A NEW DISCRETE WAVELET TRANSFORM

Transcription

1 A NEW DISCRETE WAVELET TRANSFORM ALEXANDRU ISAR, DORINA ISAR Keywords: Dscrete wavelet, Best energy concentraton, Low SNR sgnals The Dscrete Wavelet Transform (DWT) has two parameters: the mother of wavelets and the number of teratons. Selectng dfferent parameters for dfferent DWT of the same sgnal, dfferent energy concentratons n the wavelet doman are obtaned. So, for a partcular sgnal there s a better par of parameters that realzes the best energy concentraton n the wavelet transform doman. In applcatons s dffcult to fnd ths best par of parameters. Ths s the reason why the am of ths paper s to ntroduce a new DWT less senstve to the parameter selecton. Ths transform s bult usng a technque very modern n telecommuncatons, the dversty enhancement. The dversty s enhanced n the wavelet transform doman computng dfferent DWT, of the same sgnal, wth dfferent parameters. So, the nput sgnal, represented by a vector, s transformed nto a matrx. Each column of ths matrx represents the DWT of the nput sgnal, computed wth a dfferent par of parameters. Ths matrx represents the result of the new dscrete wavelet transform, named the Dversty Enhanced Dscrete Wavelet Transform, (DEDWT). The new transform can be used wth good results n denosng applcatons, especally for low SNR sgnals. 1. INTRODUCTION The dscrete wavelet transform, DWT, realzes a concentraton of the energy of the nput sgnal n a small number of coeffcents. Ths concentraton's enhancement s useful for the reducton of the number of operatons n the applcaton consdered. For a gven sgnal, usng dfferent wavelet's mothers, dfferent energy concentratons are obtaned. So, for a gven nput sgnal there s a best wavelet's mother, that realzes the hgher energy concentraton. The am of ths paper s to propose a new DWT less senstve to the selecton of the wavelet s mother. The constructon s based on the dversty enhancement s prncple. Such a transformaton s useful for the denosng of low sgnal to nose rato, SNR, sgnals. In the second secton of ths paper s presented the constructon of the new transform, the DEDWT. The computaton of ts nverse s also descrbed. The central result of ths paper, descrbed n the thrd secton, s the applcaton of the new transformaton n denosng applcatons. In the forth secton some smulaton results are presented. The last secton s dedcated to conclusons. Rev. Roum. Sc. Techn. Électrotechn. et Énerg., 47, 4, p., Bucarest, 2002.

2 Alexandru Isar, Dorna Isar 2 2. THE DEDWT, A NEW DISCRETE WAVELET TRANSFORM The parameters of the DWT are: the wavelet's mother ψ () t and the number of teratons, M. An excellent envronment for the smulaton of sgnal processng methods based on wavelets, s the Matlab toolbox WaveLab, [1]. Ths s the reason why ths toolbox wll be used n the smulatons reported n ths paper. A very modern sgnal processng method, developed n communcatons, s based on the enhancement of the dversty of the sgnal to be processed. In our case the dversty enhancement can be realzed n the DWT doman. The parameters of the DWT are the wavelet's mother and the number of teratons. So the dversty can be enhanced computng for the same sgnal, x [] n, some dfferent dscrete wavelet transforms. For each of them a dfferent par of parameters must be used. In ths way a new DWT s obtaned. Ths transform wll be called n the followng the Dversty Enhanced Dscrete Wavelet Transform (DEDWT). It s a redundant dscrete wavelet transform. Ths new transform realzes the correspondence between the vector x T [] n and a matrx DEDWT[n,m]. Every column of ths matrx represents one of the DWT of the sgnal x T [] n. Ths transform can be nverted. Its nverse wll be called IDEDWT. For every column of the matrx DEDWT[n,m] the correspondng IDWT s computed. A new matrx, E[n,m], s obtaned. Every column of ths matrx contans the sgnal x [] n. Computng the T T mean of the columns of the matrx E[n,m] the vector xo [] n = x [] n, s obtaned. Another source for the enhancement of the dversty can be the crcular translaton of the samples of the sgnal x [] n. Usng ths source another redundant DWT, named translaton-nvarant DWT, was proposed n [2]. For a better enhancement of the dversty n the wavelets transform doman ths transform can be assocated wth the transform proposed n ths paper. 3. USING THE DEDWT FOR DENOISING We wll consder n ths paper the case of addtve whte gaussan nose wth zero mean channels. We deal wth the sgnal x [ n ] = x[ n ] n [ n ], where [] n + n s a nose. To estmate the sgnal x[n], Donoho, [3], proposed the followng method: 1. The Dscrete Wavelet Transform (DWT) of the sgnal x [] n s computed obtanng the sgnal y [] n.

3 3 New dscrete wavelet transform 2. A non-lnear flterng procedure, called soft thresholdng, s appled n the wavelet transform doman: { y [] n }( y [] n t), y [] n 0, y [] n sgn t, yo [] n = (1) < t, where t s a threshold. 3. Takng the nverse DWT (IDWT), the de-nosed verson, x o [] n, s obtaned. There are some recent papers dealng wth ths denosng method, [4-7], but the case of low SNR sgnals sn t analyzed. The value of the threshold t, from the second step of the Donoho's denosng method, recommended n [3], s: t = N ln Nσ, where N represents the number 2 of samples of the sgnal x [] n and σ represents the power of n [] n. Usng ths value the mnmax mean square approxmaton error of the sgnal x[n] wth the sgnal x o [] n s obtaned, for a large class of nput sgnals wth dfferent regulartes, [3]. The frst dsadvantage of ths denosng method s the fact that t s not adaptve. In [8] s proved that the Donoho's denosng method don't works well n the case of low SNR sgnals. When the SNR of the nput sgnal decreases, the dstorton n the recovered sgnal ncreases. For very low SNR the output sgnal becomes equal wth zero (the entre nose s suppressed but the entre sgnal, x[n], s suppressed too) because the value of the threshold requred becomes superor to any wavelet coeffcent of the sgnal x[n]. So, for low SNR sgnals the use of a threshold wth the value prescrbed by the Donoho's denosng method produces unacceptable hgh dstortons. These are the reasons why n ths paper s proposed the substtuton of the DWT and IDWT wth DEDWT and IDEDWT n the Donoho s algorthm, already presented. 4. SIMULATION RESULTS The results are presented n Fg. 1. In the top (Fg. 1 a) s presented the sgnal x[n]. Under ths waveform (Fg. 1 b) s represented the acqured sgnal x [] n. Its SNR s of Under ths waveform (Fg. 1 c) s represented the result of the proposed denosng method, the sgnal x o [] n. Fnally, n the bottom of Fg. 1, (Fg. 1 d) s represented the result obtaned applyng the Donoho's denosng method wth the Haar's wavelet's mother and a number of 16 teratons. The sources for the enhancement of the dversty were the classcal nne Daubeches wavelet's mothers the Haar s wavelet s mother and the teratons numbers 1, 2, 4 and 8. The matrx DEDWT has 36 columns. The

4 Alexandru Isar, Dorna Isar 4 sgnal to dstorton rato, SDR, enhancement of the proposed denosng method s of The SDR enhancement of the classcal Donoho s denosng method s only of So the proposed denosng method realzes a better SDR enhancement. Ths concluson can be also obtaned analyzng the Fg. 1 c and 1 d. The transton dstortons from Fg. 1 d can t be observed n Fg. 1 c. Fg. 1 Smulaton results. 5. CONCLUSIONS The utlty of the new dscrete wavelet transform, proposed n ths paper, was proved. Its use n denosng applcatons has good results, outperformng the results obtaned usng the DWT. The ncreasng of the SDR can be explaned on the bass of the use of the medator, mplctly ntroduced for the computaton of the IDEDWT. The classcal Donoho s denosng method can be further mproved f another procedure for the selecton of the threshold t s used. Ths procedure s reported n [9]. Combnng ths selecton procedure and the new transform, proposed n ths paper, sgnals wth low SNR ( 1 ) can be well reconstructed on

5 5 New dscrete wavelet transform the bass of the denosng procedure. The prncple of the constructon of the new transform, the dversty enhancement, s a powerful tool n telecommuncatons. Its applcaton realzes an ncreasng of the redundancy of the sgnal to be processed. Ths redundancy s ncreasng can be exploted also n other applcatons of the wavelet s theory. An example s the mages watermarkng. Because the mage obtaned n the DEDWT doman has greater dmensons versus the mage to be watermarked, there s more room n the transformed mage to embed the watermark. Ths s the reason why the use of the new wavelet transform, proposed n ths paper, n watermarkng applcatons, permts the transparent ncluson of an ncreased amount of nformaton n the watermarked mages. ACKNOWLEDGEMENT The results reported here were partally obtaned n the framework of the CNCSIS Grant number 24, dedcated to the use of wavelet theory n data compresson. In ths framework the authors were worked n a team drected by Professor Ioan Nafornta. Receved the December 19, 2002 Unversty «Poltehnca» of Tmşoara REFERENCES 1. J. B. Buckhet, D. L. Donoho. WaveLab and Reproducble Research, n Wavelets and Statstcs, Ed. A. Antonads and G. Oppenhem, Sprnger-Verlag, New York, 1995, pp R. R. Cofman, D. L. Donoho. Translaton Invarant Denosng, In Wavelets and Statstcs, Ed. A. Antonads and G. Oppenhem, Sprnger Verlag 1995, pp D. L. Donoho. Denosng by Soft Thresholdng, Techncal Report no.409, Stanford Unversty, December M. Jansen, A. Bultheel. Asymptotc Behavor of the mnmum mean squared Error Threshold for Nosy Wavelet Coeffcents of Pece-wse Smooth Sgnals, IEEE Transactons on Sgnal Processng, 49 (6), pp , June S. Sardy, P. Tseng, A. Bruce. Robust Wavelet Denosng, IEEE Transactons on Sgnal Processng, 49 (6), June 2001, pp G. P. Nason. Choce of wavelet smoothness, prmary resoluton and threshold n wavelet shrnkage, Statstcs and Computng, 12, pp , C. Holmes, D.G.T. Denson. Perfect Samplng for the Wavelet Reconstructon of Sgnals, IEEE Transactons on Sgnal Processng 50 (2), February 2002, pp V. Katkovnk, H. Oktem, K. Egazaran. Flterng Heavy Nosed Images Usng ICI Rule for Adaptve Varyng Bandwdth Selecton, Proceedngs of ISCAS'99, pp , Orlando, Florda, June, D. Isar. L augmentaton adaptatve du rapport sgnal/brut, Rev. Roum. Sc. Techn. Électrotechnque. et Énergétque 42, 3, Bucarest (1997).

6 Alexandru Isar, Dorna Isar 6