Modified Anisotropic Diffusion Filtering Algorithm for MRI - IEEE Xplore

Modified Anisotropic Diffusion Filtering Algorithm for MRI Aditya Srivastava

Vikrant Bhateja

Harshit Tiwari

Department of Electronics and Communication Engineering, SRMGPC, Lucknow-227105 (U.P), India. Email Id: [email protected]

Member, IEEE Department of Electronics and Communication Engineering, SRMGPC, Lucknow-227105 (U.P), India. Email Id:[email protected]

Department of Electronics and Communication Engineering, SRMGPC, Lucknow-227105 (U.P), India. Email Id:[email protected]

Abstract – During the acquisition process of Magnetic Resonance Imaging (MRI), irregular bias is imposed in the intensity values of the pixels. These biases follow the Gaussian Noise distribution model and act as a constraint to the effective medical diagnosis. The conventional Anisotropic Diffusion (AD) approach is limited to preserve the structural integrity of MRI at only low noise levels. This paper proposes a modified AD algorithm aimed to improve the estimation of the diffusion constant to facilitate better edge detection and preservation of details. The proposed algorithm operates on the decomposed mask images of MRI by incorporating the domain filtering principle of the Bilateral filter(prior to the estimation of diffusion constant). Simulation trials have been conducted at different Gaussian noise variances and performance has been evaluated on the basis of Peak SignalNoise Ratio (PSNR) and Structural Similarity (SSIM). The proposed algorithm has shown stable value of evaluation parameters at higher noise variances. Also, the preservation of details has improved as compared to the conventional AD approach. Keywords – AD, Bilateral Filter, Gaussian Noise, SSIM. I.

INTRODUCTION

The recent years have seen the emergence of MRI as a powerful diagnostic technique for the detailed visualization of the internal structures of the human body. However, the image quality of the obtained MRI is constrained by the presence of large amount of noise introduced during the acquisition process [1]. This noise, usually modelled as Gaussian noise [2]-[3], introduces irregular intensity bias in the pixel values, thereby hampering the performance of the various image postprocessing modules such as enhancement [4]-[8], segmentation [9]-[10] and classification [11]-[13]. Thus, removal of this unwanted noise becomes a fundamental process in the medical image processing. Over the years, various denoising approaches [14]-[18] based on linear smoothening [19] have been developed for the suppression of the noise. However, these approaches eliminated the noise at the expense of the fine

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details and tissue edges which were lost due to blurring. This lead to the development of edge preserving AD approach introduced by Perona and Malik [20] and was based on the scale space concept introduced by [21]. This approach overcame the limitations of the linear smoothening approaches, such as blurring and loss of details, by suppressing the noise while respecting the tissue edges and small structures present within the image. Gerig et al. in their work [22] utilized this technique for noise suppression in MRI. Further improvements in the denoising ability of the AD approach were carried out in the works of [23]-[24]. An analysis on the behavior of the denoising mechanism of AD was performed in the work of [25]. This was utilized by Black et al. for the development of a robust AD filter [26] which incorporated the robust statistics in the AD filter. A fourth order partial differential equation based denoising approach was developed by [27] which utilized the concept of signal dependent noise characteristics in MRI [28]. The application of AD filter was extended for spatially varying noise levels in MRI in the work carried out by Samsonov and Johnson [29]. Recently, an extension of the AD filter based on the estimation of noise level has been developed [30]. In this paper, a modification in the estimation of the diffusion constant of the AD filter has been proposed, where the decomposed mask images have been filtered using the domain function of the bilateral filter [31]-[32] as a pre-processing step. In the remaining part of the paper, Section II details the AD process and the proposed filtering algorithm, Section III presents the simulation results and their subsequent discussions and Section IV includes the conclusion part. II. PROPOSED METHODOLOGY A. Anisotropic Diffusion (AD) Filtering AD is a widely used filtering algorithm for the biomedical images [33]-[36], specifically MRI. It is a filtering technique aimed at the noise suppression without the removal of the significant parts of the image content, typically the edges and boundaries [26]. This technique incorporates the non-linear and space-variant transformation of the original MRI, so as to

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produce a family of parameterized images [22]. These parameterized images are depicted as the convolution between the original MRI and the isotropic Gaussian filter [37] and are termed as the mask images of the original MRI. The basic AD equation as described by [22] is: It = c( x, y, t )ΔI + ∇c∇I (1) where: t is the time parameter, I is the original image, It is the filtered image, ΔI , ∇I are the gradient and the laplacian of the image respectively and c( x, y , t ) is the diffusion constant. This Eq. 1 can be further reduced to: It = c( x, y, t )ΔI (2) The diffusion constant c( x, y , t ) is the primary edge stopping parameter which controls the filtering process and leads to the edge preservation. The value of the diffusion constant can be calculated by: c( x, y, t ) = g (∇I ) (3) where: g (∇I ) is the conduction coefficient function and is represented by:

g (∇I ) = e(−(|| ∇I || / k ) 2 )

(4)

of the pixels and is therefore unaffected by the intensity bias produced by the noise. Thus, this function effectively reduces the noise present in the mask images. Mathematically, the domain filter function is represented as:

h( x) = kd −1 ¦ f ( y )c( y, x)

where: h(x) is the output image, c ( y , x ) is the parameter that measures the geometric closeness of the pixel x and its neighbour pixel y within the window Ω and kd is the normalisation parameter given by:

kd = ¦ c( y, x)

1 1 + (|| ∇I || / k )2

The resulting mask images are then utilized for the calculation of the conduction coefficient using the Eq (4) or Eq (5) which is then further utilised by the AD algorithm for the noise suppression and structure preservation of the MRI. The proposed modified AD filtering algorithm has been depicted in Fig. 1 below. START Gaussian Corrupted MRI

(5)

where: k is the gradient modulus constant and || ∇I || is the parameterized image or the decomposed mask images. Eq (4) is used when high contrast is preferred over low contrast and Eq (5) is used when wider regions are preferred over smaller regions [38]. It can be observed from Eq (3) that the value of the diffusion constant is dependent on the conduction coefficient. Further, the conduction coefficient depends on the parameterized images or the mask images. Therefore, these mask images play a crucial role in the estimation of the diffusion constant. In case of an MRI corrupted with the additive Gaussian noise, the decomposed mask images of that MRI will also contain biased pixel intensity values due to the noise. This will lead to an incorrect estimation of the diffusion constant, thereby, leading to degradation in the performance of the AD filtering. Therefore, in this paper, a modification in the AD approach has been proposed in order to remove this limitation. B. Modified AD Filtering Algorithm In the proposed algorithm, the Gaussian noise contaminated MRI is first decomposed into its respective set of mask images. In the decomposed mask images, the pixel intensities are corrupted due to the superimposition of the additive Gaussian noise leading to a bias in the intensities. Therefore, these mask images are subjected to the domain filtering function of the Bilateral Filter [31]-[32] as a pre-processing step to the evaluation of the conduction coefficient. The domain function of the Bilateral Filter solely depends on the geometric closeness

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(7)

y∈Ω

and

g (∇I ) =

(6)

y∈Ω

Initialization of constant parameters such as Integration constant and Gradient modulus Decomposition of MRI into Mask Images Application of Domain Filter on the Mask Images

Denoised Mask Images Calculation of Conduction coefficient g (∇I ) from the obtained Mask Images Evaluation of Diffusion constant c ( x , y , t ) Substitute the value of c ( x , y , t ) in the AD equation Denoised MRI START Fig.1. Flow chart of the modified AD algorithm.

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C. Performance Evaluation For the performance assessment of the algorithm, in terms of noise suppression and structure preservation, reference based parameters [39]-[40] such as PSNR and SSIM have been used in this paper. PSNR utilizes the concept of error measurement for the computation of the image quality between two images [41]. It calculates the peak value of the signal-noise ratios, in decibels (dB), between the reference image and the filtered image. Mathematically, it can be expressed as: P S N R = 1 0 log 1 0

§ ¨ ¨ ¨ 1 ¨ MN ©

· ¸ M AX I ¸ ¦ i∈ M N ( x i − y i ) 2 ¸¸¹ 2

(8)

where: MAXI is the maximum possible pixel value of the image, MN represents the size of MRI , x and y are the original and restored image respectively. The evaluation of detail preservation is carried out by computing the SSIM. It utilizes the principles of semblance and measures the structural and perceptual similarity between the reference image and the filtered image [42]. It is represented as:

SSIM ( x, y ) =

1

)

xy

+ c2 )

x + y + c1 (σ x2 + σ y2 + c2 ) 2

(9)

σ , σ are variances of x and y, x, y are the means and y, σ xy covariance of x and y, c1 and c 2 are two

where: of x

(

( 2 x y + c ) ( 2σ

2

2 x

2 y

variables to stabilize the division with weak denominator. III.SIMULATION RESULTS AND DISCUSSIONS This section deals with the application of the proposed filtering algorithm on a sample MRI, evaluation of its validity for noise suppression and comparison with conventional AD approach [22]. The experimental simulations were carried out using MATLAB (R2012a) on a computer with 2.30 GHz core-i5 processor and 4GB dynamic memory. For the simulation, a sample MRI, corrupted with synthetic Gaussian noise of different noise variance values, has been subjected to the proposed algorithm. During the simulations, it was noted that the proper optimization of the preliminary algorithm parameters resulted in an improved performance of the denoising process. Therefore, after optimization, the integration constant (ɉ) was set to 0.1429, gradient modulus (k) was set at 30. The conduction coefficient function g (∇I ) was selected on the basis of the method developed by [38]. The performance evaluation of the filtered MRI was carried out at different noise variances: 7, 13, 15, 17, 19 and the obtained values of PSNR and SSIM have been enlisted under Table I. The noisy MRI and its corresponding filtered MRI have been illustrated in Fig. 2 for the purpose of visual assessment. The introduction of intensity bias can be seen in Fig. 2(b) for the contamination of the MRI with Gaussian noise of variances: 7, 13, and 17. The corresponding filtered MRI shown in Fig. 2(c) illustrates the suppression of that intensity bias while the image contents such

as the tumor and its surrounding region have been completely retained. Further, from the high values of PSNR and SSIM obtained, it can be elucidated that the proposed algorithm was able to effectively suppress the noise contamination while preserving the structural details of the MRI. TABLE I. Computation of PSNR and SSIM of the MRI filtered by the modified AD algorithm at different noise variances.

Noise Variance

Modified AD Algorithm PSNR (dB)

SSIM

35.5483 33.2261 32.5148 31.8770 31.2602

0.9249 0.8253 0.7912 0.7572 0.7247

7 13 15 17 19

A comparison of the performance of our algorithm with the approach of [22] was done and the obtained values of the evaluation parameters have been enlisted under Table II with Fig. 3 depicting a visual comparison of the two approaches. It can be validated from the results that our algorithm has shown an improvement over the preceding approach both in terms of noise suppression and preservation of details in the MRI. TABLE II. Comparison of the modified AD algorithm with the Conventional AD approach in terms of PSNR and SSIM at different noise variances.

Noise Variance 7 9 11 13 15

Conventional AD Approach

Modified AD Algorithm

PSNR (dB)

SSIM

PSNR (dB)

SSIM

33.5701 31.0592 29.1408 27.7351 26.7979

0.7863 0.6793 0.5870 0.5173 0.4910

35.5483 34.8797 34.0835 33.2261 32.5148

0.9249 0.8916 0.8590 0.8253 0.7912

(a)

Variance = 7

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REFERENCES

Variance = 13

Variance = 17 (b) (c) Fig. 2. (a) Original sample MRI used for simulation (b) Noise corrupted MRI with different Gaussian noise variance (c) Denoised MRI obtained from the modified AD algorithm for the corresponding noisy MRI.

PSNR=26.7979 dB SSIM=0.4910

(a)

(b)

PSNR=32.5148 dB SSIM=0.7912

(c)

Fig. 3. (a) MRI corrupted with Gaussian noise of variance 15 (b) Filtered MRI using conventional approach (c) Denoised MRI obtained from the modified AD algorithm.

IV.CONCLUSION In this paper, a modified AD filtering algorithm has been proposed which aims at the improvement in the estimation of the diffusion constant of the conventional AD filtering algorithm. This improvement has been done by incorporating the domain filtering function. Simulations of the proposed algorithm were carried out on the MRI at different levels of Gaussian noise and the obtained results were compared with the existing AD approach. From the analysis of the results, it was observed that the proposed algorithm outperformed the conventional approach in terms of both PSNR and SSIM. The obtained results also validated the efficiency of the proposed algorithm in noise suppression and in its ability to recover the finer structural details even at higher noise levels. As an extension to this work, this algorithm can be merged with the non-local means technique in future.

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