Comparison of Denoising Methods in Diffusion Tensor Imagingpeople.rajagiritech.ac.in/sites/default/files/arunab/files/06726141.pdfComparison of Denoising Methods in Diffusion Tensor

Comparison of Denoising Methods in DiffusionTensor Imaging

Solwin Johnson∗, Arun A. Balakrishnan†Department of Applied Electronics and Instrumentation

Rajagiri School of Engineering and TechnologyKakkanad, Cochin, India

Email: [email protected]∗, [email protected]†

Abstract—In this paper, a non linear adaptive gaussian de-noising method for diffusion tensor imaging (DTI) is proposed.DTI image are of poor SNR and low resolution images. Inorder to improve DTI, the proposed method is applied to thediffusion weighted images (DWI) from which DTI is computed.The anisotropic flow principle is used in non linear adaptivegaussian denoising method and smoothing will vary accordingto the anisotropic flow. The proposed method is compared withthe scalar Partial Differential Equation (PDE) denoising method.The non linear adaptive gaussian denoising method shows betterperformance compared to scalar PDE. To evaluate the efficiencyof both denoising methods, image quality metrics like peaksignal-to-noise ratio (PSNR) and mean structural similarity indexmeasure (MSSIM) are used. The experimental results indicatethe good performance of proposed method and it has a betterdenoising effect in DTI compared to scalar PDE.

Index Terms—Diffusion Tensor Imaging (DTI); Diffusionweighted (DW) image; Non linear adaptive gaussian method;Peak signal-to-noise ratio (PSNR); Mean structural similarityindex measure (MSSIM).

I. INTRODUCTION

Diffusion Tensor Imaging (DTI) is recently developed non-invasive medical brain imaging technique which is moreadvanced than MRI and CT. DTI is used to image the threedimensional diffusion of water molecules occurring in thebrain. DTI gives the white matter fiber orientation detailswhich are important in diagnosis purposes and for planningsurgery. For DTI, acquisition of diffusion weighted (DW)images are needed. DW images contain diffusion information.From DW images, signal attenuation which is proportionalto the diffusion is calculated for computing diffusion tensor[1]. Diagonalization process of the diffusion tensor gives theeigen values and eigen vectors. Clinically important FractionalAnisotropic maps (FA maps) and colour images are derivedfrom the eigen values and eigen vectors [1]–[4].

DTI images are of poor SNR, low contrast and lowresolution due to the limitation accompanied by the DWimage acquisition procedures such as acquiring large volumeof data and time constraints. One method to increase thesignal strength of DTI is by increasing the number of DWdirection. Practical limitations are there for increasing DWdirection, i.e. more time, space and cost are needed. Anothermethod to increase the quality of DTI is by using pre-tensoror post-tensor based image quality enhancement methods in

DTI. Current approaches to increase the image quality ofDTI are by applying pre-tensor based structural preserveddenoising methods. Scalar PDE is commonly used structurepreserved denoising method. In scalar PDE denoising method,anisotropic diffusion principle is used and smoothing is doneby preserving structural details.

Rest of the paper is organised as follows. Section II explainsthe scalar PDE denoising method. Section III discusses theproposed method which is a non linear adaptive Gaussiandenoising method based on the principle of anisotropic flow. Insection IV image quality measures are explained. Experimentalresults are explained in section V and finally section VIconcludes the paper.

II. SCALAR PDE

Scalar PDE is a denoising method, which uses imageresidual-energy criterion to equate the degree of smoothing [5].Diffusion weighted three dimensional DW images undergoingsmoothing is represented as a time-varying function I(x, y, z,t), subjected to the diffusion equation as in (1).

∂I

∂t= ∇ · (r(|∇I|)∇I) (1)

In scalar PDE the degree of smoothing is controlled locallyby the rate constant function r(|∇I|) and globally by the timevariable, t. Smoothing processes among the individual com-ponents x, y and z must be coupled by differential geometricalnorm |G| given in (2).

|G| =

√(∂I

∂x

)2

+

(∂I

∂y

)2

+

(∂I

∂z

)2

(2)

In scalar PDE filtering, the smoothing process dependson the smoothing function r(β), where β denotes |G| or|∇I|. For edge preservation in diffusion weighted images,r(|G|) should decrease monotonically with respect to |G|. Toenhance smoothing in noise effected homogeneous regions andsuppress it in heterogeneous regions, the smoothing functionr(β) should take on a sigmoidal shape with tapered slopes atboth extremes of the |G| axis.

r(β) =1√

1 +G2(3)

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2013 Annual IEEE India Conference (INDICON)

Diffusion equation of scalar PDE filter described in (1) fora three dimensional image I=I(x, y, z, t) can be alternativelywritten [5] as given in (4).

∂I

∂t=

∂

∂x

(r (|∇I|) ∂I

∂x

)+

∂

∂y

(r (|∇I|) ∂I

∂y

)+

∂

∂z

(r (|∇I|) ∂I

∂z

)(4)

By making the step sizes in all directions equal to unity(∆x = ∆y = ∆z = 1) and replacing the variables x, y, z, andt with the discrete spatial indices i, j, k, and iteration numbern lead to the difference equation [5] as given in (5).

In+1 = In[1− ∆t

2

(rni+1 + rni−1 + rnj+1 + rnj−1

+rnk+1 + rnk−1

)]+

∆t

2(rni+1I

ni+1 + (5)

rni−1Ini−1 + rnj+1I

nj+1 + rnj−1I

nj−1

+rnk+1Ink+1 + rnk−1I

nk−1)

where r is the smoothing function. For better stability, stepsize ∆t = 0.1 was used.

III. PROPOSED METHOD

Consider two dimensional diffusion weighted noisy imagev taken from a three dimensional diffusion weighted imageI. In proposed method a non-linear adaptive gaussian filter isused as a denoising method. Gaussian filter is used to removenoise from an image. The problem of linear gaussian filteris that it will uniformly smooth the biomedical images. Theedges where cross boundaries are there in DW images, arenot preserved by linear gaussian filter. In order to preserve theedges, smoothing must be less in cross boundaries or edgesand in uniform regions of white or grey matter of the brainsmoothing must be more.

For control smoothing, a diffusion coefficient Dc is pro-posed. Dc is derived from the gradient g of the noisy imagev as in (6) based on smoothing process discussed in (2) and(3).

g =

√(∂v

∂x

)2

+

(∂v

∂y

)2

(6)

Diffusion coefficient Dc is calculated from gradient g as

Dc =1√

1 + g2(7)

Dc should decrease monotonically with respect to g. Diffusioncoefficient is used as smoothing function. Value of Dc will bemore in uniform region where more smoothing is needed andvalue of Dc will be less in regions where edges are there.In DW images the flow of diffusion in x and y direction isdenoted as fx and fy given in (8) and (9).

fx = Dc ·(∂v

∂x

)(8)

fy = Dc ·(∂v

∂y

)(9)

Total flow F can be calculated as

F =√

(fx)2 + (fy)2 (10)

The non-linear gaussian filter h(p, q) applied to the imagev(p, q) is given by (11).

h(p, q) =1

2πσ2e−

(p2 + q2

2σ2

)(11)

where p and q be the pixel location of the image. Thestandard deviation σ depends on the total flow F of diffu-sion of the image v. Thus in proposed non linear adaptiveGaussian denoising method, an anisotropic flow principle isused. Anisotropic flow depends upon the intensity differencebetween the neighbouring pixels of DW images. According tothe anisotropic flow smoothing will vary.

IV. IMAGE QUALITY METRICS

The peak-signal-to-noise ratio (PSNR) and structural sim-ilarity index measure (SSIM) are two well-known objectiveimage quality metrics. Consider a reference image x and atest image y, both of size M×N, the PSNR between x and yis defined as

PSNR(x, y) = 10 log10

[max(y)2

MSE(x, y)

](12)

where

MSE(x, y) =1

MN

M∑i=1

N∑j=1

(xi,j − yi,j)2 (13)

where x is the original image and y is the processed image.Higher PSNR [6] values gives a higher image quality sinceMSE approaches zero when PSNR value approaches infinityand smaller values of PSNR indicate high numerical differ-ences between images. Well-known quality metric known asSSIM is for measuring the similarity between two imagesutilising luminance function, contrast function and structuralfunction. The SSIM [6], [7] is defined as

SSIM(x, y) = l(x, y)c(x, y)s(x, y) (14)

The closeness of the luminance of two images measured usingthe luminance comparison function l(x,y) is maximum i.e unityonly if µx = µy .

l(x, y) =2µxµy + C1

µ2x + µ2

y + C1(15)

µx is the mean of the original image and µy is the mean ofthe processed image. µx is calculated as

µx =1

MN

M∑i=1

N∑j=1

xi,j (16)

Fig. 1. Comparison of Scalar PDE filtered FA maps with different iterations.(a) Original FA map (b) iteration=1 (c) iteration=5 (d) iteration=10.

Constant C1 is used to avoid instability when µ2x+µ2

y is veryclose to zero. C1 is selected as in (17).

C1 = (K1L)2 (17)

where L is the dynamic range of the pixel values and K1 is asmall constant(K1 � 1).

The closeness of the contrast of the two images measuredusing contrast comparison function c(x,y) is maximum i.e unityonly if σx = σy . σx is the standard deviation of the originalimage and σy is the standard deviation of the processed image.

c(x, y) =2σxσy + C2

σ2x + σ2

y + C2(18)

where C2=(K2L)2 and K2 � 1. Since covariance measuresthe correlation between the two images are measured usingthe structure comparison function represented by s(x,y). Thecovariance between the images x and y is represented by σxy .

s(x, y) =σxy + C3

σxσy + C3(19)

where positive constant C3=C2/2. The positive constants C1,C2, and C3 are used to avoid a null denominator.

Finally combining three functions given in (15), (18) and(19), SSIM can be obtained as

SSIM(x, y) =(2µxµy + C1)(2σxy + C2)

(µ2x + µ2

y + C1)(σ2x + σ2

y + C2)(20)

Mean SSIM (MSSIM) index is used to evaluate the overallimage quality of FA map as given in

MSSIM(X,Y ) =1

Q

Q∑i=1

SSIM(xi, yi) (21)

where the original and the processed images represented byX and Y has xi and yi as the image contents at the ith localwindow with Q being the number of local windows of theimage. The positive values of the MSSIM index are in therange of [0, 1]. The value 0 and value 1 of the MSSIM indexindicate zero correlation between images and high similaritybetween images respectively. For evaluating the performanceof scalar PDE and proposed method, PSNR and SSIM is used.

TABLE IQUALITY MEASURE FOR SCALAR PDE

Scalar PDE

Figure No.iterations PSNR MSSIM

Fig. 1(b) 1 19.49 dB 0.8908

Fig. 1(c) 5 19.47 dB 0.8431

Fig. 1(d) 10 19.32 dB 0.7927

TABLE IICOMPARISON OF QUALITY MEASURE FOR PROPOSED METHOD WITH

SCALAR PDE

Proposed Method Scalar PDE Method

Figure PSNR MSSIM Figure PSNR MSSIM

Fig. 3(b) 29.9 dB 0.9892 Fig. 1(b) 19.49 dB 0.8908

Fig. 2. (a), (b), (c) Diffusion weighted image (d) Original FA map (e) ScalarPDE FA map (f) Proposed method FA map (g) Original FA color map (h)Scalar PDE FA color map (i) Proposed method FA color map.

V. EXPERIMENTAL RESULTS

In this paper scalar PDE denoising method is comparedwith the proposed method. Denoising method in DTI is usedto remove noises from DTI. In Fig. 1, iterative results ofscalar PDE are compared. Number of iterations, i.e. numberof times scalar PDE denoising method has been applied toa particular three dimensional DW image and the resultsare shown in Table I. Experimental results from the TableI shows that scalar PDE method has a PSNR value of 19.49dB in iteration 1 and on increasing iterations or smoothingthe PSNR value tend to be decreasing. The MSSIM value

Fig. 3. (a) Original FA map (b) Proposed method FA map

for scalar PDE is found to be 0.8908 on iteration 1. Forincreasing smoothing the MSSIM value, i.e mean structuralsimilarity is found to be decreasing. Proposed method is anon-linear adaptive gaussian method. From Table II the resultof proposed method shows that it has a PSNR value of 29.9dB and has MSSIM value of 0.9892. Simulation results showsthat proposed method outperforms existing scalar PDE methodquantitatively as well as qualitatively. In Fig. 2 the scalar PDEand proposed denoising methods are compared and respectivecolor maps are shown. Original FA map and proposed FA mapis shown in Fig. 3. Experimental results show that proposedmethod is showing better performance compared to scalarPDE.

VI. CONCLUSION

In this paper, scalar PDE denoising method is comparedwith the proposed method using DTI dataset that has a voxelsize of 256×256×30×10. Scalar PDE is a commonly used

structural preserved denoising method in DTI. Anisotropic dif-fusion principle is used in scalar PDE and smoothing is doneby preserving structural details. Proposed method is a nonlinear adaptive gaussian method. Anisotropic flow principle isused in proposed method and smoothing will vary according tothe anisotropic flow. Performance of both denoising methodsare compared using PSNR and MSSIM quality metrices. Fromexperimental results, proposed denoising method is showingbetter results compared to scalar PDE method. From visualand numerical results, proposed method is also showing betterperformance in reducing noises in DTI compared to scalarPDE.

REFERENCES

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