A Modified Median Filter for the Removal of Impulse Noise Based on the Support Vector Machines

Transcription

1 A Modfed Medan Flter for the Removal of Impulse Nose Based on the Support Vector Machnes H. GOMEZ-MORENO, S. MALDONADO-BASCON, F. LOPEZ-FERRERAS, M. UTRILLA- MANSO AND P. GIL-JIMENEZ Departamento de Teoría de la Señal y Comuncacones Unversdad de Alcalá Crta. Madrd-Barcelona km 33,600. D.P Alcalá de Henares-Madrd SPAIN Abstract: - In ths work we present a new approach for the medan flter. We modfy ths nonlnear flter used n mpulse nose removal by applyng t only to the nosy pxels and by usng a dfferent pxel as output of the modfed flter. The decson between the nosy and no nosy pxels s mplemented by usng the Support Vector Machnes classfcaton. We use ths new classfcaton tool because of ts capablty of generalzaton and the reduced number of tranng examples needed. The results presented show that ths method slghtly outperforms prevous mpulse nose removal methods, both n reconstructon error and n edge preservaton. Besdes, ths method may be appled over hghly corrupted mages whle mantanng the hgh qualty n the recovered mages. Key-Words: - Enhancement, mpulse nose, support vector machnes, classfcaton, medan flter, nonlnear flterng. 1 Introducton Sometmes, the mages that we are usng are corrupted by mpulse nose. Ths nose may be due to camera mperfectons (saturaton) or to a nosy transmsson channel. The deal mpulse nose removal must preserve the edges and the detal nformaton nto the mages. There are several works n mpulse nose removal [1]-[4] that look for ths deal removal. The best results have been obtaned wth nonlnear technques rather than the lnear ones. One nonlnear classcal approxmaton s the medan flter [5] that gves good results when t s appled to low corrupted mages. The prncpal drawback of these methods s that they are appled to all the pxels nto the mages, nosy and not nosy. Ths fact leads to an mage blurrng and edge jtter, especally wth hghly corrupted mages. In [1], the method proposed s appled only to the nosy pxels and the detecton of nosy pxels s mplemented wth some threshold comparsons. The nosy pxels are replaced wth the output of a called "Rank ordered mean" (ROM) flter. Ths flter s a modfed medan flter whch nput are the pxels n a wndow around the nosy pxel but excludng ths one. In ths work, we present a new modfed medan flter wth an scheme smlar to [1]. Frst, we fnd the nosy pxels nto the mages by usng a Support Vector Machnes (SVM) classfer and these ones are replaced wth the output of a modfed medan flter. Ths flter does not use the nosy pxel as nput, lke n [1]. The dfference between both approaches s n the pxel nformaton used as output. We use the SVM due to ts capablty of generalzaton wth a reduced number of tranng examples. Ths property allows a tranng that can use synthetc mages nstead of real mages and then reduce the tranng complexty. Ths new method gves good results, both n "Peak Sgnal to Nose Rato" (PSNR) and edge preservaton, when appled over mages wth hgh nose ratos. Our method gves the bests results when the nose rato s below 30% (30 percent of nosy pxels nto the mage) and the results are smlar for hgher dstorton. Secton 2 of ths paper ntroduces the SVM classfer and the parameters that must be set n the tranng process to obtan the bests results. Secton 3 explans the method used for the tranng needed n the nose detecton. Secton 4 shows the modfcaton made over the medan flter and last n Secton 5 we can see and compare the results obtaned by applyng ths new method wth those presented n [1].

2 2 SVM Classfcaton There are several ways to classfy, Bayesan decson, neural networks or support vector machnes, for example. In ths work we use the SVM classfer snce ths method provdes good results wth a reduced set of data and then we do not requre an ntensve tranng lke another methods. Thus the SVM gves us a smple way to obtan good classfcaton results wth a reduced knowledge of the problem. The prncples of SVM have been developed by Vapnk and have been presented n several works as [6][7]. In the decson problem we have a number of vectors dvded nto two sets, and we must fnd the optmal decson boundary to dvde the sets. The border chosen may be anyone that dvdes the sets but only one s the optmal electon. Ths optmal electon wll be the one that maxmzes the dstance from the fronter to the data. In the two dmensonal case, the fronter wll be a lne, n a multdmensonal space the fronter wll be an hyperplane. The decson functon that we are searchng has the followng form, f ( ) w x + b = w x b x = + (1) n =1 In (1), x s a vector wth n components that must be classfed. We must fnd the vector w and the constant b that makes optmal the decson fronter. The basc classfcaton process s made by obtanng the sgn of the decson functon appled to the gven vector, a postve value represents the assgnment to one class and a negatve one represents the assgnment to the another class. In the SVM, the hyperplane w wll be a lnear combnaton of the support vectors and then (1) can be expressed as (2). f l ( ) = x α y x x + b (2) = 1 The y values that appear nto ths expresson are +1 for postve classfcaton tranng vectors and 1 for the negatve tranng vectors. Besdes, the nner product s performed between each tranng nput and the vector that must be classfed. Thus, we need a set of tranng data (x,y) n order to fnd the classfcaton functon and the α values that makes t optmal. The l value wll be the number of vectors that n the tranng process contrbute n a hgh quantty to form the decson fronter. The electon of these vectors s made by lookng at the α values, f the value s low the vector s not sgnfcant. The vectors elected are known as support vectors. Normally the data are not lnearly separable and ths scheme can not be used drectly. To avod ths problem, the SVM can map the nput data nto a hgh dmensonal feature space. The SVM constructs an optmal hyperplane n the hgh dmensonal space and then returns to the orgnal space transformng ths hyperplane n a non-lnear decson boundary. The non-lnear expresson for the classfcaton functon s gven n (3) where K s the non-lnear mappng functon. f l ( ) = α y K( x x) x + b (3) = 1 The choce of ths non-lnear mappng functon or kernel s very mportant n the performance of the SVM. The appled SVM uses the radal bass functon to perform the mappng, snce we have obtaned the bests results wth t. Ths functon has the expresson gven n (4). K ( x, y) exp γ( x y) 2 ( ) = (4) The γ parameter n (4) must be chosen to reflect the degree of generalzaton that s appled to the data used. The more avalable tranng data the less generalzaton needed n the SVM. A lttle γ reflects more generalzaton and a bg one represents less generalzaton. Besdes, when the nput data s not normalzed, ths parameter performs a normalzaton task. When some data nto the sets can not be separated, the SVM can nclude a penalty term (C) that makes more or less mportant the msmatch classfcaton. The lower C the more mportant s the msclassfcaton error. C and the kernel are the only parameters that must be chosen to obtan the SVM. 3 Nose detecton tranng In ths secton we present a way to detect mpulse nose by usng the SVM classfcaton. In ths case, the decson needed s between "the pxel s nosy" or "the pxel s not nosy". The nose type that we are detectng s known as salt&pepper. In ths nose type the corrupted pxels have values 0 or 255 wth equal probabltes. In order to obtan the classfcaton functon we must extract the nformaton needed from the mages. We cannot use the entre mage as a vector because of computatons. We must dvde the mages to obtan small and sgnfcant vectors. In ths work a vector s formed for each pxel, usng the values of the pxels n a 3x3 wndow around t. Ths way a nne components vector s calculated at each pxel except

3 for the border of the mage. These vectors are used as nputs to the tranng process and based on them we obtan the functon (2). The next step s to fnd the mages to be used nto the tranng process. The mages elected must be smple to avod an excessve tranng tme but sgnfcant enough to gve good results. The opton elected was to make controlled mages wth an eght bts gray scale and wth added random nose n a known poston. In Fg.1 an example s presented. (a) Whte pxels (b) Black pxels Fg.1. Example of tranng mages As we can see the tranng to detect whte and black nosy pxels s dvded nto two parts. Frst we use mages lke n Fg.1-a to tran the SVM for whte nosy pxels detecton and mages lke Fg.1-b for black nosy pxels detecton. Ths strategy reduces the number of support vectors and then ncreases the speed of the process. When we are searchng nosy pxels n an mage wth salt&pepper nose, really we search for whte and black pxels separately. In the tranng mages we must control the sze and the nose rato. The mages n Fg.1 have a 32x64 sze and a 30 percent nose. When we ncrease the nose rato we obtan best results over hghly corrupted mages but the number of support vectors s ncreased too. A hgher sze of the mages produces a fner gray scale but ncreases the tranng and the executon tme. 4 Modfed medan flter The classcal medan flter replaces the central pxel n a wndow wth the medan of the pxels nto ths wndow. The medan s the value of the pxel that occupes the central poston when we arrange the wndow pxels n ascendng order. In a 3x3 wndow we take the ffth value n ths ordered lst. In [1] a modfcaton s gven for the medan flter called Rank ordered mean (ROM). Ths modfcaton conssts of the excluson of the central pxel n the wndow when we calculate the medan. The central pxel has been detected as nosy prevously. The output of ths modfed medan flter s gven n (5), m ( n) ( n ) + ( n) r 4 r = 5 (5) 2 where r 4 (n) and r 5 (n) are the values of the two central pxels n the ordered lst of pxels, ( n ) [ r ( n), r ( n),, ( n) ] r 1 2 r8 = (6) The strategy used n ths work s smlar but we do not use m(n) as output. The value used as output s r 5 (n) snce we fnd emprcally that the results obtaned are better. The applcaton of the modfed medan flter can be made drectly over the nosy mage or n a recursve mplementaton that uses the prevously replaced values nto the sldng wndow,.e. the wndow contans nosy and fltered mage pxels. The recursve mplementaton mproves the results obtaned especally when the nose rato s hgh. 5 Results The results presented here have been obtaned by usng LIBSVM [8] as mplementaton for the SVM. The programs used were wrtten n C++ and compled usng the Vsual C compler. The computer has a Pentum III processor wth 128 Mbytes RAM. The 8-bt, 512x512 mages used for the experments are shown n Fg.2. They are vared and have been used n dfferent works of mpulse nose removal. The frst results are shown n Table 1. In ths table we show the PSNR obtaned over the tests mages for varous nose ratos and for recursve and nonrecursve mplementaton of the flter. The γ parameter of the kernel has been elected to obtan the best results, n ths case the value elected s γ = 5e-6. The C parameter must be elected to obtan accurate results wth a reduced number of support vectors. In ths work the parameter has a value C = A greater value ncreases the tranng tme and a more lttle value decreases the accuracy of the process. The mages used n the SVM tranng are shown n Fg.1. The number of support vectors after the tranng was 30 for whte nosy pxels and 41 for black ones.

4 (a) (b) (c) (d) Fg.2. Images used n the experments: (a) Lena; (b) Albert; (c) Brdge; (d) Sal. Lena Sal Brdge Albert Nose Rato 10% 15% 20% 25% 30% Table 1. PSNR (db) values wth nonrecursve (up) and recursve (down) mplementaton. In Table 1 s shown how the recursve mplementaton ncreases the reconstructon PSNR especally for hgh nose ratos. We can see that ths method s applcable to varous nose ratos wth success. The nonrecursve mplementaton shows a poor performance for hgh ratos. In ths case some nosy pxels reman as nose and are not well replaced. But, f we repeat the flterng over the prevously fltered mage the results are clearly mproved. For example, by flterng two tmes the Albert mage wth a 30% of nose, the PSNR obtaned s db smlar to the recursve mplementaton of the flterng process. Lena Brdge Medan flter (3x3) ROM [2] (M=2)(No tranng) ROM [2] (M=1296)(Tranng) New modfed medan flter Table 2. Comparson of methods over 20% mpulse nose mages. Table 2 shows a comparson between the method presented here and the medan flter and the ROM flter. In [1], the ROM flter s superor to other prevous method and here we can see how our method s superor to ROM flter. The man goal for an mpulse nose removal algorthm s the preservaton of the edges and mage nformaton snce ths s the prncpal drawback of the nonlnear flterng. In Fg.3 we show two examples of nose removal (recursve mplementaton) n mages wth a 20% nose rato where we can see how the edge structure s preserved and only mnmal detals makes the recovered mages dfferent from the orgnal ones.

5 (a) (b) Fg.3. Examples of recovered mages. (a) 20 % Nosy mages. (b) Reconstructed mages. (a) (b) (c) Fg.4. Results over a detal of Lena wth 40% of nose. (a) Orgnal mage (b) Nosy mage (c) Image after frst pass (d) Image after second pass (d)

6 The mages n Fg.3 present good vsual results for a hgh nose rato. Ths method may be also appled over nosy mages wth a hgher rato. In Fg.4 we present results over a detal of Lena wth a 40% nose rato. The mages n Fg.4-(c) and Fg.4-(d) show how the mage detals and the edges are preserved even when the nose rato s hgh. The mage n Fg.4-(c) s the frst obtaned when our method s appled. As we sad, f the results are not satsfactory the process can be repeated n a second teraton and we can see the results n Fg.4-(d) where most of the reman nosy pxels are recovered. In both cases the method s appled recursvely. The tranng process wth mages lke n Fg.1 takes about 10 seconds and the executon tme when the method s appled over 512x512 mages takes about 5 seconds. 6 Conclusons In ths paper, a new method for removal of mpulse nose has been presented. The nonlnear flterng operaton s made usng a modfed medan flter. Ths flter uses a classfer based on SVM to decde between nosy and no nosy pxels. A modfed medan flterng s then appled over the nosy pxels. A tranng strategy for SVM s presented that s based on synthetc mages wth a controlled nose rato. The results presented show that our new approach gves good performance both n vsual and n PSNR results for varous nose ratos even when compared wth other sgnfcant prevous methods. These good results do not ncrease the tranng tme or the executon tme. References: [1] E. Abreu, M. Lghtstone, S.K. Mtra, K. Arakawa, A new Effcent Approach for the Removal of Impulse Nose from Hghly Corrupted Images, IEEE Trans. on Image Processng, Vol. 5, No. 6, 1996, pp [2] F. Russo, Nose Removal from Image Data Usng Recursve Neurofuzzy Flters, IEEE Trans. On Instrumentaton and Measurement, Vol. 49, No. 2, 2000, pp [3] Y.H. Lee and S.A. Kassam, Generalzed medan flterng and related nonlnear flterng technques, IEEE Transactons on Acoustcs, Speech, Sgnal Processng, vol. ASSP-33, 1985, pp [4] J.P. Ftch, E.J. Coyle and N.C. Gallagher, Jr., Medan flterng by threshold decomposton, IEEE Transactons on Acoustcs, Speech, Sgnal Processng, vol. ASSP-32, 1984, pp [5] R.C. Gonzalez and R.E. Woods, Dgtal Image Processng, Addson-Wesley, Readng, MA, [6] V. Vapnk. The Nature of Statstcal Learnng Theory, New York, Sprnger-Verlag, [7] N. Crstann and J. Shawe-Taylor. An ntroducton to Support Vector Machnes and other kernel-based methods, Cambrdge, Cambrdge Unversty Press, [8] C.C. Chang and C.J. Ln. "Lbsvm: Introducton and benchmarks"