Modified Curvelet Thresholding Algorithm for Image Denoising

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1 Joural of Computer Sciece 6 (1): ISSN Sciece Publicatios Modified Curvelet Thresholdig Algorithm for Image Deoisig 1 Al-Dahoud Ali Preety D. Swami ad 3 J. Sighai 1 Departmet of Computer Sciece Faculty of Iformatio Techololgy Alzaytooah Uiversity of Jorda Amma Jorda Departmet of Electroics Samrat Ashok Techological Istitute Vidisha (Madhya Pradesh) Idia 3 Departmet of Electroics ad Commuicatio Egieerig Maulaa Azad Natioal Istitute of Techology Bhopal (Madhya Pradesh) Idia Abstract: Problem statemet: This study itroduced a adaptive thresholdig method for removig additive white Gaussia oise from digital images. Approach: Curvelet trasform employed i the proposed scheme provides sparse decompositio as compared to the wavelet trasform methods which beig ogeometrical lack sparsity ad fail to show optimal rate of covergece. Results: Differet behaviors of curvelet trasform maxima of image ad oise across differet scales allow us to desig the threshold operator adaptively. Multiple thresholds depedig o the scale ad oise variace are calculated to locally suppress the curvelet trasform coefficiets so that the level of threshold is differet at every scale. Coclusio/Recommedatios: The proposed algorithm succeeded i providig improved deoisig performace to recover the shape of edges ad importat detailed compoets. Simulatio results proved that the proposed method ca obtai a better image estimate tha the wavelet based restoratio methods. Key words: Deoisig cubic thresholdig wavelet trasform curvelet trasform. INTRODUCTION I the real world sigals do ot exist without oise which arises durig image acquisitio (digitizatio) ad/or trasmissio (Gozalez ad Woods 00). Whe images are acquired usig a camera light levels ad sesor temperature are maor factors affectig the amout of oise. Durig trasmissio images are corrupted maily due to iterferece i the chael used for trasmissio. Removig oise from images is a importat problem i image processig (Ruggeri ad Vidakovic 1999). This oise removal takes place i the origial time space domai or i a trasform domai. I trasform domai Fourier trasform is used i the timefrequecy domai ad multiresolutio trasforms like wavelet/curvelet/cotourlet trasforms are used i the time-scale domai. Deoisig a give oise corrupted sigal is a traditioal problem i both statistics ad i sigal processig applicatios. Liear deoisig methods are ot so effective whe trasiet o-statioary widebad compoets are ivolved sice their spectrum is similar to the spectrum of oise (Zhag ad Luo 1999). Noliear deoisig methods (Smith ad Agaia 004) rely o the basic idea that the eergy of a sigal will ofte be cocetrated i a few coefficiets i the trasform domai while the eergy of oise is spread amog all coefficiets i trasform domai. Therefore the oliear methods will ted to keep a few larger coefficiets represetig the sigal while the oise coefficiets will ted to reduce to zero. Deoisig methods based o multiresolutio trasforms ivolves three steps: A liear forward trasform oliear thresholdig step ad a liear iverse trasform. Wavelets are successful i represetig poit discotiuities i oe dimesio but less successful i two dimesios. As a ew multiscale represetatio suited for edges ad other sigularity curves the curvelet trasform has emerged as a powerful tool. The developig theory of curvelets predict that i recoverig images which are smooth away from edges curvelets obtai smaller asymptotic mea square error of recostructio tha wavelet methods (Cades ad Dooho 004). Curvelet trasform: A image whe aalyzed usig a -D wavelet trasform exhibits large wavelet coefficiets alog the edges i the image. At each Correspodig Author: Al-Dahoud Ali Faculty of Sciece ad Iformatio Techology Alzaytooah Uiversity of Jorda Amma Jorda 18

2 scale these edges i the image are see repeated. This requires may wavelet coefficiets to recostruct the edges i a image ad it puts a limit o wavelet deoisig. The estimatio of these large umbers of coefficiets lead to high Mea Square Error (MSE). A compariso of order of error i case of wavelets ad that of curvelet recostructio is preseted i the literature (Cades ad Dooho 004) below. No-liear approximatio of obects ca be cosidered by treatig them as fuctio of two variables oe with discotiuities alog edges ad secod which are smooth. If a obect f is represeted i a wavelet basis the the umber of wavelet coefficiets of f exceedig the threshold 1/ i absolute value icreases rapidly as c. as. This icrease idicates the requiremet of may terms to recostruct a good image. If the best partial recostructio obtaied by selectig the largest terms is represeted by f approximatio would obey: W 1 L W J. Computer Sci. 6 (1): the best -term f f (1) This result is ot optimal ad hece wavelets fail to represet obects with edges. If the edge curve is cosidered to be of legth oe the at each scale there are approximately wavelets iteractig with the edge resultig i coefficiet of size. This shows that the wavelet coefficiet decay oly as 1/. Wavelets do ot take advatage of the geometry of the edge ad hece about coefficiets are eeded to recostruct the frequecy cotet of a edge up to the subbad ξ. Wavelets beig ogeometrical move oly i horizotal ad vertical directios ad ot alog the curve ad thus caot achieve the optimal rate of covergece. The curvelet trasform has a tight frame which combies multiscale aalysis ad ideas of geometry ad ca achieve optimal rate of covergece by simple thresholdig (Starck et al. 00). This multiscale trasform has a strog directioal character i which elemets are aisotropic at fie scales. The support of these elemets is accordig to the parabolic scalig priciple legth width. Curvelets partitio the frequecy plae ito dyadic coroae. Ulike wavelets these coroa are subpartitioed ito agular wedges displayig the parabolic aspect ratio as show i Fig. 1. Curvelets at scale are of rapid decay away from a ridge of legth / ad width ad this ridge is the effective support γ µ. If coefficiets of oly those curvelets which overlap with the edge ad are early taget to it are cosidered the for a fixed scale there are at most O( / ) coefficiets of such type. 19 Fig. 1: Frequecy domai view of the curvelet tilig The size of each coefficiet ca be estimated by: θ = f γ f. γ () µ µ L µ L 1 Curvelets are L ormalized so that γ 1 ad are supported i a box of side legth / ad width. Therefore they obey: µ L1 3 4 γ B. (3) Sice f is a bouded fuctio the coefficiets θ µ verify the apriori estimate: 3 4 θ B.. f (4) µ L At each scale there are O( / ) coefficiets which are bouded by C. 3/4 ad the th largest coefficiet θ is bouded by: 3 θ C. (5) The above decay gives O( ) covergece rate for the oliear -term approximatio f defied by keepig the largest term i the curvelet expasio ad it obeys: C f f θ C. ( m) L (6) m> From the above equatio the faster rate of decay of coefficiets i a curvelet basis is obvious as compared to the decay of wavelet coefficiets i Eq. 1 for a obect with arbitrary C sigularity. µ L

3 J. Computer Sci. 6 (1): MATERIALS AND METHODS Prelimiary algorithm for image deoisig: The mathematical model of oisy image is as follows: y = x + (7) A image features a wide variety of characteristics. Hece istead of usig a sigle value as the global threshold the operator D(.) ca be desiged to produce multiple local thresholds λ adaptively for differet scales from fie to coarse. For wavelets a adaptive threshold is (Li ad He 006): y = The observed image x = The ukow origial image = The cotamiatig oise Complete curvelet deoisig procedure is performed by takig curvelet trasform of the image ad the applyig thresholdig to elimiate oisy coefficiets. Thus the iverse curvelet trasform of the thresholded coefficiets give the deoised image. The fast discrete curvelet trasform (Cades et al. 006) of the observed image is evaluated as Y usig the curvelet trasform operator C(.) usig followig equatio: Y = C(y) (8) The threshold deoted by λ for ay trasform ca be expressed i geeral terms usig the operator D(.) as: λ = D(Y) (9) For wavelet based deoisig procedures oe such threshold is the uiversal threshold (Li ad He 006) give as: λ = D (Y) = σ log N (10) g σ = The stadard deviatio of oise N = The size of image Wavelet trasform maps white oise i the sigal domai to white oise i the trasform domai. Thus i the trasform domai the sigal is cocetrated ito fewer coefficiets but the oise does ot cocetrate. The priciple behid separatio of sigal ad oise is that whe scale decreases wavelet trasform maxima of images does t icrease but at the same time wavelet trasform modulus of white oise icreases. Thus differet behaviors of wavelet trasform maxima of images ad oise across differet scales allow us to desig the operator D(.) adaptively. 0 σ log N λ = D (Y) = log( + 1) (11) where is the decompositio level of wavelet packet trasform. This modified multiple local thresholdig techique obtaied better results tha the soft ad the hard thresholdig methods which utilizes a sigle threshold value at every scale. Curvelet trasform employs the 1-D wavelet trasform as a compoet step but alog the radial variable i Rado space. Thus Eq. 11 does ot prove to be effective for thresholdig the curvelet trasform coefficiets ad requires some modificatio. Modified curvelet thresholdig for image deoisig: I this study a similar multiple threshold techique for thresholdig the curvelet coefficiets is proposed. To desig the operator D(.) it is proposed to retai all the coefficiets at the first scale sice they are the dc values ad they provide the average iformatio of the image. For the remaiig scales the coefficiets which provide the highest PSNR values seem to be correlated ad the curvelet coefficiets appear to decay i a expoetial maer as show i Fig.. I Fig. Series 1-3 are plots for λ correspodig to σ = 0 30 ad 50 respectively. Thus a scale depedet expoetial fuctio multiplied by a scale depedet logarithmic fuctio resulted i improvemet i PSNR values. Therefore the multiple local thresholds are proposed as: λ = D (Y) = σ log N.e.log( + 1) (1) ( 1) for = 3 J = The decompositio level of curvelet trasform J = The iteger correspodig to the last scale After selectig the threshold level the ext step i the process of deoisig is applyig the threshold operator T(..). This ca be expressed as: Z = T(Yλ) (13) where Z gives the thresholded coefficiets.

4 J. Computer Sci. 6 (1): This study employs cubic soft ad hard thresholdig fuctios as the threshold operator T(..) i Eq. 13 with the differece that the threshold is ot sigle valued λ but is multivalued λ as obtaied from Eq. 1. Cubic thresholdig fuctio is very flexible ad has the capability to adapt to differet types of images ad threshold operators. The cubic threshold fuctio (Li ad He 006) give as: 0 λ 3 T (Y λ ) = λ Y 1 else Y (14) The soft (Dooho 1995) ad the hard thresholds (Dooho 1994) are some simple but powerful shrikage fuctios. These thresholds select a sigle global threshold for all the scales usig Eq. 10. Employig λ of Eq. 1 with the soft thresholdig operator proposes the multiple local soft thresholdig as: T (Y λ ) = Y s λ 0 Y < λ sig(y) else (15) Similarly by combiig λ of Eq. 1 with the hard threshold operator multiple local hard thresholdig ca be give by: 0 Y < λ T k (Y λ ) = Y else (16) Fially C 1 (.) takes the fast discrete iverse curvelet trasform of the thresholded curvelet coefficiets Z as: ˆx 1 = C (Z) (17) where ˆx is the recostructed/deoised image. RESULTS AND DISCUSSION The performace of the proposed thresholdig methods is evaluated ad compared with that of soft hard ad cubic thresholdig schemes usig wavelets (Li ad He 006). Gaussia oise was added to the classical Lea ad Satur images. Multiple local thresholds are obtaied usig Eq. 1. The curvelet coefficiets are processed by thresholdig fuctios i Eq The performace of deoisig is evaluated usig Peak Sigalto-Noise Ratio (PSNR) ad Mea Square Error (MSE). PSNR is defied as the ratio of sigal power to oise power. It basically obtais the gray value differece betwee resultig image ad origial image. MSE is give by the formula: m MSE = I(i ) K(i ) m i= 0 = 0 I = The origial image K = The recostructed image m ad = The umber of rows ad colums respectively PSNR is give as: PSNR MAX MSE I = 10log10 where MAX I is the maximum pixel value of the image. Numerical values for PSNR ad MSE for the images Lea ad Satur are give i Table 1 ad Table respectively. The curvelet recostructio usig multiple local thresholds eoys superior performace over the wavelet based recostructios. The pictorial deoisig performace for images Lea ad Satur usig wavelet based soft hard ad cubic thresholds is compared with curvelet based multiple local soft hard ad cubic thresholds i Fig. 3 ad 4 respectively. Experimets show that multiple local thresholdig based o curvelets outperforms the wavelet based methods o the basis of MSE ad PSNR. Fig. : Graph idicatig relatioship of the thresholded coefficiets with the scale. Series 1 Series ad Series 3 are for stadard deviatios of 0 30 ad 50 respectively 1 Table 1: Compariso of differet thresholdig methods for Lea image Thresholdig usig Multiple local wavelets thresholdig usig curvelets Method PSNR/dB MSE PSNR/dB MSE Noisy Image Soft_thresholdig Hard_thresholdig Multiple local cubic_thresholdig

5 J. Computer Sci. 6 (1): (a) (b) (c) (d) (e) (f) (g) (h) Fig. 3: Deoisig of Lea (a) origial image; (b) oisy image; (c) soft thresholdig usig wavelets; (d) multiple local soft thresholdig usig curvelets; (e) hard thresholdig usig wavelets; (f) multiple local hard thresholdig usig curvelets; (g) multiple local cubic thresholdig usig wavelets; (h) multiple local cubic thresholdig usig curvelets (a) (b) (c) (d) (e) (f) (g) (h) Fig. 4: Deoisig of Satur. (a) origial image (b) oisy image (c) soft thresholdig usig wavelets (d) multiple local soft thresholdig usig curvelets (e) Hard thresholdig usig wavelets (f) multiple local hard thresholdig usig curvelets (g) multiple local cubic thresholdig usig wavelets (h) multiple local cubic thresholdig usig curvelets Table : Compariso of differet thresholdig methods for Satur image Thresholdig Multiple local thresholdig usig wavelets usig curvelets Method PSNR/dB MSE PSNR/dB MSE Noisy Image Soft_thresholdig Hard_thresholdig Multiple local cubic _thresholdig CONCLUSION I this study a ew deoisig techique based o adaptive selectio of thresholds to suppress oisy curvelet trasform coefficiets is preseted. Due to multiresolutioal dictioary the maxima of the curvelet trasform coefficiets vary ad so the threshold operator is desiged to produce as may local threshold values as are the scales. The proposed method efficietly adapts to oise characteristics for differet scales ad reduces the oise while preservig edges i the image. The thresholdig fuctio chose are the cubic hard ad soft thresholds ad the proposed expressio is tested agaist them. From the restored images it ca be visually depicted that the edges ad texture are well preserved takig the advatage of the fact that curvelets beig geometrical very well alig

6 J. Computer Sci. 6 (1): themselves to the cotours of the edges. Numerical experimets show the good performace of the proposed method i compariso to wavelet based decompositio. Further works ivolve extedig the proposed method to various classes of images which are differet from atural images. Aother importat issue is to test the performace o higher resolutio images. REFERENCES Cades E. L. Demaet D. Dooho ad L. Yig 006. Fast discrete curvelet trasforms. Siam Multiscale Model. Simul. 5: DOI: / X Cades E.J. ad D.L. Dooho 004. New tight frames of curvelets ad optimal represetatios of obects with piecewise C sigularities. Commu. Pure Applied Math. 57: DOI: /cpa Dooho D.L. ad I.M. Johstoe Ideal spatial adaptatio by wavelet shrikage. Biometrika 81: DOI: /biomet/ Dooho D.L De-oisig by soft thresholdig. IEEE Tras. Iform. Theor. 41: Gozalez R.C. ad R.E. Woods 00. Digital Image Processig. d. Ed. Pearso Educatio ISBN: Li Q. ad C. He 006. Applicatio of wavelet threshold to image deoisig Proceedigs of the 1st IEEE Iteratioal Coferece o Iovative Computig Iformatio Cotrol pp: DOI: /ICICIC Ruggeri F. ad B. Vidakovic A bayesia decisio theoretic approach to wavelet thresholdig Statistica Siica 9: Smith C.B. ad S.S. Agaia 004. Noliear oise suppressio usig a parametric class of wavelet shrikage fuctios Proceedigs of the SPIE Defese ad Security Symposium 5439: DOI: / Starck J.L. E.J. Cadès ad D.L. Dooho 00. The curvelet trasform for image deoisig. IEEE Tras. Image Process. 11: DOI: /0 Zhag X.P. ad Z.Q. Luo A ew time-scale adaptive deoisig method based o wavelet shrikage. Proceedigs of the IEEE Iteratioal Coferece o Acoustics Speech ad Sigal Process. pp: DOI: /ICASSP