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Rician Noise Removal Approach for Brain MR Images Using Kernel Principal Component Analysis

  • Ashish PhophaliaEmail author
  • Suman K. Mitra
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9124)

Abstract

It has been observed that the noise accumulated in medical images due to various reasons during acquisition process is Rician in nature. A Rician noise removal method of Brain Magnetic Resonance (MR) Images using Kernel Principal Component Analysis (KPCA) is proposed in this paper. The proposed approach is non-parametric in nature. It explores the image space for non-local similar patch search and clusters them accordingly. The basis vectors are then learned using KPCA for each cluster which makes the proposed method data adaptive in nature. The approach has been applied to 2D phantom Brain MR images and experimental results are comparable to the other state-of-the-art methods in terms of various quantitative measures.

Keywords

Kernel Principal Component Analysis (KPCA) Magnetic Resonance Image (MRI) Rician noise removal 

1 Introduction

Image Restoration is considered as one of the crucial ingredient of Medical Image Analysis systems. The possible sources for addition of noise are various parameters of the acquisition process such as flip angle, scan time, coil resistance, dielectric and inductive losses in sample, patient movement etc. [12]. MRI, being a non-invasive technique, offers many advantages in clinical analysis but the disturbances or noise induced in acquisition process degrade the quality of the signal. In Medical Image Denoising problem, the noise model is found to be Rician in nature which is different from commonly used distributions such as Gaussian, Poisson, etc. [8].

It has been shown that the intensities of MR images represent magnitude of underlying complex data which follows Rice distribution [7]. The real and imaginary parts are modeled as independently distributed Gaussian with means \(a_r\) and \(a_i\) respectively, with same variance \(\sigma ^2\). The probability density function (pdf) of Rician random variable \(y\) is defined as follows:
$$\begin{aligned} f_Y(y|a,\sigma )=\frac{y}{\sigma ^2}e^{\left( -\frac{y^2+a^2}{2\sigma ^2} \right) }I_0\left( \frac{ya}{\sigma ^2}\right) , y>0 \end{aligned}$$
(1)
where \(a=\sqrt{a_r^2+a_i^2}\) is underlying noise free signal amplitude and \(I_n(z)\) is \(n^{th}\) order modified bessel function of first kind. When Signal to Noise Ratio (SNR, here it is \(a/\sigma \)) is high, the Rician distribution approaches a Gaussian; when SNR approaches to zero (that is only noise is present, \(a\rightarrow 0\)) the Rician distribution becomes Rayleigh distribution and the pdf turns out to be
$$\begin{aligned} f_Y(y|a\rightarrow 0,\sigma )=\frac{y}{\sigma ^2}e^{\left( -\frac{y^2}{2\sigma ^2} \right) } \end{aligned}$$
(2)
Hence, the conventional methods for Rician noise removal first try to find the background portion in the medical images where no signal is assumed. Hence, one can use Rayleigh distribution in background portion and Gaussian distribution in the rest (where SNR is assumed to be high enough) [9, 16]. However under the noisy condition, it is difficult to find proper background in the image.

Recent methods use the principle of non-local self similarity for image restoration task, where the first step involves finding out the similar patches (in terms of some predefined criteria such as Euclidean distance) that are similar to a given reference patch from the image [1]. Thereafter, an orthonormal basis is inferred for each patch and shrinkage is performed on the coefficients when the patch is projected on that basis, coefficients are sparse in nature as described in [4, 6, 14].

Out of recently proposed techniques, BM3D [4] is most popular. BM3D technique creates a 3D stack of similar patches, projects it onto a 3D basis (tensor product of 2D-DCT and 1D-Haar), and performs hard thresholding of these coefficients followed by basis inversion, thereby allowing a coupled update of the coefficients [4]. Another class of methods such as [5, 13], first to cluster similar patches and then learn basis for each cluster instead of searching the similar patches for each underlying reference patch. However, due to nature of noise, straight forward implication of natural image denoising methods has not been advocated for medical images. The NLM method has been extended for Medical Image denoising problem in [11] where bias correction needs to be considered. BM3D has been extended using a suitable invertible transformation of the medical data into another domain where data behaves like Gaussian distributed in resultant domain. The most commonly known such kind of transformation for this purpose is Anacombe’s Transformation, also known as Variance Stabilization Technique (VST). Recently, VST has been proposed in [7] for Rician distributed data and BM3D method is referred as BM3D+VST method. The BM3D+VST method can be summarized mathematically as follows:
$$\begin{aligned} \hat{y} = VST^{-1}(BM3D(VST(z,\sigma ),\sigma _{VST}),\sigma ) \end{aligned}$$
(3)
where \(VST^{-1}\) denotes the inverse VST, \(\sigma _{VST}\) is the stabilized standard deviation induced by VST and \(z\) denotes the additive white Gaussian noise whose true intensity is represented by \(y\). However, BM3D+VST is extended to 3D medical data as BM4D method in [10]. This manuscript focuses on 2D data denoising methods only.

The aim of this article is to explore a direct technique that can handle Rician noise suitably giving rise to noise removal as good as BM3D+VST, if not better. We have extended PCA based method using Rough Set based clustering proposed in [13] to Rician noise model and bias term correction is also made, referred as ER-PCA in the paper. We have proposed a new Kernel based PCA (KPCA) method for Rician noise. However, we have adopted the clustering strategy used in [13], which is non-local approach in true-sense. As per our knowledge, KPCA has not been applied for Rician noise removal in medical image yet. The kernel based methods can find non-linearity of data in Feature Space. Recently, kernel based methods have been used in Medical imaging in [2, 15, 19]. However, choice of appropriate kernel for given data is undecidable. In the current proposal, Gaussian kernel is used and the performance of noise removal technique is at par with the state-of-the-art methods.

The paper has been arranged in following manner: Sect. 2 presents proposed method using KPCA. Section 3 compares proposed method with other state-of-the-art methods. The manuscript is concluded in Sect. 4.

2 Proposed Method Using KPCA

A non-parametric variant of PCA, known as Kernal Principal Component Analysis (KPCA) has been explored for Rician noise removal. The KPCA tries to explore structure in the data in Feature Space instead of Image Space itself and tries to capture higher-order dependencies in the data. In Fig. 1, two class data is shown in circular form and transformed to higher dimension for classification purpose, where transformation is \(\phi (x) : (x_1,x_2) \rightarrow (x_1,x_2,x^2_1+x^2_2)\). Hence, one can find a discriminating plane (linear surface) in higher dimensions which is not possible in two dimensions for given data points.
Fig. 1.

Transformation of two circular data sets into higher dimension space using kernel method where separation between them is more prominent and can be classified using linear hyper-surface.

Fig. 2.

Reconstruction using PCA and KPCA over synthetic data with Rician noise. (a) Synthetic Data, (b) Rician Noisy Data, (c) Reconstruction using PCA and (d) Reconstruction using KPCA.

In KPCA, this nonlinearity is introduced by first mapping the data into another space F using a nonlinear map \(\phi : R^N \rightarrow F\), before standard linear PCA is carried out in F using the mapped samples \(\phi (x_k)\). The map \(\phi \) and the space \(F\) are determined implicitly by the choice of a kernel function \(k\), which acts as a similarity measure. This mapping computes the dot product between two input samples \(x\) and \(y\) mapped into \(F\):
$$\begin{aligned} k(x; y) = \phi (x).\phi (y) \end{aligned}$$
(4)
One can show that if \(k\) is a positive definite kernel, then there exists a map \(\phi \) into a dot product space \(F\) such that Eq. 4 holds. The space \(F\) then has the structure of a so-called Reproducing Kernel Hilbert Space (RKHS) [2].

The identity Eq. 4 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 \(\phi (x)\) 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 \(\phi \) nor \(F\) explicitly to perform KPCA since only inner products are used in the computations. Commonly used positive definite kernel functions are polynomial kernel of degree \(d \in N, k(\mathbf x ,\mathbf y )=(\mathbf x .\mathbf y )^d\) or \(k(\mathbf x ,\mathbf y )=(\mathbf x .\mathbf y +1)^d\) or Gaussian kernel of width \(\sigma > 0\), \(k(\mathbf x ,\mathbf y ) = exp\left( -\left\| \mathbf x -\mathbf y \right\| ^2 /2\sigma ^2 \right) \). In all the experiments, Gaussian kernel has been used which is isotropic stationary in nature and also satisfies Mercer’s Theorem [19].

A synthetic experiment has been performed as shown in Fig. 2 where Rician noise added in the synthetic data. However, KPCA (with Gaussian kernel) is able to preserve orientation of the data in a better way as compared to PCA based reconstruction.
Fig. 3.

Difference comparison of KPCA with reference to BM3D+VST method (at zero level vertically) for 50 slices for noise standard deviation equal to 15 (a) T1 images with PSNR difference values, (b) T1 images with MSSIM difference values, (c) T2 images with PSNR difference values and (d) T2 images with MSSIM difference values.

The outline of present work can be described as follows:
  1. 1.

    Get the clusters of patches from the given noisy image using Rough set based method (as described in [13]).

     
  2. 2.

    For each cluster, get the basis vectors using KPCA method along pixel positions. For patches of size \(p \times p\), kernel matrix would be of size \(p^2 \times p^2\). Hence, the method is data adaptive in nature.

     
  3. 3.

    Project the noisy image patches on the obtained basis vectors in the KPCA domain.

     
  4. 4.

    Apply coefficient shrinkage method on these projected patches to get the denoised patches. Transform them back to image space.

     
  5. 5.
    Remove the bias term from each pixel of the denoised image.
    $$\begin{aligned} I_{unbiased} = \sqrt{max(\hat{I}(i,j)^2-2h^2,0)} \end{aligned}$$
    (5)
    where \(h\) is the standard deviation of noise and \(\hat{I}\) is the image obtained by step (4).
     
Fig. 4.

(a) Synthetic Noisy T1 Image with Rician noise standard deviation=15 and PSNR =22.7220 dB, Denoised image using (b) UNLM method, PSNR = 34.4622 dB, (c) BM3D+VST method, PSNR = 34.2393 dB, (d) RS-NLM method, PSNR = 32.5856 dB, (e) ER-PCA method, PSNR = 33.8155 dB, (f) KPCA method, PSNR = 34.0241 dB.

Table 1.

Performance comparison of proposed denoising strategy with different approaches on various quantitative measures under Rician Noise assumption in Brain Web database (slice = 70 & 100, Modality = T1, image size \(=181 \times 217\) and patch size \(= 5 \times 5\)). Best figures are shown in Bold.

Noise SD

Methods

Slice \(70\)

Slice \(100\)

PSNR

RMSE

MSSIM

FSIM

PSNR

RMSE

MSSIM

FSIM

\(5\)

Noisy

32.4293

37.1660

0.6134

0.9296

32.2588

38.6549

0.5564

0.8922

UNLM [11]

39.0519

8.0889

0.9832

0.9845

40.1551

6.2744

0.9882

0.9887

BM3D+VST [7]

40.9727

5.1937

0.9602

0.9843

41.4921

4.6118

0.9602

0.9857

RS-NLM [13]

39.8595

6.7163

0.9851

0.9853

41.5829

4.5164

0.9914

0.9913

ER-PCA

40.4514

5.8606

0.9791

0.9764

39.9719

6.5447

0.9689

0.9563

KPCA

40.2107

6.1946

0.9197

0.9797

41.2223

4.9073

0.9866

0.9850

\(10\)

Noisy

26.4115

148.5702

0.4717

0.8149

26.2398

154.5629

0.4183

0.7567

UNLM [11]

35.9894

16.3733

0.9608

0.9643

36.9916

12.9993

0.9707

0.9724

BM3D+VST [7]

36.3738

14.9866

0.9040

0.9607

36.8590

13.4025

0.9132

0.9653

RS-NLM [13]

35.8260

17.0011

0.9631

0.9645

37.2231

12.3246

0.9762

0.9770

ER-PCA

35.7387

17.3464

0.9389

0.9439

36.3168

15.1846

0.9597

0.9484

KPCA

36.1061

15.9395

0.9586

0.9522

36.6642

14.0172

0.9682

0.9628

\(15\)

Noisy

22.8950

333.8752

0.3744

0.7177

22.7220

347.4434

0.3331

0.6495

UNLM [11]

33.5147

28.9475

0.9299

0.9391

34.4622

23.2732

0.9453

0.9498

BM3D+VST [7]

33.7666

27.3162

0.8583

0.9368

34.2393

24.4992

0.8684

0.9447

RS-NLM [13]

32.1179

39.9292

0.9273

0.9244

32.5856

35.8523

0.9472

0.9448

ER-PCA

33.2440

30.8093

0.9133

0.9178

33.8155

27.0103

0.9377

0.9287

KPCA

33.4097

29.6557

0.9323

0.9262

34.0241

25.7438

0.9469

0.9404

Table 2.

Performance comparison of proposed denoising strategy with different approaches on various quantitative measures under Rician Noise assumption in Brain Web database (slice = 70 & 100, Modality = T2, image size \(=181 \times 217\) and patch size \(= 5 \times 5\)). Best figures are shown in Bold.

Noise SD

Methods

Slice \(70\)

Slice \(100\)

PSNR

RMSE

MSSIM

FSIM

PSNR

RMSE

MSSIM

FSIM

\(5\)

Noisy

32.4349

37.1185

0.6257

0.9365

32.2639

38.6095

0.5691

0.9052

UNLM [11]

34.4831

23.1617

0.9822

0.9813

35.2666

19.3385

0.9869

0.9858

BM3D+VST [7]

40.4738

5.8305

0.9648

0.9861

41.0752

5.0764

0.9663

0.9885

RS-NLM [13]

36.9814

13.0300

0.9856

0.9835

37.6322

11.2166

0.9915

0.9900

ER-PCA

39.8618

6.7127

0.9783

0.9727

39.1934

7.8297

0.9610

0.9473

KPCA

37.8578

10.6487

0.8002

0.9782

38.1996

9.8429

0.7610

0.9797

\(10\)

Noisy

26.4322

147.8642

0.4956

0.8356

26.2550

154.0201

0.4408

0.7757

UNLM [11]

32.9818

32.7262

0.9618

0.9623

33.8132

27.0246

0.9710

0.9687

BM3D+VST [7]

35.7377

17.3504

0.9181

0.9681

35.8044

17.0860

0.9683

0.9637

RS-NLM [13]

34.5041

23.0502

0.9691

0.9676

35.2411

19.4522

0.9799

0.9766

ER-PCA

34.8288

21.3894

0.9432

0.9323

34.5457

22.8303

0.9262

0.9008

KPCA

35.0519

20.3182

0.8527

0.9567

36.1329

15.8413

0.9184

0.9727

\(15\)

Noisy

22.9275

331.3825

0.4131

0.7519

22.7460

345.5293

0.3676

0.6776

UNLM [11]

31.4832

46.2121

0.9346

0.9408

32.1181

39.9271

0.9472

0.9456

BM3D+VST [7]

32.8504

33.7321

0.8769

0.9496

33.1694

31.3427

0.8855

0.9567

RS-NLM [13]

31.9206

41.7849

0.9446

0.9452

32.6973

34.9423

0.9601

0.9543

ER-PCA

31.7529

43.4297

0.9034

0.8973

31.7770

43.1894

0.8989

0.8718

KPCA

32.3606

37.7592

0.9363

0.9346

32.8539

33.7049

0.9516

0.9439

\(20\)

Noisy

20.4499

518.2594

0.3540

0.6871

20.2642

611.8738

0.3162

0.6059

UNLM [11]

30.0502

64.2771

0.9063

0.9205

30.5757

56.9519

0.9199

0.9216

BM3D+VST [7]

30.7168

55.1319

0.8426

0.9303

30.9691

52.0201

0.8508

0.9398

RS-NLM [13]

29.4113

74.4654

0.9109

0.9104

30.1086

63.4192

0.9293

0.9137

ER-PCA

29.6008

71.2860

0.8723

0.8785

29.6002

71.2947

0.8647

0.8527

KPCA

30.1448

62.8934

0.9129

0.9144

30.6031

56.5941

0.9316

0.9245

\(25\)

Noisy

18.5384

910.4194

0.3095

0.6362

18.3487

951.0736

0.2774

0.5520

UNLM [11]

28.6394

88.9483

0.8777

0.9012

29.1108

79.7989

0.8914

0.8987

BM3D+VST [7]

29.0589

80.7598

0.8109

0.9114

29.2912

76.5527

0.8269

0.9227

RS-NLM [13]

26.4734

146.4670

0.8599

0.8492

26.6486

140.6762

0.8696

0.8372

ER-PCA

28.0567

101.7219

0.8576

0.8824

28.1251

100.1314

0.8529

0.8685

KPCA

28.1995

98.4298

0.8792

0.8814

28.3402

95.2919

0.8932

0.8785

3 Experimental Results

This Section encompasses the qualitative and quantitative evaluations of the proposed method along with some of the state-of-the-art methods. The experiments have been carried out on 2D monochrome phantom human brain MRI images obtained from Brain Web Database [3]. The parameters are as follows: RF = 20, protocol = ICBM, slice thickness = 1 mm, volume size = \(181 \times 217 \times 181\). The experimental set up considers Rician noise model at different noise levels along with two modalities, namely T1 and T2. The simulated database provides the ground truth image for evaluating denoising performance which most of the time is unavailable with real database. The Rician noise addition and bias correction are done as suggested in [10] and [11] respectively. The evaluation measures used are Peak-Signal-to-Noise Ratio (PSNR), Root Mean Square Error (RMSE), Mean Structural Similarity Index (MSSIM) [17] and Feature Similarity Index (FSIM) [18].

For comparison purpose, several state-of-the-art methods are considered: Unbiased Non Local Means (UNLM method) presented in [11], BM3D+VST method proposed in [4], Rough Set based Non Local Means (RS-NLM) method proposed in [13] and PCA based method proposed in the [13] has been extended in this work for Rician noise, referred as Extended Rough set based PCA method (ER-PCA). The parameters of all methods are kept default as suggested by respective authors. In all the experiments, patch size is kept as \(5 \times 5\). The proposed KPCA method does not use VST method. Tables 1 and 2 represent quantitative results for two slices 70 and 100 of T1 MR and T2 MR images respectively. The ER-PCA performance is comparable to UNLM and BM3D+VST methods. The proposed KPCA method outperforms ER-PCA and preserves structure better than other state-of-the-art method. Figure 3 shows difference of PSNR and MSSIM measure for KPCA method with reference to BM3D+PCA (zero level on vertical axis) of 50 slices (from \(61^{st}\) to \(110^{th}\) slice of database mentioned above) with noise standard deviation equal to 15 for both T1 and T2 modalities. Negative value indicates BM3D+VST performs better and, in reverse, positive value is indicator of better performance of KPCA method. From Fig. 3, PSNR of KPCA fall below BM3D+VST method whereas it better preserves structure of the image in terms of MSSIM measure. This is also visually evident in Fig. 4 for the slice 100 of T1 modality at noise level 15.

4 Conclusion

In this paper, an approach for removal of Rician noise from brain MR images using Kernel PCA has been proposed. Being a manifold learning method, KPCA explores a suitable transformation for image representation through sparse bases. This method learns basis vectors from data itself unlike BM3D+VST method where basis vectors are kept fixed. The limitation of KPCA method is the selection of suitable kernel which is yet unanswered. If the nature of data is not known a-prior than one can try various kernels to find a suitable one. However, commonly used Gaussian kernel in KPCA, found to perform comparable with other state-of-the-art methods. The PCA based method proposed in [13] has also been implemented to remove Rician noise, but it fails to attain superior performance over KPCA. The proposed method is implemented on synthetic data for quantitative evaluation since ground truth data is available for the same.

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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Dhirubhai Ambani Institute of Information and Communication TechnologyGandhinagarIndia

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