An effective example based denoising method for CT images using Markov random field

The 2014 International Conference on Advanced Technologies for Communications (ATC'14) An Effective Example-based Denoising Method for CT images using Markov Random Field Dinh-Hoan Trinh Thanh-Trung Nguyen Nguyen Linh-Trung University of Engineering and Technology University of ICT Center for Informatics and Computing Vietnam National University, Hanoi Thai Nguyen University Vietnam Academy of Science and Technology Hanoi, Vietnam Thai Nguyen, Vietnam Hanoi, Vietnam Email: linhtrung@vnu.edu.vn Email: nttrungktmt@ictu.edu.vn Email: tdhoan@cic.vast.vn Abstract—We propose in this paper a novel example-based method for Gaussian denoising of CT images In the proposed method, denoising is performed with the help of a set of example CT images We construct, from the example images, a database consisting of high and low-frequency patch pairs and then use the Markov random field to denoise The proposed denoising method can restore the high-frequency band that is often lost by the traditional noise-filters Moreover, it is very effective for images corrupted by heavy noise Experimental results also show that the proposed method outperforms other state-of-the-art denoising methods both in the objective and subjective evaluations I I NTRODUCTION Computed Tomography (CT) scanning is a medical imaging technique that uses X-rays to create cross-sectional images of the body CT imaging plays an important role in a variety of diagnostic and therapeutic purposes However, the quality of CT images is often affected by random noise, resulting in a reduction of the visibility of image features especially in low contrast regions Such effects can thereby compromise the accuracy and the reliability of pathological diagnosis or surgery purposes Denoising is thus one of the essential steps that helps to improve the image quality For CT imaging, the noise can be decreased by increasing the X-ray dose However, the disadvantage of increasing the radiation dose is that high X-Ray doses may be harmful to patients As shown in [1], low radiation imaging is often associated with a number of qualitydegrading artifacts, the most prominent of them being the noise Therefore, if the noise can be removed by a robust image denoising technique, lower radiation scans become possible and thus making less damage to the patient Basically, the objective of image denoising is to estimate the true image (noise-free image) from its noisy version Many effective denoising methods have been proposed, such as the sparse representation-based methods [2]–[4], the total variation-based methods [5], [6], the Non-local Means (NLM) methods [7], [8] and the Block Matching with 3D filtering (BM3D) [9], [10] The denoising methods derive from various disciplines such as linear and nonlinear filtering, spectral and multiresolution analysis, probability theory, statistics, partial differential equations These methods rely on some explicit or implicit assumptions about the true image in order to directly denoise the noisy image As shown in [11], although some methods, such as BM3D, are considered as the state-of-the-art denoising methods, applying such methods for denoising of medical images is not easy to obtain desired results In medical 978-1-4799-6956-2/14/$31.00 ©2014 IEEE 355 imaging, edges, textures and subtle details could very well reveal crucial information about the patients Regarding the specific nature of medical images, denoising is a difficult task, and the difficulty is almost to preserve subtle details Hence, denoising of medical images still requires specific treatment Among various directions explored in studying the denoising problem for medical imaging, learning-based denoising methods seems to be a promising direction Recently, Trinh et al in [11]–[13] have proposed several novel approaches wherein the denoising is performed indirectly through learning from a training set which is constructed from a given set of standard images, called example images These methods use the assumption that the example images are taken nearly the same location with the noisy image It is shown that with a good training set, these methods can denoise very effective However, it is clear that its effectiveness highly depends on the similarity between the noisy image and the example images Inspired from this problem, we propose in this work a novel method for Gaussian denoising in CT images where the noise is removed effectively while the dependency between the noisy image and the example images is significantly reduced It is known that the classical filters such as the Gaussian filter, the anisotropic diffusion filter [14] and the Wiener filter [15], can denoise nearly perfect in homogeneous regions, but the edges and textures are often smoothed The classical filters seem to protect only the low and middle frequency components while the high frequency component is lost, resulting in a blurred image From this important observation, it can be seen that the problem of image denoising can be approached by restoring the lost high-frequency component in the image denoised by the traditional denoising methods Following this idea, we propose to define an image that consists of three bands, namely low frequency, middle frequency and high frequency The high frequency component which is lost by the classical filters will be restored by learning from a given database of examples Specifically, the learning in the proposed method is performed using the Markov random field (MRF) in [16] Unlike in the previous works [11]–[13], the database in this work is a set of high and middle frequency patch pairs from the example images This makes it possible to reduce the dependency of the method on the similarity between the example images and the image to be denoised Experimental results show that the proposed method yields excellent denoising results Hereafter the proposed method is referred to as MRFD (Markov Random Field-based Denoising) The 2014 International Conference on Advanced Technologies for Communications (ATC'14) Fig Relationship between original image and low frequency band, middle frequency band, high frequency band of a poumon image The rest of this paper is organized as follows Section II describes the proposed method Our experiments and the results are reported in Section III Finally, the conclusion and future works are presented in Section IV II E XAMPLE - BASED D ENOISING M ETHOD USING MRF As shown in [1], in general noise in CT images can be approximated by a Gaussian distribution Thus, in this work we assume that CT images are corrupted by a white Gaussian noise and the degradation model can be described as follows: Y = X + η, (1) where X is the noise-free image that we want to estimate, Y is the observed noisy image and η ∼ N (0, σ ) is the white Gaussian noise with zero mean and variance of σ In this work, we define an image X to consist of three basis frequency bands, low-band X , mid-band Xm and high-band Xh , as: X = X + Xm + Xh (2) This is demonstrated in Fig An interesting fact that although the high-band is often lost, the classical denoising methods such as Gaussian and Wiener filters could well preserve the low- and mid-bands Therefore, if denoted by Y1 the denoised image by a classical filter on Y then we can consider that X =Y and Xm = Y m (3) ˆ h for Xh Thus, estimating X becomes to find an estimate X h ˆ In this work, we focus on estimating X from Ym with the help of a database of middle and high frequency patch pairs h (um k , uk ): h (Pm , Ph ) = (um k , uk ), k ∈ I , (4) ˆ h is obtained, the final here I is the index set When X denoising result will be ˆ = X + Xm + X ˆ h = Y + Ym + X ˆ h X (5) An overview of the proposed method is illustrated in Fig The proposed method is realized in two independent phases: • Database construction: Construct a database of the middle and high frequency patch pairs from a given set of example images • Denoising: Estimate the lost high-frequency band using MRF on the constructed database In the following, we will describe in more detail each phase 356 Fig Overview of the proposed denoising method A Database Construction Phase The database in (4) is constructed from a set of standard medical images denoted by {It , t ∈ Ω} which are considered as noise-free images Before generating the patch pairs, we h first decompose It into three basis bands (It , Im t , It ) using a low-pass filter F and a bandpass filter Fm , that is Im t = Fm (It ), It = F (It ) and and the high-frequency band Iht Iht (6) is then obtained by = It − It − Im t (7) Then, similarly to [16], we normalize the contrast of Im t and Iht by ˆIm = t Im t std(Im t )+ and ˆIht = Iht , std(Ih t)+ (8) where std(·) is standard deviation operator, and is a small value added to avoid the denominator to become zero at very low contrasts The database (Pm , Ph ) stores the vectorized h m h patch pairs (um k , uk ) in which uk and uk correspond to the h m ˆ ˆ patches at the same position in It and It , respectively B Denoising Phase The main aim of this phase is to estimate Xh of X from Y with the help of the example database (Pm , Ph ) Suppose that we are given the noisy image Y with the degradation model (1) Denoising is performed in two steps as follows: m 1) Pre-process: To improve the effectiveness of the proposed method, the noisy image Y is first pre-processed by the Wiener noise-filter Fwiener [15], that is Y1 = Fwiener (Y) (9) Then, we use exactly the low-filter and the bandpass filter in (6) to extract the low-band and mid-band of Y, as given by Y = F (Y1 ), Ym = Fm (Y1 ) (10) 2) Estimate high frequency band Xh : In this step, Xh is estimated by maximizing the prior probability P r(Xh |Ym ) We divide Ym into N overlap patches yim , i = 1, 2, , N , h with patch-size of that of um i in the database Estimating X is thus performed by estimating the set of high-frequency patches xhi corresponding to yim To this end, we use the Markov Network (MN) model proposed in [16] to determine the best high frequency patches that have the best compatibility with the adjacent patches Fig shows a part of the MN used in this work In this model, one node of the network is assigned to an image patch The 2014 International Conference on Advanced Technologies for Communications (ATC'14) (a) Chest (b) Neck (c) Thorax (d) Abdomen Fig A part of an MRF model for estimating the high-frequency band Xh Nodes yi are the observed mid-frequency patches The high-frequency patch at each node xi is the quantity we want to estimate Lines in the graph indicate statistical dependencies between nodes For this MN, the joint probability has a factorized form: P r(Xh |Ym ) = Z Ψ(xhi , xhj ) Φ(xhi , yim ), (11) i (i,j)∈E where Z is a normalization constant such that the probability sum to one, E is the set of edges in the MN denoted by the neighboring nodes xhi and xhj , Ψ and Φ are the potential functions In the proposed method, we determine N high-frequency patches {xhi }N i=1 as a subset of N high-frequency patches of the database Ph such that {xhi }N i=1 = arg max N {xh i }i=1 ⊂Ph Φ(xhi , yim ) i where and Φ(xhi , yim ) = Ψ(xhi , xhj ) = N {ˆ xhi }N i=1 = are defined as in [16]: m − xm i − yi 2β12 − Oij (xhi ) − Oji (xhj ) exp 2β22 exp yim − xm i arg N h {xh i }i=1 ⊂Ph ,xi ∈Ωi 2 i=1 Oij (xhi ) − Oji (xhj ) +λ (16) 2 j:(i,j)∈E,xh j ∈Ωj (i,j)∈E Ψ(xhi , xhj ) Original images for evaluating proposed method are determined by, Ψ(xhi , xhj ), (12) Φ(xhi , yim ) Fig The approximate solution of this problem is found by using the belief propagation algorithm [16] The estimated high ˆ hi is then applied to the inverse of the contrast frequency patch x normalization that we have used in the pre-processing step (13) 2 (14) h where (xm i ,xi ) is a patch pair in (Pm ,Ph ), β1 and β2 are positive parameters, Oij is an operator which extracts a vector consisting of the pixels of patch xhi in the overlap region between patches xhi and xhj It is easy to see that (12) can be rewriten as follows: {xhi }N i=1 = arg N {xh i }i=1 ⊂Ph yim − xm i 2 i Oij (xhi ) − Oji (xhj ) +λ 2 (15) , j:(i,j)∈E where (i, j) denotes an edge in set E of edges in the MN, λ is a positive parameter To solve this problem, we use the algorithm proposed by Freeman et al in [16] The algorithm has two steps as follows: Step 1: For each patch yim (i = 1, 2, , N ), its K nearest K m neighbors {um k }k=1 of yi is first searched from the data set Pm The set of K corresponding high frequency patch Ωi = {uhk }K k=1 in Ph is used as the set of candidates for estimating xhi at the hidden node of the MN h N Step 2: The estimates {ˆ xhi }N i=1 of desired patches {xi }i=1 357 Fig Some noise-free images used to construct the database III P ERFORMANCE E VALUATION In this section, we present several experimental results on CT images to show the performance of the proposed MRFD method The MRFD method is compared to three state-ofthe-art denoising methods, namely, Wiener filter (WN) [15], Non-local means (NLM) [7], and Total Generalize Variation The 2014 International Conference on Advanced Technologies for Communications (ATC'14) TABLE I CT Chest Neck Thorax Abdomen σ 10 20 30 10 20 30 10 20 30 10 20 30 SSIM COMPARISON ON CT SCANS SSIM WN 0.8758 0.7565 0.6364 0.9228 0.7722 0.6228 0.8792 0.7701 0.6703 0.8976 0.7561 0.6164 TGV 0.8617 0.8045 0.7360 0.8820 0.8550 0.7942 0.8663 0.8268 0.7721 0.8640 0.8181 0.7528 NLM 0.9128 0.8070 0.7094 0.9323 0.8537 0.7688 0.9200 0.8347 0.7449 0.9167 0.8260 0.7342 MRFD 0.9226 0.8630 0.7865 0.9378 0.8711 0.8102 0.9223 0.8708 0.7892 0.9371 0.8724 0.7833 (TGV) [6] We use the image quality metric namely Structural SIMilarity (SSIM) index [17] for objective evaluation We report here the experimental results on four test CT images in Fig with three noise levels σ = 10, 20 and 30 For the proposed MRFD method, the database (Pm , Ph ) is constructed from 20 example images (five of them are shown in Fig 5) We use the Wiener filter Fwiener in (9) (wiener2 function in Matlab) with neighborhoods of size × for the pre-process step, the Gaussian filter is used to extract the middle and low frequency bands In all the experiments, we use the patch size of 11 × 11, λ in (16) is set to 0.5, and the parameter K in Step is set to 30 For subjective comparison, we show in Fig the experimental results on the CT image of the chest with noise level of σ = 20 Visually, the result obtained by MRFD in Fig 6(f) shows that the noise was effectively removed while maintaining small details and image structure (see in the enlarged rectangle region) Moreover, Table I shows the objective evaluation using SSIM Clearly, the SSIM of our method (MRFD) is the highest, especially in high level noise cases This confirms that MRFD outperforms the other methods in preserving image structure As it can be seen, the result obtained by MRFD is much better than the other results IV C ONCLUSION R EFERENCES [4] [5] [6] [7] [9] [10] [11] [12] H Lu, I.-T Hsiao, X Li, and Z 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Trinh, M Luong, F Dibos, J.-M Rocchisani, C D Pham, and T Q Nguyen, “Novel example- based method for super-resolution and denoising of medical images, ” IEEE Transactions on Image Processing, vol 23,... in Fig An interesting fact that although the high-band is often lost, the classical denoising methods such as Gaussian and Wiener filters could well preserve the low- and mid-bands Therefore,... comparison on the CT image of chest with noise level σ = 20 [8] In this paper, an effective example- based method using MRF has been proposed The proposed method uses a database of example patch pairs