An Application of Second Generation Wavelets for Image Denoising

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ACEEE Int. J. on Signal and Image Processing , Vol. 4, No. 3, Sept 2013

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An Application of Second Generation Wavelets forImage Denoising using Dual Tree Complex Wavelet

TransformSK.Umar Faruq1, Dr.K.V.Ramanaiah2, Dr.K.Soundara Rajan3

1Quba college of Engineering &Technology, Nellore, A.P, IndiaEmail : [email protected]

2NBKR Institute of Science & Technology, Nellore, A.P, India.Email :{ [email protected], [email protected]}

Abstract—The lifting scheme of the discrete wavelet transform(DWT) is now quite well established as an efficient techniquefor image denoising. The lifting scheme factorization ofbiorthogonal filter banks is carried out with a linear-adaptive,delay free and faster decomposition arithmetic. This adaptivefactorization is aimed to achieve a well transparent, moregeneralized, complexity free fast decomposition process inaddition to preserve the features that an ordinary waveletdecomposition process offers. This work is targeted to getconsiderable reduction in computational complexity and powerrequired for decomposition. The hard striking demerits ofDWT structure viz., shift sensitivity and poor directionalityhad already been proven to be washed out with an emergenceof dual tree complex wavelet (DT-CWT) structure. The wellversed features of DT-CWT and robust lifting scheme aresuitably combined to achieve an image denoising with prolificrise in computational speed and directionality, also with adesirable drop in computation time, power and complexity ofalgorithm compared to all other techniques.

Index Terms— Lifting scheme, Dual Tree Complex Wavelettransform (DT CWT), Denoising, Computational complexity,coding gain and PSNR.

I. INTRODUCTION

One of the fundamental challenges in the field of imageprocessing and computer vision is image denoising, wherethe underlying goal is to produce an estimate of the originalimage by suppressing noise from a noise contaminatedversion of the image. Image noise may be caused by differentintrinsic (i.e., sensor) and extrinsic (i.e., environment)conditions which are Often not possible to avoid. Images aremost frequently bear upon by noise evincing the facts likeimage capturing sensor internal imperfections, scarce ofproper illumination of the object to be captured, during theprocess of its acquisition mean by digitization and itstransmission. In fact the performance of the image acquiringsensors are generally affected by a wide variety of factorssome among which are the factors such as environmentalconditions during its acquisition process and the quality ofthe sensing elements deployed in image capturing systems,i.e., image acquisition sensors like Charge Coupled Device(CCD) cameras. For instance in acquiring images with a cameracapturing object illumination and lighting levels and sensortemperature are major factors affecting the amount of noise

in the resulting image. Another major possibility of imagecorruption with noise will be its transmission to a point ofinterest from a point of its perception by a sensor, principallydue to the interference in the channel used for itstransmission. Image noise is usually with reference to widestochastic variations as opposed to deterministic distortionssuch as shading or lack of focus. It is actually the degree ofvariation of pixel values caused by the statistical nature ofradioactive decay and detection processes. Even if we acquirean image of a uniform (flat) source on an ideal gamma camerawith perfect uniformity and efficiency the number of countsdetected in all pixels of the image will not be the same. Inaddition to noise added inherently by a sensor, imageprocessing techniques also corrupt the image with noise [1].There are a wide variety of noise models which can degradethe image, among which the most common is the additiveone i.e., the additive white Gaussian noise will show a seri-ous impact on visual perception of image by its intensivedegradation. And the remaining possible noise models whichhave a probability of corrupting an image are salt and pepper,speckle, poison and thermal noises in the communicationmedium. This undesirable corruption of image by noise isinherent in any image acquisition device. Certainly it is pos-sible to distinguish between two regimes when the degrada-tion structure was observed by photo sensors: in the firstscenario, the measured intensities are sufficiently high andthe noise is assumed to be signal-independent degradation.In the second scenario, only a few photons are detected,leading to a strong signal –dependent degradation. Thisimage degradation by noise will make the image to be nullinformation conveying object. Removing such noise is ofgreat benefit in many applications and this may explain vastinterest in this problem and its solution. The ultimate goal ofimage denoising technique is to improve the degraded imagein some sense by suppressing the random noise, which hascorrupted the image, while preserving the most importantvisual features of the image, such as edges. Denoising is anessential step prior to any higher-level image-processingtasks such as segmentation, photo restoration, visual track-ing where obtaining the original image content is crucial.Many algorithms have been proposed for image denoising,and there has been a fair amount of research on waveletbased image denoising, because wavelet provides an

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appropriate basis for image denoising. But this single treewavelet based image denoising has poor directionality, lossof phase information and shift sensitivity as its limitations.Hence we proposed dual tree wavelet structure for an im-proved quality image denoising while getting rid of all abovelimitations and lifting scheme structure for faster, complexfree and efficient decomposition process.

II. DUAL TREE AND LIFTING FRAME WORK

The conceptual idea central to a wavelet or sub-bandtransform [2],is illustrated in figure (1). The real wavelettransform makes an use of analysis filters h 0(n) andh1(n),followed by sub sampling, while the reverse transformfirst up-samples and then uses two synthesis filters f0(n) andf1(n).Emulating same process with { and{ as analysis and synthesis filter pairs[3]withsuitable sub sampling and up- sampling processes will leadto an imaginary tree wavelet transform. where the analysisand synthesis filter pairs in one tree will differ with those inanother tree with a half sample delay. ; (1)Since the filters in both trees can be made to offset by halfsample, the wavelets resulting from the filter pair satisfies theHilbert transform condition, given as

(2)

For real and imaginary trees, the modulation Matrices [1]indicated by and , respectively , defined as

(3)

(4)

The condition for perfect reconstruction can be representedin z-domain asFor real

(5)For imaginary

(6)All the filters are assumed to be causal for simplicity and isthe cause for introducing the term .secondly, and

are the dual versions of and respec-tively. We can write, for perfect reconstruction as

(7)

Fig 1(a): Real tree wavelet transform

(8)

Fig 1(b): Imaginary wavelets transform.

Where I being an identity matrix of order ‘2’.If all filters areFIR, then , and , belongs toGL(2;R[z, ]).As a special case in a orthogonal wavelettransform, , ,and , the modulation matrices

and ,which are intern times a unitary matrix. In poly phase notation, these filterscan be written interims of their even and odd phases,according to the following relations

(9)

(10)

(11)

(12)Where contains even coefficients and con-tains odd coefficients for i=0,1.Thus

(13)

(14)

(15)

(16)

We then assemble the polyphase matrices

(17)

(18)

So that

(19)

(20)

We now define and as dual version of

and ,then the perfect reconstruction propertyyields that

(21)

(22)

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Where P(z) and contain only Laurent polynomials. Theresultant real and imaginary wavelet transforms in polyphasenotation can be represented schematically in figure (2)

Fig 2.a:Polyphase representation of real tree wavelet transform.

Fig 2b.Polyphase representation of imaginary tree wavelettransform.

In both cases first the image has been sub sampledinto its even and odd parts, then apply the dual polyphasematrices. For inverse transform case, first apply the polyphasematrix and then join the even and odd parts. The major prob-lem of an FIR wavelet transform resides in finding the matri-ces and with determinants as one. Once we

have such a matrices, and the four filters forthe wavelet transform will be derived immediately frompolyphase matrices and . From Cramer’s rule

(23)

(24)This implies

(25)(26)

The most trivial example of a polyphase matrix is P(z)=I.This will result in, and

. Thus the wavelet transform doesnothing else but sub sampling even and odd samples. Thistransform is generally called as a polyphase transform, but inthe context of lifting it is often referred to as the lazy wavelettransform. This lazy wavelet transform [4] incorporates anumerous complex computations, which will consume a largeamount of power, and time for computation, which results inhigh coding and computational complexity.

To avoid these hard striking shortcomings due to processcomplexity and delay, the natural wavelets are furtherconfigured to yield a simple, flexible and computationallyefficient second generation wavelet structure..The robustlifting scheme [4] [5] was introduced to build secondgeneration wavelets referred commonly as lifted wavelets,which are not necessarily translates and dilates of one fixedfunction [6].This construction is entirely spatial and thereforeideally suited for building second generation wavelets, that

are more general in the sense that all the classical waveletscan be generated by the lifting scheme. For fasterimplementation of wavelet transform,the lifting scheme makesan optimal use of the similarities between the high pass andthe low pass filters. The flexibility afforded by the liftingscheme allows the basis functions associated with waveletcoefficients near the window’ boundaries to change theirgeneral shape .This new basis function can be used tominimize boundary effects. Being a new method forbiorthogonal wavelet construction, the lifting scheme allowsthe faster implementation of wavelet transform. The numberof flops can be reduced by a factor of two. The basic blockdiagram of lifting steps is illustrated in figure .3

Fig. 3: Representation of lifting steps.

Where is a predict[7] operator and U is an updateoperator[8].The lifting scheme can also be used in situa-tions, such as wavelets on bounded domains, wavelets oncurves and surfaces, weighted wavelets, and wavelets withirregular sampling. Conceptually the lifting scheme originateswith a ‘lazy wavelet’, a function which essentially doesn’t doa thing , but having a formal properties of a wavelet. Thelifting scheme then generally builds a new wavelet, with im-proved properties by adding in, a several new basis func-tions to the lazy wavelet to make it faster as shown in fig-ure(4). This is the inspiration behind the name ‘LiftingScheme’.

Fig.4: Fast wavelet transform using lifting scheme

which first does a lazy wavelet transform and then computes and finally lifts the wavelet coefficients of the input

image The lifting scheme uses the relationship between theperfect reconstruction filter pairs ,that have thesame low pass or high pass filter. The filter pairs{ } and { } to be complementary if incase the corresponding polyphase matrices and have their determinant as 1. and if the filter pair,{} and {} arecomplementary, then consequently the filter pairs {} and{}are also complementary. Then any other finite filters and,complementary to and respectively are of the form

(27)(28)

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Where S(z) is a Laurent polynomial. Similarly as per as duallifting concerned, there exists any finite filters of the form,

(29)(30)

Where t(z) is a Laurent polynomial. The resulted liftedwavelet tree structure for the case of classical sub bandscheme ,when lifted the lowpass sub band with the help ofthe high pass sub band for both real and imaginary trees isshown in figure(5).And in a similar way the dual lifted wave-let tree structure for the case of classical sub band scheme,when lifted the high pass sub band with the help of the lowpass sub band for both real and imaginary trees is shown infigure(6).

(a)

(b)

Fig .5.The Lifting scheme (a) real (b) imaginary trees: first aclassical sub band filters scheme and then lifting the low pass sub

band with the help of the high pass sub band.

(a)

(b)Fig. 6:The dual lifting scheme (a) real,(b) imaginary wavelet trees:First a classical sub band filter scheme and later lifting the high pass

sub band with the help of the low pass sub band.

III. PROPOSED WORK

In this paper, it is proposed to reduce the computationalcomplexity, computational time, power and complexity ofcoding structure, with a substantial growth in averagecomputational speed, and coding gain in a traditional dualtree complex wavelet transform, which employs two separateQuadrature distinct discrete wavelet structures in which oneis taken as the real tree wavelet structure[9][10],and the otheris taken as the imaginary tree wavelet structure. Each discrete

wavelet transform structure irrespective of its nature will havea pair of low pass and high pass analysis filters and a similarpair of low pass and high pass synthesis filters. The analysisand synthesis filter pair in one tree will differ with those inother tree by an half sample delay. But the wavelets in thetraditional system are lazy wavelets(i.e., computationallycomplex, slower and consumes a high quantity ofpower).Hence to modernize the wavelets and to cater thehard striking inabilities of traditional wavelets. The traditionalwavelets are further processed and modified, to lift them, in aview to get a more simple, flexible, linear, complexity free andfast second generation wavelets[11]. These secondgeneration wavelets are constructed, first by lifting the lazywavelet which involves bi orthogonal factorization [12][13][14] [15] of lazy wavelet filters to eradicate delay artifacts andto remove the coefficients in the filter transfer functionsresponsible for delayed processing, or operational complexity,consuming higher processing power, to yield the lifted filtersfor second generation wavelets, called lifted wavelet filtersfor computationally faster lifted wavelet transform.

The process of the lifting wavelet construction involvesfactorization of wavelet filters to get simple lifted waveletfilters. This activity involves running Euclidean algorithmstarting from polyphase components of and as{ and { } respectively and theirgcd’s are monomials. With the non-uniqueness of the divi-sion, we can always choose the quotients ,so that thegcd is a constant. Let this constant be then we thus have

(31)

(32)

in case ,the first quotient is zero.We can always assume that ‘n’ is even. If we know the filters

and ,then we can always find the complementary

filters and by letting

(33)

(34)

Here the final diagonal matrix follows from the fact thatdet ( =det( )=1 and ‘n’ is even, then by rewrit-ing the above equations, we observe that

(35)

Using the first equation of above in case ‘i’ is odd and thesecond in case is even yields

(36)

As an example to this factorization process to builtcomputationally speed lifting wavelets, let us consider a lazy

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wavelet of haar type. Now our task is to factor the filtersassociated with lazy haar wavelet transform to build ,activelifting wavelet filters there by to build lifted haar wavelets. Inthe case of unnormalized wavelets we know that for a singletree

using the aforementioned Euclidean algorithm, we can thuswrite the polyphase matrices as

(37)

(38)

Thus on the analysis side we have

(39)

(40)

Thus the forward and reverse wavelet transformimplementation structure using the alternating lifting and duallifting steps are better illustrated in figure.7

Fig .7a .Forward wavelet transform using the lifting scheme.

Fig.7.b:The inverse wavelet transform using the lifting scheme

(41)Similarly the inverse factorization yields,

(42)

Thus the lifting wavelets are designed for both real andimaginary trees ,and incorporated to get real and imaginarydiscrete wavelet transform[11] trees, which are suitablycombined to form the dual tree complex wavelet transform toerase the limitations of the single tree discrete wavelettransform, such as shift variance, poor directionality, loss ofphase information and aliasing.Thus for the case of a boir 6.8 wavelet type, the correspondinglifting scheme LS = {... ‘p’ [ -2.65899636 0.99715069] [0]

‘d’ [ 0.27351197 0.27351197] [0] ‘p’ [ 3.87782215 -3.26868661] [2] ‘d’ [ -0.28650326 -0.28650326] [-2] ‘p’ [ -0.54859417 2.94176754] [4] ‘d’ [ 0.09982322 -0.34381326 -0.34381326 0.09982322] [ [-2] [ 0.86857870] [ 1.15130615] [73] };

The transfer functions of the corresponding lifted waveletanalysis filters for the dual tree complex wavelet transformare given in figure (8)

Fig .8:Transfer functions of lifted wavelet dual tree analysis filters.

Similarly the transfer functions of the corresponding liftedwavelet synthesis filters for the dual tree complex wavelettransform are given in figure .9

Fig. 9:Transfer functions of lifted wavelet dual tree synthesisfilters.

The corresponding lifted wavelet coefficients are givenfor the dual tree complex lifting wavelet transform both foranalysis and synthesis separately as in table(1) and table (2).

Thus a computationally fast dual tree complex wavelettransform is constructed based on lifted wavelets .Now thisefficient and faster DTCWT tool is employed as a tool ofinterest for an image denoising. In this paper a noise degradedimage is first applied to both real and imaginary discretewavelet transform trees to obtain real and imaginarycoefficients. These coefficients are suitably combined to getcomplex wavelet coefficients.

The corresponding dual tree wavelet structure is shownin figure.10

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TABLE I. ANALYSIS LIFTED WAVELET COEFFICIENTS FOR DTCWT WITH LS.

??0(??) ??1( ??) ??0( ??) ??1 (??) 0

0.0019 -0.0019 -0.0170 0.0119 0.0497 -0.0773 -0.0941 0.4208 0.8259 0.4208 -0.0941 -0.0773 0.0497 0.0119 -0.0170 -0.0019 0.0019

0 0 0

0.0144 -0.0145 -0.0787 0.0404 0.4178 -0.7589 0.4178 0.0404 -0.0787 -0.0145 0.0144

0 0 0 0

0 0 0

0 .0019 -0.0019 -0.0170 0 .0119 0 .0497 -0.0773 -0.0941 0 .4208 0 .8259 0 .4208 -0.0941 -0.0773 0 .0497 0 .0119 -0.0170 -0.0019 0 .0019

0 0 0 0 0

0.0144 -0.0145 -0.0787 0.0404 0.4178 -0.7589 0.4178 0.0404 -0.0787 -0.0145 0.0144

0 0 0 0

TABLE II. SYNTHESIS LIFTED WAVELET COEFFICIENTS FOR DTCWT WITH LS

?? ??1(??) ??0 (??) ??1 (??)

0 0 0

0.0144 0.0145 -0.0787 -0.0404 0.4178 0.7589 0.4178 -0.0404 -0.0787 0.0145 0.0144

0 0 0 0

0 -0.0019 -0.0019 0.0170 0.0119 -0.0497 -0.0773 0.0941 0.4208 -0.8259 0.4208 0.0941 -0.0773 -0.0497 0.0119 0.0170 -0.0019 -0.0019

0 0 0 0 0

0.0144 0.0145 -0.0787 -0.0404 0.4178 0.7589 0.4178 -0.0404 -0.0787 0.0145 0.0144

0 0 0 0

0 0 0

-0.0019 -0.0019 0.0170 0.0119 -0.0497 -0.0773 0.0941 0.4208 -0.8259 0.4208 0.0941 -0.0773 -0.0497 0.0119 0.0170 -0.0019 -0.0019

Fig.10:Dual tree complex lifted wavelet structure.

The dual tree complex wavelet transform embedded withlifting scheme is initially aimed to achieve a noteworthyprogress in the computational speed up in the image analy-sis. This computationally outstanding tool integrated withthe globally accepted adapted wavelet threshoulding

techniques have been employed as the key techniques todevelop a modified denoising algorithm for image denoising.Since the dual tree complex wavelet transform with liftingstructure incredibly accelerate the decomposition processwith the conserved shift invariance and directional selectiveproperties as earlier , adaptive soft thresholding techniquecentral to the adaptive wavelet threshoulding has been inte-grated to denoise the image. The algorithm is intelligibly pre-sented and implemented as1. Select the test image of size 256X256.2. Select the type of the wavelet to be lifted.3. Lift the selected wavelet with sufficient predict and update

operations to generate the lifted wavelet structure of theselected wavelet .

4. Design and generate the lifted wavelet filters associatedwith the lifted wavelet(Both for analysis and synthesis ).

5. Apply the dual tree based complex wavelet transform onthe test image using the lifted wavelet filter pair foranalysis.

6. Grouping the connected components using theneighborhood based statistical modeling of the complexwavelet coefficients.

7. Apply the adaptive soft thresholding technique with athreshold value of 40.

8. Applly the inverse dual tree complex wavelet transformon the thresholded complex wavelet coefficients usingthe lifted wavelet filters for synthesis.

9. Compare the results obtained in terms of PSNR and RMSE.

A. Computational complexityIn this section ,we used to focus on the computational

complexity of the wavelet transform .As a comparison basewe used the standard algorithm(DTCWT), whichcorresponds to applying the polyphase matrix. This will takethe advantage of the fact that the filters will be sub sampledand thus avoids computing samples that will be sub sampledimmediately. For each filter either it may be low pass or highpass,the number of multiplications and additions requiredare and respectively.Thus in a standardalgorithm without applying lifting scheme, the number ofmultiplications and additions required for a single discretewavelet transform tree are and( respectively. Similarly for an imaginarytree, the ( )number of multiplicationsand( ) number of additions are required.Thus for dual tree complex wavelet transform, the total numberof multiplications and additions neededare( ) and( ) respectively. Forthe case of symmetric filters let

, , a n dalso be assumed that .Then for a standard algorithm,the number of multiplications and additions required for areal tree are 2 +2 +2 and 2 .Similarly for an

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imaginary tree the total number of multiplications andadditions required are 2+2+2 and 2.Then for the dual treecomplex wavelet transform, the total number of multiplicationsand additions required are 4(4 and 4() respectively. Similarlyif the lifting scheme is incorporated, then for a single tree, 2scaling operations are required, for lifting steps,4 liftingoperations are required and for final lifting steps, the numberof operations required are 2(+1).Then the total number ofoperations required are 2(),among which the total number ofmultiplications and additions are and respectively. Thus forthe dual tree complex wavelet transform using the liftingscheme the total number of 2() multiplications and 2(additionsare required. Correspondingly the computational operations

required for various transform cases are investigated intable.3.

Graphically the computational complexity of the proposedalgorithm with DTCWT-LS is compared with DTCWT asshown in fig.11. As computational complexity increases, thecomputational time increases and vice-versa. Thus an algo-rithm with higher computational complexity will take highertime for its computation and that with lower computationalcomplexity will take less time for its computation on the sameprocessor. Thus the computational time with several wave-lets are compared in terms of no of computations required asin table .4

TABLE III. COMPARISON OF COMPUTATIONAL COMPLEXITY BETWEEN STANDARD ALGORITHM AND LIFTING ALGORITHM.

Fig.11

TABLE IV COMPARISON OF COMPUTATIONAL TIME BETWEEN STANDARD ALGORITHM AND LIFTING ALGORITHM FOR VARIOUS WAVELET FILTERS

B. Computational Time:As the number of operations are increases then the

computational time will also increases. Thus thecomputational time of a standard and lifting algorithms arecompared as in table.5,.Inversely the computational time ofthe dual tree complex wavelet transform with lifting schemewill be compared graphically with that of DTCWTas shownin figure.12.

C. Coding gain:As the number of operations in the lifting scheme based

real ,imaginary and dual tree complex wavelet transform are

TABLE V

exactly half of those in standard real, imaginary and dual treecomplex wavelet transforms respectively, then the overallcoding gain will increases by 100%.Graphically the coding

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gain of the DTCWT-LS is compared with that of the DTCWTas shown in figure 13

Figure 13

IV. RESULTS AND DISCUSSION

The denoising results are presented and compared withthe former algorithms employing the conventional DTCWTand adaptive wavelet thresholding. The simulation resultsobtained have ascertained that there is a remarkable progressin the PSNR value and notable drop in the RMSE value of thealgorithm compared to the former algorithm. Practically aPSNR value of 40.04 and an RMSE value of 05.0267 are pos-sible for additive white Gaussian noise(AWGN),but with theformer algorithm the maximum PSNR value of 22.8709 andminimum RMSE value of 14.9928 are only possible withAWGN. With an impulse noise ,the modified algorithm offersa maximum PSNR of 47.4560 and minimum RMSE of04.5656,where as the former algorithm can only offer a maxi-mum PSNR of 24.2356 and a minimum RMSE of 12.5612..Animportant note here is that all the results are obtained with aBarbara.jpg image of 256X256 as a test image.

A comparative analysis of the performance offered bythe DTCWT without lifting scheme(LS) and the DTCWT withlifting scheme are done in terms of the no of computationsrequired and the amount of time elapsed to perform them fora particular image format. The performance of the both the

Fig(12):Compaison of computational time of DTCWT-LS withthat of the DTCWT

algorithms have been compared with several images of dif-ferent size and format and the results are given in table.7 .Forinstance with the DTCWT without LS algorithm the no ofcomplex computations required to process a cameraman im-age of size 256X256 and of tif format, a real tree requires9568256 computations ,for an imaginary tree 9568256 com-putations are needed and for dual tree on a whole 19136512computations are essential. But with DTCWT with LS algo-rithm ,to process the same image it requires 4849664 real,4849664 imaginary and 9699328 dual tree computations aresufficient. Similarly to process a Barbara image of jpg formatand of 256X256 size the DTCWT without LS algorithms takesa time of 0.1486 seconds for real tree, 0.024 seconds for imagi-nary tree and on a whole 0.2106 seconds are essential fordual tree. Where as to process the same image the DTCWTwith LS consumes a time of 0.1239 seconds for real tree,0.0188seconds for imaginary tree and on a total 0.1436 seconds fordual tree is enough. The testing results have strongly wit-nessed the computational dominance of the DTCWT withLS algorithm over the DTCWT without LS algorithm, andoutstandingly judged the computational efficiency of theDTCWT with LS algorithm In order to compare the qualityof the denoised image signals obtained with both the algo-rithms such as DTCWT without lifting scheme and DTCWTwith lifting scheme, the percent root mean square formula forboth original and reconstructed image signals is

(43)

Where is the denoised image signal obtained with the

DTCWT without LS and is the resulting denoised imagesignal obtained by the DTCWT. Similarly the outputs ob-tained with the DTCWT with LS and DTCWT without LS arehere subtracted from the original input and differences arecomputed as RMS error (RMSE).The generalized formula forRMSE is

(44)

where is denoised image signal and is the originalimage signal without any noise.

(a) (b) (c)Fig.14:Visual Results obtained with Gaussian noise and (a).Originalimage(b)Noisy image (c)Denoised image.

The denoising results obtained with Gaussian noise andimpulse noise are illustrated in figure.14 & 15.

Image quality performance metrics PSNR and MSE arecomputed and are tabulated as given in below table.6.

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(a) (b) (c)Fig.15Visual Results obtained with an Impulse noise (a).Original

image, (b)Noisy image (c)Denoised image.

TABLE VIN oise Type Technique PSNR RMSE

AW GN DTCWT 32. 0637 6 .3585 D TCWT-LS 47. 8222 15. 5608

Impulse DTCWT 22. 8946 18. 2731

D TCWT-LS 48. 4823 12. 9214

Image No of Operations Computation time

Dual Tree CWT For Lifting scheme DTCWT Dual Tree CWT DTCWT-LS Real Imaginary Dual Real Imaginary Dual Real Imaginary Dual Real Imaginary Dual

cameraman 9568256 9568256 19136512 4849664 4849664 9699328 0.547 0.547 0.547 0.1293 0.0202 0.1562 peppers 28704768 28704768 57409536 14548992 14548992 29097984 0.1712 0.044 0.3051 0.1051 0.0309 0.1203 Lena 5840000 5840000 11680000 2960000 2960000 5920000 0.1455 0.0194 0.1951 0.1212 0.0162 0.1629 Brandyrose 8380400 8380400 16760800 4247600 4247600 8495200 0.1515 0.0217 0.2073 0.1241 0.0175 0.1402 Barbara 9568256 9568256 19136512 4849664 4849664 9699328 0.1486 0.024 0.2106 0.1239 0.0188 0.1436 Bear 8906000 8906000 17812000 4514000 4514000 9028000 0.1536 0.0229 0.212 0.1233 0.0182 0.1313 Kid 8438800 8438800 16877600 4277200 4277200 8554400 0.1455 0.0219 0.2047 0.1229 0.0177 0.1763 Watch 4380000 4380000 8760000 2220000 2220000 4440000 0.1454 0.0189 0.1928 0.1205 0.0154 0.1564 Parrot 9568256 9568256 19136512 4849664 4849664 9699328 0.1479 0.0236 0.2088 0.125 0.0201 0.1451 Baboon 5840000 5840000 11680000 2960000 2960000 5920000 0.1442 0.0201 0.1942 0.103 0.0162 0.1192 House 9568256 9568256 19136512 4849664 4849664 9699328 0.1582 0.0247 0.2199 0.1254 0.0196 0.1396

TABLE VII TESTED RESULTS FOR COMPUTATIONAL COMPLEXITY

V. CONCLUSIONS

In this paper we developed a algorithm which suitablycombines the salient features of nearly shift invariant,directionally selective dyadic decomposition tree based dualtree complex wavelet transforms deploying quadrature dis-tinct ,perfect reconstruction filter banks and linear adaptive,and faster decomposition arithmetic based robust liftingscheme for an image denoising in a view to get a substantialspeed up in the decomposition and reconstruction structureof the dual tree complex wavelet transform while preservingall performance salient features that results from the dual treecomplex wavelet transform itself. This algorithm provides asubstantial drop in the average decomposition time, averagepower required for decomposition as a major distinction fac-tor from an ordinary dual tree complex wavelet decomposi-tion algorithm. With this lifted Dual tree structure we achievedan accountable growth in the coding efficiency and codinggain. This algorithm first performs the factorization of thefilters associated with the selected wavelet type based onthe Euclidean algorithm and then constructs its associatedlifting scheme from which the filters associated with the liftedwavelet will be derived. These filters are employed to achievean in-place-implementation of the fast Dual tree Waveletstructure. This computationally efficient, dual tree complexwavelet structure has been employed as a tool of interest foran image denoising and achieved a considerable growth inimage quality and performance metrics viz., PSNR and RMSE.

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