3D MR image denoising using rough set and kernel ...

Magnetic Resonance Imaging 36 (2017) 135–145

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Original contribution

3D MR image denoising using rough set and kernel PCA method Ashish Phophalia a, Suman K. Mitra b,⁎ a b

Indian Institute of Information Technology, Vadodara, India Dhirubhai Ambani Institute of Information and Communication Technology, Gandhinagar, India

a r t i c l e

i n f o

Article history: Received 5 May 2016 Received in revised form 13 September 2016 Accepted 5 October 2016 Keywords: Image denoising Magnetic resonance imaging Rough set theory Kernel Principle Component Analysis

a b s t r a c t In this paper, we have presented a two stage method, using kernel principal component analysis (KPCA) and rough set theory (RST), for denoising volumetric MRI data. A rough set theory (RST) based clustering technique has been used for voxel based processing. The method groups similar voxels (3D cubes) using class and edge information derived from noisy input. Each clusters thus formed now represented via basis vector. These vectors now projected into kernel space and PCA is performed in the feature space. This work is motivated by idea that under Rician noise MRI data may be non-linear and kernel mapping will help to define linear separator between these clusters/basis vectors thus used for image denoising. We have further investigated various kernels for Rician noise for different noise levels. The best kernel is then selected on the performance basis over PSNR and structure similarity (SSIM) measures. The work has been compared with state-of-the-art methods under various measures for synthetic and real databases. © 2016 Elsevier Inc. All rights reserved.

1. Introduction

Gaussian with means ar and ai respectively, with same variance σ2. The pdf of Rician random variable y can be defined as follows:

Being non-invasive technique, magnetic resonance imaging (MRI) is a widely used modality (along with x-ray, CT, etc.) for clinical diagnosis. The acquisition process of medical images is highly sensitive and gets accumulated with noise or undesired signals. In general, the noise realization is assumed to be Gaussian in nature. However, it has shown that the noise in magnetic resonance (MR) image could be Rician in nature [1]. But Gaussianity assumption is also valid for actual Ricianity in low SNR (signal to noise ratio) MRI. Be it Gaussian or Rician, removal of noise from MR images is essential for further analysis. There have been many denoising/restoration methods proposed for medical images ranging from diffusion filters [2] to dictionary [3] and clustering based filters [4]. Research on designing new techniques of denoising is still going on. These techniques are then implemented on different modalities of MR images with different functionalists. In this paper, an attempt is made to design a new technique of denoising to cater to Rician noise model in 3D volumetric MR images. It has been shown that the intensities of MR images are magnitude of underlying complex data following rice distribution [1,5]. The real and imaginary part are modeled as being independently distributed


 2 2
ya  −y þa2 y 2σ f Y ðyja; σ Þ ¼ 2 e I0 2 ; yN0 σ σ

⁎ Correspoding author at: DA-IICT Post Bag 4, Near Indroda Circle, Gandhinagar, Gujarat 382007, India. Tel.: +91 79 30510584 (Office); fax: +91 79 30520010. E-mail addresses: [email protected] (A. Phophalia), [email protected] (S.K. Mitra). URL: http://intranet.daiict.ac.in/~suman_mitra/newSite/ (S.K. Mitra).

http://dx.doi.org/10.1016/j.mri.2016.10.010 0730-725X/© 2016 Elsevier Inc. All rights reserved.

where a ¼

ð1Þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2r þ a2i is underlying noise free signal amplitude and In(z)

is nth order modified Bessel function of first kind. Let SNR be the signal to noise ratio (here, it is a/σ). When SNR is high, the Rician distribution approaches a Gaussian; when SNR is approaches to zero (that is only noise is present, a → 0) the Rician distribution becomes Rayleigh distribution with the pdf f Y ðyja→0; σ Þ ¼


 2 −y2 y 2σ e σ2

ð2Þ

Many state-of-the-art methods for 3D image denoising are extensions of their 2D counterpart version. However, computational complexity becomes a crucial factor during the extension. The 3D MR image denoising was introduced in modern literature in Ref. [6] and then followed by [3,7–11]. Instead of playing with patches in 2D images, here a voxel is defined as 3D cube centered at location (i, j, k) in ℛ3. Hence, a voxel is simply counterpart of a patch with size w × w × w (say). Consequently, exploring relationship among intensity values in ℛ3 is highly sought and thus leads to computational intensive process. The optimized blockwise NLM (OBNLM [6]) is extension of nonlocal means method [12]. OBNLM tweaks the computation of similarity

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weight between voxels by having predefined criteria for mean and variance of both the voxels. At the same time, it adopts the blockwise strategy to drop restoration of adjacent n voxels which effectively reduces the computation load by n3 times instead of processing each voxel in the image space. The same has been adopted in Ref. [7,10]. In case of ABONLM method [7], denoising is performed under wavelet framework using adaptive soft wavelet coefficient mixing (ASCM) approach. A nonparametric kernel regression framework has been adopted for 3D MR image denoising in unbiased kernel regression method [9]. UKR is rooted on a zeroth order 3D kernel regression and similarity weight between voxels is derived on small sized feature vectors based on image intensity and gradient information. The sparseness and selfsimilarity has been unified in PRI-NLM method [8]. PRI-NLM incorporated rotational invariant version of NLM [13] and discrete cosine transform hard thresholding for sparsity purpose. One of the most well-known image denoising methods, BM3D has been extended to 3D MR image denoising as BM4D method [10]. Like BM3D, BM4D is also equipped with collaborative filtering notion where similar voxels are arranged in fourth dimension. It is also a two stage method where the output of first stage guides the second stage and uses hard thresholding in first pass and wiener filtering in second pass. Ideally, BM4D is designed for Gaussian noise whereas variance stabilization technique (VST) has been adopted to deal with Rician noise in 3D MRI data. The VST based scheme is also adopted in dual phase HOSVD-R method [11] which is also a natural extension of HOSVD method proposed in Ref. [14]. Recently, another two stage method PRI-NL-PCA [3] was proposed based on sparsity and self-similarity of voxels. It encompasses PCA thresholding strategy in first stage whereas rotational invariant NLM method is deployed in second stage. In this work, we present rough set theory (RST) based method for MR image denoising. Rough set [15] is a mathematical model to deal

with uncertainty present in the granular form of data. RST is well studied and applied in various domains. However, the same is not explored much as far as medical image application is concerned. In the current proposal, we are working on medical image denoising where we are uncertain about the nature of noise accumulated in the image. The main motivation of using RST is to handle this uncertainty present. While using RST on various applications, it is observed that RST can well handle the uncertainty. This motivated us to use RST. A nonparametric variant of PCA, known as kernal principal component analysis (KPCA) has been explored for Rician noise removal. The KPCA tries to explore the structure of the data in feature space instead of data space and tries to capture higher-order dependencies in the data. In Fig. 1, a two class data is shown in circular form and transformed to higher dimension for classification purpose, where transformation is ϕ(x): (x1, x2) → (x1, x2, x21 + x22). Hence, one can find a plane (linear surface) in higher dimensions which is not possible in two dimensions for classifying given data points. The concept of KPCA is utilized for voxel classification and thereby leading towards 3D MR image denoising. It has been observed that KPCA works much better compared to classical PCA in case data possess as a non-linear characteristic. Many of the denoising techniques, the noise is assumed to be Gaussian. However, in the current scenario, it has been observed by other researchers that noise could be non-Gaussian in nature. Now to handle non-Gaussian characteristic of noise, KPCA is used. The current proposal aims to resolve uncertainty by RST followed by KPCA which is expected to handle non-Gaussian nature of noise. The current work is an extension of the previous work presented in Ref. [16]. We have incorporated following extensions: • We have experimented with more set of kernels in comparison to our previous work (only one kernel was used).

Fig. 1. The figure on left shows two circular data on (x, y) plane where finding a linear separation is impossible. However, in figure on right, can easily separate the data in (x, y, z) space where function is f: (x, y) → (x, y, x2 + y2). The function f is called the kernel mapping causing non-linear input space to project on feature space where linearity can be identified.

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2. Material & methods 2.1. Rough set based clustering formation

Fig. 2. Image granules with upper and lower approximation of an object as conceptualized in rough set theory.

• With more set of kernels, we have shown exhaustive set of experiments for parameter estimation of those kernels. • We have experimented on various noise levels on T1, T2 and PD modalities whereas only basic experiments presented in previous work with T1 modality only. This paper is organized as follows: Section 2 discusses RST based clustering approach followed by overview of kernel principal component analysis. The proposed methods in presented in the last subsection of Section 2. Section 3 presents simulation results of proposed method along with state-of-the-art-method on phantom and real MRI data. The manuscript is concluded in Section 4.

Rough set theory (RST) can be utilized to explore structural similarity between pixel and set of pixels even in the presence of noise. This classification is defined on predefined attribute(s), (say Θ), by forming granules within the image space, Ω. This indeed establishes an equivalence classes among the data, based on attributes. Rough sets define a class by approximating two sets, namely lower approximation and upper approximation sets of a class with respect to an attribute(s). The lower approximation set of class with respect to attribute(s), (say C Θ ), consists of certainly classified member of Ω whereas the upper approximation set of class with respect to attribute(s), (say C Θ ), constructed by possible members of that class defined by attribute(s), Θ. The figurative description is given in Fig. 2. The other properties of the class such as accuracy, roughness measure, rough entropy etc. can be derived as mentioned in Ref. [17]. Rough set based derivation of class label (RCL) information and edge details (REM) have been derived in Ref. [18]. For given attribute(s), a pixel or set of pixels called granules can be classified in either lower or upper approximation of an object. The attribute considered is the intensity values at each location in the image space. The objects present in the image are categorized in intensity range by optimizing image histogram. The rough set based entropy criteria [17] was used in optimizing intensity thresholds. The class label can be assigned by comparing intensity values at each location against the intensity ranges of all the objects. Similarly, a granule (set of adjacent pixels) is assigned to an object's lower (and upper also) if all the pixels fall in its intensity range. Otherwise it will be assigned in object's upper approximation only. The union of difference of both the approximations of all objects will generate pixels which are very near to possible edges in the image. We have extended the method proposed in Ref. [4] to 3D imaging. A voxel (cube of size p × p × p) is converted in one dimensional representation. We used the same approach for deriving lower and upper approximation of all the objects. Unlike, finding similar patches for a candidate patch, pools of similar patches are constructed based on whether the patch lies in the object or on the boundary of two or more objects. Hence for K number of objects present in the image, the

Fig. 3. Flowchart of proposed method.

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Fig. 4. Comparison of PSNR values for three different noise levels for T1, T2 and PD modalities.

A. Phophalia, S.K. Mitra / Magnetic Resonance Imaging 36 (2017) 135–145

Fig. 5. Comparison of SSIM values for three different noise levels for T1, T2 and PD modalities.

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(a) T1 volume

(b) T2 volume

(c) PD volume

Fig. 6. Original volumes of T1, T2, PD data from BrainWeb Database.

(a) Noisy volume

(b) UKR

(a) Noisy volume

(b) UKR

(c) PRI-NLM

(d) BM4D

(c) PRI-NLM

(d) BM4D

(e) PRINLPCA

(f) Proposed

(e) PRINLPCA

(f) Proposed

Fig. 7. Comparison of various methods on T1 from BrainWeb Database.

Fig. 8. Comparison of various methods on T2 from BrainWeb Database.

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(a) Noisy volume

141

(a) Noisy volume

(b) UKR

(c) PRI-NLM

(d) BM4D

(e) PRINLPCA

(f) Proposed

(b) UKR

(c) PRI-NLM

(d) BM4D

Fig. 10. Comparison of various methods on subject 012 from OASIS Database.

(e) PRINLPCA

(f) Proposed

Fig. 9. Comparison of various methods on PD from BrainWeb Database.

function k, which acts as a similarity measure. This mapping computes the dot product between two input samples x and y mapped into F via kðx; yÞ ¼ ΦðxÞ:ΦðyÞ

 K . Here, we have coni sidered 4 objects in 3D image: cerebrospinal fluid (CSF), white matter (WM), gray matter (GM) and background. Hence, the number of pools constructed is 15. K



number of pools constructed would be ∑i¼1

2.2. Kernel principal component analysis In KPCA, the nonlinearity is introduced by first mapping the data into another space F using a nonlinear map Φ: RN → F, before a standard linear PCA is carried out in F using the mapped samples ϕ(xk). The map Φ and the space F are determined implicitly by the choice of a kernel

ð3Þ

One can show that if k is a so-called positive definite kernel, then there exists a map Φ in a dot product space F such that Eq. (3) holds. The space F then has the structure of a so-called reproducing kernel Hilbert space (RKHS) [19]. The identity of Eq. (3) is important for KPCA since PCA in F can be formulated entirely in terms of inner products of the mapped samples. Thus, we can replace all inner products by evaluations of the kernel function. This has two important consequences. First, inner products in F can be evaluated without computing Φ(x) explicitly. This allows to work with a very high-dimensional, possibly infinite-dimensional RKHS F. Second, if a positive definite kernel function is specified, we need to know neither Φ nor F explicitly to perform KPCA since only inner products are used in the computations.

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If projected dataset ϕ(xi) does not have zero mean, one can use Gram

~ F to substitute the kernel matrix CF which is given by matrix C ~ F ¼ C F −1N C F −C F 1N þ 1N C F 1N C

ð7Þ

where 1N is the N × N matrix with all elements equal to 1/N. 2.3. Proposed method

(a) Noisy volume

Based on the REM and RCL as discussed in Section 2.1 and using KPCA as discussed in Section 2.2, the proposed method is built in. The outline of present work can be described as follows:

(b) UKR

1. Get the clusters of patches from the given noisy image using rough set based method (as described in Ref. [4]). Note that here patch implies cubes of size p × p × p. 2. For each cluster, get the basis vectors using KPCA method. 3. Project the noisy image patches on the obtained basis vectors in the KPCA domain. 4. Apply coefficient shrinkage method on these projected patches to get the denoised patches. Transform them back to image space. 5. Remove the bias term from each pixel of the denoised image. ^Iunbiased ði; j; kÞ ¼

(c) PRI-NLM

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
 2 2 max ^I ði; j; kÞ −2h ; 0

ð8Þ

where h is the standard deviation of noise and^I is the image obtained by step (4). (Refer Fig. 3) The next section deals with the details of experiments performed.

(d) BM4D

3. Experimental section Experiments are conducted on two sets of data. One is phantom data and the other is real data. The methods used for comparison with proposed methods are: (a) UKR [9], (b) PRI-NLM [8], (c) BM4D [10] and (d) PRINLPCA [3]. 3.1. Validation on phantom database

(e) PRINLPCA

(f) Proposed

Fig. 11. Comparison of various methods on subject 018 from OASIS Database.

1 In PCA, the covariance matrix is defined as C ¼ N−1 X T X where, X is called data matrix containing samples in columns. The covariance matrix in case of KPCA of size M × M, is calculated by

CF ¼

N 1X ϕðxi Þϕðxi ÞT N i¼1

ð4Þ

its eigenvalues and eigenvectors are given by C F vk ¼ λk vk

ð5Þ

where k = 1, 2, …, M. Putting the value of CF in Eq. (5) and mathematical N

simplification lead to vk ¼ ∑i¼1 aki ϕðxi Þ and hence ak (N-dimensional column vector of aki) can be solved by C F ak ¼ λk Nak

ð6Þ

This Section encompasses the qualitative and quantitative evaluation of the proposed methods along with some of the state-of-the-art methods. The experiments have been carried out on 3D monochrome phantom human brain MR images obtained from Brain Web Database as shown in Fig. 6 [20]. The parameters are as follows: RF = 20, protocol = ICBM, slice thickness = 1 mm, volume size = 181 × 217 × 181. The experimental set up considers Rician noise model at different noise levels for three modalities, T1, T2 and PD. The simulated database provides the ground truth images for evaluating denoising performance which most of the time is unavailable with real database. The evaluation measures used are peak-signal-to-noise ratio (PSNR), root mean square error (RMSE), structural similarity index (SSIM) [21] and Bhattacharya coefficient (BC). The kernels used in this work are mentioned in Table 1 with their respective parameters. In this work, exhaustive experimental results are presented in Figs. 4 and 5 for finding optimized kernel parameter values over all the three modalities. The parameter values are evaluated in the range 10−3 to 106 with incremental step 10. The Fig. 4 shows performance of various kernels over PSNR measure. The Fig. 5 shows performance of various kernels over SSIM measure. Both the figures reveal that simple kernel is consistent with respect to both the measure in comparison to other kernels. Hence, in all the experiments, simple kernel was used. This gives an additional advantage as no parameter learning is required for optimal performance. The proposed method with single threaded MATLAB implementation takes around 45 min on core i7 processor, 2.10 GHz and 8 GB

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143

(a) Noisy volume

(b) UKR

(c) PRI-NLM

(d) BM4D

(e) PRINLPCA

(f) Proposed

Fig. 12. Comparison of various methods on 2D image patch of subject 018 from OASIS Database.

RAM machine. The UKR is observed to have same computational time whereas BM4D with MATLAB/C implementation takes 11 min and others run in less than five minutes. Table 2 represents results on noise-free data to identify artifact introduced by proposed method and others. The proposed method is able to attain best values in all the measures showcase that it is not adding artifact in the data. Tables 3, 4 and 5 present denoising results for T1, T2 and PD modality respectively. All the tables present results with various noise levels (h = 5, 10, 15, 20, 25). In Tables 3, 4 and 5, 2 × 2 block represents PSNR (top-left), RMSE (top-right), SSIM (bottom-left) and BC (bottom-right). The best result along each noise is represented in bold figure for all the measures. At lower noise level, KPCA method performed comparable well to other methods in all the modalities. At higher level, PSNR of proposed method falls very down whereas it is able to preserve the structure of medical data far better than other state-of-the-methods. At the same time, proposed method also stood best in probabilistic measure BC. The kernel method may not be able to project data on manifold under high noisy condition that might cause the fall in PSNR values. However, a simple pre-processing such as mean or median filtering can be applied at higher noise level as done in Ref. [9]. The figurative comparison has been shown in Figs. 7, 8 and 9 for T1, T2 and PD modalities respectively. 3.2. Results on real databases The proposed method has been tested on real OASIS database [22]. The subject details of selected data from OASIS dataset are as follows:

subject ID: 12, 18, age: 30 and 39 (both male) respectively, scan number: mpr-1, type: MPRAGE, voxel resolution: 1.0 mm × 1.0 mm × 1.25 mm, orientation: sagittal, TR (ms)=9.7, TE (ms)=4.0, TI(ms)= 20.0, flip angle=10. In all these experiments, noise level is estimated using method proposed in Ref. [23]. The standard deviation of noise level estimated are 5.16 and 8.92 for subject 12 and 18 respectively. The results are shown in Figs. 10 and 11 for subject 012 and 018 respectively. Fig. 12 shows 2D patch for the subject 018 for subjective evaluation of all the methods.

4. Conclusions RST is well studied and applied in various domains. However, the same is not explored much as far as medical image application is concerned. In the current proposal, we are working on medical image denoising where we are uncertain about the nature of noise accumulated in the image. KPCA is also a well-studied area. It has been observed that KPCA works much better compared to classical PCA in case possess a non-linear characteristic. Many of the denoising techniques, the noise

Table 2 Results of state-of-the-art methods for noise free data for T1, T2 and PD modalities (represented row-wise against each method).

PRI-NLM Table 1 Various kernel used in this work.

UKR

Kernel name

Expression

Simple/linear Gaussian

xTy

Laplacian Multiquadratic Sigmoid

Parameter 2

; expð− kx−yk 2 Þ

2h Þ ; expð− kx−yk h

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 kx−yk2 þ h tanh(‖x − y‖2 + h2)

– h

BM4D

h h h

KPCA

PSNR

RMSE

SSIM

BC

12.45 11.57 7.25 12.45 11.55 7.24 47.64 40.57 54.76 296.48 200.14 263.69

60.83 67.31 11.61 60.83 67.45 110.73 1.06 2.39 0.47 0.0 0.0 0.0

0.7064 0.7053 0.7060 0.7063 0.7045 0.7044 0.9976 0.9966 0.9989 1.0 1.0 1.0

0.8548 0.8594 0.8618 0.8519 0.8819 0.8519 0.9955 0.9923 0.9529 0.9988 0.9963 0.9901

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Table 3 Results of state-of-the-art methods for T1 modality (represented row-wise against each method). In each 2 × 2 block, top-left figure is PSNR, top-right is RMSE, bottom-left is SSIM and bottom-right is BC measure. The figure against each noise level is represented in bold face. h→

5

Noise

31.78 0.3473 36.04 0.5084 45.16 0.9527 43.81 0.9452 40.36 0.6640 44.09 0.9904

UKR PRI-NLM BM4D PRI-NL-PCA KPCA

10 06.57 0.3497 04.02 0.5373 01.41 0.9009 06.57 0.9008 02.45 0.4699 01.59 0.9818

25.76 0.2170 30.59 0.3359 39.95 0.8769 39.17 0.8729 35.01 0.4918 39.45 0.9770

15 13.13 0.2910 07.53 0.2760 02.56 0.8291 02.81 0.8690 04.53 0.5332 02.72 0.9760

20

22.25 0.1583 33.72 0.6507 36.91 0.8129 36.50 0.8167 32.0 0.4243 36.83 0.9535

19.69 0.2859 05.25 0.7809 03.64 0.8173 03.81 0.8597 06.41 0.5705 03.67 0.9641

19.75 0.1209 32.11 0.6364 34.84 0.7734 34.61 0.7735 29.90 0.3864 32.93 0.9446

25 26.24 0.3033 06.32 0.8210 04.62 0.8276 04.74 0.8549 08.16 0.5930 05.76 0.9505

17.82 0.0949 32.85 0.7762 33.27 0.7297 33.14 0.7390 28.32 0.3662 32.11 0.9391

32.77 0.3207 05.81 0.8902 05.53 0.8225 05.62 0.8518 09.78 0.6173 06.33 0.9446

Table 4 Results of state-of-the-art methods for T2 modality (represented row-wise against each method). In each 2 × 2 block, top-left figure is PSNR, top-right is RMSE, bottom-left is SSIM and bottom-right is BC measure. The figure against each noise level is represented in bold face. h→

5

Noise

31.78 0.3540 35.79 0.5141 43.87 0.9537 42.95 0.9460 40.1 0.6818 43.46 0.9916

UKR PRI-NLM BM4D PRI-NL-PCA KPCA

10 06.57 0.3496 04.14 0.5411 01.63 0.8942 01.82 0.8971 02.52 0.4872 01.71 0.9797

25.76 0.2341 30.12 0.3434 39.41 0.8819 38.11 0.8746 34.45 0.4745 38.65 0.9798

15 13.11 0.2894 07.95 0.3204 02.73 0.8247 03.17 0.8660 04.83 0.5154 02.98 0.9732

20

22.26 0.1850 31.03 0.4739 36.62 0.8216 35.21 0.8201 31.21 0.3980 35.58 0.9564

19.65 0.2671 07.16 0.6155 03.76 0.8148 04.43 0.8566 07.02 0.5359 04.24 0.9597

19.77 0.1552 29.87 0.5732 34.25 0.7845 33.09 0.7790 28.90 0.3624 26.97 0.9434

25 26.17 0.2259 08.19 0.7732 04.94 0.8244 05.65 0.8525 09.15 0.5536 11.42 0.9638

17.84 0.1344 28.74 0.4727 32.09 0.7414 31.41 0.7458 27.03 0.3361 26.39 0.9364

32.67 0.2559 09.32 0.6726 06.34 0.8184 06.86 0.8500 11.34 0.5518 12.22 0.9585

measure. A simple kernel having no parameter was found best over all the three MRI modalities used in this work. However, it can be exercised with other known kernel with suitable parameter estimation method or data adaptive kernel for Rician model can be thought of. The proposed method is non-iterative and single stage method in comparison to some of predecessor methods like BM4D, PRI-NL-PCA etc. In this work, intensity value is used as a feature in clustering step and in kernel space however more features can be considered like gradient information as in UKR. The predecessor methods restrict the search space for searching similar voxels. However, current method exploits the whole volume and thereby form clusters for similar voxels. The experimental results are appeared to be either comparable or better than that of the state-of-the-art methods.

as Gaussian. However, in the current scenario, it has been observed by other researcher that noise could be non-Gaussian in nature. Now to handle non-Gaussian characteristic of noise, KPCA is used. Hence, this work explores Kernel method in the presence of Rician noise in the MRI data. Being signal dependent noise, applicability of linear denoising operation such as PCA is not advisable. It is expected in the present work that kernel method may project the data in the feature space where linear separation is possible. We have an extended rough set based clustering method to collect similar voxels conditioned on class and edge information. These similar voxels then used to define kernel matrix via kernel function. We have experimented with a small set of kernels having at most single parameter. The selection of best kernel is made by PSNR and SSIM

Table 5 Results of state-of-the-art methods for PD modality (represented row-wise against each method). In each 2 × 2 block, top-left figure is PSNR, top-right is RMSE, bottom-left is SSIM and bottom-right is BC measure. The figure against each noise level is represented in bold face. h→

5

Noise

31.78 0.3430 36.22 0.5112 45.67 0.9528 44.46 0.9455 40.64 0.6879 44.88 0.9898

UKR PRI-NLM BM4D PRI-NL-PCA KPCA

10 06.57 0.3455 03.94 0.5422 01.33 0.8922 01.53 0.8946 02.37 0.4927 01.45 0.9770

25.78 0.2074 36.28 0.6616 41.04 0.8800 39.38 0.8731 34.99 0.4935 40.02 0.9756

15 13.10 0.2842 03.91 0.7192 02.26 0.8205 02.74 0.8629 04.53 0.5350 02.54 0.9717

22.28 0.1493 34.26 0.7614 38.02 0.8181 36.22 0.8175 31.58 0.4049 36.83 0.9608

20 19.61 0.2609 04.94 0.8462 03.20 0.8079 03.94 0.8520 06.72 0.5457 03.67 0.9671

19.81 0.1163 30.71 0.6648 35.61 0.7793 33.88 0.7754 29.04 0.3513 27.34 0.9468

25 26.07 0.2466 07.43 0.8133 04.23 0.8162 05.16 0.8458 09.01 0.5364 10.95 0.9705

17.89 0.0955 28.98 0.4693 33.45 0.7340 32.01 0.7419 27.02 0.3203 26.94 0.9399

32.48 0.2347 09.07 0.6645 05.42 0.8094 06.40 0.8414 11.37 0.5325 11.47 0.9564

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