OPTIMISED PERMUTATION FILTER

Transcription

1 50 Acta Electotechnca et Infomatca o, Vol, 200 OPTIMISED PERMUTATIO FILTER * Rastslav LUKÁČ, ** Ján LIZÁK * Depatment of Electoncs and Multmeda Communcatons, Techncal Unvesty of Košce, Pak Komenského 3, Košce, Slovak Republc, tel: , fax: , E-mal: lukac@tukesk ** Depatment of Compute Gaphcs and Image Pocessng, Comenus Unvesty, Mlynská dolna, Batslava, Slovak Republc, tel: , E-mal: anlzak@stonlnesk SUMMARY Ths pape focuses the use of pemutaton theoy n mage pocessng, namely we pesent new esults elated to optmsed pemutaton fltes Thus, to fom an estmate n the envonment coupted by mpulse nose, an optmsed set of ank ules povdes the best tool fo outles eecton onlnea fltes based on pemutaton theoy, called pemutaton fltes, ncopoate to a flte desgn both tempoalode and ank-ode nfomaton ncluded n a pemutaton goup Unlke a maoty of ank-ode based fltes, whee only ank-ode nfomaton of an odeed nput set s consdeed, by usng the full potental of the pemutaton goup tansfomaton, an estmate accuacy gven by optmsed pemutaton fltes s extemely hgh In geneal, an optmsaton algothm does not exclude the use both mean absolute eo and mean squae eo ctea Howeve, n the case of the mean absolute eo ctea, the optmsed pemutaton flte should povde the sgnal-detals pesevaton athe than the nose attenuaton and contawse the achevng of the optmsed pemutaton flte unde mnmsaton the mean squae eo ctea s chaactesed by the nose suppesson athe than the pesevaton of mage featues Based on above dscusson, the novelty of ths pape les at the analyss the optmsaton paametes such as fogettng facto and optmsaton ctea We wll show that n the case of small wndow szes (eg a wndow sze equal to fve), thee s necessay to acheve the best balance between the nose suppesson and the sgnal-detals pesevaton Thus, we povde an optmsaton ctea compomse ext, the nfluence of used optmsaton ctea s deceased wth an nceased wndow sze, snce a pemutaton goup sze s lage consdeably than an nput sequence Fo ths statc case, we povde the dependence of estmaton accuacy on the fogettng paamete All above conclusons ae suppoted by a numbe of fgues and tables Keywods: nosy mage flteng, mpulse nose, pemutaton flte, ank nfomaton, tempoal nfomaton IDRODUCTIO The employment of nonlnea fltes [,2,5,9,] can makedly mpove the estmaton effcency n a wde ange of smoothng applcatons, whee the envonment s modelled as nonlnea o when the nose coupton s nongaussan The typcal example of the stuaton elated to both nonlnea envonment and nongaussan nosy pocess, smultaneously, s epesented by useful mage nfomaton coupted by the mpulse nose o outles When somethng s nonlnea, t means that the supeposton popety cannot be appled and thus, the use of nonlnea fltes on uncoupted sgnal (useful nfomaton) and the nose, each sepaately, s mpossble To suppess the mpulse nose, usually the obust ode-statstcs theoy [6,8,3] s used The nonlneaty of ode-statstc fltes [3,5,7,2] s caused by an odeng of nput set A class of odestatstc fltes avods the falues of lnea fltes that tend to ntoduce a blung of sgnal-detals In addton, lnea fltes poduce new samples n a esultng mage In the case of ode-statstc fltes, a flte output s a sample fom the nput set The pefomance of chosen flteng method depends especally on the measue of the accuacy of the soluton Of couse, thee s necessay to acheve a compomse between the nose suppesson and the sgnal-detals pesevaton Snce flte nonlneaty, the optmal flteng stuaton, when only affected samples ae pocessed, cannot be fully obtaned Fo that eason, the seachng of effcent estmaton algothms s stll actual One way how to come nea to the optmal stuaton wth smultaneous use the obust odestatstcs theoy les at the consdeaton both ankode and tempoal ode nfomaton contaned n the nput set A class of ode-statstc pemutaton fltes [2,3], utlses the full potental of the pemutaton goup and thus, t gves the assumpton fo successful use n smoothng applcatons Ths pape focuses the analyss of optmsaton paametes n the acquston pocess of optmsed pemutaton fltes [] Followng analyss povde smplfcaton fo addtonal use of optmal set of ank ules and the futue mplementaton of pemutaton fltes, too 2 PERMUTATIO FILTERS Unlke some ode-statstc fltes, namely ankode fltes, a class of pemutaton fltes utlses both ank ode and tempoal ode nfomaton ncluded n the nput set x Rank-ode fltes [8,,3] ae based on the elatonshp between nput

2 Acta Electotechnca et Infomatca o, Vol, vecto x and ts ank-odeed veson x and thus, these fltes gnoe tempoal nfomaton contaned n x The tempoal nfomaton les at changes of sample postons n x By ths way, thee s possble to geneate! dffeent pemutatons of x, howeve, all pemutatons esult to the same ankodeed vecto x Fo that eason, the output of pemutaton fltes s a functon of the pemutaton that maps x to x [2,3,4] Let n be a tme poston of nput sequence and x ( n) = [ x( n K),, x( n),, x( n+ K)] () = [ x( n), x2( n),, x (2) an nput set n ths tme poston ote that the nput set x ( n) R s detemned by a symmetcal wndow shape wth a sze = 2K + The tme poston of the wndow shape s gven by cental sample xn ( ) = x( + )/2( n) The odeng of the nput set x ( n) means that samples x ( n), x 2 ( n),, x ( n ) ae mapped to the ank-odeed vecto [3] x ( n) = [ x() ( n), x(2) ( n),, x( ) (3) whee x() ( n), x(2) ( n),, x( )( n ) defned by x() ( n) x(2) ( n) x( )( n) (4) epesent ode-statstcs Fo bette undestandng of pemutaton flteng opeaton, we consde followng smplcty Let x = [ x, x2,, x ] be nput set o tempoal-odeed vecto and x = [ x(), x(2),, x( )] ank-odeed vecto Then the obsevaton pemutaton p x s defned as the pemutaton that maps x to x Above opeaton can be expessed as [2,4] x px = [ x, x,, x ] (5) px 2px px = [ x(), x(2),, x( )] (6) = x (7) whee the mappng fom the tempoal-odeed ndces to ank-odeed ndces detemnes the output ank decson l x It s clea that the set of tempoal-odeed ndces Ω = [,2,, ] of the nput set x s mapped to the same set Ω whch s chaactesed by the change of element poston, only, caused by the sample odeng n x ote that all pemutatons of Ω fom the goup of pemutatons [2,3,0] Snce the output of pemutaton fltes s always the ode-statstc fom { x (), x (2),, x ( ) }, then t s possble to sepaate the goup of pemutatons to subsets H, fo =,2,,, called blocks of patton It s clea that each H s assocated wth the ode-statstc x () Snce H, H2,, H have to be pawse dsont and the unon s the goup of pemutatons, the set H = { H, H2,, H } s smply called patton on the goup of pemutatons In the case of sample equvalence, some blocks H can be empty subsets The set of all pattons s descbed as Ω H The output of pemutaton flte s defned by F ( xh ; ) x (8) P = ( lx ) whee x s nput set and H Ω H s a patton Block assocated wth the l x th ode-statstc contans the obsevaton pemutaton To come nea to the optmal flteng stuaton, thee s necessay to acheve the optmal set of! (fo each possble pemutaton) output ank ules l x 3 OPTIMISATIO The am of the optmsaton s to set the decson vecto l = ( l, l2,, l!), whee p H l fo =,2,,! (9) The optmal flte can be obtaned by optmsng each of the elements l n vecto l ndependently Let {} o be ognal and { x } nosy tanng sequence, both of Q elements In geneal, the sum of L γ nomed eos to be mnmsed can be expessed as [2,3] Q ( Q n) γ λ on ( ) FP ( xh ; ) (0) n= whee λ (0,] s fogettng facto, and FP ( xh ; ) s the output of pemutaton flte If γ =, then the optmsed flte wll be chaactesed by pesevng chaactestcs, whle γ = 2 detemnes that the pemutaton flte povdes the nose attenuaton chaactestcs Howeve, as wll be shown n the next secton, the nfluence of used metcs γ to a flte behavou deceases wth an nceasng wndow sze To povde bette undestandng of above optmsaton, we smplfy above optmsaton n dependence on tme n Let P( n) = [ P( n), P2( n),, P be a vecto of L γ nomed eos at tme n, whee each [3,4] Pn ( ) = on ( ) x() ( n) γ fo =,2,, () chaacteses the eo between the desed and the th ode statstc ote that x () s one of possble outputs Mak R ( n) = [ R, ( n), R,2 ( n),, R, as a vecto contanng estmaton (cumulatve) eos so that chaacteses obsevaton wth pemutaton ndex The accodance between and pemutaton ndex detemnes the update of each R ( n), fo =,2,,!, wth the consdeaton P ( n), e vecto of L γ nomed eos Mathematcally, above opeaton can be descbed by λr ( n ) + P( n) f = R ( n) = (2) λr ( n ) othewse The optmal l tem s found by smply seachng R ( n) fo the mnmum cumulatve eo, e l = k fo Rk, ( n) = mn{ R, ( n), R,2 ( n),, R, ( n)} (3) The ntalsaton ncludes followng opeatons such as the set-up R (0) = 0 and l = ( + )/2, both fo =,2,,! ote that sequence poston stats wth n = Although ntal settng of each l s abtay fom the set {,2,, }, thee can occu

3 52 Optmsed Pemutaton Flte (a) (c) (e) (b) (d) (f) Fg Acheved esults (a) ognal mage Lena (b) ognal mage Bdge (c) 0% andom-valued nose (d) 0% andom-valued mpulse nose (e) medan wth +3 wndow shape (+3) (f) medan (+3) stuatons, whee some l s neve updated Fo that υ wth pobablty pυ xn ( ) = (4) eason, t s appopate to set each l to the obust on ( ) wth pobablty -pυ output ank, namely medan ank ( + )/2 whee xn ( ) s nosy mage sgnal, on ( ) descbes The optmsaton conssts of the epeatng steps ognal mage sgnal, both n the tme poston n (9), (), (2) and (3) and t s fnshed at the end of and υ epesent an andom value, e mpulse, wth tanng sequences, e fo n= Q, o when flte s the occuence pobablty p υ suffcently taned The dffeence fom ognal was evaluated by two ctea, namely the mean absolute eo (MAE) 4 SIMULATIO RESULTS and mean squae eo (MSE) The fst ctea evaluates the sgnal-detals pesevaton well, As the test mages, two gay-scale mages of wheeas the second one s a mo of the nose dffeent statstcal popetes wee used (Fga,b) suppesson Two-dmensonal defntons of MAE Each mage has a esoluton of pxels wth 8-bts/pxel gay-scale quantzaton The complexty of an mage s evaluated wth egads to the poblem aeas such as mage detals and edges The fst mage, well-known Lena (Fga), ncludes a numbe of detals and lage monotonous feld, too Fom the flteng aspect, the poblem aeas ae epesented by women ha and eyes Image Bdge (Fgb) s moe complex, thee ae many edges Small obects such as vegetaton, tees and bush wll epesent places, whee the flte behavou can be vey poblematc To llustate the degee of damage, we use the model of andom-valued mpulse nose Ths nose type (Fgc,d) eplaces some of the mage pxels by gay pxels, n the case of 8 bt-quantzed mage by the value fom 0 to 255 The mathematcal fomula fo vaable valued mpulse nose s gven by [8] and MSE ae expessed as [8] A B AB = =,, (5) A B ( AB = =, ) 2, (6) MAE = o x MSE = o x whee A, B epesent mage dmensons and, detemne tme poston, e n= B+ ow, we pesent some smulaton esults At fst, we obseved the flte behavou n dependence on L γ nomed eo (Fg3) In the case of +3 wndow shape (t s coss wndow of fve samples), the best balance between the nose suppesson and the sgnal-detals pesevaton was acheved fo γ = 5 Fo squae wndow 3x3, e nne samples, the flte behavou s elatvely constant It means, that n ths case the used nom can obtan the abtay value

4 Acta Electotechnca et Infomatca o, Vol, (a) (c) (e) (b) (d) (f) Fg 2 Acheved esults (a) pemutaton flte (+3, λ =, γ = ) (b) pemutaton flte (+3, λ =, γ = ) (c) medan (3x3) (d) medan (3x3) (e) pemutaton flte (3x3, λ =, γ = ) (f) pemutaton flte (3x3, λ =, γ = ) Optmsaton Optmsed on Lena Optmsed on Bdge Image Lena Bdge Lena Bdge Method MAE MSE MAE MSE MAE MSE MAE MSE Identty Medan (+3) Medan (3x3) Pemutaton flte (+3) Pemutaton flte (3x3) Table Pefomance of poposed methods (pemutaton flte: λ =, γ = ) To obtan the obectve compason of the flte pefomance (Fg2a,b and Fg2e,f), acheved esults wee compaed wth a medan flte (Fge,f and Fg2c,d), e wth the ntal set of output ank ules The optmsed pemutaton flte (Table ) smoothes (Fg2e) the nose excellent and peseves sgnal-detals, smultaneously Howeve, the optmsed pemutaton flte povdes above excellent behavou fo the tanng sequence, only Sequences wth dffeent statstcal popetes (Fg2f) ae pocessed nsuffcently It means that optmsed set of output ank ules s not obust When ou attenton s focused on the nfluence of paamete λ (Fg4), the best flte behavou s obseved fo the maxmal possble value, e λ = Exponental chaacte of λ execses especally at the begnnng of the tanng sequence, snce λ < nvolved by a hgh numbe esults to close zeo 5 COCLUSIO In ths pape, new esults and paametc analyss of the optmsaton fo pemutaton fltes wee pesented Smulaton esults showed us that fo a wndow sze = 9, the choce of mnmsaton ctea (MAE, MSE) s not mpotant In the case of fogettng facto λ, the best esults wee acheved fo λ = The only poblem s to optmse the set of output anks so that the futue eseach should be oented to the seachng fo faste optmsaton Consdeng wose esults not acheved on tanng sequences, the addtonal eseach task s elated to obustness mpovng of optmsed pemutaton fltes so that these fltes should epesent the most effcent flteng algothm fo nongaussan nosy pocesses Fo that eason, the key ole wll be played by smplfcaton of the flte mplementaton

5 54 Optmsed Pemutaton Flte omalsed magntude 3x3 wndow (MAE) of eo ctea wndow (MSE) omalsed 09 magntude of eo ctea nomalsed MAE MAE MSE nomalsed MSE wndow (MAE) x3 wndow (MSE) γ Fg 3 Dependence of nomalsed eo ctea (MAE, MSE) on paamete γ λ Fg 4 Dependence of nomalsed eo ctea (MAE, MSE) on paamete λ REFERECES [] Ace, GR Hall, TA Bane, KE: Pemutaton Weghted Ode Statstc Flte Lattces, IEEE Tansactons on Image Pocessng, Vol 4, o 8, August 995, pp [2] Bane, KE Ace, GR: Pemutaton Fltes: A Class of onlnea Fltes Based on Set Pemutatons, IEEE Tansactons on Sgnal Pocessng, Vol 42, o 4, Apl 994, pp [3] Bane, KE Ace, GR: Desgn of Pemutaton Ode Statstc Fltes though Goup Colong, IEEE Tansactons on Ccut and Systems-II Analog and Dgtal Sgnal Pocessng, Vol 44, o 7, July 997, pp [4] Hade, RC Bane, KE: Extended Pemutaton Fltes and the Applcaton to Edge Enhancement, IEEE Tansactons on Image Pocessng, Vol 5, o 6, June 996, pp [5] Lukáč, R: ew Stuctues of LUM Smoothes and Impulse Detectos fo osy Images Dssetaton, FEI TU Košce, Febuay 200, (n Slovak) [6] Lukáč, R: Vecto LUM Smoothes as Impulse Detecto fo Colo Images Poceedngs of Euopean Confeence on Ccut Theoy and Desgn ECCTD '0 "Ccut Paadgm n the 2 st Centuy" n Espoo, Fnland, August 28-3, 200, pp III-37 III-40 [7] Lukáč, R: Boolean LUM Smoothe Jounal of Electcal Engneeng, Vol 5, o 9-0, 2000, pp [8] Lukáč, R Machevský, S: LUM Smoothe wth Smooth Contol fo osy Image Sequences EURASIP Jounal on Appled Sgnal Pocessng, Vol 200, o 2, 200, pp 0-20 [9] Lukáč, R Machevský, S: Boolean Expesson of LUM Smoothes IEEE Sgnal Pocessng Lettes, Vol 8, o 2, Decembe 200, pp? [0] Lukáč, R Machevský, S: Reduced Pemutaton Goup by Mean-Based Colong Jounal of Electcal Engneeng, Vol 5, o 5-6, 2000, pp [] Schmulevch, I Melnk, V Egazaan, K: The Use of Sample Selecton Pobabltes fo Stack Flte Desgn IEEE Sgnal Pocessng Lettes, Vol 7, o 7, July 2000, pp [2] Paedes, JL Ace, GR: Optmzaton of Stack Fltes Based on Moed Theshold Decomposton, IEEE Tansactons on Sgnal Pocessng, Vol 49, o 6, June 200, pp [3] Ye, W Zhou, X: Ctea of Convegence of Medan Fltes and Petubaton Theoem, IEEE Tansactons on Sgnal Pocessng, Vol 49, o 2, Febuay 200, pp BIOGRAPHY Rastslav Lukáč eceved the MSc (Ing) degee wth dstncton at the Techncal Unvesty of Košce, Slovak Republc, at the Depatment of Electoncs and Multmeda Communcatons n 998 In 200 he fnshed PhD study Cuently, he s an assstant pofesso at the Depatment of Electoncs and Multmeda Communcatons at the Techncal Unvesty of Košce Hs eseach nteest ncludes mage flteng, mpulse detecton, neual netwoks and pemutatons Ján Lzák was bon on n Staá Ľubovňa In 997 he gaduated fom gymnasum D Tataku n Popad Recently, he s a student of the fnal class at the Faculty of mathematcs, physcs and nfomatcs at the Comenus Unvesty n Batslava, scope compute gaphcs The feld of hs dploma wok focuses the class of pemutaton fltes