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

An Effective Example-based Denoising Method forCT images using Markov Random Field Dinh-Hoan Trinh Center for Informatics and Computing Vietnam Academy of Science and Technology Hanoi, V

An Effective Example-based Denoising Method for

CT images using Markov Random Field

Dinh-Hoan Trinh

Center for Informatics and Computing

Vietnam Academy of Science and Technology

Hanoi, Vietnam Email: tdhoan@cic.vast.vn

Thanh-Trung Nguyen University of ICT Thai Nguyen University Thai Nguyen, Vietnam Email: nttrungktmt@ictu.edu.vn

Nguyen Linh-Trung University of Engineering and Technology Vietnam National University, Hanoi

Hanoi, Vietnam Email: linhtrung@vnu.edu.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 INTRODUCTION

Computed Tomography (CT) scanning is a medical

imag-ing 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

quality-degrading 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

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 denois-ing problem for medical imagdenois-ing, learndenois-ing-based denoisdenois-ing 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 fre-quency and high frefre-quency The high frefre-quency 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 Experi-mental results show that the proposed method yields excellent denoising results Hereafter the proposed method is referred to

as MRFD (Markov Random Field-based Denoising)

Fig 1 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 EXAMPLE-BASEDDENOISINGMETHOD USINGMRF

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:

where X is the noise-free image that we want to estimate, Y

is the observed noisy image and η ∼ N (0, σ2) is the white

Gaussian noise with zero mean and variance of σ2

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:

This is demonstrated in Fig 1 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 Y1the denoised

image by a classical filter on Y then we can consider that

Thus, estimating X becomes to find an estimate ˆXh for Xh

In this work, we focus on estimating ˆXh from Ym with the

help of a database of middle and high frequency patch pairs

(um

k, uh

k):

(Pm, Ph) =(um

k, uhk), k ∈ I , (4) here I is the index set When ˆXh is obtained, the final

denoising result will be

ˆ

X = X`+ Xm+ ˆXh= Y`+ Ym+ ˆXh (5)

An overview of the proposed method is illustrated in Fig 2

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

us-ing MRF on the constructed database

In the following, we will describe in more detail each phase

Fig 2 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 first decompose Itinto three basis bands (I`, Imt , Iht) using a low-pass filter F` and a bandpass filter Fm, that is

I`t= F`(It) and Imt = Fm(It), (6) and the high-frequency band Iht is then obtained by

Iht = It− I`t− Imt (7) Then, similarly to [16], we normalize the contrast of Im

t and

Iht by

ˆm

m t

std(Im

t ) +  and

ˆh

h t

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 patch pairs (umk , uh

k) in which um

k and uhk correspond to the patches at the same position in ˆImt and ˆIht, respectively

B Denoising Phase The main aim of this phase is to estimate Xh of X from

Ymwith 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: 1) Pre-process: To improve the effectiveness of the pro-posed method, the noisy image Y is first pre-processed by the Wiener noise-filter Fwiener [15], that is

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 Yminto N overlap patches ymi , i = 1, 2, , N , with patch-size of that of um

i in the database Estimating Xhis thus performed by estimating the set of high-frequency patches

xh

i corresponding to ym

i 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 3 shows a part of the MN used in this work In this model, one node of the network is assigned to an image patch

Fig 3 A part of an MRF model for estimating the high-frequency band

Xh Nodes y i are the observed mid-frequency patches The high-frequency

patch at each node x i 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) = 1

Z Y

(i,j)∈E

Ψ(xhi, xhj)Y

i

Φ(xhi, ymi ), (11)

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 xh

i and xh

j, Ψ and Φ are the potential functions

In the proposed method, we determine N high-frequency

patches {xh

i}N

i=1 as a subset of N high-frequency patches of

the database Ph such that

{xhi}Ni=1= arg max

{x h

i } N i=1 ⊂P h Y

i

Φ(xhi, ymi ) Y

(i,j)∈E

Ψ(xhi, xhj),

(12) where Φ(xhi, yim) and Ψ(xhi, xhj) are defined as in [16]:

Φ(xhi, ymi ) = exp−kxm

i − ym

i k2

Ψ(xhi, xhj) = exp−kOij(xh

i) − Oji(xh

j)k2

where (xmi ,xhi) 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 xh

i in the overlap region between patches xhi and xhj It is easy to see that (12) can

be rewriten as follows:

{xh

i}N

i=1= arg min

{x h

i } N i=1 ⊂Ph

X

i



kym

i − xm

i k2

j:(i,j)∈E

kOij(xhi) − Oji(xhj)k22, (15)

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 ym

i (i = 1, 2, , N ), its K nearest neighbors {umk}K

k=1of yimis first searched from the data set

Pm The set of K corresponding high frequency patch Ωi =

{uh

k}K

k=1in Ph is used as the set of candidates for estimating

xh

i at the hidden node of the MN

Step 2: The estimates {ˆxhi}N

i=1 of desired patches {xhi}N

i=1

Fig 4 Original images for evaluating proposed method.

are determined by, {ˆxhi}N

i=1= arg min

{x h

i } N i=1 ⊂P h ,x h

i ∈Ω i

N

X

i=1



kym

i − xm

i k2 2

j:(i,j)∈E,x h

j ∈Ωj

kOij(xhi) − Oji(xhj)k22

(16)

The approximate solution of this problem is found by using the belief propagation algorithm [16] The estimated high frequency patch ˆxh

i is then applied to the inverse of the contrast normalization that we have used in the pre-processing step

Fig 5 Some noise-free images used to construct the database.

III PERFORMANCEEVALUATION

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-of-the-art denoising methods, namely, Wiener filter (WN) [15], Non-local means (NLM) [7], and Total Generalize Variation

TABLE I SSIM COMPARISON ON CT SCANS

Chest

10 0.8758 0.8617 0.9128 0.9226

20 0.7565 0.8045 0.8070 0.8630

30 0.6364 0.7360 0.7094 0.7865

Neck

10 0.9228 0.8820 0.9323 0.9378

20 0.7722 0.8550 0.8537 0.8711

30 0.6228 0.7942 0.7688 0.8102

Thorax

10 0.8792 0.8663 0.9200 0.9223

20 0.7701 0.8268 0.8347 0.8708

30 0.6703 0.7721 0.7449 0.7892

Abdomen

10 0.8976 0.8640 0.9167 0.9371

20 0.7561 0.8181 0.8260 0.8724

30 0.6164 0.7528 0.7342 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 4 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 3 × 3 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 1 is set to 30

For subjective comparison, we show in Fig 6 the

ex-perimental 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

meth-ods in preserving image structure As it can be seen, the result

obtained by MRFD is much better than the other results

IV CONCLUSION

In this paper, an effective example-based method using

MRF has been proposed The proposed method uses a database

of example patch pairs to restore the high frequency band

which is lost by the common filters The experimental results

on the CT images are very promising, demonstrating the

ability of the method for a potential improvement of diagnosis

accuracy In the future works, we are going to study solutions

for optimizing the database as well as for improving the

computing speed of the proposed algorithm

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