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

## 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].

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].

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

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].

- 1.
Get the clusters of patches from the given noisy image using Rough set based method (as described in [13]).

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

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] | | | 0.9602 | 0.9843 | 41.4921 | 4.6118 | 0.9602 | 0.9857 | |

RS-NLM [13] | 39.8595 | 6.7163 | | | | | | | |

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] | | | 0.9040 | 0.9607 | 36.8590 | 13.4025 | 0.9132 | 0.9653 | |

RS-NLM [13] | 35.8260 | 17.0011 | | | | | | | |

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 | | | | 0.9453 | | |

BM3D+VST [7] | | | 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 | | 0.9448 | |

ER-PCA | 33.2440 | 30.8093 | 0.9133 | 0.9178 | 33.8155 | 27.0103 | 0.9377 | 0.9287 | |

KPCA | 33.4097 | 29.6557 | | 0.9262 | 34.0241 | 25.7438 | 0.9469 | 0.9404 |

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] | | | 0.9648 | | | | 0.9663 | 0.9885 | |

RS-NLM [13] | 36.9814 | 13.0300 | | 0.9835 | 37.6322 | 11.2166 | | | |

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] | | | 0.9181 | | 35.8044 | 17.0860 | 0.9683 | 0.9637 | |

RS-NLM [13] | 34.5041 | 23.0502 | | 0.9676 | 35.2411 | 19.4522 | | | |

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 | | | 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] | | | 0.8769 | | | | 0.8855 | | |

RS-NLM [13] | 31.9206 | 41.7849 | | 0.9452 | 32.6973 | 34.9423 | | 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] | | | 0.8426 | | | | 0.8508 | | |

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 | | 0.9144 | 30.6031 | 56.5941 | | 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] | | | 0.8109 | | | | 0.8269 | | |

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 | | 0.8814 | 28.3402 | 95.2919 | | 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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