value obtained for the proposed adaptive fuzzy hexagonal bilateral filter is higher than the fuzzy hexagonal NLM filter. For higher rician noise level, NLM filter generates the artifacts on its own and fails to restore the noisy MRI accurately leads to the low PSNR value than the noisy image.

Table 1 PSNR (RMSE) comparison on synthetic MR image at various noise level for the Median filter, NLM filter, Adaptive filter, Bilateral filter, Wiener filter, Fuzzy trapezoidal NLM filter, Fuzzy hexagonal NLM filter with the proposed adaptive Fuzzy hexagonal bilateral filter
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The bilateral filter in the proposed method has less computation cost than the NLM filter. Median filter preserves edges but fails to retain the structural information. Adaptive filter and wiener filter are unable to smooth the noise effectively. The NLM filter can able to preserve distinct edge features but it excessively smoothens the homogeneous regions and reduce the contrast between gray and white scale image [15] whereas the proposed adaptive fuzzy hexagonal bilateral filter preserves the edges and structural details of the image. Table 1 shows that mean of the proposed adaptive fuzzy hexagonal bilateral filter has 71, 72 and 72% for T1 weighted, T2 weighted and PD weighted respectively on average compared to fuzzy hexagonal NLM filter [5].

The differences in RMSE values of the proposed adaptive fuzzy hexagonal bilateral filter with the existing methods effectively characterize the performance of the restoration methods. Table 1, also shows the RMSE improvement of the processed image obtained by the proposed adaptive fuzzy hexagonal bilateral filter and existing filters. The proposed adaptive fuzzy hexagonal bilateral filter reduces RMSE by 99.6% for T1, 99.6% for T2 and 99.5% for PD compared to existing fuzzy hexagonal with NLM filter. SSIM quality index confirms the better visual quality of the proposed method at all noise levels. Figure 3 illustrates SSIM comparison of the proposed adaptive fuzzy hexagonal bilateral filter with other methods for the synthetic MRI. The proposed method gives much better SSIM improvement by retaining the structural details both at low and high level noise. The bilateral filter smoothen the local variations without affecting edges and the classification have been done using fuzzy hexagonal membership function improves the SSIM of noisy MRI both at low and high level.

Fig. 3
figure 3

SSIM comparsion for the synthetic MR data a T1 weighted b T2 weighted c PD weighted

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Figure 4 shows the comparison of the synthetic MRI at 10% noise level. From the figure it is observed that the visual quality improvement achieved by the proposed adaptive fuzzy hexagonal bilateral filter reduces the noise and restore the original MRI effectively than existing methods. From the visual perception, it has been observed that the wiener filter is unable to smooth noise completely. NLM filter removes the noise but the structural information has been lost. When the noise level increases, the NLM filter with hexagonal MF and trapezoidal MF produced extra blur in the restoration. The proposed adaptive fuzzy hexagonal bilateral filter has the ability to reduce noise by retaining fine structural details and better edge preserving capabilties than existing methods.

Fig. 4
figure 4

Synthetic MRI with 10% noise a Original image b Rician noise image (PSNR = 22.76) c Median filter (PSNR = 23.84) d NLM filter (PSNR = 18.96) e Adaptive filter (PSNR = 21.15) f Bilateral filter (PSNR = 68.49) g Wiener filter (PSNR = 23.90) h Fuzzy trapezoidal MF with NLM filter (PSNR = 21.12) i Fuzzy hexagonal MF with NLM filter (PSNR = 21.42) j Proposed adaptive fuzzy hexagonal bilateral filter (PSNR = 69.94)

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4.2 Application to clinical MRI data

To evaluate the consistency of the proposed method on clinical MRI T1 weighted, T2 weighted and PD weighted images for varying rician noise ratio from 0.05 to 0.30 with a scale of 0.05 has been used. The quantitative metrics comparison for varying noise rates are tabulated in Tables 2 and 3. The proposed adaptive fuzzy hexagonal bilateral filter restores the clinical MRI in both data sets better than existing methods, as it is easier to tune the decay of distance function in bilateral filtering. The edge preservation in the proposed method is higher, as it averages within smooth regions of the image than averaging across image edges are shown in Tables 2 and 3 respectively.

Table 2 PSNR (RMSE) comparison on clinical MR image at various noise level for the Median filter, NLM filter, Adaptive filter, Bilateral filter, Wiener filter, Fuzzy trapezoidal NLM filter, Fuzzy hexagonal NLM filter with the proposed adaptive Fuzzy hexagonal bilateral filter
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Table 3 PSNR (RMSE) comparison on clinical MR image for public dataset at various noise level for the Median filter, NLM filter, Adaptive filter, bilateral filter, Wiener filter, Fuzzy trapezoidal NLM filter, Fuzzy hexagonal NLM filter with the proposed adaptive Fuzzy hexagonal bilateral filter
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The proposed adaptive fuzzy hexagonal bilateral filter has been quantitatively compared with existing methods, to analyze the SSIM for all noise levels for clinical dataset as shown in Fig. 5. The qualitatively analysis is shown for the clinical dataset in Fig. 6 and from the Fig. 6 it gives residue significantly less compared to other methods. In Figs. 6 and 7 the clinical public dataset are analysed and it result shows high performance for the proposed method. The proposed method shows better edge preserving potential due to adaptive weight adjustment compared to existing methods.

Fig. 5
figure 5

SSIM comparison for the clinical MRI a T1 weighted b T2 weighted c PD weighted

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Fig. 6
figure 6

Clinical MRI with 10% noise a Original image b Rician noise image (PSNR = 19.53) c Median filter (PSNR = 24.85) d NLM filter (PSNR = 20.46) e Adaptive filter (PSNR = 25.98) f Bilateral filter (PSNR = 67.75) g Wiener filter (PSNR = 25.56) h Fuzzy trapezoidal MF with NLM filter (PSNR = 23.45) i Fuzzy hexagonal MF with NLM filter (PSNR = 25.68) j Proposed adaptive fuzzy hexagonal bilateral filter (PSNR = 75.29)

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Fig. 7
figure 7

SSIM comparison for the clinical MRI for public dataset a T1 weighted b T2 weighted c PD weighted

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The enhanced accuracy of the proposed adaptive hexagonal bilateral filter can be understood by considering the two facts; First, fuzzy classification has been done adaptively using hexagonal membership function. Second, the implementation to classifying the pixel with additional scalar parameters to tune the proposed method effectively than the trapezoidal membership functions. Although the parameter is more but the effectiveness of the restoration obtained by proposed adaptive fuzzy hexagonal bilateral filter method makes reliable. The restored MRI image is noiseless due to the adaptive nature of noisy image in hexagonal fuzzy membership function.

5 Conclusion

In this paper, adaptive fuzzy hexagonal bilateral filter has been presented and analyzed for Brain MRI denoising application. This method increases the PSNR without affecting the significant structures in the image. Although RMSE is a quantitative parameter, the visual perception sense and SSIM guarantee have been analysed. The visual inspection of the proposed method based on image residuals evaluates the efficiency. In future, the estimation of the noise level will be determined before restoration process and then analysis the restored MRI.