Research Article Adaptively Tuned Iterative Low Dose CT ...downloads.hindawi.com/journals/cmmm/2015/638568.pdfdenoising method in CT imaging [ ], nonlocal mean [ ], and BMD [ ]. In

Research ArticleAdaptively Tuned Iterative Low Dose CT Image Denoising

SayedMasoud Hashemi1 Narinder S Paul12 Soosan Beheshti3 and Richard S C Cobbold1

1 Institute of Biomaterials and Biomedical Engineering University of Toronto Toronto ON Canada M5S 3G92Joint Department of Medical Imaging Toronto General Hospital University Health Network Toronto ON Canada M5G 2N23Department of Electrical and Computer Engineering Ryerson University Toronto ON Canada M5B 2K3

Correspondence should be addressed to SayedMasoud Hashemi sayedmasoudhashemiamroabadimailutorontoca

Received 23 January 2015 Revised 2 May 2015 Accepted 3 May 2015

Academic Editor Hugo Palmans

Copyright copy 2015 SayedMasoud Hashemi et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Improving image quality is a critical objective in low dose computed tomography (CT) imaging and is the primary focus of CTimage denoising State-of-the-art CT denoising algorithms are mainly based on iterative minimization of an objective functionin which the performance is controlled by regularization parameters To achieve the best results these should be chosen carefullyHowever the parameter selection is typically performed in an ad hocmanner which can cause the algorithms to converge slowly orbecome trapped in a local minimum To overcome these issues a noise confidence region evaluation (NCRE)method is used whichevaluates the denoising residuals iteratively and compares their statistics with those produced by additive noise It then updates theparameters at the end of each iteration to achieve a better match to the noise statistics By combining NCRE with the fundamentalsof block matching and 3D filtering (BM3D) approach a new iterative CT image denoising method is proposed It is shown that thisnew denoising method improves the BM3D performance in terms of both the mean square error and a structural similarity indexMoreover simulations and patient results show that this method preserves the clinically important details of low dose CT imagestogether with a substantial noise reduction

1 Introduction

While X-ray computed tomography (CT) enables ultrafastacquisition of patient images obtained with excellent spatialresolution the dose needed to achieve diagnostic imagequality can result in a significant increase in the risk ofdeveloping cancer [1] Consequently low-dose CT imaging isclinically desired and has been under investigation for severalyears Lowering the radiation dose may seriously degradediagnostic performance or undermine physician confidenceby producing noisier images [2 3] Several different algorith-mic approaches have been proposed to reduce the effect ofnoise in the CT images including projection data denoising[4ndash6] optimizing the reconstruction algorithms to includethe noise statistics [7ndash9] and CT image denoising [10ndash12]The latter is the focus of this paper where an adaptively tunediterative CT image denoising algorithm is presented

The main source of noise in X-ray projection data isquantum noise caused by statistical fluctuations of X-rayquanta reaching the detectors so that the CT projection

noise follows the Poisson distribution [3] However becauseof the use of different reconstruction algorithms and signalprocessing steps in CT reconstruction the noise statistics ofthe processed CT images are usually unknown and hard tomodel and are spatially changingMoreover directional noisein the form of streak artifacts is present in many CT imagesAs a result incorporating accurate noise statistics into image-basedCTdenoising can be very challengingWhen denoisingis based on the projection data and its statistics otherdifficulties arise Specifically such denoising methods andthe associated iterative reconstructions require access to theCT raw data which is often unavailable Furthermore thesemethods have a high computational complexity making itchallenging to obtain a final image in a reasonable length oftime depending on the available computational resourcesOn the other hand image-based denoising methods arefast and can be applied directly on the CT images withoutchanging the clinical workflow

A simplified noise model is usually used in image baseddenoising algorithms in which following the Central Limit

Hindawi Publishing CorporationComputational and Mathematical Methods in MedicineVolume 2015 Article ID 638568 12 pageshttpdxdoiorg1011552015638568

2 Computational and Mathematical Methods in Medicine

Theorem (CLT) [13] the final noise in each voxel follows aGaussian distribution [11 14ndash17] The CLT can be used sinceeach voxel in CT images is computed by adding values frommany different projectionsWith this assumption a noisy CTimage y can be modeled by

y = x +n (1)

where x is the noiseless image and n is a zero mean additiveanisotropic Gaussian noise with variance of 1205902

119899 which varies

with the pixel location and its valueDifferent image based denoising algorithms have been

used to estimate the noiseless CT images such as anisotropicdiffusion [10] total variation (TV) [18] bilateral filtering[19] or wavelet-based techniques [11 12 20] These methodscan usually be formulated as an unconstrained Lagrangianmultiplier optimization problem [18 21ndash23] that is

119909 = arg min119909

121003817100381710038171003817

119909 minus y1003817100381710038171003817

22 +120582ℎ (119909) (2)

in which 120582 is a regularization parameter that controls thetradeoff between the data fidelity and the regularization termℎ(119909) and 119909

119902= sum

119899

119894=1 |119909119894|119902 Different regularization terms

ℎ(119909) lead to different denoising methods For example inTV-based methods [24ndash26] ℎ(119909) = radic120575

ℎ119909

2+ 120575V119909

2 where 120575ℎ

and 120575V are the gradient in horizontal and vertical directionsand inwavelet soft thresholdingmethods ℎ(119909) = sum

119894|Ψ2D(119909)119894|

where Ψ2D(119909) is the 2D wavelet transform [27]There is a strong dependence of the quality of the result

on the regularization parameter It is a challenging task tofind the regularization parameter 120582 that provides the bestbalance between signal smoothing and feature preservation[26] Specifically if 120582 is not appropriately adjusted theoptimization is trapped in a local minimum that is if 120582 is toosmall noise is only partially removed and if it is too large theimage may be oversmoothed [25] Some methods have beenproposed to update the regularization parameters iterativelysuch as use of the discrepancy principle [28] generalizedcross-validation [26] and L-curve [29] These methods failin certain situations are problem specific and generallyincrease the computational complexity of the algorithms

One straight forward approach used in many algorithmsis to use a heuristic 120582 value combined with a criterion to stopthe algorithm before the estimated signal is oversmoothedDifferent stopping criteria have been proposed for iterativedenoising problems For instance Akkoul et al [30] useda switching median filter algorithm to stop the iterativeprocess when the number of changed pixels in the denoisingiterations is a minimum In [20 31] the statistical propertiesof high frequency wavelet subbands were used to stop the TViterations However such methods are unable to differentiateoversmoothed data from well-denoised data As a result toavoid oversmoothing the updating steps are typically chosento be small which decreases the convergence speed

In this paper a noise confidence region evaluation(NCRE) method is used to address the regularization selec-tion and the algorithm stopping problems It adaptivelyupdates the regularization parameters at the end of each

iteration by validating the result of that iteration The algo-rithm stops when the statistical properties of the denoisingresidual resemble those of the additive white Gaussian noiseUsing NCRE a new iterative block matching and 3D filtering(BM3D)method is proposed which outperforms BM3D [32]itself The proposed method is compared with anisotropicdiffusion denoising which is generally regarded as a standarddenoising method in CT imaging [10] nonlocal mean [33ndash35] and BM3D [32] In addition we study the noise prop-erties of CT images and show that the noise in small imageblocks has an additive white Gaussian model which justifiesthe success of the nonlocal based denoising algorithms inimage based CT denoising [33 36ndash39]

2 Problem Formulation andCT Noise Properties

Recently it has been shown that nonlocal patch basedalgorithms outperform others in CT image denoising [33 3436ndash41] For example in [41] a nonlocal means (NLM) basedmethod which takes advantage of the presence of repeatingstructures in a given image was compared with a principlecomponent analysis based denoising method and a highlyconstrained backprojection method It was shown that theNLM method outperformed both of the methods in termsof the contrast to noise ratio noise standard deviation andsquared error Another class of algorithms looks for similarblocks in the whole 2D image and stacks them together in3D arrays Denoising is then performed through transformdomain shrinkage of the 3D arrays An algorithm called K-SVD [37 38 42] uses these 3D patches to train an optimumdictionary This method which assumes that each 3D blockcan sparsely be represented by the trained dictionary atomsuses shrinkage algorithms to denoise the patches

Our proposed algorithmmakes use of the blockmatchingand 3D filtering (BM3D) technique [32 36] This is a nonit-erative denoising method that currently outperforms manynewer algorithms [40] It is composed of two major filteringsteps In both stages collaborative filtering is utilized whichitself has four stages (1) grouping similar patches with areference patch (2) calculation of the 3D wavelet coefficientsof each stack of patches (3) denoising the wavelet coefficients(thresholding in step (1) or Wiener filtering in step (2))and (4) recovering the denoised image by calculating theinverse 3Dwavelet transformationTheBM3Dapproach aimsto denoise the patches by Wiener filtering which is done instep (2) To find best match to similar patches in the step (2)and to reliably estimate the Wiener coefficients the methodrequires a reliable estimate of the noiseless image which is themain purpose of step (1) The input for this step which is ahard thresholding block is the 3D noisy wavelet coefficientsof similar patches located by block matching applied to theavailable noisy image A hard threshold with a heuristicallydetermined value of 27120590

119899is used in step (1) The resulting

denoised coefficients are then transformed back to a spatialdomain to be used as the initial estimate of the noiseless dataused for calculating the Wiener filter coefficients

Computational and Mathematical Methods in Medicine 3

To denoise the images patch basedmethods generally usea model based on

y119866119895= x119866119895+n119866119895 119895 isin [1 119898] (3)

in which 119866 is the patch grouping information 119898 is thenumber of 3D patches y

119866119895denotes the noisy patches x

119866119895

denotes the noiseless 3D patches and n119866119895

sim N(0 1205902119866119894)

is the noise at each 3D patch Conventional BM3D usesan independent identical additive Gaussian noise model inwhich the noise variances 1205902

119866119894are similar in all patches Using

this assumption its regularization term is a nonlocal waveletℓ1-norm ℎ(119909) = NLW(119909 119866) where [43]

NLW (119909 119866) =

119898

sum

119895=1

100381710038171003817100381710038171003817

Ψ3D (119909119866119895)1003817100381710038171003817100381710038171 (4)

in which Ψ3D is the 3D wavelet transform Using this reg-ularization in (2) BM3D solves the following optimizationproblem in its first step

x = arg min119909

121003817100381710038171003817

119909 minus y1003817100381710038171003817

22 +120582119898

sum

119895=1

100381710038171003817100381710038171003817

Ψ3D (119909119866119895)1003817100381710038171003817100381710038171 (5)

In our approach we modify the BM3D formulation forCT image denoising by incorporating a more realistic noisemodel to include the nonstationarity of the noise and itsdependence on the position and value of the pixels Ourproposed method uses the noise properties of the patchesx119866119895 studied in the Appendix to improve the performance of

BM3D for CT image denoising

21 Noise in CT Images Although a reasonable statisticalmodel for the CT projection data is the independent Poissondistributions [3] it has been shown that the corrected polyen-ergetic X-ray projections can be modeled more accuratelyby a Gaussian distribution with the following relationshipbetween its mean and variance

120590

2119894119899=radic(119869119894times exp(120585 (119894 119899)

119904

))

2

+ (120590

2119890)

119894119895

(6)

where 120585 is the mean and 1205902119894119899is the variance of the projections

at the 119894th projection angle (120593119894) and the 119899th detector bin whose

distance is 119897119899from the detectors center 119904 is a scaling factor

120590

2119890is the electronic noise variance and 119869

119894is a parameter

adaptive to different detector channels [6] During the recon-struction process the noise distribution is changed by thereconstruction algorithm and filters As a result due to thecomplicated dependencies of noise on scan parameters andon spatial position the noise distribution in the reconstructedCT images is usually very difficult to determine (a detailedstudy of the noise model in CT can be found in [44 45])Using the discrete filtered back projection relation

119891 (119909 119910) =

120587Δ119905

119873120587119892

119873120587119892

sum

119894=1

119873119892

sum

119899=1119888 (119909 cos120593

119894minus119910 sin120593

119894minus 119897119899) 119892119894119899 (7)

the noise variance in the reconstructed images can bedescribed by [38]

120590

2119899(119909 119910)

= (

120587Δ119905

119873120587119892

)

2 119873120587119892sum

119894=1

119873119892

sum

119899=1119888

2(119909 cos120593

119894minus119910 sin120593

119894minus 119897119899) 120590

2119894119899

(8)

where Δ119905 is the distance between the center of two adjacentdetectors 119892

119894119899is the parallel projection at the 119894th angle and

the 119899th detector bin and 119888(sdot) is the ramp filter in the spatialdomain This can be interpreted as the backprojection of theprojection noise variances making it nonstationary objectdependent and correlated Moreover since the variance ofeach voxel is the summation of the variances from manyangles if the variance in one direction is significantly largerthan that at another direction then the variances alongthat direction will be more correlated than that for otherdirections [46] producing what is known as a streak artifactin the reconstructed images It should be noted that thiseffect is not included in our model In (8) the ramp filter 119888(sdot)is symmetric about the center of rotation and 120590

2119894119899

dependson the attenuation of the media through which the X-raybeams pass Therefore we assert that the noise variances ofsmall neighborhoods with similar attenuations and similarradial distances from the center of rotation should be verysimilar The results of experimental tests of this assertion arepresented in the Appendix

22 Modified Formulation Based on the above discussionand our experimental results presented in the Appendix itcan be assumed that the noise in 3D similar patches of the CTimages follows a white additive Gaussian distribution withdifferent variances for different 3D patches 1205902

119866119894 Using (3) the

modified optimization problem used in this paper is given by

119869 (y 119866 Λ) = x

= arg min119909

121003817100381710038171003817

119909 minus y1003817100381710038171003817

22

+

119898

sum

119895=1120582119895

100381710038171003817100381710038171003817

Ψ3D (119909119866119895)1003817100381710038171003817100381710038171

(9)

in which Λ = [1205821 1205822 120582119898] is a set of regularizationparameters that are functions of 120590

119866119894 To improve the regu-

larization parameter selection we used an adaptive updatingmethod based on an evaluation of the noise statistics Itautomatically avoids the oversmoothing without loweringthe convergence speed The NCRE method validates thestatistical properties of the error residuals at the end of eachiteration and categorizes the result as well denoised partiallydenoised or oversmoothed This information is then usedto update the parameters in the next iteration or to stop thealgorithm at the end of the iteration for which the similaritybetween error residual and Gaussian noise is satisfied


3 Proposed Denoising Method BM3D-NCRE

If x119894denotes the signal recovered at 119894th iteration the denoising

residual at the end of 119894th iteration can be expressed byΔy119894= y minus x

119894which ideally should be the noise (3) n =

n119866119894 forall119894 = 1 119898 Here we provide a quantitativemeasure

that verifies the similarity between the structure of Δy119894and

that of n

31 Noise Confidence Region Evaluation (NCRE) In [47] itwas shown that the following function of zero mean whiteGaussian noise n with length 119899 and for any given scalarvalue of 119911

G (119911n) = 1119899

119899

sum

119895=1G (119911n

119895)

G (119911n119895) =

1 if 10038161003816100381610038161003816

n119895

10038161003816100381610038161003816

le 119911

0 if 10038161003816100381610038161003816

n119895

10038161003816100381610038161003816

gt 119911

(10)

is equivalent to sorting the absolute value of the noise ele-ments n

119895 The expected value of this function is 119864(G(119911n)) =

119865(119911) and its variance is var(G(119911n)) = 119865(119911)(1minus119865(119911))119899 where119865(119911) = 2120601(119911120590

119899) minus 1 and 120601(sdot) is the cumulative distribution

function (CDF) of a Gaussian distributionThereforeG(119911n)is bounded by the following lower (119871

119899) and upper (119880

119899) values

119871119899(119911) = 119865 (119911) minus 120577

radic

1119899

119865 (119911) (1 minus 119865 (119911))

119880119899(119911) = 119865 (119911) + 120577

radic

1119899

119865 (119911) (1 minus 119865 (119911))

(11)

with probability of Pr|G(119911n) minus 119865(119911)| le

120577radic(1119898)119865(119911)(1 minus 119865(119911)) asymp 2120601(120577) minus 1 If the sorted absolutevalues of a signal lie between these two boundaries for alarge enough 120577 that signal will follow a white Gaussiandistribution with a confidence probability close to one

As shown in Figure 1 these boundaries divide the[119911G(119911n)] space into three regions At the end of eachiteration G(119911 Δy

119894) the sorted absolute value of the residual

is calculated If this sequence falls into Region II (in ourproposed algorithm being a subset of a region is evaluated byhaving a high fraction ofG(119911 Δy

119894) in that region for example

in our simulations this fraction is 90) it means that theresidual has a Gaussian-like structure and denoising stopsOn the other hand if the denoising at the 119894th iteration hasremoved not only the noise but also parts of the noiseless dataitself Δy

119894will have some of the image informationmaking its

samples larger than Gaussian noise Therefore for a specificvalue of 119911 G(119911 Δy

119894) (average number of Δy

119894s with absolute

values smaller than 119911) is smaller than G(119911 119899) and falls inRegion III This will enforce continuation of the denoisingto the (119894 + 1)th step with changing of the regularizationparameters 120582

119895such that the denoising algorithm extracts less

noise in the next iteration that is decreasing forall120582119895= 120582119895119904

119904 gt 1 If G(119911 Δy119894) falls in Region I it indicates that the noise

is partially removed In this case the algorithm continues tothe (119894+1)th step and changes the regularization parameters 120582

119895

0 1 2 3 4 5

010203040506070809

1

Region I

Region II

Region III

z

g(zn)

Figure 1 Three possible regions for the residual at the end of eachiteration If it lies in Region II (noise confidence region) denoisingis stopped

such that more noise is extracted by the denoising algorithmthat is increasing forall120582

119895= 119904 times 120582

119895 119904 gt 1 In summary at

each iteration when G(119911 Δy119894) is in either Region I or III

the regularization parameter is updated such that it movestoward Region II The value of 119904 can be tuned as a fixed or anadaptively changing variable based on the euclidean distancebetween G(119911 Δy

119894) and 119865(119911) 119889

119894= G(119911 Δy

119894) minus 119865(119911)

2 Inour proposed method to update Λ = [1205821 120582119898] a global119904 is used (similar to [33 41]) which is updated based onthe placement of the denoising residual of the patches indifferent Regions IndashIII The algorithm is stopped when thedenoising residual of the soft tissue around the lung is placedin Region II An example of a soft tissue mask for a thoracicphantom denoted by M is shown in Figure 2 The pixels inthis region have very similar CT and have almost the sameradial distances from the center of the rotation Therefore itcould be assumed that the noise in this region has a whiteGaussian distribution

32 Summary of the Proposed Method BM3D-NCREAlgorithm 1 shows the proposed iterative Λ updatingscheme in which sdot denotes an element-wise multiplicationThe updating method uses a memory strategy for recoveryof possible lost edges and fine details This process isrepresented by

y119894= (1minus120572

119896) y +120573

119896x119894+ (120572119896minus120573119896) x119894minus1 (12)

in which 119896 isin [1 2] and 120572119896and 120573

119896are positive scalars

chosen based on the conditions given in [48]This stage of thealgorithmwas inspired by the second order iterative methods[48 49] that improve the convergence rate of the iterativemethods

To denoise CT images the fundamentals of BM3D wereused in an iterative scheme the output of theWiener filter is abetter estimate of the original image than the input ofWienerfilter from the first step Therefore this output can be fedinto the first step to provide better Wiener coefficients in thesecond iterationThemodified BM3D formulation (9) is usediteratively in Algorithm 1 to denoise the CT images whereNCRE adjusts the threshold values applied on 3D waveletcoefficients of the first step The parameters of the BM3D


(a)

minus150 minus100 minus50 0 50 100 1500

0002

0004

0006

0008

001

0012

0014

CT values (HU)

Occ

uran

ce p

roba

bilit

y

(b)

Figure 2 Noise statistics of the soft tissue region surrounding the lung (a) Top the thoracic phantom which is used to evaluate the noisecharacteristics and bottom the soft tissue region of the phantom (denoted byM in Algorithm 1) (b) The statistical distribution of the noisein the soft tissue region is shown by the blue experimental values and these are compared with Gaussian distribution with the same variance(red dashed line)

Initialize 119894 = 1M Soft tissue maskx0 = 119869(y 119866119894 Λ 119894) Δy0 = y minus x0G0 = sort(|M sdot Δy0|)while M sdotG

119894sube Region II and 119894 ltmaxiter do

if G119894sub Region I then

increase forall120582119895isin Λ 119895 = [1 119898] towards Region II

x119894= 119869(y119894 119866119894 Λ119894)

y119894= (1 minus 1205721)y + 1205731x119894 + (1205721 minus 1205731)x119894minus1

end ifif G119894sub Region III then

decrease forall120582119895isin Λ119894 119895 = [1 119898] towards Region II

x119894= 119869(y119894 119866119894 Λ119894)


end ifΔy119894= y minus x

119894G119894= sort(|M sdot Δy

119894|) 119894 larr 119894 + 1

end while

Algorithm 1 Proposed Iterative Regularization Parameter Updat-ing

algorithm are chosen based on the ones that resulted in thebest performance in [50]The initial value forΛ is 27120590

119866119894with

noise variances estimated independently for each 3D stackusing the median-absolute deviation method as describedin [51] In each iteration if the sorted absolute value of thedenoising residual of the soft tissues around the lungM sdotΔy

119894

falls into Region I the threshold value will be increased andif it falls into Region III the threshold values are decreased sothat in the next iteration the residual moves towards RegionII

4 Results and Discussion

Three test methods were used to evaluate the performanceof the proposed algorithm The first method consisted of

5 10 15 20 25 30 35 40

001002003004005006007008

Nor

mal

ized

squa

red

erro

r

086088090920940960981

SSIM

Iteration number

Region I

Region IIRegion III

Stopping point

Figure 3 Squared error (blue line) and the Structural SimilarityIndex (SSIM) (green dashed line) showing the changes with eachiteration when using BM3D-NCRE The shading colors show theregion in which Δy

119894rsquos are placed after each iteration The algorithm

stops when Region II is reached

a simulated Shepp-Logan phantom corrupted by addingPoisson noise to the fan beam X-ray projections The imageswere reconstructed using the ifanbeam command in MatlabThe number of unattenuated photons in each projection wastaken to be 1times1013 which led to a noise variance similar to theimages reconstructed from the tube current of 50mAs andpeak voltage of 120 kVp White Gaussian noise with standarddeviation of 100 was added to all the projections to simulatethe presence of electronic noise The second method used aCATPHAN phantom (Phantom Laboratory Greenwich NYUSA) This is a standard phantom widely used for CT imagequality evaluation and contains spheres of differing contrastas well as line pairs with differing spacing that can be used totest the spatial resolution Ideally the denoising algorithmsshould enable us to distinguish between smaller spheres withlower contrasts in the low contrast slice and should keep theline pair resolution unchanged The third method uses axialchest CT images froma clinical patient All three testmethodsused the following parameters in the Algorithm 1 120577 = 6


(a) (b) (c) (d)

Figure 4 Shepp-Logan phantom simulations (a) original phantom (b) noisy reconstructed phantom (c) denoised by BM3D and (d)denoised by BM3D-NCRE

Figure 5 Top line pair slice of the CATPHAN phantom scanned with 50mAs and 120 kVp (window widthwindow level = 40060HU)Bottom red rectangular ROI of the images with the red line showing the cut-off line pair resolution (window widthwindow level =400500HU) Left to right image reconstructed with FC52 (STD = 64HU) and reconstructed with AIDR3D (STD = 41HU) FC52 denoisedwith the proposed method (STD = 22HU) FC52 denoised with nonlocal mean (STD = 34HU) and FC52 denoised with BM3D method(STD = 27HU)

Figure 6 Low contrast study using CATPHAN low contrast slice Top images scanned with 50mAs120 kVp and bottom scanned with300mAs120 kVp Left to right image reconstructed with FC52 reconstructed with AIDR3D reconstructed with FC52 and denoised byBM3D-NCRE reconstructed with FC52 and denoised by nonlocal mean and reconstructed with FC52 and denoised by BM3D In all imageswindow widthwindow level = 10070HU

Computational and Mathematical Methods in Medicine 7minus1

000

minus500 0

500

(a)

minus100

0minus8

00minus6

00minus4

00minus2

00 020

040

0

(b)

minus100

0

minus500 0

500

(c)

minus100

0

minus500 0

500

(d)

minus100

0

minus500 0

500

(e)

(f) (g) (h) (i) (j)

minus50

minus40

minus30

minus20

minus10 0

10 20 30 40 50

(k)

minus60

minus40

minus20 0 20 40 60

(l)

minus20

minus30

minus40

minus50

minus10 0 10 20 30 40 50

(m)minus2

0minus3

0minus4

0minus5

0

minus10 0 10 20 30 40 50

(n)

Figure 7 Comparison of the effects of anisotropic diffusion nonlocal mean BM3D and BM3D-NCRE (a) In the original image the circularregions show the area from which the noise variance is measured the dashed rectangular region is the area which is shown in (d)-(e) andaverage noise of the three regions is around 55HU (b) Denoised by anisotropic diffusion noise is around 25HU (c) Denoised by BM3D-NCRE noise is around 25HU (d) Denoised by nonlocal mean noise is around 21HU (e) Denoised by BM3D noise is around 28HU((f)ndash(j)) The zoomed-in region shown by dashed rectangle in image (a) The difference between original image and the image denoised by(k) anisotropic diffusion (l) BM3D-NCRE (m) nonlocal mean and (n) BM3d are shown The window widthwindow level in (a)ndash(j) is1600minus300HU and is 1000HU in (k)ndash(n)

1205721 = 09 1205731 = 08 1205722 = 04 and 1205732 = 03 The parametersof BM3D are chosen similar to the ones proposed in [50] thesize of the blocks is 8 times 8 sliding step to process every nextreference block is 3 maximum number of similar blocks is16 and the size of the search neighborhood for full-searchblockmatching is 39 times 39

Figure 3 shows the mean square error (MSE) and thestructural similarity index (SSIM) [52] resulting from succes-sive iterations of BM3D-NCRE applied to the reconstructednoisy Shepp-Logan phantom images The first iteration isequivalent to the result of BM3D As shown the MSEdecreases and the SSIM increases in successive iterations toa point where the algorithm is stopped by falling into RegionII The results of denoising the Shepp-Logan phantom withBM3D and BM3D-NCRE are shown in Figure 4 As can beseen the noise is removed more effectively by BM3D-NCRE

However the streak artifacts are still visible in both denoisedimages

In the second test the CATPHAN phantom was scannedusing a lowdose (50mAs 120 kVp) and a high dose (300mAs120 kVp) protocol Image reconstructions were performedwith a Toshiba Aquilion One CT using the proprietary lungkernel FC52 and the proprietary iterative reconstructionalgorithm Adaptive Iterative Dose Reduction 3D (AIDR3D)[53]The latter uses anisotropic diffusion denoising as its baseto improve the image quality at each iteration Our proposeddenoising method was applied to the images reconstructedwith the high spatial resolution filter algorithm FC52 Theseare compared to the images reconstructed with AIDR3Dand the FC52 reconstructed images denoised by nonlocalmean and BM3D The nonlocal mean package provided byGabriel Peyre on Mathwork File Exchange was used here


Figure 8 Phantom scanned with eight different X-ray source currents with the same peak voltage top left to right 5 10 25 and 50mAs andbottom left to right 100 150 200 and 250mAs

minus1200 minus1000 minus800 minus600 minus400 minus200 0 200 4000

0005

001

0015

002

0025

003

0035

004

CT values (HU)

Occ

uran

ce p

roba

bilit

y


0005

001

0015

002

0025

003

0035

CT values (HU)

Occ

uran

ce p

roba

bilit

y

minus1200minus1000 minus800 minus600 minus400 minus200 0 200 400 6000

0002

0004

0006

0008

001

0012

0014

0016

0018

CT values (HU)

Occ

uran

ce p

roba

bilit

y

minus1200minus1000 minus800 minus600 minus400 minus200 0 200 400 600 8000

1

2

3

4

5

6

7

8

9

CT values (HU)

Occ

uran

ce p

roba

bilit

y

Correlation = 099773 current = 250mAsCorrelation = 099828 current = 150mAs

Correlation = 099491 current = 50mAsCorrelation = 099257 current = 10mAstimes10minus3

Figure 9 Comparison of the noise distribution in the six regions of the phantom shown in Figure 8 with white Gaussian noise Blue dots arethe measured values and the red dashed lines are the fitted Gaussian distributions


minus1000 minus800 minus600 minus400 minus200 0 200CT of the ROIs

Varia

nce o

f the

RO

Is

Dose increased from50mAs to 250mAs

102

103

Figure 10 Showing the noise variance changes for the backgroundand the five circular regions of the phantom as the dose wasincreased from 5mAs to 250mAs

(httpwwwmathworkscommatlabcentralfileexchange13619-toolbox-non-local-means) This package is based onthe method described in [35] The BM3D code is also basedon the package provided by Alessandro Foi on his homepage(httpwwwcstutfisimfoiGCF-BM3D) It should be notedthat the parameters of nonlocal mean and BM3D wereheuristically adjusted to achieve the best performancebased on visual inspection of the results In addition theparameters are adjusted to keep the spatial resolution inthe line pair resolution slice the same Figure 5 shows theline pair slice reconstructed by FC52 AIDR3D and FC52denoised by BM3D-NCRE nonlocal mean and BM3DAs can be seen all these methods have the same spatialresolution as the original image whose resolution was notimproved by the proprietary iterative reconstruction methodused (AIDR3D)

Figure 6 shows that the detectability of low contrastobjects is improved with our method and outperforms thatachieved by AIDR3D nonlocal mean and BM3D Visibilityof the spheres with higher contrasts in the images denoisedby BM3D is very close to the ones denoised by BM3D-NCREHowever the visibility is significantly less for the spheres withlower contrasts Both BM3D and BM3D-NCRE outperformAIDR3D and nonlocal mean while the number of visiblespheres is almost the same as in the images denoised bynonlocal mean and the images reconstructed by AIDR3D

Thefinal comparisonwasmade using a lowdose (50mAs120 kVp) lung CT of a patient reconstructed using FC52and processed by anisotropic diffusion denoising BM3D-NCRE nonlocal mean and BM3D A single axial slice ofthe images is shown in Figure 7 As can be seen anisotropicdiffusion removes some fine details and reduces the contrastof the small features Nonlocal mean BM3D and BM3D-NCRE outperform anisotropic diffusion denoising in senseof preserving the small structures Comparing the resultsof nonlocal mean with BM3D-NCRE it can be seen thatthe low contrast features are kept perfectly unchanged inBM3D-NCRE while they are removed or their contrasts arereduced in the image denoised by nonlocalmean In additionthe noise is not homogeneously removed from the imageComparing BM3D with BM3D-NCRE it can be seen that

the image denoised by BM3D-NCRE has less noise and theremoved noise is more homogeneous It should be notedthat the parameters of the anisotropic diffusion denoisingare adjusted in such a way that the noise variance in thereconstructed images is the same as the images denoised byBM3D-NCRE

5 Conclusions

An iterative denoising scheme was proposed for low doseCT images which adjusts the denoising parameters at eachiteration based on the effect of the denoising method inthe previous iteration Noise confidence region evaluation(NCRE)was used to compare theGaussian noisewith denois-ing residual to determine if the denoising was effectivelyweakly or strongly executed Based on this information thedenoising parameters were adjusted for the next iterationBM3D was used in the new proposed iterative scheme Thephantom study showed that our proposed method improvedlow contrast detectability The patient study demonstratedthat the image was efficiently denoised and the visibility ofsmall objects was preserved However it should be noted thatthe modified optimization model is not accurate when theelectronic noise dominates the photon fluctuations as couldoccur for very low doses In addition the streak artifactswould still be present in images denoised by the proposedmethod

Appendix

Phantom Study of the Noise Statistics inReconstructed CT Images

Thenoise statistics in CT images were studied using a Toshibaphantom containing five circular regions of differing CT[minus1000 minus100 100 150 350] (HU) and a Toshiba AquilionOne CT scanner (this should be applicable to other scannersand can be easily tested in a similar way) As shown inFigure 8 six regions are considered in the phantom aregion in the background (with CT of 0HU) and the fiveregions inside the circles Scanning was performed witheight different doses controlled by changing the tube current[5 10 25 50 100 150 200 250] (mAs) and with a fixed peakvoltage of 120 kV The noise distribution in each region wascompared with white Gaussian additive noise As can be seenin Figure 9 the noise distribution matches white Gaussiannoise with high accuracy provided that the dose is sufficientlyhigh that is greater than 50mAs as in our experimentsThe noise distribution of four selected X-ray tube currents isshown in Figure 9 Figure 10 summarizes the noise variancechanges for each region with different CTrsquos as a functionof the dose When tube current is less than 10mA theelectronic noise dominates the photon fluctuations causingthe measured CT to be inaccurate Moreover the noisevariances of the six regions differ even for the same tubecurrentsMore realistically we also studied the noise statisticsof soft tissue surrounding the lung in a commercially availableadult thoracic anthropomorphic phantom (Lungman Kyoto


minus40 minus30 minus20 minus10 0 10 200

001

002

003

004

005

006

CT values (HU)

Occ

uran

ce p

roba

bilit

y

minus25 minus20 minus15 minus10 minus5 0 5 10 15 200

001

002

003

004

005

006

007

CT values (HU)

Occ

uran

ce p

roba

bilit

y

Figure 11 Two small soft tissue regions in the lung of a thoracic anthropomorphic phantom are shownThe noise distribution of these regions(blue dots) which could be used in patch based denoising is compared to that of the fitted Gaussian distribution (dashed red lines)

Kakagu Japan) These pixels have very close CT values andalmost the same radial locations It was observed that thenoise of these regions follows a Gaussian distribution withan acceptable accuracy Two examples are shown in Figure 11Based on these results we assume that the noise in smallneighborhoods of the image with similar CT follows a whiteGaussian distribution This assumption enables patch basedmethods such as BM3D and K-SVD to use a Gaussian noisemodel in the similar patches of the image and to perform thedenoising on each 3D stack independently

Conflict of Interests

The authors declare that there is no conflcit of interestsregarding the publication of this paper

Acknowledgments

This study was partially supported by Toshiba CanadaMedical Systems Group and NSERC Canada Grant no 3247-2012

References

[1] D J Brenner and E J Hall ldquoComputed tomographymdashanincreasing source of radiation exposurerdquo The New EnglandJournal of Medicine vol 357 no 22 pp 2277ndash2284 2007

[2] L Yu X Liu S Leng et al ldquoRadiation dose reduction incomputed tomography techniques and future perspectiverdquoImaging in Medicine vol 1 no 1 pp 65ndash84 2009

[3] J Hsieh Computed Tomography Principles Design Artifactsand Recent Advances vol PM188 SPIE Press Book BellinghamWash USA 2nd edition 2009

[4] Z Liao S HuM Li andW Chen ldquoNoise estimation for single-slice sinogram of low-dose X-ray computed tomography usinghomogenous patchrdquoMathematical Problems in Engineering vol2012 Article ID 696212 16 pages 2012

[5] S Hu Z Liao and W Chen ldquoSinogram restoration for low-dosed X-ray computed tomography using fractional-orderPerona-Malik diffusionrdquo Mathematical Problems in Engineer-ing vol 2012 Article ID 391050 13 pages 2012

[6] T Li X Li J Wang et al ldquoNonlinear sinogram smoothing forlow-dose X-ray CTrdquo IEEE Transactions on Nuclear Science vol51 no 5 pp 2505ndash2513 2004

[7] H Lee L Xing R Davidi R Li J Qian and R Lee ldquoImprovedcompressed sensing-based cone-beamCT reconstruction usingadaptive prior image constraintsrdquo Physics in Medicine andBiology vol 57 no 8 pp 2287ndash2307 2012

[8] J Tang B E Nett and G-H Chen ldquoPerformance compar-ison between total variation (TV)-based compressed sensingand statistical iterative reconstruction algorithmsrdquo Physics inMedicine and Biology vol 54 no 19 pp 5781ndash5804 2009

[9] K Li J Tang and G-H Chen ldquoStatistical model based iterativereconstruction (MBIR) in clinical CT systems experimentalassessment of noise performancerdquo Medical Physics vol 41 no4 Article ID 041906 2014


[10] A M Mendrik E-J Vonken A Rutten M A Viergever andB Van Ginneken ldquoNoise reduction in computed tomographyscans using 3-D anisotropic hybrid diffusion with continuousswitchrdquo IEEE Transactions on Medical Imaging vol 28 no 10pp 1585ndash1594 2009

[11] H Rabbani R Nezafat and S Gazor ldquoWavelet-domainmedicalimage denoising using bivariate laplacianmixture modelrdquo IEEETransactions on Biomedical Engineering vol 56 no 12 pp2826ndash2837 2009

[12] A Borsdorf R Raupach T Flohr and J Hornegger ldquoWaveletbased noise reduction in CT-images using correlation analysisrdquoIEEE Transactions on Medical Imaging vol 27 no 12 pp 1685ndash1703 2008

[13] L Scharf and C Demeure Statistical Signal Processing Detec-tion Estimation and Time Series Analysis Addison-WesleyReading Mass USA 1991

[14] F Zhu T Carpenter D R Gonzalez M Atkinson and JWardlaw ldquoComputed tomography perfusion imaging denoisingusing Gaussian process regressionrdquo Physics in Medicine andBiology vol 57 no 12 pp N183ndashN198 2012

[15] S Hyder Ali and R Sukanesh ldquoAn efficient algorithm fordenoising MR and CT images using digital curvelet transformrdquoAdvances in Experimental Medicine and Biology vol 696 pp471ndash480 2011

[16] R Sivakumar ldquoDenoising of computer tomography imagesusing curvelet transformrdquo ARPN Journal of Engineering andApplied Sciences vol 2 no 9 pp 21ndash26 2007

[17] H Rabbani ldquoImage denoising in steerable pyramid domainbased on a local Laplace priorrdquo Pattern Recognition vol 42 no9 pp 2181ndash2193 2009

[18] E Y Sidky Y Duchin X Pan and C Ullberg ldquoA constrainedtotal-variation minimization algorithm for low-intensity x-rayCTrdquoMedical Physics vol 38 no 1 pp S117ndashS125 2011

[19] A R Al-Hinnawi M Daear and S Huwaijah ldquoAssessment ofbilateral filter on 12-dose chest-pelvis CT viewsrdquo RadiologicalPhysics and Technology vol 6 no 2 pp 385ndash398 2013

[20] Y Wang and H Zhou ldquoTotal variation wavelet-based medicalimage denoisingrdquo International Journal of Biomedical Imagingvol 2006 Article ID 89095 6 pages 2006

[21] K Hamalainen L Harhanen A Hauptmann A Kallonen ENiemi and S Siltanen ldquoTotal variation regularization for large-scale X-ray tomographyrdquo International Journal of Tomographyamp Simulation vol 25 no 1 pp 1ndash25 2014

[22] Z Zhu K Wahid P Babyn D Cooper I Pratt and Y CarterldquoImproved compressed sensing-based algorithm for sparse-view CT image reconstructionrdquo Computational and Mathemat-ical Methods in Medicine vol 2013 Article ID 185750 15 pages2013

[23] S Hashemi S Beheshti P R Gill N S Paul and R SC Cobbold ldquoFast fanparallel beam CS-based low-dose CTreconstructionrdquo in Proceedings of the 38th IEEE InternationalConference on Acoustics Speech and Signal Processing (ICASSPrsquo13) pp 1099ndash1103 IEEE May 2013

[24] K Chen E Loli Piccolomini and F Zama ldquoAn automatic reg-ularization parameter selection algorithm in the total variationmodel for image deblurringrdquo Numerical Algorithms vol 67 no1 pp 73ndash92 2014

[25] S Osher A Sole and L Vese ldquoImage decomposition andrestoration using total variation minimization and the 119867

1rdquoMultiscale Modeling and Simulation vol 1 no 3 pp 349ndash3702003

[26] H Liao F Li and M K Ng ldquoSelection of regularizationparameter in total variation image restorationrdquo Journal of theOptical Society of America A vol 26 no 11 pp 2311ndash2320 2009

[27] D L Donoho ldquoDe-noising by soft-thresholdingrdquo IEEE Trans-actions on Information Theory vol 41 no 3 pp 613ndash627 1995

[28] G M Vainikko ldquoThe discrepancy principle for a class ofregularization methodsrdquo USSR Computational Mathematicsand Mathematical Physics vol 22 no 3 pp 1ndash19 1982

[29] P C Hansen ldquoAnalysis of discrete ill-posed problems by meansof the l-curverdquo SIAM Review vol 34 no 4 pp 561ndash580 1992

[30] S Akkoul R Harba and R Ledee ldquoAn image dependentstopping method for iterative denoising proceduresrdquo Multidi-mensional Systems and Signal Processing vol 25 no 3 pp 611ndash620 2014

[31] S Hashemi S Beheshti R S Cobbold and N S Paul ldquoNon-local total variation based low-dose Computed Tomographydenoisingrdquo in Proceedings of the 36th Annual InternationalConference of the IEEE Engineering in Medicine and BiologySociety (EMBC 14) pp 1083ndash1086 Chicago Ill USA August2014

[32] K Dabov A Foi V Katkovnik and K Egiazarian ldquoImagedenoising by sparse 3-D transform-domain collaborative filter-ingrdquo IEEE Transactions on Image Processing vol 16 no 8 pp2080ndash2095 2007

[33] Z Li L Yu J D Trzasko et al ldquoAdaptive nonlocal meansfiltering based on local noise level for CT denoisingrdquo MedicalPhysics vol 41 no 1 Article ID 011908 2014

[34] K Lu N He and L Li ldquoNonlocal means-based denoising formedical imagesrdquo Computational and Mathematical Methods inMedicine vol 2012 Article ID 438617 7 pages 2012

[35] A Buades B Coll and JMMorel ldquoA reviewof image denoisingalgorithms with a new onerdquo SIAM Journal on MultiscaleModeling and Simulation vol 4 no 2 pp 490ndash530 2005

[36] D Kang P Slomka R Nakazato et al ldquoImage denoising oflow-radiation dose coronary CT angiography by an adaptiveblock-matching 3D algorithmrdquo inMedical Imaging 2013 ImageProcessing vol 8669 of Proceedings of SPIE February 2013

[37] K Abhari M Marsousi J Alirezaie and P Babyn ldquoCom-puted Tomography image denoising utilizing an efficient sparsecoding algorithmrdquo in Proceedings of the 11th InternationalConference on Information Science Signal Processing and theirApplications (ISSPA rsquo12) pp 259ndash263 July 2012

[38] D Bartuschat A Borsdorf H Kostler R Rubinstein andM Sturmer A Parallel K-SVD Implementation for CT ImageDenoising Fridrich-Alexander University Erlangen Germany2009

[39] D H Trinh M Luong J M Rocchisani C D Pham H DPham and F Dibos ldquoAn optimal weight method for CT imagedenoisingrdquo Journal of Electronic Scince and Technology vol 10no 2 pp 124ndash129 2012

[40] P Chatterjee and P Milanfar ldquoPatch-based near-optimal imagedenoisingrdquo IEEE Transactions on Image Processing vol 21 no4 pp 1635ndash1649 2012

[41] J Dutta R M Leahy and Q Li ldquoNon-local means denoisingof dynamic PET imagesrdquo PLoS ONE vol 8 no 12 Article IDe81390 2013

[42] F Yu Y Chen and L Luo ldquoCT image denoising based on sparserepresentation using global dictionaryrdquo in Proceedings of the 7thICME International Conference onComplexMedical Engineering(CME rsquo13) pp 408ndash411 May 2013


[43] X Shu J Yang andN Ahuja ldquoNon-local compressive samplingrecoveryrdquo in Proceedings of the IEEE International Conferenceon Computational Photography (ICCP rsquo14) pp 1ndash8 IEEE SantaClara Calif USA May 2014

[44] C Won Kim and J H Kim ldquoRealistic simulation of reduced-dose CT with noise modeling and sinogram synthesis usingDICOM CT imagesrdquo Medical Physics vol 41 no 1 Article ID011901 2014

[45] B R Whiting P Massoumzadeh O A Earl J A OrsquoSullivanD L Snyder and J F Williamson ldquoProperties of preprocessedsinogramdata in x-ray computed tomographyrdquoMedical Physicsvol 33 no 9 pp 3290ndash3303 2006

[46] J Hsieh ldquoAdaptive streak artifact reduction in computedtomography resulting from excessive x-ray photon noiserdquoMedical Physics vol 25 no 11 pp 2139ndash2147 1998

[47] S Beheshti M Hashemi X-P Zhang and N Nikvand ldquoNoiseinvalidation denoisingrdquo IEEE Transactions on Signal Processingvol 58 no 12 pp 6007ndash6016 2010

[48] J M Bioucas-Dias and M A Figueiredo ldquoA new TwISTtwo-step iterative shrinkagethresholding algorithms for imagerestorationrdquo IEEE Transactions on Image Processing vol 16 no12 pp 2992ndash3004 2007

[49] O Axelsson Iterative Solution Methods Cambridge UniversityPress Cambridge UK 1994

[50] M Lebrun ldquoAn analysis and implementation of the BM3Dimage denoising methodrdquo Image Processing On Line vol 2 pp175ndash213 2012

[51] S G Chang B Yu andMVetterli ldquoAdaptivewavelet threshold-ing for image denoising and compressionrdquo IEEETransactions onImage Processing vol 9 no 9 pp 1532ndash1546 2000

[52] ZWang A C Bovik H R Sheikh and E P Simoncelli ldquoImagequality assessment from error visibility to structural similarityrdquoIEEE Transactions on Image Processing vol 13 no 4 pp 600ndash612 2004

[53] T Yamashiro T Miyara O Honda et al ldquoAdaptive iterativedose reduction using three dimensional processing (AIDR3D)improves chest CT image quality and reduces radiation expo-surerdquo PLoS ONE vol 9 no 8 Article ID e105735 2014

Submit your manuscripts athttpwwwhindawicom

Stem CellsInternational

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


MEDIATORSINFLAMMATION

of


Behavioural Neurology

EndocrinologyInternational Journal of



Disease Markers


BioMed Research International

OncologyJournal of



Oxidative Medicine and Cellular Longevity


PPAR Research

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Immunology ResearchHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

ObesityJournal of



Computational and Mathematical Methods in Medicine

OphthalmologyJournal of


Diabetes ResearchJournal of



Research and TreatmentAIDS


Gastroenterology Research and Practice


Parkinsonrsquos Disease

Evidence-Based Complementary and Alternative Medicine

Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom


Theorem (CLT) [13] the final noise in each voxel follows aGaussian distribution [11 14ndash17] The CLT can be used sinceeach voxel in CT images is computed by adding values frommany different projectionsWith this assumption a noisy CTimage y can be modeled by

y = x +n (1)

where x is the noiseless image and n is a zero mean additiveanisotropic Gaussian noise with variance of 1205902

119899 which varies

with the pixel location and its valueDifferent image based denoising algorithms have been

used to estimate the noiseless CT images such as anisotropicdiffusion [10] total variation (TV) [18] bilateral filtering[19] or wavelet-based techniques [11 12 20] These methodscan usually be formulated as an unconstrained Lagrangianmultiplier optimization problem [18 21ndash23] that is

119909 = arg min119909

121003817100381710038171003817

119909 minus y1003817100381710038171003817

22 +120582ℎ (119909) (2)

in which 120582 is a regularization parameter that controls thetradeoff between the data fidelity and the regularization termℎ(119909) and 119909

119902= sum

119899

119894=1 |119909119894|119902 Different regularization terms

ℎ(119909) lead to different denoising methods For example inTV-based methods [24ndash26] ℎ(119909) = radic120575

ℎ119909

2+ 120575V119909

2 where 120575ℎ

and 120575V are the gradient in horizontal and vertical directionsand inwavelet soft thresholdingmethods ℎ(119909) = sum

119894|Ψ2D(119909)119894|

where Ψ2D(119909) is the 2D wavelet transform [27]There is a strong dependence of the quality of the result

on the regularization parameter It is a challenging task tofind the regularization parameter 120582 that provides the bestbalance between signal smoothing and feature preservation[26] Specifically if 120582 is not appropriately adjusted theoptimization is trapped in a local minimum that is if 120582 is toosmall noise is only partially removed and if it is too large theimage may be oversmoothed [25] Some methods have beenproposed to update the regularization parameters iterativelysuch as use of the discrepancy principle [28] generalizedcross-validation [26] and L-curve [29] These methods failin certain situations are problem specific and generallyincrease the computational complexity of the algorithms

One straight forward approach used in many algorithmsis to use a heuristic 120582 value combined with a criterion to stopthe algorithm before the estimated signal is oversmoothedDifferent stopping criteria have been proposed for iterativedenoising problems For instance Akkoul et al [30] useda switching median filter algorithm to stop the iterativeprocess when the number of changed pixels in the denoisingiterations is a minimum In [20 31] the statistical propertiesof high frequency wavelet subbands were used to stop the TViterations However such methods are unable to differentiateoversmoothed data from well-denoised data As a result toavoid oversmoothing the updating steps are typically chosento be small which decreases the convergence speed

In this paper a noise confidence region evaluation(NCRE) method is used to address the regularization selec-tion and the algorithm stopping problems It adaptivelyupdates the regularization parameters at the end of each

iteration by validating the result of that iteration The algo-rithm stops when the statistical properties of the denoisingresidual resemble those of the additive white Gaussian noiseUsing NCRE a new iterative block matching and 3D filtering(BM3D)method is proposed which outperforms BM3D [32]itself The proposed method is compared with anisotropicdiffusion denoising which is generally regarded as a standarddenoising method in CT imaging [10] nonlocal mean [33ndash35] and BM3D [32] In addition we study the noise prop-erties of CT images and show that the noise in small imageblocks has an additive white Gaussian model which justifiesthe success of the nonlocal based denoising algorithms inimage based CT denoising [33 36ndash39]

2 Problem Formulation andCT Noise Properties

Recently it has been shown that nonlocal patch basedalgorithms outperform others in CT image denoising [33 3436ndash41] For example in [41] a nonlocal means (NLM) basedmethod which takes advantage of the presence of repeatingstructures in a given image was compared with a principlecomponent analysis based denoising method and a highlyconstrained backprojection method It was shown that theNLM method outperformed both of the methods in termsof the contrast to noise ratio noise standard deviation andsquared error Another class of algorithms looks for similarblocks in the whole 2D image and stacks them together in3D arrays Denoising is then performed through transformdomain shrinkage of the 3D arrays An algorithm called K-SVD [37 38 42] uses these 3D patches to train an optimumdictionary This method which assumes that each 3D blockcan sparsely be represented by the trained dictionary atomsuses shrinkage algorithms to denoise the patches

Our proposed algorithmmakes use of the blockmatchingand 3D filtering (BM3D) technique [32 36] This is a nonit-erative denoising method that currently outperforms manynewer algorithms [40] It is composed of two major filteringsteps In both stages collaborative filtering is utilized whichitself has four stages (1) grouping similar patches with areference patch (2) calculation of the 3D wavelet coefficientsof each stack of patches (3) denoising the wavelet coefficients(thresholding in step (1) or Wiener filtering in step (2))and (4) recovering the denoised image by calculating theinverse 3Dwavelet transformationTheBM3Dapproach aimsto denoise the patches by Wiener filtering which is done instep (2) To find best match to similar patches in the step (2)and to reliably estimate the Wiener coefficients the methodrequires a reliable estimate of the noiseless image which is themain purpose of step (1) The input for this step which is ahard thresholding block is the 3D noisy wavelet coefficientsof similar patches located by block matching applied to theavailable noisy image A hard threshold with a heuristicallydetermined value of 27120590

119899is used in step (1) The resulting

denoised coefficients are then transformed back to a spatialdomain to be used as the initial estimate of the noiseless dataused for calculating the Wiener filter coefficients



y119866119895= x119866119895+n119866119895 119895 isin [1 119898] (3)



119866119895


sim N(0 1205902119866119894)




NLW (119909 119866) =

119898

sum

119895=1

100381710038171003817100381710038171003817

Ψ3D (119909119866119895)1003817100381710038171003817100381710038171 (4)


x = arg min119909

121003817100381710038171003817

119909 minus y1003817100381710038171003817

22 +120582119898

sum

119895=1

100381710038171003817100381710038171003817

Ψ3D (119909119866119895)1003817100381710038171003817100381710038171 (5)




120590

2119894119899=radic(119869119894times exp(120585 (119894 119899)

119904

))

2

+ (120590

2119890)

119894119895

(6)




120590




119891 (119909 119910) =

120587Δ119905

119873120587119892

119873120587119892

sum

119894=1

119873119892

sum

119899=1119888 (119909 cos120593

119894minus119910 sin120593

119894minus 119897119899) 119892119894119899 (7)


120590

2119899(119909 119910)

= (

120587Δ119905

119873120587119892

)

2 119873120587119892sum

119894=1

119873119892

sum

119899=1119888

2(119909 cos120593

119894minus119910 sin120593

119894minus 119897119899) 120590

2119894119899

(8)




2119894119899



119866119894 Using (3) the


119869 (y 119866 Λ) = x

= arg min119909

121003817100381710038171003817

119909 minus y1003817100381710038171003817

22

+

119898

sum

119895=1120582119895

100381710038171003817100381710038171003817

Ψ3D (119909119866119895)1003817100381710038171003817100381710038171

(9)











that of n


G (119911n) = 1119899

119899

sum

119895=1G (119911n

119895)

G (119911n119895) =

1 if 10038161003816100381610038161003816

n119895

10038161003816100381610038161003816

le 119911

0 if 10038161003816100381610038161003816

n119895

10038161003816100381610038161003816

gt 119911

(10)






119899) and upper (119880

119899) values

119871119899(119911) = 119865 (119911) minus 120577

radic

1119899

119865 (119911) (1 minus 119865 (119911))

119880119899(119911) = 119865 (119911) + 120577

radic

1119899

119865 (119911) (1 minus 119865 (119911))

(11)

















119895

0 1 2 3 4 5

010203040506070809

1

Region I

Region II

Region III

z

g(zn)



119895= 119904 times 120582




119894) and 119865(119911) 119889

119894= G(119911 Δy

119894) minus 119865(119911)



y119894= (1minus120572

119896) y +120573

119896x119894+ (120572119896minus120573119896) x119894minus1 (12)






(a)


0002

0004

0006

0008

001

0012

0014

CT values (HU)

Occ

uran

ce p

roba

bilit

y

(b)






x119894= 119869(y119894 119866119894 Λ119894)




x119894= 119869(y119894 119866119894 Λ119894)




119894|) 119894 larr 119894 + 1

end while



119866119894with


119894




5 10 15 20 25 30 35 40

001002003004005006007008

Nor

mal

ized

squa

red

erro

r

086088090920940960981

SSIM

Iteration number

Region I

Region IIRegion III

Stopping point






(a) (b) (c) (d)





000

minus500 0

500

(a)

minus100

0minus8

00minus6

00minus4

00minus2

00 020

040

0

(b)

minus100

0

minus500 0

500

(c)

minus100

0

minus500 0

500

(d)

minus100

0

minus500 0

500

(e)

(f) (g) (h) (i) (j)

minus50

minus40

minus30

minus20

minus10 0

10 20 30 40 50

(k)

minus60

minus40

minus20 0 20 40 60

(l)

minus20

minus30

minus40

minus50

minus10 0 10 20 30 40 50

(m)minus2

0minus3

0minus4

0minus5

0

minus10 0 10 20 30 40 50

(n)









0005

001

0015

002

0025

003

0035

004

CT values (HU)

Occ

uran

ce p

roba

bilit

y


0005

001

0015

002

0025

003

0035

CT values (HU)

Occ

uran

ce p

roba

bilit

y


0002

0004

0006

0008

001

0012

0014

0016

0018

CT values (HU)

Occ

uran

ce p

roba

bilit

y


1

2

3

4

5

6

7

8

9

CT values (HU)

Occ

uran

ce p

roba

bilit

y






Varia

nce o

f the

RO

Is


102

103






5 Conclusions


Appendix





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Acknowledgments


References






























































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of


















y119866119895= x119866119895+n119866119895 119895 isin [1 119898] (3)



119866119895


sim N(0 1205902119866119894)




NLW (119909 119866) =

119898

sum

119895=1

100381710038171003817100381710038171003817

Ψ3D (119909119866119895)1003817100381710038171003817100381710038171 (4)


x = arg min119909

121003817100381710038171003817

119909 minus y1003817100381710038171003817

22 +120582119898

sum

119895=1

100381710038171003817100381710038171003817

Ψ3D (119909119866119895)1003817100381710038171003817100381710038171 (5)




120590

2119894119899=radic(119869119894times exp(120585 (119894 119899)

119904

))

2

+ (120590

2119890)

119894119895

(6)




120590




119891 (119909 119910) =

120587Δ119905

119873120587119892

119873120587119892

sum

119894=1

119873119892

sum

119899=1119888 (119909 cos120593

119894minus119910 sin120593

119894minus 119897119899) 119892119894119899 (7)


120590

2119899(119909 119910)

= (

120587Δ119905

119873120587119892

)

2 119873120587119892sum

119894=1

119873119892

sum

119899=1119888

2(119909 cos120593

119894minus119910 sin120593

119894minus 119897119899) 120590

2119894119899

(8)




2119894119899



119866119894 Using (3) the


119869 (y 119866 Λ) = x

= arg min119909

121003817100381710038171003817

119909 minus y1003817100381710038171003817

22

+

119898

sum

119895=1120582119895

100381710038171003817100381710038171003817

Ψ3D (119909119866119895)1003817100381710038171003817100381710038171

(9)











that of n


G (119911n) = 1119899

119899

sum

119895=1G (119911n

119895)

G (119911n119895) =

1 if 10038161003816100381610038161003816

n119895

10038161003816100381610038161003816

le 119911

0 if 10038161003816100381610038161003816

n119895

10038161003816100381610038161003816

gt 119911

(10)






119899) and upper (119880

119899) values

119871119899(119911) = 119865 (119911) minus 120577

radic

1119899

119865 (119911) (1 minus 119865 (119911))

119880119899(119911) = 119865 (119911) + 120577

radic

1119899

119865 (119911) (1 minus 119865 (119911))

(11)

















119895

0 1 2 3 4 5

010203040506070809

1

Region I

Region II

Region III

z

g(zn)



119895= 119904 times 120582




119894) and 119865(119911) 119889

119894= G(119911 Δy

119894) minus 119865(119911)



y119894= (1minus120572

119896) y +120573

119896x119894+ (120572119896minus120573119896) x119894minus1 (12)






(a)


0002

0004

0006

0008

001

0012

0014

CT values (HU)

Occ

uran

ce p

roba

bilit

y

(b)






x119894= 119869(y119894 119866119894 Λ119894)




x119894= 119869(y119894 119866119894 Λ119894)




119894|) 119894 larr 119894 + 1

end while



119866119894with


119894




5 10 15 20 25 30 35 40

001002003004005006007008

Nor

mal

ized

squa

red

erro

r

086088090920940960981

SSIM

Iteration number

Region I

Region IIRegion III

Stopping point






(a) (b) (c) (d)





000

minus500 0

500

(a)

minus100

0minus8

00minus6

00minus4

00minus2

00 020

040

0

(b)

minus100

0

minus500 0

500

(c)

minus100

0

minus500 0

500

(d)

minus100

0

minus500 0

500

(e)

(f) (g) (h) (i) (j)

minus50

minus40

minus30

minus20

minus10 0

10 20 30 40 50

(k)

minus60

minus40

minus20 0 20 40 60

(l)

minus20

minus30

minus40

minus50

minus10 0 10 20 30 40 50

(m)minus2

0minus3

0minus4

0minus5

0

minus10 0 10 20 30 40 50

(n)









0005

001

0015

002

0025

003

0035

004

CT values (HU)

Occ

uran

ce p

roba

bilit

y


0005

001

0015

002

0025

003

0035

CT values (HU)

Occ

uran

ce p

roba

bilit

y


0002

0004

0006

0008

001

0012

0014

0016

0018

CT values (HU)

Occ

uran

ce p

roba

bilit

y


1

2

3

4

5

6

7

8

9

CT values (HU)

Occ

uran

ce p

roba

bilit

y






Varia

nce o

f the

RO

Is


102

103






5 Conclusions


Appendix





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006

CT values (HU)

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roba

bilit

y


001

002

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005

006

007

CT values (HU)

Occ

uran

ce p

roba

bilit

y





Acknowledgments


References






























































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of























that of n


G (119911n) = 1119899

119899

sum

119895=1G (119911n

119895)

G (119911n119895) =

1 if 10038161003816100381610038161003816

n119895

10038161003816100381610038161003816

le 119911

0 if 10038161003816100381610038161003816

n119895

10038161003816100381610038161003816

gt 119911

(10)






119899) and upper (119880

119899) values

119871119899(119911) = 119865 (119911) minus 120577

radic

1119899

119865 (119911) (1 minus 119865 (119911))

119880119899(119911) = 119865 (119911) + 120577

radic

1119899

119865 (119911) (1 minus 119865 (119911))

(11)

















119895

0 1 2 3 4 5

010203040506070809

1

Region I

Region II

Region III

z

g(zn)



119895= 119904 times 120582




119894) and 119865(119911) 119889

119894= G(119911 Δy

119894) minus 119865(119911)



y119894= (1minus120572

119896) y +120573

119896x119894+ (120572119896minus120573119896) x119894minus1 (12)






(a)


0002

0004

0006

0008

001

0012

0014

CT values (HU)

Occ

uran

ce p

roba

bilit

y

(b)






x119894= 119869(y119894 119866119894 Λ119894)




x119894= 119869(y119894 119866119894 Λ119894)




119894|) 119894 larr 119894 + 1

end while



119866119894with


119894




5 10 15 20 25 30 35 40

001002003004005006007008

Nor

mal

ized

squa

red

erro

r

086088090920940960981

SSIM

Iteration number

Region I

Region IIRegion III

Stopping point






(a) (b) (c) (d)





000

minus500 0

500

(a)

minus100

0minus8

00minus6

00minus4

00minus2

00 020

040

0

(b)

minus100

0

minus500 0

500

(c)

minus100

0

minus500 0

500

(d)

minus100

0

minus500 0

500

(e)

(f) (g) (h) (i) (j)

minus50

minus40

minus30

minus20

minus10 0

10 20 30 40 50

(k)

minus60

minus40

minus20 0 20 40 60

(l)

minus20

minus30

minus40

minus50

minus10 0 10 20 30 40 50

(m)minus2

0minus3

0minus4

0minus5

0

minus10 0 10 20 30 40 50

(n)









0005

001

0015

002

0025

003

0035

004

CT values (HU)

Occ

uran

ce p

roba

bilit

y


0005

001

0015

002

0025

003

0035

CT values (HU)

Occ

uran

ce p

roba

bilit

y


0002

0004

0006

0008

001

0012

0014

0016

0018

CT values (HU)

Occ

uran

ce p

roba

bilit

y


1

2

3

4

5

6

7

8

9

CT values (HU)

Occ

uran

ce p

roba

bilit

y






Varia

nce o

f the

RO

Is


102

103






5 Conclusions


Appendix





001

002

003

004

005

006

CT values (HU)

Occ

uran

ce p

roba

bilit

y


001

002

003

004

005

006

007

CT values (HU)

Occ

uran

ce p

roba

bilit

y





Acknowledgments


References






























































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of

















(a)


0002

0004

0006

0008

001

0012

0014

CT values (HU)

Occ

uran

ce p

roba

bilit

y

(b)






x119894= 119869(y119894 119866119894 Λ119894)




x119894= 119869(y119894 119866119894 Λ119894)




119894|) 119894 larr 119894 + 1

end while



119866119894with


119894




5 10 15 20 25 30 35 40

001002003004005006007008

Nor

mal

ized

squa

red

erro

r

086088090920940960981

SSIM

Iteration number

Region I

Region IIRegion III

Stopping point






(a) (b) (c) (d)





000

minus500 0

500

(a)

minus100

0minus8

00minus6

00minus4

00minus2

00 020

040

0

(b)

minus100

0

minus500 0

500

(c)

minus100

0

minus500 0

500

(d)

minus100

0

minus500 0

500

(e)

(f) (g) (h) (i) (j)

minus50

minus40

minus30

minus20

minus10 0

10 20 30 40 50

(k)

minus60

minus40

minus20 0 20 40 60

(l)

minus20

minus30

minus40

minus50

minus10 0 10 20 30 40 50

(m)minus2

0minus3

0minus4

0minus5

0

minus10 0 10 20 30 40 50

(n)









0005

001

0015

002

0025

003

0035

004

CT values (HU)

Occ

uran

ce p

roba

bilit

y


0005

001

0015

002

0025

003

0035

CT values (HU)

Occ

uran

ce p

roba

bilit

y


0002

0004

0006

0008

001

0012

0014

0016

0018

CT values (HU)

Occ

uran

ce p

roba

bilit

y


1

2

3

4

5

6

7

8

9

CT values (HU)

Occ

uran

ce p

roba

bilit

y






Varia

nce o

f the

RO

Is


102

103






5 Conclusions


Appendix





001

002

003

004

005

006

CT values (HU)

Occ

uran

ce p

roba

bilit

y


001

002

003

004

005

006

007

CT values (HU)

Occ

uran

ce p

roba

bilit

y





Acknowledgments


References






























































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of

















(a) (b) (c) (d)





000

minus500 0

500

(a)

minus100

0minus8

00minus6

00minus4

00minus2

00 020

040

0

(b)

minus100

0

minus500 0

500

(c)

minus100

0

minus500 0

500

(d)

minus100

0

minus500 0

500

(e)

(f) (g) (h) (i) (j)

minus50

minus40

minus30

minus20

minus10 0

10 20 30 40 50

(k)

minus60

minus40

minus20 0 20 40 60

(l)

minus20

minus30

minus40

minus50

minus10 0 10 20 30 40 50

(m)minus2

0minus3

0minus4

0minus5

0

minus10 0 10 20 30 40 50

(n)









0005

001

0015

002

0025

003

0035

004

CT values (HU)

Occ

uran

ce p

roba

bilit

y


0005

001

0015

002

0025

003

0035

CT values (HU)

Occ

uran

ce p

roba

bilit

y


0002

0004

0006

0008

001

0012

0014

0016

0018

CT values (HU)

Occ

uran

ce p

roba

bilit

y


1

2

3

4

5

6

7

8

9

CT values (HU)

Occ

uran

ce p

roba

bilit

y






Varia

nce o

f the

RO

Is


102

103






5 Conclusions


Appendix





001

002

003

004

005

006

CT values (HU)

Occ

uran

ce p

roba

bilit

y


001

002

003

004

005

006

007

CT values (HU)

Occ

uran

ce p

roba

bilit

y





Acknowledgments


References






























































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of

















000

minus500 0

500

(a)

minus100

0minus8

00minus6

00minus4

00minus2

00 020

040

0

(b)

minus100

0

minus500 0

500

(c)

minus100

0

minus500 0

500

(d)

minus100

0

minus500 0

500

(e)

(f) (g) (h) (i) (j)

minus50

minus40

minus30

minus20

minus10 0

10 20 30 40 50

(k)

minus60

minus40

minus20 0 20 40 60

(l)

minus20

minus30

minus40

minus50

minus10 0 10 20 30 40 50

(m)minus2

0minus3

0minus4

0minus5

0

minus10 0 10 20 30 40 50

(n)









0005

001

0015

002

0025

003

0035

004

CT values (HU)

Occ

uran

ce p

roba

bilit

y


0005

001

0015

002

0025

003

0035

CT values (HU)

Occ

uran

ce p

roba

bilit

y


0002

0004

0006

0008

001

0012

0014

0016

0018

CT values (HU)

Occ

uran

ce p

roba

bilit

y


1

2

3

4

5

6

7

8

9

CT values (HU)

Occ

uran

ce p

roba

bilit

y






Varia

nce o

f the

RO

Is


102

103






5 Conclusions


Appendix





001

002

003

004

005

006

CT values (HU)

Occ

uran

ce p

roba

bilit

y


001

002

003

004

005

006

007

CT values (HU)

Occ

uran

ce p

roba

bilit

y





Acknowledgments


References






























































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of



















0005

001

0015

002

0025

003

0035

004

CT values (HU)

Occ

uran

ce p

roba

bilit

y


0005

001

0015

002

0025

003

0035

CT values (HU)

Occ

uran

ce p

roba

bilit

y


0002

0004

0006

0008

001

0012

0014

0016

0018

CT values (HU)

Occ

uran

ce p

roba

bilit

y


1

2

3

4

5

6

7

8

9

CT values (HU)

Occ

uran

ce p

roba

bilit

y






Varia

nce o

f the

RO

Is


102

103






5 Conclusions


Appendix





001

002

003

004

005

006

CT values (HU)

Occ

uran

ce p

roba

bilit

y


001

002

003

004

005

006

007

CT values (HU)

Occ

uran

ce p

roba

bilit

y





Acknowledgments


References






























































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of


















Varia

nce o

f the

RO

Is


102

103






5 Conclusions


Appendix





001

002

003

004

005

006

CT values (HU)

Occ

uran

ce p

roba

bilit

y


001

002

003

004

005

006

007

CT values (HU)

Occ

uran

ce p

roba

bilit

y





Acknowledgments


References






























































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of


















001

002

003

004

005

006

CT values (HU)

Occ

uran

ce p

roba

bilit

y


001

002

003

004

005

006

007

CT values (HU)

Occ

uran

ce p

roba

bilit

y





Acknowledgments


References






























































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of




































































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of

































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of





















of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of
















Documents

Research Article Adaptively Tuned Iterative Low Dose CT ...downloads.hindawi.com/journals/cmmm/2015/638568.pdfdenoising method in CT imaging [ ], nonlocal mean [ ], and BMD [ ]. In