20th Iranian Conference on Electrical Engineering, (ICEE2012), May 15-17,2012, T ehran, Iran

Non-local Means Denoising using An Adaptive Kernel

Amir Ali Tahmouresi*, Saeid Saryazdi**, and Saeid Reza Seydnejad*** *Shahid Bahonar university of Kennan, aatahmouresi@eng.uk.ac.ir ** Shahid Bahonar university of Kerman , saryazdi@mail.uk.ac.ir

*** Shahid Bahonar university of Kerman , sseydnejad@mail.uk.ac.ir

Abstract: Non-local means algorithm is one of the powerful image denoising methods. Maintaining noise near edges and textural parts of a noisy image, is one of the main drawbacks of NLM. In this paper we introduce an adaptive kernel derived from image structure to remove maintained noise. Experimental results show superiority of our algorithm in comparison with original NLM as well as a method based on shape adaptive patches.

Keywords: Non-local means, Denoising, Shape adaptive patches, Adaptive kernel.

1. Introduction

In the image acquisition process, noise is inevitably added to images. This noise is a hannful factor for image processing procedures. Consequently image denoising is one of the most important issues in image processing publications. Through the years, authors examined several aspects of images to handle this problem. For example, Perona and malik proposed a scale space based approach [1], Norbert Wiener presented an adaptive filter [2], which is employed in several image denoising methods [3], furthennore, various versions of wavelets [4], [5], and spatial filters [6], [7], suggested interesting results.

Recently, Buades et al. proposed Non-Local Means (NLM) filter which has received a lot of attention from image processing community [S]. An NLM filter, regardless to the spatiality, uses the redundancies in an image. Providing remarkable results, NLM suffers from some drawbacks: excessive computational complexity, showing artefactual effects [9], and spatial dependent behavior, that is to say NLM remains noise near high contrasted edges, which is called noise halo.

In order to overcome latter problem, authors of shape adaptive patches NLM (NLM-SAP) suggested a group of predefined kernels [10]. In original non-local means algorithm an isotropic Gaussian kernel is used which in most publications is removed because of its suboptimal properties. In NLM-SAP it is proved well shaped kernels provides NLM with more similar patches, which helps , , The work is supported by Young Researchers Society (e -mail: yrs@mail ­

.uk .ac .ir) .

NLM to cope with the problem of shortage of similar patches near high contrasted edges. In [10] a group of flat, shaped and oriented kernels are proposed to be used instead the Gaussian kernel proposed by Buades. Any of these shaped kernels are implemented in an iteration of NLM, and then a criterion called SURE is used to choose the best kernel for each part of image. Although this method improves NLM performance in some cases like elimination of noise halo, but suffers from some drawbacks.

First, choosing a limited number of pre-defmed kernels do not suffice for denoising of images with different structures, moreover in a single image there are different structures and textures which needs a large number of predefined kernels [11]. Second, big group of kernels leads to numerous iterations of NLM which is computationally impractical. Third, due to lack of robustness of SURE, particularly in intensive noise, NLM-SAP has a defective performance and removes details of image.

Taking all into account, it seems if we use image structure as a reference for kernel design, better results are attained. In other words, we should design kernels based on the neighborhood around every single pixel of image rather than some predefined kernels.

Benefits of this approach are of three folds: these kernels are well adapted to every parts of image. By means of this manner NLM can be implemented in only one iteration. There is no need to criterions like SURE.

In this paper we propose an adaptive kernel which contrary to NLM-SAP, is extracted from the image structure itself, and being implemented in an iteration of NLM, perfonns greatly in different images with different levels of noise. this means proposed kernel removes noise halo while preserves details of image.

The structure of paper is as follows; in the next section we briefly review the non-local means approach and some of its improvements. The method NLM-SAP is surveyed in section 3. In section 4 the proposed adaptive kernel is presented. Section 5 is devoted to an experimental study. Finally we conclude in section 6.

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2. Non-local Means and Its Improvements

Conventionally, a noisy image is regarded as addition of a white Gaussian noise to the original image:

v(i)=u(i)+n(i) (1) Where U ( i ) , V ( i) and n (i) are the values

of the original image, noisy image and Gaussian white noise in the position i ,respectively. Methods based on spatial neighborhood denoising use the pixels close to the given pixel for denoising process. Denoised pixel is computed as follows

u(i)= L w(i,j)v(j) JEW, L W(i, j) (2)

JEW,

Where Wi is neighborhood around i which its pixels are used for denoising process and W ( i , j) are weights of the pixels in this neighborhood.

w(i,j)=e Ilpi-p,II;"

JI' (3)

Filtering parameter, h controls the smoothness of result. Buades suggested to use the Euclidean norm of two patches Pi' P J which are multiplied by a Gaussian kernel with standard deviation a . Despite of the robustness of the weights, large search area does not guarantee a better result [12], besides, from computational point of view, it leads to impracticality of algorithm. h plays a crucial role in NLM method, thus several methods are presented to locally [13], [14] and globally [15] determine it.

There are two main types of speed up methods: 1. Methods based on neglecting irrelevant patches

from weight computing process [16], [17]. 2. Methods based on exploiting Summed Area

Tables and FFT which can be computed very rapidly [18], [19].

BM3D probably is the most evolved version of NLM frame work. Despite of its complicating implementation it provides high quality results [20].

3. NLM-SAP

One of the ambiguous aspects of NLM is patch size selection. Depending on the patch size two draw backs arise. On one hand, if we choose a small patch size, noise remains in homogenous parts of image (figure l.a). On the other hand, choosing large patch size, leads to noise halo near the edges (figure l.b). This phenomenon arises when patches similar to the current patch are rare. Some efforts have been made to remove noise halo [14], but a better technique can be using shaped and directional patches.

a Fig. l. Images denoised by NLM with 3x3 and 7x7 patches.

Basically, in NLM, thanks to plentiful similar patches in homogenous parts of image, noise can be easily removed, but it is not the case for textural or high detailed parts. As figure 2.a suggests pixels which patches around of them are similar to showed patch, are only on the gray line. This is called rare patch effect. It means patches similar to current patch are hard to find. This small group of similar patches is against the philosophy of NLM

(taking advantage of redundant similarities in image). In figure 2.b it is showed if we use a well-shaped patch, all pixels in gray area can be used to denoise pixel of

a b Fig. 2. Well-shaped patches help to find more similar patches.

To our knowledge, there are only two shape adaptive kernel based methods in NLM oriented algorithms. First, authors of [21] extended 8M3 D in this way, to present better results. Second, in NLM-SAP [10] a number of predefined kernels (like one showed in figure 2.b) with different sizes and directions are proposed to be implemented in several iteration of NLM (i.e. 15 iterations), in order to suppress rare patch effect. Limited number of patches in NLM-SAP are not enough for denoising of several images with different structures.

Furthermore this method performs poorly in denoising of intensively noisy images. As it is mentioned in [24] SURE criterion is sensitive to noise. It results to lack of robustness of SURE, especially in intensive noise. Authors of [10] use this measure for kernel selection which leads to unfavorable results.

In the next section we introduce our adaptive kernel

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which is simply derived using the structure of image itself and being implemented in only one iteration, out perfonns the NLM-SAP from objective and subjective points of view.

4. Proposed Method

In [10], to overcome noise halo difficulty, it is recommended to use predefined patches, which needs several iterations of NLM to be accomplished. Contrarily, we propose an adaptive kernel, simply derived from the own image structure; moreover, it can be implemented in

only one iteration of NLM. Flowchart of proposed method is depicted in figure 3.

No

� �

�oisy imag�

, Get pixel i

....... from image

, Makek;

, � Get pixelj

From Wi

, Make kj

, k=k/\kj

, Calculation

Of (3) using k

Are all pixels Of W;

Processed?

Yes ' Calculation

Of(2)

, Are all pixels

Of image Processed?

Yes y_ ---- ---

.. QeTl()ised il11ag� ... Fig. 3. Flowchart of proposed method

No

It is obvious from figure 2 that central pixel together with other pixels of the kernel should lay in the same part of image (in one side of an edge). In other words, pixels which their gray level values are close can construct a suitable kernel.

The proposed kernel is based on a thresholding on the difference between the central pixel and the others of patch, to convert them to binary kernels.

k,(i)� (� Ip;(i)-pJi)I>th Ip;(i)-pJl)I<th (4)

Where I E Pi . Suppose one patch is around the current pixel i ,and other around the pixel j ,in the neighborhood Wi ' Two kernels k; and kj are built for patches p; and Pj , respectively. Subsequently, two binary kernels are ANDed together. The obtained kernel could be used for determining distance between pixels i and j by multiplying by difference of tow patches. Figure 4 presents the flowchart for construction of the proposed adaptive kernel.

According to the original fonnulation of NLM, OJ ( i , i) = 1 . Therefore, in some cases, other weights

in neighborhood W; get suppressed. Some authors suggested using fixed or adaptive weights [12], [13]. We use the weight for central pixel proposed by V. Dore et al. as follows:

( . .) ( _0'; h I ( . .) . ·1' (5) OJ 1, } = max e , max \ OJ 1, } ,1 *- } J •

In the next section, experimental results show, by taking advantage of this weighting, and robust kernel mentioned above, proposed method leads to remarkable results.

5. Experimental Results

We implemented our method and NLM using a search window of size 15 X 15 and a 7 X 7 kernel. Five images, Cameraman, Barbara, Mandrill, House and Lena, with different contents, are denoised in three noise levels. In the proposed method, binary kernels are firstly built for all pixels in the image then AND operation is done in core of NLM algorithm, thus from computational time point of view, difference between our algorithm and NLM method is negligible.

Four criteria are used to show objective quality of compared methods.

PSNR: As a classic image quality assessment criterion, it is derived as follows.

R2 PSNR=IOloglO MSE

(6)

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k

ki Fig. 4. Process of making the adaptive kernel for weighting pixel j in neighborhood Wi centered on pixel i.

where R is maximum gray level value of image, and MSE shows mean square error.

I [11 (m, n)-12(m, n)]2

MSE= m,nE! (7)

11 and 12 are the two images to be compared and M and N are the length and width of the images, respectively.

MSSIM: under the assumption that human visual perception is highly adapted for extracting structural information from a scene, MSSIM provides an alternative framework for quality assessment based on the degradation of structural information. MSSIM is ranged from 0 to 1, and a higher MSSIM means a higher quality of the denoised image [22].

FOM: figure of merit uses the distances between edge pixels in the filtered and original images to build a reliable criterion [23].

FOM=_l t 1

1Ni=ll+ad2 (8)

Where 1 N = max { 1 [ ' 1 A } and 1 A and 1 J

represent the number of edge map points in original and filtered images, respectively. a is a scaling constant

(1/9), and d is the separation distance of two edge points in original and filtered images,

Difference between original and noisy image: it is described as follows.

D = loriginal-denoisedl (9) For better perception D undergoes gamma

correction with y =0.8 ,

GC=(l-y)DY ( 10) This measure shows removed structure and texture

from noisy image in comparison to original one, To state the matter differently the less image structure and texture represented in D image, the better detail preserving property presented by denoising method,

We adjust h empirically to maximize PSNR for NLM and proposed method and also it is the case for threshold value used in the adaptive kernels,

In table 1 we compared our method with NLM and NLM-SAP. Although, in low level of noise there is a slight difference between proposed method and NLM­SAP, but in intensively noisy images, our method, completely outperforms other ones, in all three bench marks.

Figure 5 shows the image denoising capability and detail preserving ability of proposed method in comparison with naive non-local means and NLM-SAP. NLM maintains noise in high contrasted edges of Cameraman image and also in fur and face of Mandrill. NLM-SAP removes details in addition to noise; this is on account of bad performance of SURE in detection of bias and variance, especially in strong noise [24], and use of limited number of kernels which are not appropriate for all parts of image. The proposed kernel leads to sake of the maximum homogeneity and results in a powerful tool to denoise textural parts of image.

In addition detail preserving property of proposed method in comparison to two other methods is perceivable in the difference between original and denoised images,

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Table I. Comparison of proposed method with NLM and NLM-SAP based on PSNR dB), MSSIM and FOM.

noise NLM NLM SAP Proposed method -Image PSNR MSS[M FOM PSNR MSS[M FOM PSNR MSS[M FOM

10 32.6632 0.9806 0.6508 33.5505 0.9829 0.6890 33.4326 0.9824 0.6689 C.man 20 28.8439 0.9497 0.4691 29.7560 0.9563 0.4971 29.4793 0.9566 0.4886

40 25.7153 0.9[29 0.3596 26.2439 0.9[08 0.367[ 26.2535 0.9151 0.3672

10 33. [996 0.9798 0.6405 33.7610 0.9838 0.6807 33.5638 0.9820 0.6586 Barbara 20 29.6114 0.9577 0.4620 30.3146 0.9610 0.4720 30.3275 0.9633 0.5024

40 25.8450 0.9080 0.3217 25.9295 0.9049 0.2951 26.5513 0.9167 0.3404

10 29.3557 0.970[ 0.6680 29.5633 0.9736 0.6778 29.4857 0.9711 0.6678 Mandrill 20 25.2990 0.9[26 0.4680 25.7337 0.9231 0.4788 25.5497 0.9215 0.4805

40 22.2979 0.8242 0.2760 22.0669 0.7968 0.2604 22.5162 0.8384 0.3364

10 34.7799 0.9731 0.7921 35.5191 0.9766 0.7902 35.2543 0.9764 0.7895 House 20 31.8237 0.9599 0.6590 32.6241 0.9605 0.6575 32.4589 0.9617 0.6625

40 27.6691 0.9[5[ 0.4571 27.9847 0.9[46 0.4577 28.3538 0.9208 0.4740

10 34.3637 0.9737 0.6[55 35.0721 0.9781 0.6232 34.7909 0.9774 0.61 [[ Lena 20 31.3159 0.95[3 0.4205 31.9729 0.9550 0.4300 3 [.8038 0.9567 0.4431

40 28.0031 0.9042 0.2547 28.2897 0.9056 0.2499 28.3970 0.9099 0.2731

Average 29.3858 0.9382 0.5010 29.8921 0.9389 0.5084 29.8812 0.9433 0.5176

Due to the fact that parameter h is not [ocally adapted for proposed method, it remains noise in the homogenous parts of image, like the sky in cameraman image and also face of Mandrill. Setting this parameter locally is a good food for thought.

6. Conclusion

In this paper a new adaptive kernel is proposed for non-local means in order to remove residual noise near high contrasted edges of denoised image by NLM. Our adaptive kernel is derived by taking the structure of image into account. Additionally proposed method is implemented in one iteration of NLM, and Experimental results confirm its superiority.

References

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a b c d Fig. 5. (a) Cameraman image, result of implementation of (b) NLM, (c) NLM-SAP and (d) Proposed method

e f g h on Cameraman image with noise standard deviation 20, (e) cropped Cameraman image. (f-h) Difference between (b-d) and (e). (i) Mandrill image, result of implementation of U) NLM, (k) NLM-SAP and (I)

k Proposed method on Mandrill image with noise standard deviation 40, (m) cropped Mandrill image. (n-p)

m n 0 p Difference between (i-I) and (m).

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[23] W.K. Pratt, "Digital Image Processing," New York: Wiley, 1977.

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