Comparison of robust M estimator, S estimator & MM estimator with Wiener based denoising filter for gray level image denoising with Gaussian noise

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1 Comparson of robust M estmator, S estmator & MM estmator wth Wener based denosng flter for gray level mage denosng wth Gaussan nose Mr. Pratyaksh A Maru Electroncs and Communcaton Department, Dr. Jvraj Mehta Insttute of Technology,Mogar, Anand, Gujarat, Inda. pratyax01maru@gmal.com Abstract In any mage processng system denosng of mages s an mportant step. The mages can be corrupted by dfferent noses wth dfferent levels. There are three types of noses avalable: mpulse, Gaussan and Speckle noses wth mxture of them. Many algorthms are proposed to remove salt & pepper (mpulse) nose as well as Gaussan nose. The Robust statstcs based flter s also proposed to remove ether mpulse or Gaussan nose usng Lorentan rho functon based robust M estmator. In ths paper we evaluate the performance of MM-estmator, S-estmator, M- estmator, medan flter and Wener flter based mage Denosng flters for Gaussan nose. The Result shows that for Gaussan nose wener based flter gves good nose reducton compare to other. Index Terms - Image Denosng, M-estmator, MM-estmator, S- estmator, Medan flter, Wener flter. I. INTRODUCTION Image processng Denosng s an mportant step for removng nose from mages. The Denosng technque gves the fnal result of any mage processng algorthm and ths result may vary wth dfferent technques. The mages can be degraded by three basc noses whch nclude mpulse nose, Gaussan nose and speckle nose. Ths degradaton of mages s because of acquston or transmsson of mages [1]. Images are degraded by the sensng envronment when acqured through optcal, electro optcal or electronc means. The result of degradaton s n the form of sensor nose; blur the mages due to camera msfocus, relatve object-camera moton, change n atmosphere, etc. Robust statstcs based flters are already proposed by many researchers for mage denosng problem. In 2001, Hamza and Krm proposed three flterng schemes; the mean medan flter, the mean-relaxed medan flter, and Mean-Log- Cauchy flter for mage denosng [3]. MM-estmator proposed by Yoha [5] n possesses both hgh break down pont and effcency. And t s proved to be better compared to robust M-estmators n terms of outler removal. Hence, n ths paper we check the performance of wener based flter to remove nose from the mages. The paper s organzed as follows: In secton 2 we dscuss Image denosng usng robust statstcs. Secton 3 dscusses wth estmators. In secton 4, the proposed method s dscussed. In secton 5, results of M-estmator, S-estmator, MM-estmator and wener based mage flterng are dscussed. Secton 6 concludes the paper. II. IMAGE DENOISING USING ROBUST STATISTICS In ths secton we dscuss the general scheme for mage denosng usng robust estmator. We select a wndow from the mage and then replace ts center value by applyng robust estmator. An llustratve example s depcted n Fg-1. An example 6 6 mage matrx s shown n Fg-1. From that 3 3 mage s extracted as shown n Fg-2. The data s arranged n a vector form and then robust lne fttng s done as shown n Fg-3. Now center value of 3 3 mage s replaced by the ntercept value of the ftted lne as shown n Fg-4. In the llustratve example the orgnal center ntensty value of 125 s replaced by the ntercept value 16. The gray value 125 s treated as an outler. The process s repeated for the whole mage[1] Fg-1 Image wth gray value Fg wndow of mage Fg-3 Robust lne fttng result on mage pxels

2 Fg-4 Image wth replaced value III. BASICS OF ROBUST M AND MM ESTIMATORS Robust lne fttng s mmensely used nowadays n outler rejecton. Robust statstcs solve the problem of outler and fnd the best ft model of the data. In robust lne fttng the goal s to fnd the regresson parameter values of the lne model that mnmze the resdual errors. Many robust estmators are avalable for robust lne fttng. For example robust M-estmator, S-estmator, MM-estmator and CMestmator. M estmator: One popular robust technque s the M- estmators. For a lne model y mx c, Let r be the resdual of the th datum, the dfference between the th observaton y, and ts ftted value, y ˆ. The standard leastsquares method tres to mnmze r, whch s unstable 2 f there are outlers present n the data. Outlyng data gve an effect so strong n the mnmzaton that the parameters thus estmated are dstorted. The M-estmators try to reduce the 2 effect of outlers by replacng the squared resduals r by another functon of the resduals, yeldng. mn ( r ) (1) Where s a symmetrc, postve-defnte functon wth a unque mnmum at zero. functon can be Cauchy, Talwar, Welsch, Far, Huber etc. Cauchy functon Huber functon MM estmator: Frst proposed by Yoha (1987), MMestmators have become ncreasngly popular and are perhaps now the most commonly employed robust regresson technque. They combne a hgh breakdown pont (50%) wth good effcency (approxmately 95%). The MM n the name refers to the fact that more than one M-estmaton procedure s used to calculate the fnal estmates. It has both the hgh breakdown property and a hgher statstcal effcency than S estmaton [6]. n ˆMM r ( ) arg mn (2) S ˆ 1 and ˆ S may be chosen n order to attan both a hgh breakdown pont and a hgh effcency. Functon can be Tukey Bweght functon. The steps to compute MM-estmator are as follows [5]. 1. Compute an ntal consstent estmate ˆ 0 wth hgh breakdown pont but possbly low normal effcency. 2. Compute a robust scale ˆ S of the resduals r ( ). 3. Fnd a soluton ˆ. S-estmators for lnear regresson were ntroduced by Rousseeuw and Yoha (1984) as an alternatve to M- estmators that do not suffer that much from outler ponts and at the same tme have a hgh breakdown pont and do not requre an auxlary scale estmator. The S-estmator ˆs mnmzes the scale functon, that s ˆ argmn ˆ ( ) s p R IV. PROPOSED METHOD The problem of lnear regresson methods s that a sngle outler value may cause severe error n the estmaton so we are usng wener based flter for mage denosng. In table 1 proposed algorthm s shown. No. Steps 1 Select mage 2 Apply the nose Use a robust estmator based flter for example M-,Sand MM- estmator. 3 Fnd the MSE (mean square error), MAE (mean 4 absolute error) 5 Fnd the PSNR and compare the results n Table-1 proposed algorthm Welsch functon Tukey functon Fg-5 functon for M estmator [2] To check the performance of the proposed method we fnd MSE (mean square error), MAE (mean absolute error) and PSNR (peak sgnal to nose raton). For better result PSNR should be hgh, MSE and MAE should be low. MSE, MAE and PSNR are calculated usng the followng equatons.

3 MSE X ˆ j Xj ( m n) MAE X ˆ j Xj ( m n) PSNR ( ) 10log MSE V. EXPERIMENTAL RESULT In ths secton the results obtaned usng all flters are dscussed. The values of PSNR, MSE, MAE for all estmators 2 (3) (4) (5) are shown for dfferent nose levels. Expermental result shows that at 0.9 nose level for Impulse nose M estmator gves PSNR of 3.30 for talwar functon. MM estmator based flter gves PSNR of Wener based flter gves PSNR of 3.38 whch s hgher than all estmators. The resultant mages for test4 wth 0.9 nose level are shown n Fg-8, whch clearly shows the effcacy of wener based flter to remove Gaussan nose as compared to M-,S-,MM-estmator based flter. The comparson of MSE values for dfferent nose levels are gven n Table-2. We get almost the same results for other dfferent natural mages for whch wener based flter gves hgher PSNR and lower MAE values as compared to other estmator based denosng flter. PSNR Nose Level andrew bsquare cauchy M far huber logstc talwar welsch medan wener S-estmator MM-estmator Table-2 PSNR value of M-estmator, medan, wener, S-estmator, MM-estmator for test4.png MSE Nose Level andrew bsquare cauchy M far huber logstc talwar welsch medan wener S-estmator MM-estmator Table-3 MSE value of M-estmator, medan, wener, S-estmator, MM-estmator for test4.png

4 MAE Nose Level andrew bsquare cauchy M far huber logstc talwar welsch medan wener S-estmator MM-estmator Table-4 MAE value of M-estmator, medan, wener, S-estmator, MM-estmator for test4.png Fg-6: PSNR comparson between M-estmator, Medan, Wener, S-estmator, MM- estmator for test1.png Fg-7: MSE comparson between M-estmator, Medan, Wener, S-estmator, MM estmator for test1.png

5 Here comparson of PSNR values for dfferent nose levels are gven n Table-2 and graph s shown n Fg-6, MSE values are shown n Table-3 and MAE values are shown n Table-4. Expermental result shows that for Gaussan nose Wener based flter gves hgh PSNR. (a) (b) (c) (d) (e) (f)

6 (g) (h) () (j) (k) (l)

7 (m) (n) Fg-8: (a) orgnal sgnal (b) Gaussan nose wth 10% nose level (c) M-estmated output(andrew rho functon) (d) M- stmated output(bsquare rho functon) (e) M-estmated output(cauchy rho functon) (f) M-estmated output(far rho functon) (g) M-estmated output(huber rho functon) (h) M-estmated output(logstc rho functon) () M-estmated output(talwar rho functon) (j) M-estmated output(welsch rho functon) (k) medan flter (l) wener output (m) S- estmated output (n) MM-estmated output at 10% nose level for test4.png. VI. CONCLUSION We conclude that for Gaussan nose wener based flter gves good nose reducton. Scentfc Research ISSN X Vol.39 No.3 (2010), pp EuroJournals Publshng, Inc VII. FUTURE WORK Future work can be to check the effcacy of other robust estmators for the mage denosng problem. REFERENCES [1] Pratyaksh A Maru, Chntan K. Mod, and P S V Nataraj, Comparson of robust MM estmator and robust M estmator based denosng flters for gray level mage denosng, IEEE Conference on Communcaton System and Network Technology (CSNT), May, [2] Mchael J. Black, On the Unfcaton of Lne Processes, Outler Rejecton, and Robust Statstcs wth Applcatons n Early Vson Internatonal Journal of Computer Vson 19(1), 57-91, [3] A. Ben Hamza and Hamd Krm, Image Denosng: A Nonlnear Robust Statstcal Approach, IEEE transactons on sgnal processng, vol.49, no.12, December [4] Tamer Rabe,- Robust Estmaton Approach for Blnd Denosng, IEEE transactons on mage processng, vol. 14, no. 11, November, [5] Rcardo A. Maronna, R. Douglas Martn, and Vctor J. Yoha, Robust Statstcs: Theory and Methods, John Wley & Sons Ltd, ISBN: , 2006, pp [6] Vncenzo Verard, and Chrstophe Croux, "Robust regresson n Stata". FBE Research Report KBI_0823, pp. 1-13, [7] Gurpreet Chahal and Harmnder Sngh, Robust Statstcs based Flter To Remove Salt and pepper Nose In Dgtal Images, Internatonal Journal of Informaton Technology and Knowledge Management July-December 2010, Volume 2, No. 2, pp [8] R. K. Kulkarn, S. Meher, Mrs. J. M. Nar, An Algorthm for Image Denosng by Robust Estmator, European Journal of