JOURNAL OF ELECTRONIC SCIENCE AND TECHNOLOGY, VOL. 10, NO. 2, JUNE 2012 1

AbstractThis paper proposes a novel exemplar

-based method for reducing noise in computed tomography (CT) images. In the proposed method, denoising is performed on each block with the help of a given database of standard image blocks. For each noisy block, its denoised version is the best sparse positive linear combination of the blocks in the database. We formulate the problem as a constrained optimization problem such that the solution is the denoised block. Experimental results demonstrate the good performance of the proposed method over current state-of-the-art denoising methods, in terms of both objective and subjective evaluations.

Index TermsConstrained quadratic programming, CT image, exemplar-based denoising.

1. Introduction Many applications require reliable and accurate images,

especially in medical fields where images play an increasingly important role in the pathological diagnosis or surgical intervention. However, image quality is often affected by various artifacts, such as noise which could make it difficult to analyze or to extract useful information. This can seriously affects the quality of pathological diagnosis. Therefore, denoising plays an important role for improving the quality of medical imaging. Basically, the goal of image denoising is to reduce the noise as much as possible, while retaining important features such as edges and fine details. To this end, many noise filters have been proposed which come from various disciplines such as linear and nonlinear filtering, spectral and multi-resolution analysis, probability theory, statistics, and partial differential equations. The classical filters such as the Gaussian filter,

Manuscript received June 1, 2012; revised June 11, 2012. D. H. Trinh and F. Dibos are with the LAGA Laboratory, Université

Paris 13, Sorbonne Paris Cité, F-93430, Villetaneuse, France (e-mail: trinh@univ-paris13.fr ; dibos@math.univ-paris13.fr ).

M. Luong is with the L2TI Laboratory, Université Paris 13, Sorbonne Paris Cité, F-93430, Villetaneuse, France (e-mail: marie.luong@univ-paris13.fr )

J. M. Rocchisani is with the Hopital Avicenne-Medicine Nucleaire, 93000 Bobigny, France (e-mail: jean-marie.rocchisani@avc.aphp.fr)

C. D. Pham and H. D. Pham are with the CIID Laboratory, Vietnam Academy of Science and Technology, Hanoi, Vietnam (e-mails: pcduong@ciid.vast.ac.vn ; phdien@ciid.vast.ac.vn)

the anisotropic filter, the total variation filter, etc. often suffer from blurring effect in high frequency regions[1].

In the last decade, many efficient state-of-the-art denoising methods have been proposed. Among them, Nonlocal means (NLM) was proposed by Buades et al. in 2005[1] for Gaussian denoising. It takes advantage of the presence of repeating structures in a given image. Then, denoising is performed by computing a weighted average of pixels with similar neighborhoods. Since 2005, NLM has been cited more than 500 times thanks to its efficiency and simplicity. Many extensions of this method to medical image denoising have been later introduced[2],[3]. Another effective method, namely K-SVD, was proposed within the sparse representation framework by Elad and Aharon et al.[4],[5]. In this method, an optimal over-complete dictionary of image blocks adapted for the noisy image is first determined. By assuming that each block is sparsely represented over the dictionary of atoms, denoising is carried out by coding each block as a linear combination of only a few atoms in the dictionary. This method has been proven to be very effective in removing Gaussian noise. A more recent method was proposed by Dabov et al. is BM3D[6]. This method finds similar blocks in the whole 2-D image and stacks them together in 3-D arrays, and then performs denoising through transform-domain shrinkage of the 3-D arrays. Some applications relying on the BM3D for removing noise in medical images have been proposed, for example in [7], [8].

With CT images, it has been proven that the noise was often found to have a Gaussian distribution[9]. This is important information. However, noise may come from different sources, such as image acquisition modes, transmission, storage, and display devices. Hence, it is not easy to determine exactly the nature of noise. In such cases, the denoising methods that are based on the assumption of independent identically distributed additive noise might not be efficient enough. These methods may affect seriously regions that have no noise or very slight noise and where there may be no need to denoise. So, it remains necessary to propose a new solution more consistent with medical images.

We introduce in this paper a novel block-based denoising method for removing noise in CT images with the help of a given database of standard image blocks (noise-free or very little noise) used as prior. The noisy image is considered as an arranged set of small blocks and denoising will be performed on each block. Here, noise on

An Optimal Weight Method for CT Image Denoising

Dinh-Hoan Trinh, Marie Luong, Jean-Marie Rocchisani, Canh-Duong Pham, Huy-Dien Pham, and Françoise Dibos

JOURNAL OF ELECTRONIC SCIENCE AND TECHNOLOGY, VOL. 10, NO. 2, JUNE 2012 2

the block is assumed to have locally an additive Gaussian distribution. Note that noise level of different blocks may be different. For a noisy block as input, its output (denoised block) is defined as a sparse positive linear combination of the standard blocks in the database such that the output is closest to the input. We formulate the problem of block denoising as a constrained optimization formulation where the similarity between blocks is considered. In particular, a measure of similarity between blocks is used as penalization function to enforce sparsity. Our main contribution is that, unlike previous methods, the proposed method can effectively remove noise with uniform or non-uniform distribution. Furthermore, the proposed method offers a simple formulation from the denoising problem within the sparse representation framework into a common quadratic programming (QP) problem.

The rest of this paper is organized as follows. In Section 2 we describe the proposed algorithm. Our experiments and results are reported in Section 3. The conclusion and future works are presented in Section 4.

2. Optimal Weight Method As mentioned above, the proposed method is

established based on the database of standard blocks. For a given noisy image, the question is how to establish a reliable database? Fortunately, in medical imaging, it is interesting to observe that many images are acquired at nearly the same location, and some of them can be considered as good images (noise-free images or with very little noise) by experts. Therefore, the standard blocks in the database can be collected selectively from those good images. The idea using the standard images for denoising has been proposed before in [13]. In order to obtain a good database, the selection of these images should be such that they would have a variety of intensities as well as shapes. Since the standard images and the noisy image are taken from nearby locations and thanks to the repetition of local structures of images, small blocks tend to recur many times inside these images. We can suppose that, the database is adaptive for a given noisy image, in the sense that, for a given block in the noisy image, a rich amount of similar blocks can be extracted from the database. The database can thus be established from such standard images.

Let , 1, 2, , niu i MΩ = ∈ =ℝ ⋯ denote the database

of vectorized standard. Assume that Y is a noisy image that needs to be denoised. Our aim is to estimate the true image, denoted by X, from Y with the help of the database Ω. To this end, we consider Y as a set of small overlapping blocks

, 1, 2, , i iY y R Y i M= = = ⋯ , (1)

where niy ∈ℝ represent the vectorized n n× block

centered at location i, while iR is a linear operator that

extracts block at i. Overlap is often taken for these blocks

for two reason: to avoid blockiness artifacts and to better restore in denoising process. The proposed method is performed block-wise with two main steps as follows.

Step 1. Block denoising: For each noisy block iy ,

estimate the corresponding noise-free block ix of X. This is realized based on determining the best sparse positive representation of iy over the database Ω via an optimization problem where a measure of similarity between blocks is used as penalization function to enforce sparsity.

Step 2. Aggregation: Using the result obtained from the

first step, construct the final estimate X from X. The following subsections present these steps in detail.

Hereafter, we refer our method as denoising by optimal weight (DOW) method.

Step 1. Block denoising: Consider a noisy block iy , with the noise component

2(0, )i iη σ∼N . Thanks to the repetition of local structures of images, we can believe that there exists a subset of similar blocks (in Ω) which can be considered as the candidates for each noise-free block ix . Such blocks will play an important role in finding an estimate ˆix of ix

from iy . In this work, ix is assumed as a weighted sum of the blocks in Ω,

u , 0 .i k k kk

x kα α= ≥ ∀∑ (2)

To avoid the influence of the non-candidate blocks, it is better if only a few blocks which are the good candidates for ix are involved for the estimation of ix . Then, ix can be seen as a sparse positive linear combination of the elements in Ω where most of zero coefficients correspond to the elements ku which are not good candidates for ix .

That is why we try to estimate ix based on a sparse

positive linear representation over Ω with the weights kα

depending on the similarity between ku and iy . Under

these conditions, denoising a block iy implies to solve the following sparse decomposition problem:

( )

2

00 2

1arg min || ||

2

( , ) ,

i k kk

i i k kk

y u

d y u

αα α λ α

φ α

∗

≥= − +

+

∑

∑

(3)

where 0|| ||⋅ stands for the 0ℓ -norm which counts the

non-zero entries in α, λ is a positive constant, :iφ →ℝ ℝ

is a non-negative increasing function, and ( , )i kd y u is the block-distance (dissimilarity).

To define the dissimilarity ( , )i kd y u , we consider the

residual block, i ky u− . Since ku is considered as noise-free, ku is similar to ix if i k iy u η− ≃ ( iη ∼

2(0, )iσN ). Thus ( ) 0i kE y u− ≃ and 2var( ) 0i k iy u σ− − ≃ . Therefore,

TRINH et al.: An Optimal Weight Method for CT Image Denoising 3

2, var( ) ( ) 0.i k i k i i ky u E y uλ σ= − − + − ≃ (4)

The parameter ,i kλ may help to evaluate the property of

noise in the residual block. So, in this work, the dissimilarity is computed by:

2

,2( , ) ,i k i k i kd y u y u λ= − + (5)

where 2|| ||⋅ is 2ℓ -norm. The function iφ in (3) is

defined by the reversed Huber function in [12]:

2 2, if 0

( ), if

2

i

iii

i

t ttt

t

ρρφ ρ

ρ

≤ ≤ += >

, (6)

where iρ is a preset threshold corresponding to iy .

In (3), ( ( , ))i i kd y uφ may be viewed as the penalty

coefficients, in the sense that if the value of ( ( , ))i i kd y uφ

is suitably large, and if kα is large, the term

( ( , ))i i k kd y uφ α will be penalized a heavy cost. Thus, in the

cases where ku and iy are very dissimilar (i.e.

( ( , ))i i kd y uφ is large), the objective function in (3) can be

minimized with kα often very small or null. Moreover,

from (6) we see that ( )i tφ strongly increases when it ρ> .

So, in the sparse solution α ∗ of (3), the non-zero components often correspond to the small penalty coefficients ( ( , ))i i kd y uφ (i.e., ku is similar to iy ). In

other words, ˆi k kkx uα ∗= ∑ is determined from several

candidate blocks ku in the database Ω.

It is easy to see that the objective in (3) is not a convex function, since 0ℓ -norm is not a true norm. This problem

is too complex to solve in general. Thus, we replace

0ℓ -norm by 1ℓ -norm. Now, (3) is convex and rewritten as

( )

2

10 2

1arg min

2

( , )

i k kk

i i k kk

y u

d y u

αα α λ α

φ α

∗

≥= − +

+

∑

∑

. (7)

Let us denote 0

( ) k

i kS y uα ∗ >

= ∪ as the support set of iy .

As analyzed above, ( )iS y includes ku where ( , )k id u y is not very large. Thus, with a suitable value of the threshold ir , we have

( ) : ( , ) i i k k i iS y u d u y rΩ Ω⊆ = ∈ ≤ . (8)

So, to save computing time, problem (7) should be considered on the subset Ωi of Ω,

( )( )

2

0 : 2

:

1arg min

2

( , )

k i

k i

i k kk u

i i k kk u

y u

d y u

α Ω

Ω

α α

λ φ α

∗

≥ ∈

∈

= −

+ +

∑

∑

. (9)

Obviously, (9) is a convex quadratic programming (QP) problem with convex constraint set, which can be

effortlessly solved. With the solution α ∗ of (10), the true

block ix can be estimated by :ˆ

k ii k kk ux uΩ α ∗∈= ∑ .

Step 2. Aggregation Suppose that we have obtained the estimation ˆix of

ix from iy for all i . The final denoised image X is

then determined from ˆ , 1, 2, , ix i N= ⋯ as the solution of the following minimization problem:

2

21

ˆ ˆminN

i iX i

X x R X=

= −∑ . (10)

Problem (8) has a closed-form least-squares solution, given by

1

1 1

ˆ ˆN N

T Ti i i i

i i

X R R R x−

= =

= ∑ ∑ . (11)

As in [4], [5], we put ˆix in their proper locations and

perform averaging in overlap regions to get the final image

X , enforcing then the consistency between neighbouring blocks. Indeed, since adjacent blocks with overlap are often similar, averaging in the overlap region proposes satisfactory result.

3. Performance Evaluation We have carried out several experimental results of the

proposed method in the two cases: images with simulated noise and real noisy images. Here, the proposed method is compared with some well-known denoising methods, including the NLM [1], the K-SVD[5], and the BM3D[6].

In our method, we use 5×5 patches in the case of noise standard deviation 10σ = and 7×7 patches in the other

cases, λ in (9) is set to 1, iρ in (6) is set to 2inσ , ir in

(8) is chosen such that Ω is the set of 50 first elements ku

which have the shortest distance toiy . In this paper, we use

the method in [14] to estimate the noise standard deviation on the block.

In this paper, the noisy images are generated from some CT images by adding Gaussian noise with σ =20 and 30. The database of standard patches is established from two standard images (images of nearly the same location as the test image). For objective evaluation, we use two image quality metrics, namely PSNR and SSIM[10]. Due to the lack of space, we only report here the objective evaluations on three test images (Fig. 1). As can be seen from Table 1, the PSNR of the proposed method (DOW) is the highest. The SSIM of our method is higher than the NLM and KSVD. In some case the SSIM of the BM3D is higher. For subjective comparison, we only show on Fig. 2 the results of CT image of chest (Fig. 1 (a)). Visually, DOW efficiently removes the noise and preserves most of the original structures while the results of NLM, KSVD, and BM3D are mostly blurred.

JOURNAL OF ELECTRONIC SCIENCE AND TECHNOLOGY, VOL. 10, NO. 2, JUNE 2012 4

(a) (b) (c)

Fig. 1. Test images for (a) CT of Chest, (b) CT of Pelvis, and (c) CT of Abdomen.

Table 1: Objective comparison on quality metrics

Standard image 1 Standard image 2

Noisy image NLM

K-SVD BM3D

DOW Original image

Fig. 2. Experimental results on CT image of chest with Gaussian noise (σ = 20).

(a) (b)

(c) (d)

(e) (f)

Fig. 3. Experimental results on CT image of abdomen with real noise: (a) noisy image, (b) standard image, (c) result by NLM, (d) result by K-SVD, (e) result by BM3D, and (f) result by the presented DOW.

The proposed method is also performed on CT images with real noise. Fig. 3 (a) presents a noisy CT image of abdomen. It is strongly corrupted by tomographic noise. Here, we use a standard image, Fig. 3 (b), to establish the database Ω. The block size in this case is 7×7. Denoised images by the four methods NLM, K-SVD, BM3D, and DOW are illustrated in Fig. 3 (c)−(f), respectively. We can see in Fig. 3 (f), that our method’s result is very interesting. By effectively denoising while slightly enhancing contrast, the result has its quality improved, although it has less contrast than the standard image (Fig. 3 (b)) due to the poor contrast of the noisy image. The results of NLM, K-SVD and BM3D are blurred, e.g. bones in white. We can see the limitation of these methods in the case of instable noise.

Another experiment on a CT image of the thorax is illustrated in Fig. 4, where the noisy image (Fig. 4 (b)) is acquired with optimized reconstruction filter, 2 mm thickness and different values of parameters for tomographic reconstruction. In this experiment, DOW’s result (Fig. 4 (c)) is compared with a standard image at the same position Fig. 4 (d). This is an image of the thorax at the level of the pulmonary arteries, of 3 mm thickness, being acquired with standard reconstruction filter and hence noise-free (or low noise). We use another noise-free CT image at nearly location to establish the database Ω of standard blocks (selected randomly). The size of block used in this experiment is 5×5. As it can be seen in Fig. 4 (c) and Fig. 4 (d), the denoised image preserves well small details

Test images σ

PSNR

NLM KSVD BM3D DOW

(a) 20 32.08 33.71 34.44 36.09

30 28.77 32.74 33.60 35.89

(b) 20 34.34 34.50 36.12 37.35 30 30.81 33.18 34.99 36.64

(c) 20 31.90 32.81 33.37 36.40 30 30.27 31.61 32.43 36.27

σ SSIM

NLM KSVD BM3D DOW

(a) 20 0.792 0.821 0.854 0.851

30 0.575 0.739 0.819 0.817

(b) 20 0.896 0.884 0.917 0.918 30 0.775 0.835 0.895 0.894

(c) 20 0.782 0.816 0.815 0.822 30 0.725 0.729 0.760 0.795

TRINH et al.: An Optimal Weight Method for CT Image Denoising 5

(a) (b)

(c) (d)

Fig. 4. CT image of the thorax: (a) image used to establish the block-database, (b) noisy image, (c) result by DOW, and (d) standard image.

Fig. 4 (d), the denoised image preserves well small details and maintains the same resolution as for the standard noise-free image (Fig. 4 (d)). Furthermore, the noise is very effectively removed and the contrast is enhanced compared with the standard image.

4. Conclusions In this paper, a novel example-based method has been

proposed. The method is based on finding a sparse positive linear representation of the input noisy image patch over a database of standard image patches. The experimental results have demonstrated the effectiveness of the method on medical images. In the future works, we are going to study optimal solutions for establishing the database as well as for improving the computing speed of the proposed algorithm. Moreover, we will apply this model to the problem of image super-resolution.

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denoising algorithms with a new one,” SIAM J. Multiscale Modeling & Simulation, vol. 4, no. 2, pp. 490–530, 2005.

[2] J. V. Manjon, J. Carbonell-Caballero, J. J. Lull, G. Garcia-Marti, L. Marti-Bonmati, and M. Robles, “MRI denoising using nonlocal means,” Medical Image Analysis, vol. 12, no. 4, pp. 514–523, 2008.

[3] P. Coupe, P. Hellier, C. Kervrann, and C. Barillot, “Nonlocal means-based speckle filtering for ultrasound images,” IEEE Trans. on Image Processing, vol. 18, no. 10, pp. 2221–2229, 2009.

[4] M. Elad and M. Aharon, “Image denoising via sparse and redundant representations over learned dictionaries,” IEEE Trans. on Image Processing, vol. 15, no. 12, pp. 3736–3745, 2006.

[5] M. Aharon, M. Elad, and A. M. Bruckstein, “The K-SVD: an algorithm for designing of overcomplete dictionaries for sparse representation,” IEEE Trans. on Signal Processing, vol. 54, no. 11, pp. 4311–4322, 2006.

[6] K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, “Image denoising by sparse 3D transform-domain collaborative filtering,” IEEE Trans. Image Processing, vol. 16, no. 8, pp. 2080–2095, 2007.

[7] A. Foi, “Noise estimation and removal in MR imaging: the variance-stabilization approach,” in Proc. of 2011 IEEE Int. Symposium on Bio. Imaging, Chicago, 2011, pp. 1809–1814.

[8] M. Mäkitalo and A. Foi, “Optimal inversion of the Anscombe transformation in low-count Poisson image denoising,” IEEE Trans. Image Processing, vol. 20, no. 1, pp. 99–109, 2011.

[9] H. Lu, X. Li, I. T. Hsiao, and Z. Liang, “Analytical noise treatment for low-dose CT projection data by penalized weighted least squares smoothing in the K-L domain,” Proc. SPIE, vol. 4682, pp. 146–152, May 2002.

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[12] P. J. Huber, Robust Statistics, New York: Wiley, 1981. [13] D. H. Trinh, M. Luong, J. M. Rocchisani, C. D. Pham, and

F. Dibos, “Medical image denoising using kernel ridge regression,” in Proc. of 2011 IEEE Int. Conf. on Image Processing, Brussels, 2011, pp. 1597–1600.

[14] D. D. Muresan and T. W. Parks, “Adaptive principal components and image denoising,” in Proc. of Int. Conf. on Image Processing, Barcelona, 2003, pp. 101–104.

Dinh-Hoan Trinh was born in Thanh-Hoa Province, Vietnam, in 1983. He received the B.S. degree from the Vietnam National University, Hanoi in 2005 and the M.S. degree from the Institute of Mathematics, Hanoi in 2008. He is currently pursuing the Ph.D. degree with the LAGA Laboratory, Université Paris 13, Sorbonne Paris Cité,

France. His research interests include optimization, statistical learning theory, and medical image processing.

Marie Luong received the Engineer Diploma from the Ecole Nationale Supérieure d’Electricité et de Mécanique de Nancy, France in 1991 and the Ph.D. degree from the L'Institut National Polytechnique de Lorraine (Lorraine University), France, in 1996. She is currently associate professor at the Université Paris 13, Sorbonne Paris Cité,

France. Her research interests include image restoration, super- resolution, segmentation, watermarking, and medical applications.

Jean-Marie Rocchisani was born in Oran, Algeria, in 1949. He received the M.S. degree in mathematics in 1976, a Doctorate in Medicine in 1980, and a Ph.D. in applied mathematics in 1985. He is currently working as a specialist with Avicenne University Hospital, Bobigny, France, and as an assistant professor in biophysics with

JOURNAL OF ELECTRONIC SCIENCE AND TECHNOLOGY, VOL. 10, NO. 2, JUNE 2012 6

Université Paris 13. His research interests are medical images processing.

Canh Duong Pham was born in Hanoi, Vietnam, in 1950. He received the M.S. degree from the State University of Byelorussia, USSR in 1973 and the Ph.D. degree in mathematics from the Computer Center of the Academy of Science USSR, Moscow, Russia in 1983. From 1973 to 1979, he worked at the Institute of

Mathematics, Hanoi, Vietnam. From 1980 to 1984, he was a researcher and postgraduate study at the Computer Center of the Academy of Sciences USSR, Moscow. From 1984 to 2009, he was a researcher with the Institute of Mathematics, Hanoi, Vietnam. Since 2009, he has been a specialist at the Center for information infrastructure development, Vietnam Academy of Science and Technology. His research interests include control theory, mathematical programming, and signal processing.

Huy Dien Pham was born in Hai-Phong, Vietnam, in 1952. He received the M.S. degree from the Moscow State University, in 1976, and the Ph.D. degree in mathematics from Hanoi Institute of Mathematics of the Vietnam Academy of Science & Technology in 1983. From 1976 to 2009 he worked at the Hanoi Institute of Mathematics, where

he was the Head of Department of Software Research and Development from 2005 to 2009. Since 2009 he has been the Deputy Director of the Center for Information Infrastructure Development, Vietnam Academy of Science and Technology. His research interests include optimization theory, numerical methods, and signal processing.

Françoise Dibos is currently full professor in applied mathematics with the Université Paris 13, Sorbonne Paris Cité, France. Her research interests include mathematical modelization for image processing, projective analysis and video analysis in real time, and medical imaging.