Performance Improvement in the Bivariate Models by using Modified Marginal Variance of Noisy Observations for Image-Denoising Applications

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1 PROCEEDING OF WORLD ACADEM OF CIENCE, ENGINEERING AND ECHNOLOG VOLUME 5 APRIL 005 IN Performace Improvemet i the Bivariate Models by usig Modified Margial Variace of Noisy Observatios for Image-Deoisig Applicatios R. ethilkumar Abstract Most simple oliear thresholdig rules for wavelet- based deoisig assume that the wavelet coefficiets are idepedet. However, wavelet coefficiets of atural images have sigificat depedecies. his paper attempts to give a recipe for selectig oe of the popular image-deoisig algorithms based o Visuhrik, urehrik, Oraclehrik, Bayeshrik ad Bihrik ad also this paper compares differet Bivariate models used for image deoisig applicatios. he first part of the paper compares differet hrikage fuctios used for image-deoisig. he secod part of the paper compares differet bivariate models ad the third part of this paper uses the Bivariate model with modified margial variace which is based o Laplacia assumptio. his paper gives a experimetal compariso o six 5x5 commoly used images, Lea, Barbara, Goldhill, Clow, Boat ad toehege. he followig oise powers 5dB, 6dB, 7dB, 8dB ad 9dB are added to the six stadard images ad the correspodig Peak igal to Noise Ratio (PNR) values are calculated for each oise level. eywords Bihrik, Image-Deoisig, PNR, hrikage fuctio. A I. INRODUCION N image is corrupted by oise i its acquisitio or trasmissio. he goal of deoisig is to remove the oise while retaiig as much as possible the importat sigal features. raditioally, this is achieved by liear processig but for the past few years image- deoisig has bee doe usig oliear techiques. A simple deoisig algorithm that uses the wavelet trasform cosist of the followig three steps, ()Calculate the wavelet trasform of the oisy sigal () Modify the oisy wavelet coefficiets accordig to some rule. (3) Compute the iverse trasform usig the modified coefficiets. Oe of the most well-kow rules for the secod step is soft thresholdig aalyzed by Dooho [, ]. Due to its effectiveess ad simplicity, it is frequetly used i the literature. he mai idea is to subtract threshold value from all coefficiets larger tha ad to set all other coeffi- Mauscript received March 5,005. his work was supported i part by the V.L.B Jaakiammal Egieerig College. R. ethilkumar is with the V.L.B. Jaakiammal Egieerig College, amilnadu, 6404 INDIA (phoe: +9 (04) ; rsethil_976@yahoo.com). ciets to zero. Alterative approaches ca be foud i, for example, [],[3],[4] ad [5]. Visuhrik[] uses oe of the well-kow thresholdig rules: the Uiversal threshold. I additio, subbad adaptive systems have superior performace, such as urehrik[3], which is a data-drive system. Recetly, bayeshrik[5], which is also a datadrive subbad adaptive techique, is proposed ad outperforms Visuhrik ad sometimes urehrik. he orgaizatio of this paper is as follows. I ectio II, the basic idea of image deoisig usig Visuhrik[], urehrik[3], Bayeshrik[5] ad Oraclehrik[5] are described briefly. he PNR values obtaied for six deoised images for differet algorithms are compared with Bivaraite Model. hese models try to capture the depedecies betwee a coefficiet ad its paret. ectio III, compares the Bivariate Model ad Bivariate Model 3. he, ectio IV describes the Bivariate model with modified margial variace which use the Laplacia assumptio. II. COMPARION OF BIVARIAE MODEL WIH OHER IMAGE-DENOIING MODEL he followig subsectios explai differet image deoisig shrikage fuctios A. Visuhrik For image deoisig, Visuhrik [] (Visually calibrated adaptive smoothig) is kow to yield smoothed images. he threshold choice for Visuhrik is log M (called the uiversal threshold ad is the oise variace). he soft-threshold fuctio [] (also called the hrikage fuctio), η ( x) sg( x).max( x,0) () takes the argumet ad shriks it toward zero by the threshold. B. urehrik he urehrik[3] uses a hybrid of the uiversal threshold ad the URE threshold, derived from miimizig stei s ubiased risk estimator [4], ad has show to perform well. he URE threshold choice is depedet o the eergy of the particular sub bad [3]. he threshold o subbad to be used with a soft shrikage fuctio is, PWAE VOLUME 5 APRIL 005 IN WAE.ORG

2 PROCEEDING OF WORLD ACADEM OF CIENCE, ENGINEERING AND ECHNOLOG VOLUME 5 APRIL 005 IN URE (, arg mi[ URE (, )] () 0 ) N + [mi(, )] [: N where, is the detail coefficiets from subbad ad N is the umber of coefficiets i { }. C. Oraclehrik he Oraclehrik, which is truly optimal softthresholdig estimator assumig the origial image is kow. he threshold of Oraclehrik i each sub bad is O with ij kow. i, j ] (3) arg mi ( η ( ) ) (4) D. Bayeshrik he Bayeshrik rule uses a Bayesia mathematical framework for images to device subbad depedet thresholds that are early optimal for soft threshold. he formula for the threshold o a give subbad is [5], ij (5) where is the estimated oise variace ad ij is the estimated sigal variace o the sub bad cosidered. he oise variace is estimated as the media of the absolute deviatio of the diagoal detail coefficiets o the fiest level (i.e., subbad HH). he estimate of the sigal stadard deviatio o subbad is where N max(,0) (6) k ˆ is a estimate of the variace of the observatios, with N beig the umber of the wavelet coefficiets o the subbad uder cosideratio. Combiig equ. (5) ad equ. (6), the threshold choice for Bayeshrik is are oisy observatios of ad also k ad are the oise samples. he estimator of is [7] which ca be iterpreted as a bivariate shrikage fuctio. Here (g) + is defied as [6] (8) 0, if g < 0 ( g )+ { (9) g, otherwise his estimator requires the prior kowledge of the oise variace ad the margial variace for each wavelet coefficiet [8]. o estimate the oise variace from the oisy wavelet coefficiets, a robust media estimator is used from the fiest scale wavelet coefficiets []. ( ) media i, i ε subbad HH (0) he algorithm is summarized as follows: ) Calculate the oise variace usig (0). ) Calculate ˆ give i subsectio.4. 3) Calculate ˆ usig (6). 4) Estimate each coefficiet usig ad i (8). Fig. ad able I show that the Bihrik shrikage fuctio which gives better PNR (Peak igal to Noise Ratio) whe compared to the models described i subsectios A,B, C ad D. (a) (b) ˆ max(, ), < (7) E. Bivariate Model Let represets the paret of ( is the wavelet coefficiet at the same positio as the th wavelet coefficiet, but at the ext coarser scale). he problem formulated i the wavelet domai as + ad + to take ito accout the statistical depedecies betwee a coefficiet ad its paret. k ad Fig.. (a) Origial image, (b) Noisy image with PNR 9.5dB. Power of Noise added 9dB. PWAE VOLUME 5 APRIL 005 IN WAE.ORG

3 PROCEEDING OF WORLD ACADEM OF CIENCE, ENGINEERING AND ECHNOLOG VOLUME 5 APRIL 005 IN r + (5) ˆ ˆ r + (6) ˆ ˆ Model 3 outperforms Model i most of the cases. Four stadard images are take ad the differet bivariate models are applied to these test images. he PNR values obtaied are tabulated i table II. IV. MODEL WIH LAPLACIAN AUMPION I our experimets, we obtaied better PNR values with Bivariate Model if we use Laplacia assumptio. he correspodig margial variace of oisy observatios is give by [9], N ˆ (7) k k Fig. ad Fig. 3 compare the Bivariate Model, Bivariate Model 3 ad Bivariate Model with Laplacia assumptio. able II shows that the Bivariate Model with Laplacia assumptio outperforms Bivariate Model ad Bivariate Model 3. Fig..(c) Compariso of deoised stoehege image obtaied for differet shrikage models III.COMPARION OF DIFFEREN BIVARIAE MODEL he Bivariate Model already described i sectio II. he algorithm for Bivariate Model 3 is summarized as follows: ) Calculate the oise variace usig (0). ) For each subbad, (a) Calculate ˆ ad ˆ usig () ad (); (b) Calculate ˆ ad ˆ usig (3) ad (4); (c) Estimate each coefficiet usig the successive approximatio method. ˆ k () N k ˆ k () N k Fig. compariso of differet Bivariate Models usig Barbara stadard image. max(,0) (3) max(,0) (4) PWAE VOLUME 5 APRIL 005 IN WAE.ORG

4 PROCEEDING OF WORLD ACADEM OF CIENCE, ENGINEERING AND ECHNOLOG VOLUME 5 APRIL 005 IN Fig.3.(a) Origial image (b) Noisy image with PNR 8.5 db (c) Model with PNR 3.0 db (d) Model 3 with PNR 3.43 db (e) Model with Laplacia assumptio PNR 3.77 db. ABLE I FOR VARIOU E IMAGE AND P VALUE, LI PNR OF ()Visuhrik, () urehrik, (3) Oraclehrik (4) Bayeshrik ad (5) Bihrik Noisy Visuhrik urehrik Oraclehrik Bayeshrik Bihrik Lea P5dB P6dB Clow P5dB P6dB Boat P5dB P6dB Goldhill P5dB P6dB PWAE VOLUME 5 APRIL 005 IN WAE.ORG

5 PROCEEDING OF WORLD ACADEM OF CIENCE, ENGINEERING AND ECHNOLOG VOLUME 5 APRIL 005 IN ABLE -II FOR VARIOU E IMAGE AND P VALUE, LI PNR OF () Model, () Model 3, (3) Model with Laplacia Assumptio Noisy Model Model 3 Model with Laplacia Assumptio PNR (db) PNR (db) PNR (db) PNR(dB) Lea P7dB P8dB Barbara P7dB P8dB Boat P7dB P8dB Goldhill P7dB P8dB V. CONCLUION I this paper the differet shrikage fuctios used for image deoisig were discussed ad the results were verified experimetally usig MALAB 6.5 image processig toolbox. I order to show the effectiveess of the Bihrik estimator, six examples were preseted ad compared with other effective techiques. he improved PNR was obtaied i the Bayeshrik estimator, whe simultaeous deoisig ad compressio is used [5]. I Bihrik estimator, i additio to the orthogoal wavelet trasform, dual-tree Complex Wavelet trasform (CW) [7, 8] is used to improve PNR. REFERENCE [] D.L. Dooho ad I.M. Johstoe, Ideal spatial adaptatio via wavelet shrikage, Biometrika, vol. 8, pp , 994. [] D.L. Dooho, De-oisig by soft-thresholdig, IEEE ras. Iform. heory, vol.4, pp 63-67, May 995. [3] D.L. Dooho ad I.M. Johstoe, Adaptig to ukow smoothess via wavelet shrikage, Joural of the America tatistical Assoc., vol. 90, o. 43, pp 00-4, December 995. [4] C.M. tei, Estimatio of the mea of a multivariate ormal distributio, A. tatist., vol.9, o.6, pp 35-5, 98. [5].G. Chag, Bi u ad M. Vetterli, Adaptive wavelet thresholdig for image deoisig ad compressio, IEEE ras. Image processig, vol. 9, pp , eptember 000. [6] L. edur ad I. W. elesick, A bivariate shrikage fuctio for wavelet based deoisig, IEEE It. Cof. Acoust., peech, igal Processig (ICAP), pp 6-64, 00. [7] L. edur ad I. W. elesick, Bivariate shrikage with local variace estimatio, IEEE igal Processig Letters, vol.9, o., pp , December 00. [8] L. edur ad I. W. elesick, Bivariate shrikage fuctios for wavelet-based deoisig exploitig iterscale depedecy, IEEE ras. o igal Processig, vol.50, o., pp , November 00. [9] L. edur ad I.W. elesick, ubbad Adaptive Image Deoisig Via Bivaraite hrikage, IEEE It. Cof. Image Processig (ICIP), pp , 00. R.ethilkumar received the B.E. degree i electroics ad commuicatio egieerig i 999 from the P..N.A Egieerig College, Madurai amaraj Uiversity, amiladu, Idia ad the M.E. degree i electrical ad electroics egieerig i 003 from the Coimbatore Istitute of echology, Bharathiar Uiversity, amiladu, Idia. He has put i four years teachig experiece. He is curretly workig as a Lecturer i V.L.B. Jaakiammal Egieerig College, Coimbatore. PWAE VOLUME 5 APRIL 005 IN WAE.ORG