Magnetic Resonance Imaging 31 (2013) 1599–1610

Contents lists available at ScienceDirect

Magnetic Resonance Imaging

j ourna l homepage: www.mr i journa l .com

Laplacian based non-local means denoising of MR images with Rician noise

Hemalata V. Bhujle ⁎, Subhasis ChaudhuriDepartment of Electrical Engineering, Indian Institute of Technology, Bombay, Mumbai 400076, Maharashtra, India

⁎ Corresponding author. Tel.: +91 22 2576 4439; fax:E-mail addresses: hemalata@ee.iitb.ac.in, hemalatha@

0730-725X/$ – see front matter © 2013 Elsevier Inc. Alhttp://dx.doi.org/10.1016/j.mri.2013.07.001

a b s t r a c t

a r t i c l e i n f o

Article history:Received 16 January 2013Revised 26 May 2013Accepted 2 July 2013

Keywords:Magnetic Resonance ImagingLaplacian of GaussianNonlocal-meansRician noise

Magnetic Resonance (MR) image is often corrupted with a complex white Gaussian noise (Rician noise)which is signal dependent. Considering the special characteristics of Rician noise, we carry out nonlocalmeans denoising on squared magnitude images and compensate the introduced bias. In this paper, wepropose an algorithm which not only preserves the edges and fine structures but also performs efficientdenoising. For this purpose we have used a Laplacian of Gaussian (LoG) filter in conjunctionwith a nonlocalmeans filter (NLM). Further, to enhance the edges and to accelerate the filtering process, only a few similarpatches have been preselected on the basis of closeness in edge and inverted mean values. Experimentshave been conducted on both simulated and clinical data sets. The qualitative and quantitative measuresdemonstrate the efficacy of the proposed method.

+91 22 2572 3707.bvb.edu (H.V. Bhujle).

l rights reserved.

© 2013 Elsevier Inc. All rights reserved.

1. Introduction

Magnetic Resonance (MR) images are affected by different kindsof noise during the acquisition process. Frequency coils as well aspreamplifiers which are part of the acquisition system tend tointroduce thermal noise in MR imaging. MR images are sampled inthe frequency domain and extracted from both real and imaginarychannels. Signals in these channels are corrupted with a complexwhite Gaussian noise and hence, in general, noise in the MR image ismodeled as Rician distributed. As Rician noise is signal dependent,diagnoses from MR images are adversely affected by such kind ofnoise. It affects visual inspection and image analysis procedures suchas segmentation and registration.

Noise removal can be dealt with in two different ways. Multipleimages of the same data can be acquired and averaging on these datacan bedone. But this procedure is quite slowand some time introducesmotion artifacts. Further, this doesnot help in reducingbias in thedata,if there is any. In the second method, after image acquisition somesuitable image denoising techniques are applied which providereliable and fast results. Many denoisingmethods have been proposedin the literature. A few of them include Bayesian approaches [1],anisotropic diffusion filter [2], total variation minimization [3],adaptive smoothing [4] and wavelet thresholding [5,6].

In natural and medical images many repeated patterns exist atdifferent locations of the same image. These redundancies in the imagehave been exploited for the first time for better denoising by Buades etal. using a nonlocal means filter (NLM) [7]. Originally, NLM filter wasdesigned for Gaussian distributed noise, but this filter has been

modified appropriately to adapt for Rician distributed noise for MRdenoising in [8]. Though this filter provides promising results,oversmoothing is observed in some regions which results in a loss ofedges and fine structures in the image. In this paper, we havedeveloped a filter to denoise Rician noise perturbed magnitude MRimages. In the proposed method, NLM denoising is carried out on thesquared magnitude of the image. Bias which is introduced by Riciannoise is compensated further. The proposed filter tends to preservefine details and edges in the MR image because of the way theweighing term of NLM has been modified. The weighing term ismodified such that it is a combined function of both edge and intensitysimilarities. In the proposedmethod candidate neighborhoods are alsocompared for edge similarity, which otherwise would have comparedonly for the intensity similarity as in [7]. However, it is required toreduce the effect of noise before extracting the edges. The Laplacian ofGaussian (LoG) filter combines Gaussian filtering with a Laplacianoperator and hence we have used an LoG filter to extract the edges.Further, to speedup the computational process, strategies similar to [9]but with proper modifications are applied. All our experiments havebeen validated both quantitatively and qualitatively. We have usedpeak signal to noise ratio (PSNR), Bhattacharya coefficient (BC) anduniversal quality index (UQI) as quantitative measures. Though PSNRis a widely used quality measure, as far as human visual system (HVS)is concerned, it is not so reliable as argued in [10,11] and hence wehave additionally taken the structural similarity index metric (SSIM)for an objective evaluationwhich is believed to be a better indicator ofthe perceived image quality.

The paper is organized in the following manner. In section 2, wediscuss the relevant literature. In section 3, we study the character-istics of MR images. Section 4 starts with the introduction of NLMfilter followed by the methodology adopted for the proposed

1600 H.V. Bhujle, S. Chaudhuri / Magnetic Resonance Imaging 31 (2013) 1599–1610

method. In section 5, we discuss the validation strategies followed inour experiments. In the same section we discuss the results andcompare them with other state-of-the-art denoising methods.Section 6 concludes the paper.

2. Literature survey

The conventional filters, like Wiener filter [12], anisotropicdiffusion [13,14] and trilateral filter [15] have been extensivelyused in MR image denoising. But most of these filters fail to reduceRician noise in MR images as these are designed for the removal ofGaussian noise. Denoising Rician noise perturbed MR data using amaximum likelihood approach is addressed in [16,17]. Waveletshave also been extensively applied for denoising MR data [18–21], inwhich a few methods model noise as Gaussian distributed and quitea few as Rician distributed. Wavelet soft thresholding has beenapplied to MR data in [18]. But small details of the signal are lost inthis technique. Rician noise removal forMRI usingwavelet transformwas first proposed by Nowak [19]. This technique has been furtherextended using wavelet packets for denoising low SNRMRI byWoodet al. [22]. In [23], the authors have employed undecimated wavelettransform to provide effective representation of noisy coefficients.Further bilateral filtering has been performed on the approximatecoefficients to improve the filter performance. Awate et al. [24] intheir approach have used a Bayesian estimator to denoise MRimages. In [25], the authors have proposed a speckle reducing,anisotropic diffusion based filtering of MRI images that is moresuited to deal with Rician distributed noise. The filtered image is biascompensated for an improved performance. Authors in [26] haveproposed automatic parameter selection strategy for anisotropic

ig. 1. Illustration of residual images. (Top to bottom and left to right): Rician noise added (σ = 9%) MR image, UNLM denoised image, residual image of UNLM, denoised imagesing the proposed method and the corresponding residual image.

Fu

diffusion filtering of MR images. In [27], the authors have proposeddenoising of diffusion tensor images by adapting second order semi-implicit Craig–Sneyd scheme in anisotropic filtering. Nonlocal meansfilter is applied for Gaussian distributed MR images by Coupe` et al.[28,29]. They tried to speed up the nonlocal means filter bypreselecting a few similar patches based on mean and variancemeasures, as well as parallelizing the computation using 8 CPU's.

The application of nonlocal means filter for Rician noiseperturbed MR image was initially proposed by Manjo`n et al. [8]. Inthis approach authors have compensated the bias introduced byRician noise (UNLM). The method proposed in [30] does model thenoise distribution as a Rician process and offers a maximumlikelihood estimation of pixels to be denoised. The method offersimproved results over standard NLM method based solution. Themethod proposed in [31] is quite similar to the previous method. Butthese methods suffer from a common drawback of not preservingfine structures and details in an image. Effect of oversmoothing isobserved for some of these filters, which results in a substantialdegradation in performance at a higher noise level. We attempt tosolve this problem by explicitly considering the patch similarity inedge map also for a better edge preserving denoising.

Medical images usually comprise of more features and finestructures compared to other type of images. These features shouldbe better preserved as these provide important information tophysicians for a better diagnosis. Preserving edges and structures forimages corrupted with Rician noise becomes more difficult as Riciannoise is signal dependent. In the original NLM filter, weights formatching a candidate patch are assigned solely on the basis ofintensity similarity between corresponding pixels. If we consider twopatches which have a similarity only in terms of the edge pattern, i.e.,

Fig. 2. Block diagram of the proposed method. Here ES and IM are the thresholdselection functions for edge and intensity patches as given in Eq. (17). α1, α2, η1 andη2 are the upper and lower limits of thresholds for edge and inverted mean similarity

Fig. 3. Simulated brain images used in experimentat

1601H.V. Bhujle, S. Chaudhuri / Magnetic Resonance Imaging 31 (2013) 1599–1610

.

at other locations in these patches there exist dissimilarities withrespect to pixel gray values, such patches suffer while computingdistances as the distance is computed based on the intensitysimilarity between corresponding pixels at all locations in the twopatches. These patches are given a less weight compared to others asweight is not a function of edge similarity. We improve this weighingterm in such a way that both intensity and edge similarities are given

ion (l

an equal importance. We have appropriately modified the weighingterm by adding an edge preserving term to serve this purpose.

3. The distribution of noisy MRI data

An image is usually assumed to be corrupted with an additivewhite Gaussian noise which can be effectively removed by an NLMmethod. But noise in MR image is signal dependent, unlike additivenoise, and is usually modeled by a Rician distribution. Detectabilityof features inMR images gets drastically reduced due to bias which isintroduced by a Rician distribution.

MR images are reconstructedby computing inversediscrete FourierTransform (IDFT) of the measured signal components from real andimaginary channels. The raw MR data are corrupted by a complexGaussian noise and during the process of reconstruction by IDFT, thenoise characteristics do not change due to linearity and orthogonalityproperties of Fourier Transform (FT). Thus noise in the reconstructeddata is still a complex white Gaussian noise. For any kind of imageanalysis and visual inspection this datum is taken as a magnitudeimage, when the distribution of noise changes to a Rician one.

The complex MR datum X is given by

X ¼ XRe þ jXIm : ð1Þ

where XRe and XIm are real and imaginary components of the data.These components are independently affected by ξ1 and ξ2, where ξ1and ξ2 are AWGN with zero mean and standard deviation σ.

XRe ¼ Scosθþ ξ1 ð2Þ

XIm ¼ Ssinθþξ2 ð3Þ

Here S is the original MR image and θ is the phase. A noisy MRimage can be represented as the magnitude of the noisy raw data,

Xj j ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiScosθþ ξ1Þ2 þ Ssinθþ ξ2Þ2

��rð4Þ

It can be shown that the distribution of |X| becomes Rician [32,33]and is represented as

P X S ;σÞ ¼ X

σ2 e−X

2þS2

2σ2 I 0XS

σ2

� ������

ð5Þ

eft to right): T1-w image and T2-w MS lesions image.

Fig. 4. Comparison of denoising results on an enlarged portion of the simulated T1-w image (left to right, top to bottom): Images of original, Rician noise added (17%), results ofNLM, FUNLM, UNLM and the proposed method.

1602 H.V. Bhujle, S. Chaudhuri / Magnetic Resonance Imaging 31 (2013) 1599–1610

Here I0 denotes the modified Bessel function of the first kind withorder zero, S is the noiseless signal, σ2 is the noise variance and X isthe observed MR magnitude image. It may be mentioned that MRimages collected with multi-channel coils do not suffer from Riciandistributed noise [34]. Hence, in this paper we restrict our studies toonly those MR images, such as single coil data, for which the noisedistribution is known to be Rician.

Estimation of noise in a magnitude image |X| is a difficult task.Nowak [19] has shown that by squaring the magnitude, bias in theMR images can be made additive and signal independent. Taking theexpectation of |X|2 as given in Eq. (4) yields,

E Xj j2h i

¼ E Scosθþ ξ1ð Þ2 þ Ssinθþ ξ2ð Þ2h i

¼ μ2S þ 2σ2

:ð6Þ

Hence bias in the squared magnitude domain is obtained as 2σ2.It may be argued here that squaring the data may introduceadditional zero crossings in the image, thus producing spuriousedges. However the image being a non-negative function, themonotonicity property is preserved and no new edges are formed.Let the squared magnitude image be denoted by I = |X|2. Thus theproblemwe attempt to solve in this paper is as follows. Given a noisyMR image |X|, obtain an estimate of the original magnitude image S.

The noise in the MRI is estimated from the background regions.The estimate of noise in the MRI is given by σ ¼

ffiffiμ2

q, where μ is

the mean value of the pixels from the selected background regionof the squared magnitude image I. The background is selectedusing the Ostu [35] thresholding technique. Here we assume thatthe image pixels are uncorrelated in the background region as theyare mostly due to noise. If there is a correlation, one can use thestatistical method proposed in [36,37]. Some other techniques ofestimating the noise variance σ2 for Rician perturbed MR imageinclude estimators based on local variance and skewness [38] andmaximum absolute difference (MAD) in wavelet-domain [39].We use the estimate σ ¼

ffiffiμ2

qdue to its simplicity and the fact

that experimentally it was found to offer a good estimate of thenoise variance.

4. Methodology

4.1. Laplacian based enhanced nonlocal means filter

4.1.1. Nonlocal means filterA nonlocalmeans filter replaces each pixel in the noisy image by a

weighted average of all other pixels in the image. This can berepresented as

1603H.V. Bhujle, S. Chaudhuri / Magnetic Resonance Imaging 31 (2013) 1599–1610

NLM I ið Þð Þ ¼ ∑j∈Niw Ni;Nj

� �I jð Þ ð7Þ

with 0≤w Ni ;Nj

� ≤1;∑j∈Ni

w Ni;Nj

� ¼ 1, where I (i) is theparticular pixel being filtered, Ni is the neighborhood of ith pixelin the image and w (Ni, Nj) is the weight with which each noisypixel is multiplied with. This weight function depends on thesimilarity between neighborhoods (Ni and Nj) of pixels i and j. Toreplace each noisy pixel I (i), the neighborhood Ni of pixel I (i) iscompared with neighborhood Nj of each pixel I (j) in the wholeimage. Although according to the NLM principle, the search forsimilar patches should be done for the entire image, for computa-tional efficiency it is restricted to a user defined search windowwhich is usually much smaller than the size of the image. In theoriginal NLM paper for image denoising [7], the search window sizeis limited to 21×21. However in all NLM based MR image denoisingmethods mentioned in [8,28,29,40], the search window is chosen as11×11 and the patch size as 5×5 pixels. We also use the same set ofparameters in all our experiments. Pixels which are near to thecenter of the neighborhood will contribute more in distancecalculation than pixels that are far and hence this distance functionis weighted by a Gaussian kernel of standard deviation a. This canbe represented as

Fig. 5. Comparison of denoising results on an enlarged portion of the simulated T2-w MS leresults of NLM, FUNLM, UNLM and the proposed method.

d ¼ ‖I Nið Þ−I Nj

� �‖22;a: ð8Þ

The term ‖ ⋅ ‖2,a2 represents the Gaussian weighted distancefunction and a is the standard deviation of the Gaussian kernel.Higher values of d imply a lesser similarity between neighborhoodsand the corresponding center pixels are weighted less. The overallweight for the jth patch during the NLM averaging process is given by

w Ni ;Nj

� �¼ 1

z ið Þ e−d

h2ð9Þ

where h is the filtering parameter and Z is a normalization constantgiven by

Z ið Þ ¼ ∑j e−d jð Þh2 : ð10Þ

4.1.2. Modified weighing termAll nonlocal means based MR image denoising methods men-

tioned in literature [28,29,40] use the same strategy to computedistance between two patches as explained above. However thesefilters tend to oversmooth an image and hence small details and

sions image (left to right, top to bottom): Images of original, Rician noise added (9%)

,

Fig. 6. Comparative plots of PSNR (dB) values obtained for the simulated (A) T1-w and(B) T2-w (MS) lesions brain images with varying levels of noise perturbation.

Fig. 7. Comparative plots of SSIM (%) values obtained for the simulated (A) T1-w and(B) T2-w (MS) lesions brain images with varying levels of noise perturbation.

1604 H.V. Bhujle, S. Chaudhuri / Magnetic Resonance Imaging 31 (2013) 1599–1610

edges are lost. This problem ismore severe in case of Rician distributedMRI as Rician noise is signal dependent. Performance of any denoisingalgorithm can be tested using the principle of residual noise. Residualnoise can be defined as the difference between noisy and denoisedimages. For an ideal denoising method, the residual noise should notcontain any structure left from the original image. This is illustrated inFig. 1. Fig. 1 shows the outputwhen UNLMmethod [8] is applied on anMR image corrupted with a Rician noise. We observe that quite a fewstructures and details still remain in the residual image, instead ofbeing a white noise sequence. One can clearly see a pattern which hasa similarity with the original image in the residual image, signifyingthat one can possibly do a better denoising of the given image. Topreserve these structures and edges more efficiently, we modify theweighing term of NLM such that two neighborhoods with similarintensities along with similar edge patterns will get more weight. Theblock diagram of the proposed method is shown in Fig. 2. We applyLoG filter on the squaredmagnitude image to extract the edges as thisfilter is less sensitive to noise due to the presence of Gaussian low passfiltering of the signal before the spatial derivatives are computed. TheLoG kernel with a spread parameter σw has the form

LoG x; yð Þ ¼ − 1πσ4

w

1− x2 þ y

2

2σ2w

" #e− x

2 þ y2

� �2σ2

w

: ð11Þ

In this study we have used σw = 1 empirically. Edges for theinput image are extracted only once at the outset and search forsimilar edge patterns is carried out over an 11×11 search window.Patch similarity between edge maps is computed as

D1 ¼ ‖LoG Nið Þ−LoG Nj

� �‖2;b: ð12Þ

LoG (Ni) and LoG (Nj) are the extracted edge maps for theneighborhoods I (Ni) and I (Nj), respectively. The value of D1 iscomputed as sum of the element wise absolute differences betweentwo edge maps. To provide more weights to the center pixels of thepatches, this function is weighted by a Gaussian kernel with astandard deviation b, as in the case of standard NLM. This distance isfurther combined with the distance computed for the intensitysimilarity to form a newweighing term. The computation of distancebased on intensity similarity is the same as that of the original NLMmethod as given in Eq. (8)

D2 ¼ ‖I Nið Þ−I Nj

� �‖22;a : ð13Þ

1605H.V. Bhujle, S. Chaudhuri / Magnetic Resonance Imaging 31 (2013) 1599–1610

The new weighing term is formed after combining these twodistances as below.

wm i; jð Þ ¼ 1z ið Þ e

− D2h2

� �− D1

h′2

� �ð14Þ

where h′ is the filtering parameter for the edge term and Z is theoverall normalization term given by

Z ið Þ ¼ ∑j e− D2

h2

� �− D1

h′2

� �ð15Þ

Since the proposed method requires matching of both intensityand edge patches, computationally it requires double the time forsearching. Hence a suitable speed up strategy is required. Also fromEq. (14), we observe that the filter weights are related to twodifferent quantities as is usually done in bilateral filtering [41].However, it is not similar to a bilateral filter as the weighting term isnot related to range. Rather it depends on the derivative. It may alsobe noted that in homogeneous regions, the edge map does not playmuch role and the intensity based patch matching prevails.

Fig. 8. Comparative plots of SSIM (%) values obtained for the two speed-up strategiesfor simulated (A) T1-w and (B) T2-w (MS) lesions brain images with varying levels ofnoise perturbation.

Fig. 9. Comparative plots of (A) BC and (B) UQI values obtained for the simulatedT1-w brain image.

Now we revisit the motivation of this work as illustrated inFig. 1. We also provide the result of denoising using the proposedmethod, further details of which will follow in subsequentsubsections, for the input image in Fig. 1. When we look at theresidual image, we hardly see any structural features, suggestingthat the proposed method has been able to extract informationfrom the image better.

4.1.3. Speed-up strategySeveral techniques have been proposed in the literature to

speed up the computation process of NLM. Some methodseliminate unrelated neighborhoods from the search window bysetting appropriate thresholds [9,28], whereas quite a few [42,43]use fast computation techniques such as Fast Fourier Transform(FFT) and parallelized computation for speed up. We have used thestrategy similar to [9] but with a different threshold selectioncriterion. To enhance the edge, only a few neighborhoods havingsimilar edge patterns are considered for the distance calculation.Hence the first value of threshold for the proposed method is basedon edge similarity which is selected such that most neighborhoodswith very different edge patterns are easily filtered out. In [9] theauthors have used the local mean as the threshold. We use inverted

Fig. 10. Comparative plots of (A) BC and (B) UQI values obtained for the simulated T2-w (MS) lesions brain image.

1606 H.V. Bhujle, S. Chaudhuri / Magnetic Resonance Imaging 31 (2013) 1599–1610

mean instead of the local mean as suggested in [44], the reasonbeing that the local mean is more intensity sensitive and hencehigh and low intensity pixels are treated differently whereas theinverted mean reduces such a difference. The inverted mean iscomputed as

Fig. 11. Comparison of denoising results on a clinical T1-w head image (left to right): L

Inv I Ni

� � ¼ max Ið Þ− I Ni

� � ; ð16Þ

where I Ni

� � is the local mean. Combining all, the modified

weighing term for the proposed NLML (with L for Laplacian) filter isgiven as

wm i; jð Þ ¼ 1z ið Þ e

−D2

h2

� �−

D1

h′2

� �

¼0

if α1⟨∇2

I Nið Þ��� ���∇2I Nj

� ���������⟨α2

and η1⟨Inv I Nið Þð ÞInv I Njð Þð Þ ⟨η2

otherwise

ð17Þ

where α1, α2, η1 and η2 are upper and lower limits of thresholds foredge similarity and invertedmean gray value similarity, respectively.The denoising process is now carried out with themodifiedweighingterm as given by

NLML I ið Þð Þ ¼ ∑j∈Ωwm i; jð ÞI jð Þ: ð18Þ

4.1.4. Bias correctionNLM method which is originally designed for the removal of

white noise yields poor results if applied without any modificationfor Rician distributed MR data due to the bias term. According toEq. (6)

E I½ � ¼ μ2S þ 2σ2 ð19Þ

where σ2 is the Gaussian noise variance of complex MRI, μS2 is theweighted average of squared magnitude of original image S. In theproposed method we adopt the bias removal strategy where theeffect of bias is compensated by subtracting the value of bias fromthe denoised result as suggested by Nowak [19]. The term σ isestimated from the background region as explained in Section 3.Hence the final denoised magnitude MR image S is obtained bycomputing the square root of the resultant. We call the proposedmethod as UNLML (unbiased NLML). The denoised image S isobtained as

UNLML S ið Þð Þ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiNLML I ið Þð Þ−2σ2

qif NLML I ið Þð Þ≥2σ2

¼ 0 else:

ð20Þ

ow SNR input MR image, results of NLM, FUNLM, UNLM and the proposed method.

1607H.V. Bhujle, S. Chaudhuri / Magnetic Resonance Imaging 31 (2013) 1599–1610

As mentioned earlier, since an image is a non-negative functioneverywhere, squaring of the observation and subsequent processing,including bias correction, do not introduce any spurious edges.

4.2. Overall description of the algorithm

a) Compute the square of noisy image to obtain its squaredmagnitude I.

b) Compute the noise variance σ2.c) Apply LoG filter and extract the edges of the input image I.

Compute the distance between patches based on edge similarity(Section 4.1.2).

d) Modify the weighing term of NLM such that it is a combinedfunction of both intensity and edge similarities (Section 4.1.2).

e) To speed up the computation, remove unrelated patches byselecting appropriate threshold values based on average invertedmean gray and edge similarity using Eq. (17).

f) Compensate bias by subtracting the value 2σ2 from thedenoised result. Square root of the resultant gives the denoisedmagnitude MR image S (Section 4.1.4).

5. Experimental validation and results

In this section we investigate the performance of the proposedfilter. Optimal parameter selection for the filter is a crucial step forany denoising algorithm and hence impact of each parameter isstudied thoroughly. Finally our method is compared with recentlydeveloped Rician based NLM algorithms for MR images. In order tovalidate the performance of the proposed method, we need tosimulate Rician noise corrupted MR images. We consider a good

Fig. 12. Comparison of denoising results on an enlarged portion of the clinical T1-w head iand the proposed method.

quality (high SNR) MR image as the original image S and use Eqs. (2)and (3) with θ = 0° to generate the noisy MR data |X| as given inEq.(4).

Several levels of noise have been added: 5%, 9%, 13%, 17% and 21%.Here z% of noise implies z

100 t, where t is the value of the brightestpixel in the image.

5.1. Validation strategy

The efficacy of the proposed method is verified both qualitativelyand quantitatively. As quantitative measures we have used peaksignal to noise ratio (PSNR), Bhattacharya coefficient (BC) [45] anduniversal quality index metrics (UQI) [46]. In UQI, distortion in anyimage is modeled as a combination of loss of correlation, luminancedistortion and contrast distortion and hence is claimed to be betterthan other widely used distortionmetrics. Structural similarity indexmeasure (SSIM) and visual comparisons are used as qualitativemeasures. Though PSNR is a commonly used objective metric, thisdoes not necessarilymatchwell with respect to the perceived quality[47]. However, it does provide a quick understanding of departurefrom the true value from a signal processing perspective. SSIMwithin a local window is calculated as

SSIM x; yð Þ ¼2μxμy þ C1

� �2σxy þ C2

� �μ2x þ μ2

y þ C1Þ σ2x þ σ2

y þ C2

� �� ð21Þ

where x and y are the image patches extracted from the localwindows of the original and noisy images, respectively. μx, σx

2 andσxy are the mean, variance and cross correlation computed within

mage (left to right, top to bottom): Original MR image, results of NLM, FUNLM, UNLM

Fig. 13. Comparison of denoising results on an enlarged portion of the clinical T1-w knee image (left to right, top to bottom): Original MR image, results of NLM, FUNLM, UNLMand the proposed method.

1608 H.V. Bhujle, S. Chaudhuri / Magnetic Resonance Imaging 31 (2013) 1599–1610

the local window, respectively. C1 and C2 are numerical constants asgiven in [11]. Finally local SSIMs are averaged to compute SSIM forthe whole image.

Bhattacharya coefficient (BC) is a statistical measure for objectivequality assessment. BCmeasures thedistancebetween twohistograms

BC k; lð Þ ¼ ∑255b¼0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik bð Þl bð Þ

p: ð22Þ

where k and l are the two histograms and b is the bin index.Histograms of the denoised images are compared with those ofground truth images. The value of BC close to 1 indicates k and l beingvery similar.

5.2. Performance comparison

The proposed filter is compared with the following methods bothqualitatively and quantitatively.

a] Original NLMmethod for Rician distributed MR image [7] withthe filtering parameter h = 0.87 σ.

b] Unbiased NLM method with h = 1.2 σ as suggested in [8] toachieve the best accuracy.

c] Coupe` method for Rician noise perturbed MR image [28] withh = 0.87 σ. For a fair comparison we have additionally compen-sated the bias here.We refer to this method as fast unbiased nonlocalmeans (FUNLM) filter.

For all methods, the patch size is chosen to be 5×5 and the searchwindow is restricted to 11×11 pixels, as this is the most commonchoice in all literature. For the proposed UNLML method, we also useh = 0.87 σ and the additional parameter h′ is empirically chosen tobe h′ ¼

ffiffiffiffiffiffiffiffiffiffi3 � h

p.

5.2.1. Validation on simulated data setIn our experiments we have taken simulated data from the Brain

Web Database [48] which is widely used in literature. Two differenttypes of images, i.e., T1-wMR image and from a pathological context

T2-w multiple sclerosis (MS) lesions MR image are considered. Bothimages have a spatial resolution of 217×181 pixels. Fig. 3 shows theexamples of the simulated MR images used in our experimentation.

The denoised results obtained for T1-w and T2-w (MS) lesionsMR images corrupted with 17% and 9% Rician noise are shown inFigs. 4 and 5, respectively. From the figures it is observed that theNLM method removes the noise quite well but some ringing effectand artifacts are observed in the denoised result for T1-w image. Inthe case of T2-w (MS) lesions image, where a part of the backgroundcan be viewed, we observe that the background in the resultobtained by the NLM method is quite away from that of the originalimage. This is due to the non-removal of the bias term from thesmoothed image when there is a small shift in background intensity.FUNLM method preserves small structural details better than theNLMmethod but the result is still quite noisy and there is quite a bitof blurring of finer details. These are evident from both Figs. 4 and 5.UNLM method works better than both of these methods and is ableto remove the noise quite substantially. However, an excesssmoothing is observed which results in a loss of some edges andsmall structural features. Edge preserving capability of the proposedmethod can be clearly observed from the results of both T1-w andT2-w MS lesions images. The small details and edges which are lostin the UNLM method are preserved well in the proposed method. Inaddition, it removes the noise very well.

The PSNR values obtained for the simulated dataset using theaforementioned denoising techniques are given in Fig. 6. Theproposed method performs better than all other methods almosteverywhere and more so as the noise perturbation increases. Quite alow PSNR value is observed for the NLM method compared to othermethods, which is due to non removal of the bias term. When thebias is removed, as in UNLM and FUNLM methods, the proposedmethod still provides 0.5–1.0 dB improvement over these existingmethods. Comparative SSIM values obtained for the simulated dataset are given in Fig. 7. From the plots, it is clearly observed that theproposed method yields much higher SSIM values at all noise levels

1609H.V. Bhujle, S. Chaudhuri / Magnetic Resonance Imaging 31 (2013) 1599–1610

compared to other methods. Fig. 8 illustrates the comparative resultsbetween the two speed-up strategies adopted in denoising thesimulated data set. In [9], the authors have used average mean andgradient as thresholds. As gradient computation is sensitive to noise,significant degradation is observed at higher noise levels. However,the speed-up strategy used in Eq. (17) leads to better results whichcan be clearly observed from the figure.

From Figs. 9 and 10 we observe that the BC and UQI values arehigher for the proposed method compared to others. Higher BCvalues indicate that there is a strong correlation between theoriginal and the denoised image. Higher UQI values for theproposed method suggest existence of a strong similarity betweenoriginal and denoised images in terms of luminance, contrast andcorrelation. Thus Figs. (6, 7) and Figs. (9, 10) substantiate our claimthat the proposed method performs better than other NLM typedenoising techniques.

5.2.2. Validation on clinical data setFor this part of our experimentation, we have taken clinical MR

data acquired with a single coil from mr-tip.com. Two images T1-weighted head MR image and a T1-weighted knee MR image with aresolution of 251×301 pixels are considered. As there is no groundtruth for the real data, quantitative evaluation is no longer feasible.We can only perform a visual comparison of results obtained fromcompeting methods. Fig. 11 demonstrates the results obtained forthe clinical head MR image. Noise in the image is estimated usingNowak's method [19] which is required to select the parameter h =0.87 σ. For better clarity, in Fig. 12 results are shown for an enlargedportion of this image. From the denoised results of the head image itis observed that both NLM and UNLM methods tend to oversmooththe image and hence some edges and small details are lost. Resultsobtained with FUNLM method are quite noisy. However, theproposed method preserves the small details better compared toother methods. In addition, it removes the noise very well. In Fig. 13we show an enlarged portion of the knee image. One can definitelysee the noise around the bone regions in the original image.Although both NLM and UNLM do a good job in denoising, somedetails are lost. For example, the crack in the lower bone, just belowthe joint, is not properly visible. This is visible in FUNLM result, but itis still quite noisy. However, the denoising is quite good in theproposed method and yet the crack is visible.

6. Conclusions

In this paper, we have proposed an edge preserving NLMdenoising method for Rician noise perturbed MR image. In most ofthe Rician based NLMmethods, we observe an oversmoothing effectdue to which edges and fine structures are not preserved well. In ourmethod, we are able to preserve the edges by modifying theweighing term of NLM such that it depends on edge similarity aswell. Further, to accelerate the process we have preselected a fewsimilar neighborhoods based on closeness in edge and invertedmean values. The proposed algorithmworks very well for simulated,clinical and pathological MR data set corrupted with a wide range ofinput noise. Performance of the proposed method has beenevaluated qualitatively and quantitatively with PSNR, SSIM, BC,UQI measures. Through these quality metrics it is observed that theproposed method works better than several existing methods.

In this paper we have provided an experimental validation ofthe proposed method to deal with Rician noise perturbed images.A mathematical validation of the proposed technique for the givennoise distribution remains to be performed. Further, as mentionedearlier, some of the current generation MR images no longersatisfy the assumption of Rician noise. A general purpose NLM

denoising scheme to deal with a wide variety of MR imagesrequires further studies.

Acknowledgments

The authors are thankful to the anonymous reviewers for theirvery constructive comments and suggestions.

References

[1] Geman S, Geman D. Stochastic relaxation, Gibbs distribution, and the Bayesianrestoration of images. IEEE Trans Pattern Anal Mach Intell 1984;6:721–41.

[2] Perona P, Malik J. Scale-space and edge detection using anisotropic diffusion.IEEE Trans Pattern Anal Mach Intell 1990;12:629–39.

[3] Rudin LI, Osher S, Fatemi E. Nonlinear total variation based noise removalalgorithms. Physica D 1992;60:259–68.

[4] Saint-Marc P, Chen J-S, Medioni G. Adaptive smoothing: a general tool for earlyvision. IEEE Trans Pattern Anal Mach Intell 1991;13(6):514–29.

[5] Donoho DL. De-noising by soft-thresholding. IEEE Transact InformatTheory1995;41:613–27.

[6] Donoho DL, Johnstone IM. Ideal spatial adaptation by wavelet shrinkage.Biometrika 1994;81(3):425–55.

[7] Buades A, Coll B, Morel JM. A non-local algorithm for image denoising. IEEEComput Vision Pattern Recognit 2005;2:60–5.

[8] Manjo`n JV, Carbonell-Caballero J, Lull JJ, Garcia-Mart G, Mart-Bonmati L, RoblesM. MRI denoising using non-local means. Med Image Anal 2008;12:514–23.

[9] Mahmoudi M, Sapiro G. Fast image and video denoising via nonlocal means ofsimilar neighbourhoods. IEEE Signal Process Lett 2005;12:839–42.

[10] Eskicioglu AM, Fisher PS. Image quality measures and their performance. IEEETransact Commun 1995;43(12):2959–65.

[11] Wang Z, Bovik AC, Sheikh HR, Simoncelli EP. Image quality assessment: from errorvisibility to structural similarity. IEEE Trans Image Process 2004;13(4):600–12.

[12] Martin-Fernandez M, Alberola-Lopez C, Ruiz-Alzola J, Westin CF. Sequentialanisotropic Wiener filtering applied to 3D-MRI data. Magn Reson Imaging2007;25:278–92.

[13] Gerig G, Kubler O, Kikinis R, Jolesz FA. Nonlinear anisotropic filtering of MRI data.IEEE Trans Med Imaging 1992;11(2):221–33.

[14] Sijbers J, denDekker AJ, der Linden AV, VerhoyeM, DyckDV. Adaptive anisotropicnoise filtering for magnitude MR data. Magn Reson Imaging 1999;17:1533–9.

[15] WongW, Chung A. Trilateral filtering: a non-linear noise reduction technique forMRI. Proc Int Soc Magn Reson Med 2004;11:2218.

[16] Sijbers J, den Dekker AJ, Scheunders P, Dyck DV. Maximum-likelihood estimationof Rician distribution parameters. IEEE Trans Med Imaging 1998;17(3):357–61.

[17] Rajan J, Veraart J, Audekerke JV, Verhoye M, Sijbers J. Nonlocal maximumlikelihood estimation method for denoising multiple coil magnetic resonanceimages. Magn Reson Imaging 2012;30(10):1512–8.

[18] Weaver JB, Xu Y, Healy DM, Cromwell LD. Filtering noise from images withwavelet transforms. Magn Reson Med 1991;21:288–95.

[19] Nowak RD. Wavelet-based Rician noise removal for magnetic resonanceimaging. IEEE Trans Image Process 1999;8:1408–19.

[20] Alexander ME, Baumgartner R, Summers AR, Windischberger C, Klarhoefer M,Moser E, et al. A wavelet-based method for improving signal-to-noise ratio andcontrast in MR images. Magn Reson Imaging 2000;18:169–80.

[21] Zaroubi S, Goelman G. Complex denoising of MR data via wavelet analysis:applications for functional MRI. Magn Reson Imaging 2000;18:59–68.

[22] Wood JC, Johnson KM. Wavelet packet denoising of magnetic resonance images:importance of Rician noise at low SNR. Magn Reson Med 1999;41:631–5.

[23] Anand CS, Sahambi JS. Wavelet domain non-linear filtering for MRI denoising.Magn Reson Imaging 2010;28:842–61.

[24] Awate SP, Whitaker RT. Feature-preserving MRI denoising: a nonparametricempirical Bayes approach. IEEE Trans Med Imaging 2007;26(9):1242–55.

[25] Krissian K, Aja-Fernández S. Noise driven anisotropic diffusion filtering of MRI.IEEE Trans Image Process 2009;18:2265–74.

[26] Tong C, Sun Y, Payet N, Ong S. A general strategy for anisotropic diffusion in MRimage denoising and enhancement. Magn Reson Imaging 2012;30(10):1381–93.

[27] Xu Q, Anderson AW, Gore JC, Ding Z. Efficient anisotropic filtering of diffusiontensor images. Magn Reson Imaging 2010;28(2):200–11.

[28] Coupe` P, Yger P, Barillot C. Fast non local means denoising for 3D MR images.Med Image Comput Comput Assist Interv 2006:33–40.

[29] Coupe` P, Yger P, Prima S, Hellier P, Kervrann C, Barillot C. IEEE Trans MedImaging 2008;27(4):425–41.

[30] Lili H, Greenshields IR. A nonlocal maximum likelihood estimation method forRician noise reduction in MR images. IEEE Trans Med Imaging 2009;28:165–72.

[31] Rajan J, Jeurissen B, Verhoye M, Audekerke JV, Sijbers J. Maximum likelihoodestimation-based denoising of magnetic resonance images using restricted localneighbourhoods. Phys Med Biol 2011;56:5221–34.

[32] Henkelman RM.Measurement of signal intensities in the presence of noise inMRimages. Med Phys 1985;12(2):232–43.

[33] Gudbjartsson H, Patz S. The Rician distribution of noisy MRI data. Magn ResonMed 1995;34:910–4.

1610 H.V. Bhujle, S. Chaudhuri / Magnetic Resonance Imaging 31 (2013) 1599–1610

[34] Dietrich O, Raya JG, Reeder SB, IngrischM, Reiser MF, Schoenberg SO. Influence ofmultichannel combination, parallel imaging and other reconstruction tech-niques on MRI noise characteristics. Magn Reson Imaging 2008;26:754–62.

[35] Otsu N. A threshold selection method from gray-level histograms. IEEE TransSyst Man Cybern 1979;9:62–6.

[36] Aja-Ferna`ndez S, Brion V, Trista`n-Vega A. Effective noise estimation and filteringfrom correlated multiple coil MR data. Magn Reson Imaging 2013;31:272–85.

[37] Aja-Ferna`ndez S, Trista`n-Vega A, Alberola-Lo`pez C. Noise estimation in singleand multiple coil magnetic resonance data based on statistical models. MagnReson Imaging 2009;27(10):1397–409.

[38] Rajan J, Poot D, Juntu J, Sijbers J. Noise measurement from magnitude MRI usinglocal estimates of variance and skewness. Phys Med Biol 2010;55:N441–9.

[39] Coupé P, Manjón JV, Gedamu E, Arnold D, Robles M, Collins DL. Robust Riciannoise estimation for MR images. Med Image Anal 2010;14:483–93.

[40] Wiest-Daessle' N, Prima S, Coupe` P, Morrissey SP, Barillot C. Rician noiseremoval by non-local means filtering for low signal-to-noise ratio MRI:applications to DT-MRI. Med Image Comput Comput Assist Interv 2008:171–9.

[41] Tomasi C, Manduchi R. Bilateral filtering for gray and color images. Proc. 6th Int.Conf. Computer Vision. Washington; 1998.

[42] Wang J, Guo Y-W, Ying YT, Liu YL, Peng QS. Fast non-local algorithm for imagedenoising. IEEE Int Conf Image Process 2006:1429–32.

[43] Liu YL, Wang J, Chen X, Guo Y-W, Peng QS. A robust and fast nonlocal meansalgorithm for image denoising. J Comput Sci Technol 2008;23:270–9.

[44] Manjo`n JV, Coupe` P, Mart-Bonmat L, Collins DL, Robles M. Adaptive non-localmeans denoising of MR images with spatially varying noise levels. J Magn ResonImaging 2010;31(1):192–203.

[45] Bhattacharyya A. On a measure of divergence between two statisticalpopulations defined by their probability distributions. Bull Calcutta Math Soc1943;35:99–109.

[46] Wang Z, Bovik AC. A universal image quality index. IEEE Signal Processing Letters2002;9(3):81–4.

[47] Wang Z, Bovik AC. Mean squared error: love it or leave it? A new look at signalfidelity measures. IEEE Signal Process Mag 2009;26:98–117.

[48] http://www.bic.mni.mcgill.ca/brainweb/.