Median Filter based wavelet transform for multilevel noise

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1 Medin Filter bsed wvelet trnsform for multilevel noise H S Shuk Nrendr Kumr *R P Tripthi Deprtment of Computer Science,Deen Dyl Updhy Gorkhpur university,gorkhpur(up) INDIA *Deptrment of Mthemtics,Grphic Er University Dehrdun profhsshuklgkp@rediffmil.com, nrendr98@yhoo.co.in, tripthi_rp031@rediffmil.com Abstrct: In digitl imge different kinds of noises exist in n imge nd vriety of noise reduction techniques re vilble to perform de-noising. Selection of the de-noising lgorithm depends on the types of noise. Gussin noise, speckle noise, slt & pepper noise, shot noise re types of noises tht re present in n imge. The principle pproch of imge de-noising is filtering. Avilble filters to de-noise n imge re medin filter, Gussin filter, verge filter, wiener filter nd mny more. A prticulr noise cn be de-noising by specific filter but multilevel noise re chllenging tsk for digitl imge processing. In this pper we propose medin filter bsed Wvelet trnsform for imge de-noising. This technique is used for multilevel noise. In this pper three noise model Gussin noise, Poisson noise nd slt nd pepper noise for multilevel noise hve been used. In the end of pper we compre our technique with mny other de-noise techniques. Key-Words: Gussin noise, Multilevel noise, Threshold, Wvelet trnsform, Threshold rtio, Poisson noise. removl, in which the wvelet coefficients re thresholded in order to remove their noisy prt, Wvelet thresholding methods first time introduced by Donoho in 1993.Wvelet thresholding methods do not require ny prticulr ssumptions bout the nture of the signl nd exploits the sptilly dptive multiresolution of the wvelet trnsform. Before sometimes wvelet trnsforms used in signl nd imge processing, especilly in the field of signl denoising. Donoho et. l [1-5].In the 1990s, the field ws dominted by wvelet shrinkge nd wvelet thresholding methods.vrious methods re vilble[6].. Noise model The noise model is sptil invrint, i.e., independent of sptil loction. The noise model is uncorrelted with the object function. Noise models [7] cn be ctegorized into two groups: dditive noise nd multiplictive noise. ()Additive noise models: In dditive noise model, the noise is superimposed upon the imge, which resulted in vrition of the imge signl. Some common noise distributions re 1. Introduction Noisy imge due to errors in the imge cquisition process so tht pixel vlues do not reflect the true intensities of ctul picture. The presence of noise gives n imge with blur nd snowy ppernce. Wvelets trnsforms re bsed on smll wves, clled wvelet, of vrying frequency in limited durtion. Wvelet thresholding methods re used for noise Gussin noise distribution: Gussin noise is sttisticl noise. It is distributed over the signl. The probbility density function (PDF) of Gussin noise is equl to tht of the norml distribution, lso known s Gussin Distribution.PDF of Gussin rndom vrible, z is given by ( z µ ) 1 σ pz ( ) = e πσ Where: ISBN:

2 z = gry level. µ = men σ = stndrd devition p(z) = probbility density function Ryleigh noise distribution: The Ryleigh distribution of the probbilities of the rndom vrible X is chrcterized by the probbility density function ( p/ σ ) exp( p/ σ ), when p 0 f( p) = 0, when p <0 The distribution function is x ( ) = ( ) = 1 exp( / ) 0 F x f p dp x σ The mthemticl expecttion is EX = π /σ nd the vrince is DX = (4 - π)σ 4 /. The mximum vlue of the density function is equl to 1/ σ e nd is reched when x = σ. Gmm (,b) noise distribution: PDF of Gmm(,b) noise distribution rndom vrible, z is given by b b 1 z z pz ( ) = e, for z 0. b 1 where the prmeter re such tht >0,b is positive integer,nd indictes fctoril. The men nd vrince of this density re given by b b z = nd σ = Exponentil noise distribution: PDF of Exponentil noise distribution is p( z) = e z, for z 0. Where > 0.The men nd vrince of this density function re 1 z = nd σ = p(z) = probbility density function z = gry level. (b)multiplictive noise models b In this model the noise is signl dependent, nd is multiplied to the imge. Two commonly multiplictive noise models re: Slt-nd-Pepper (impulse): The slt-nd-pepper noise re lso clled shot noise, impulse noise or spike noise tht is usully cused by fulty memory loctions,mlfunctioning pixel elements in the cmer sensors, or there cn be timing errors in the process of digitiztion.in the slt nd pepper noise there re only two possible vlues exists tht is nd b nd the probbility of ech is less thn 0..If the numbers greter thn this numbers the noise will swmp out imge. For 8-bit imge the typicl vlue for 55 for slt-noise nd pepper noise is 0. Probbility density function of impulse noise is given by p, for z = pz ( ) = pb, for z = b 0, otheerwise If b >, intensity b will pper s light dot in the imge. level will pper like drk dot, if either nd is zero, the impulse noise is clled unipolr. If neither probbility is zero, nd especilly if they re pproximtely equl,impulse noise vlues will resemble slt nd pepper grnules rndomly distributed over the imge. For this reson, bipolr impulse noise lso is clled slt nd pepper noise. Speckle noise = R + jl Where, re independent Gussin, with zero men Poisson Noise Poisson noise provides noise source whose probbility density function is not continuous. For exmple, rndom number tht cn tke on only discrete vlues hs probbility function tht lso is discrete. 3. Filter If only the noise is presented in the digitl imge [8], i.e., without considering the ISBN:

3 degrdtion function, following techniques cn be used to reduce the noise effect: Men filter: For every pixel in the imge, the pixel vlue is replced by the men vlue of its neighboring pixels ( NxM ) with weight wk = 1/( NM ). This will resulted in smoothing effect in the imge. Medin filter: For every pixel in the imge, the pixel vlue is replced by the sttisticl medin of its neighboring pixels ( NxM ). Although medin filter lso provides smoothing effect, it is better in preserving detiled imge informtion, for exmple: edges. Homomorphism filter: In the cse of multiplictive noise, one cnnot simply pply smooth filter to the observed noisy imge f(x,y), s the Fourier trnsform of the product of two functions is not seprble. To overcome this issue, logrithmic representtion of the imge model is used insted, i.e. In( f ( x, y) = In( o( x, y)) + In( n( x, y)) Where the Fourier trnsforms of the logrithmic function is PSNR = 10log ( R / MSE) Since the noise model, through logrithmic opertion, becomes dditive, smooth filter cn thus be pplied to remove the noise effect. 4. Prmetric description The prmeters considered for imge processing re pek signl to noise rtio (PSNR) nd men squre error (MSE). 10 PSNR = R MSE 10log 10 ( / ) Where R is mximum vlue of the pixel present in n imge nd MSE is men squre error between the originl nd de-noised imge with size A B. Men squre error is defined s: MSE 1 M N = i 1 j 1 * x i j A B y i j = = [ (, ) (, )] Where, (,) is originl imge nd (,) is de-noised imge. Root men squre error is defined s: = ( ) i.e. root men squre error is squre root vlue of men squre error. 5. Proposed system Multilevel Noise: The noise considered in this pper is multilevel noise which is combintion of Gussin noise, Slt & Pepper noise nd Poisson noise. Thus the reduction of multilevel becomes n importnt spect in the ppliction of digitl imge. In the first level noisy imge is given s X(k,l) = O(k,l)*G(k,l) Where G is Gussin noise, O is originl imge nd X is the noisy imge with Gussin noise.(k,l) re the vrible of sptil loction (k represents the rw nd l represents the column). In the second level we re using Gussin noisy imge s n input imge in Poisson noise. Y(k,l)=X(k,l)*P(k,l) Where Y is noisy imge ssocited with Gussin nd Poisson noise. X is Gussin noisy imge nd P is Poisson noisy imge.(k, l) re the vrible of sptil loction (k represents the rw nd l represents the column). In the third level we re using previous level output noisy imge s input to Z imge. Z(k,l)=Y(k,l)*S(k,l) Where Z is the output imge with Gussin noise, Poisson noise Slt & Pepper noise. Y is the finl noisy imge output. ISBN:

4 Wvelet trnsforms using medin filter: Wvelet trnsformtion is used for reduction of noise but if we used it for multilevel noise not give better result. Medin filter is nonliner filter.during medin filtering first sorting ll the pixel vlues from the surrounding neighborhood into numericl order nd then replcing the pixel being considered with middle pixel vlue. Medin vlue must be written to seprte rry or buffer.medin filter replce the vlue of imge pixel by the medin of intensity level in the neighborhood of tht pixel. f( xy, ) = medin{ Z ( k, l)} ( kl,) Sxy Vlues of the pixel t (x, y) is included in the computtion of the medin, pixel in region defined by S. xy It is possible to improve the result by soft thresholding defined s 1 1 F = ST ( f) = ST (( f, ψm)) ψ m m T Where S T1 ( α) = mx 0,1 α α S T1 ( α) is soft thresholding function Proposed Procedure: To remove the noise of imge by incresing the PSNR nd MSE vlue Fig 1 explin the proposed concept. The noise used multilevel which is leveled by Gussin noise, pssion noise nd slt & pepper noise nd for filtering pplied medin filter for the multilevel noise in initil filtering nd then pplied wvelet trnsform, hrd thresholding nd soft thresholding Hrd thresholding in wvelet bses: f is the output imge fter medin filtering. efficient non liner de-noising estimtor is obtined by threshholding the coefficient of f, which is selected by orthogonl bsis B = N { ψ } m of R m De-noise the piecewise regulr imges on bsis of wvelet. The hrd thresholding opertor with threshold T 0 pplied to sme imge f is defined s S 0 0 ( f ) = < f, ψ > ψ = S ( < f, ψ ) ψ T < f, ψ > T m m T m m m> Where hrd threshold opertor is, 0 α α > T ST ( α) = 0, otherwise The de-noise estimtor is then defined s f = S 0 T ( f) Wvelet denoising with soft thresholding the estimted imge f using hrd thresholding. Fig 1: imge reconstruction process using medin nd wvelet trnsform 6. Result ISBN:

5 b k l c e g i d f h j Fig: the bove figure explin s follows: ().Originl imge (b).multilevel noisy imge (c).wiener filtered imge (d). 3 X 3 medin filtered imge (e).5 X 5 medin filtered imge (f).7 X 7 medin filter imge (g).noisy coefficient (h).threshold coefficient(i).hrd filtered imge without medin filter (j).soft filtered imge without medin filtered (k).hrd filtered imge with medin filtered (l).soft filtered imge with medin filtered. Tble 1: Comprison of MSE,SNR nd PSNR vlues for multilevel noise. IMAG E Multilevel noise imge Wiener Filter Imge Medin Filter 3 X 3 imge Medin Filter 5 X 5 imge Medin Filter 7 X7 Men- Squre Error(MS E) Signl-to- Noise Rtio(SN R) Pek Signl-to- Noise Rtio(PSN R) ISBN:

6 imge Hrd g without medin Soft g without Medin Hrd g with medin filter Soft g with medin filter The bove tble show tht medin filter bsed wvelet trnsform method is more efficient for removing multilevel noise thn other filters. 7. Conclusion In this pper efficient techniques for de-noising for the imge hs been proposed combined medin filter nd wvelet trnsform.this method is verified on the imge where these imge re corrupted by noise t different density.experimentl result show tht the combined with medin nd wvelet trnsform method is more efficient for removing multilevel noise. [3] Dvid L. Donoho nd Iin M. Johnstone. Adpting to unknown smoothness vi wvelet shrinkge. Journl of the Americn Sttisticl Assocition, pges 100{14, , 11, 13, 15, 4. [4] Dvid L. Donoho, Iin M. Johnstone, Gerrd Kerkychrin, nd Dominique Picrd.Wvelet shrinkge: symptopi. Journl of the Royl Sttisticl Society, Ser. B, pges 371{394, [5] Dvid L. Donoho. De-noising by softthresholding. IEEE Trnsctions on Informtion Theory, 41(3):613 {67, My , 4. [6] Anestis Antonidis, Jeremie Bigot, nd Theofnis Sptins. Wvelet estimtors in nonprmetric regression: A comprtive simultion study. Journl of Sttisticl Softwre,6(6):1,{83, June , 0, 4. [7] S. O. Rice "Mthemticl nlysis of rndom noise", Bell Syst. Tech. J., vol. 3, no. 3, pp [8] Y. H. Lee nd S. A. Kssm "Generlized medin filtering nd relted nonliner filtering techniques", IEEE Trns. Acoust., Speech, Signl Processing, vol. ASSP-33, pp References: [1] Dvid L. Donoho nd Iin M. Johnstone. Minimx estimtion vi wvelet shrinkge.technicl report, [] Dvid L. Donoho nd Jin M. Johnstone. Idel sptil dpttion by wvelet shrinkge.biometrik, 81(3):45{455, , 10, 1, 13. ISBN: