VALIDATING THE NEUTROSOPHIC APPROACH OF MRI DENOISING BASED ON STRUCTURAL SIMILARITY

This paper focuses on validating the proposed Neutrosophic Set (NS) approach of Magnetic Resonance Image (MRI) denoising based on structural similarity such as Structural Similarity Index (SSIM) and Quality Index based on Local Variance (QILV). The Neutrosophic Set approach of median filter is used to reduce the Rician noise in MR image. The experiments have conducted on real MR image with Rician noise added. The visual and the diagnostic quality of the denoised image is well preserved. The performance of this filter is compared with median filter and non local mean filter (NLM).


Introduction
Magnetic resonance imaging (MRI) is the most powerful imaging technique [22] developed to study the structural features and the functional characteristics of the internal body parts.The diagnostic and visual quality of the MR images are affected by the noise added while acquisition.This is problematic for further tasks such as segmentation of important features; classification of images for computer aided diagnostics, three dimensional image reconstruction and image registration.Therefore, denoising should be performed to improve the image quality for more accurate diagnosis.
The MR image is commonly reconstructed by computing the inverse Discrete Fourier Transform of the raw data.The signal component of the measurement is present in both real and imaginary channels.Each of the orthogonal channels is affected by additive white Gaussian noise.The noise in the reconstructed complex valued data is thus complex white Gaussian noise.Most commonly, the magnitude of the reconstructed MRI image is used for visual inspection and for automatic computer analysis.Since the magnitude of the MRI signal is the square root of the sum of the squares of two independent Gaussian variables, it follows Rician distribution [8].The Rician noise is signal dependent and is therefore difficult to separate from the signal.Numerous approaches of denoising MR images have been proposed including approaches based on anisotropic diffusion [7,10,19,20,26], the wavelet transform [2,6,15,16,17,25], bilateral and trilateral filtering [9,23,24], the non-local means algorithm [3][4][5][11][12][13] in the literatures.This paper is an extended work of [14].The Neutrosophic Set approach of MRI denoising is validated in terms of structural similarity (SSIM) [21] index and quality index based on local variance (QILV) [1].As already mentioned in [1,21], peak signal to noise ratio (PSNR) is not the reliable quality metric with respect to human visual system (HVS).SSIM is an objective quality measure based on the structural content of the image that engulfs the properties of HVS.Medical images contain delicate structural features that contribute more to the diagnosis.QILV will provide the statistical similarity between the denoised and the original image.

Neutrosophic MRI denoising
Neutrosophy, a branch of philosophy introduced in [18] as a generalization of dialectics, studies the origin, nature and scope of neutralities, as well as their interactions with different ideational spectra.Neutrosophy theory considers proposition, theory, event, concept or entity, <A> is in relation to its opposite <Anti-A> and the <Neut-A> which is neither <A> nor <Anti-A>.The neutrosophy is the basis of the neutrosophic logic, neutrosophic probability, and neutrosophic set and neutrosophic statistics [18].In neutrosophic set, the indeterminacy is quantified explicitly and the truth-membership, indeterminacy-membership and falsity-membership are independent.The neutrosophic set is a general formal frame work which generalizes the concept of the classic set, fuzzy set, interval valued fuzzy set, intuitionistic fuzzy set, and interval valued intuitionistic fuzzy set, paraconsistent set, dialetheist set, paradoxist set and tautological set [18].The neutrosophic set had been applied into image processing such as denoising [27], segmentation [28,29].The definition of a neutrosophic set and its properties are described briefly.

Neutrophic Set
Definition 1 (Neutrosophic Set).Let U be a Universe of discourse and a neutrosophic set A is included inU .An element x in set A is noted as T, I and F are called the neutrosophic components.The element belongs to A in the following way.It is % t true in the set, % i indeterminate in the set, and % f false in the set, where t varies in T , i varies in I and f varies in F .

Transform the image into neutrosophic set Definition 2 (Neutrosophic image).
Let U be a Universe of discourse and W is a set of U , which is composed by bright pixels.A neutrosophic image NS P is characterized by three membership sets , .a pixel P in the image is described as and belongs to W in the following way: It is t true in the set, i indeterminate in the set, and f false in the set, where t varies in T , i varies in I and f varies in F .  ) , ( j i is the absolute value of difference between intensity ) , ( j i g and its local mean value ) , ( j i g .

Neutrosophic image entropy
For a gray image, the entropy is utilized to evaluate the distribution of the gray levels.If the entropy is the maximum, the intensities have equal probability.If the entropy is small, the intensity distribution is non-uniform.

Definition 3 (Neutrosophic image entropy).
Neutrosophic entropy of an image is defined as the summation of the entropies of three subsets I T , and F : where T En , I En and F En are the entropies of sets I T , and F respectively. ) (i p T , ) (i p I and ) (i p F are the probabilities of elements in I T , and F respectively, whose values equal to i .

-median filtering operation
The values of ) , ( j i I is employed to measure the indeterminate degree of element ) , ( j i P NS .To make the set I correlated with T and F , the changes in T and F influence the distribution of element in I and vary the entropy of I .The summary of neutrosophic set approach of MRI denoising is described as below (see figure 1):

Definition 4 ( -median filtering operation
Step 1: Transform the image into NS domain; Step 2: Use -median filtering operation on the true subset T to obtain T ˆ; Step

Results and Validation
Two brain MR images of 5 th slice, T2 Axial with repetition/ echo time (TR/TE) 9000/87 ms and T1 Sagittal with TR/TE 552/17 ms which are obtained using SIEMENS 1.5 T MRI scanner for a 2 years old male child are considered for the experiments.This method is compared with median filter and Non Local Mean (NLM) filter proposed by Buades et al [3].The results of the different denoising filters on a brain MRI corrupted with Rician noise is shown in Figure 2.  The performance of the denoising algorithm is measured by using the peak-signal-to-noise ratio (PSNR), the Structural Similarity (SSIM) index] and the Quality Index based on Local Variance (QILV).The PSNR will give the pixel by pixel similarity.The peak signal to noise ratio in decibel (dB) is measured using the following formula: where j i I , and j i I d , represent the intensities of pixels j i, in the original image and denoised image respectively.The higher the PSNR, the better the denoising algorithm is.
SSIM and QILV give the measure of the structural similarity between the original and the denoised images and are in the range of 0 to 1.The SSIM works as follows: Let x and y be two non negative images, where as one has perfect quality.Then, the SSIM can serve as a quantitative measure of the similarity of the second image.The system separates the task of similarity measurement into three comparisons: luminance, contrast and structure.It can be defined as (19) where x and y are the estimated mean intensity and x and y are the standard deviations respectively.
xy can be estimated as 1 C and 2 C in eqn. 19are constants and the values are given as is a small constant and L is the dynamic range of the pixel values (255 for 8 bit gray scale images).
The QILV between two images x and y can be defined as where Vx and Vy are the estimated means of the local variance and Vx and Vy are the standard deviations of the local variance respectively.VxVy is the covariance between the two images.The performance comparison of Median, NLM, and NS Median Filters based on the PSNR in dB, SSIM and QILV values for T2 Axial and T1 Sagittal slices of brain MRI corrupted with different levels of Rician noise are given in Figure 3 and Figure 4 respectively.The numerical values of the quality measures: PSNR in dB, SSIM and QILV are listed in Table 1 for the different denoising filters.As already discussed in [14], According to the values of PSNR, at High SNR (low noise level), NLM performs well than NS filter and at low SNR (high noise level), NS filter performs better than that of NLM.This result is justified by the quality measures SSIM and QILV.That is, According to the structural similarity measures, by applying the Neutrosophic Set approach of median filter, the detailed structures are well preserved for low SNR compared with Median filter and Non local mean filter.

Conclusion
The proposed denoising technique based on Neutrosophic Set for reducing Rician noise from MR image have been validated in this article by using the structural similarity measures (SSIM, QILV).Experiments have been carried out on real MR images.This filtering method tends to produce good denoised image not only in terms of visual perception but also in terms of the quality measures such as PSNR, SSIM and QILV.This filter performs better than Median filtering method for reducing the Rician noise with different noise levels.Further, and also it outperforms the Non Local Mean approach when the noise level is high (low SNR).

3 :
Compute the entropy of the indeterminate subset go to Step 2;Step 5: Transform subset T ˆfrom the neutrosophic domain into the gray level domain.

Figure 2 :
Figure 2: Results of different denoising filters on a brain n =15.T2 Axial (left) and T1 Sagittal (right) slices.From top to bottom: (a) the original image, (b) the noise corrupted image, and the results of (c) median filter, (d) the non local mean filter and (e) the neutrosophic set filter.

Figure 3 :
Figure 3: Comparison of Median, NLM, and NS Median Filters based on the PSNR in dB values: T2 Axial (a) High SNR (b) Low SNR; T1 Sagittal (c) High SNR (d) Low SNR.

Table 1 :
Performance Comparison of Median, NLM and NS Median Filters