Mixed Noise Suppression in Color Images by Signal-Dependent LMS L-Filters

Transcription

1 46 R. HUDEC MIXED OISE SUPPRESSIO I COLOR IMAGES BY SIGAL-DEPEDET LMS L-FILTERS Mxed ose Suppresson n Color Images by Sgnal-Dependent LMS L-Flters Róbert HUDEC Dept. of Telecommuncatons Unversty of Žlna Veľý del Žlna Slova Republc robert.hudec@fel.utc.s Abstract. The paper s devoted to the sgnal-dependent (SD) desgn of adaptve LMS L-flters th margnal data orderng for color mages. The same stem of SD processng of nosed grayscale mages as appled on nosy color mages. Component-se and multchannel modfcatons of SD LMS L-flter n R G B (gamma corrected RGB sgnals) color space ere developed. Both modfcatons for flterng to-dmensonal statc color mages degraded by mxed nose consstng of addtve Gaussan hte nose and mpulsve nose ere used. Moreover sngle-channel spatal mpulse detectors as detectors of mpulses and detals ere used too. Consderng expermental results SD modfcatons of L-flters for nosy color mages can be concluded to yeld the best results. Keyords Order-statstcs sgnal-dependent flter color mage flterng L-flter margnal data orderng.. Introducton Many applcatons n telecommuncatons employ fltraton of sgnals that are corrupted by varous types of nose. If noses are of a nonlnear nature nonlnear flters have to be used to execute ther effectve suppresson. On the other hand lnear flters have to be exploted hen addtve nose s to be cancelled. If addtve mxng of nonlnear and addtve noses appear the queston hat type of a flter s optmal arses. The approprate soluton offers the lnear combnaton of order statstcs. Ths class of flters s called L-flters and they approxmate flter coeffcents consderng to the nose model ell [] [4] [7]. In the paper to modfcatons of adaptve SD LMS L-F (Sgnal-Dependent Least Mean Square L-flter) for nosed color mages are desgned usng sngle- or multchannel versons of adaptve L-flters and SID (Spatal Impulse Detector). Sgnal-dependent (SD) processng processes homogeneous nput observatons apart from detaled ones separately. The SMD detector (Spatal Medan Detector) s used as one of SID detectors and serves as a stch beteen outputs of partal flters [4]. In Secton 2 the smple theory of the sngle- and mult-channel adaptve LMS L-flters s ntroduced. Secton 3 s devoted to the desgn of adaptve component se and multchannel sgnal-dependent LMS L-flters th SMD. The expermental results are presented n Secton 4. In Secton 5 obtaned results and further development that can mprove the fltraton results are dscussed. 2. Adaptve LMS L-Flters 2. Sngle-Channel Adaptve LMS L-Flter The component-se fltraton of nosed color mages s a method that employs flters desgned for processng nosed grayscale statc mages [] [2]. The dsadvantage of ths method s hdden n the fact that t doesn t explot correlaton beteen channels. On the other hand ths feature s advantageous f color mages degraded by non-correlated nose are processed. In ths case the fltraton by multchannel flters nexactly estmates the reference sgnal. Flters based on lnear combnaton of order statstc (L-flters) desgned for grayscale mages can be used as partal flters for component-se processng of color mages. The output of a L-flter s gven by y = T xr () here = [ ] T s a vector flter coeffcents and xr s a vector of ordered nput pxels (ascendng or descendng order) ( x x x x ) T xr = K. (2) 2 Invertng autocorrelaton matrx for non-statonary sgnals s computatonally dffcult (t s tme varyng) and therefore the adaptve algorthm can offer good estmaton of flter coeffcents. Adaptve algorthms mnmze a crteron functon (Mean Absolute Error Mean Square Error total poer ). LMS algorthm that s based on nose gradent and Steepest Descent method belongs to frequently used adaptve algorthms: 2 µ ε xr = +. (3) + Eqn. (3) s equvalent to the ell-non lnear LMS algorthm (Wdro one) hch s very popular n lnear adap-

2 RADIOEGIEERIG VOL. 2 O. 3 SEPTEMBER tve flterng. Whereas eqn. 3 uses the vector of ordered observatons to update the adaptve L-flter coeffcents LMS algorthm explots non-ordered ones. Adaptve component-se LMS L-flter (LMS C L-flter) arses by usng adaptve sngle channel LMS L-flters n the componentse fltraton of color mages. 2.2 Multchannel Adaptve LMS L-Flter Recently attenton has been gven to the non-lnear processng of vector-valued sgnals. It s non that there s no unversally accepted method to order multvarate data. Therefore the margnal (M-orderng) reduced (R-orderng) partal condtonal orderng etc. as sub-orderng prncples ere developed [3] [7]. Flters based on these prncples employ correlaton beteen channels and hence they have more nformaton about processng sgnals. ~ T ~ [ X ] = + 2 d X + µ (9) ~ = + 2µ e X. Eqn. (9) s an adaptaton equaton of an adaptve multchannel LMS L-flter (LMS P L-F) [3] [7]. LMS algorthm offers smple recursve equaton of L-flter coeffcents updatng. Convergence propertes of general LMS L-flter depend on egenvalues of autocorrelaton matrx dstrbuton. 3. Adaptve Sgnal-Dependent LMS L-Flters Very good fltraton results ere acheved by sgnaldependent processng of nosed grayscale mages redound to ther applcaton on color mages. The prncple of SD processng as appled n conjuncton th sngle- and mult-channel adaptve LMS L-flters. Color mages n R G B color space are defned by three components and mpulse detecton can be performed for any of them. Ths case spatal mpulse detectors (SID) developed for nosy grayscale mages can be used [4]. Fg.. The adaptve sgnal-dependent LMSP L-flter. Let x x be a random sample of observatons of a p- dmensonal random varable X. Then each nput observaton x = x x K x (4) j ( ) T j j2 j p belongs to the p-dmensonal space denoted as R P [3]. M- orderng orders components of observatons separately by the next la x x L x = p. (5) 2 K Output of p-dmensonal L-flter that processes p-dmensonal vectors x s gven as p y = A ~ x (6) = here A are p matrces of flter coeffcents and x ~ s vector of order statstcs n -th channel of dmenson ( x x K x ) T ~ x =. (7) 2 Let l T l = p denotes the l-th ro of the matrx A. Then flter coeffcents = p that mnmze MSE T T T [ ] T 2 K p = (8) can be recursvely computed by the steepest descent algorthm as follos 3. SID for Grayscale Images The SD prncple s based on a separate processng of detaled and homogeneous nput observatons. SID serves for the determnaton of the nput type. The determnaton result defnes the flter hch ll process nput data. The decson rule for the general SD L-flter n combnaton th SID s gven by IF D Level (0) = THE HL-flter ELSE LL-flter Here D s the result of mpulse detecton for -th mage pxel n the -th nput observaton. The value of the level defnes the number of detected mpulses n the observed samples. The H L-flter processes the detaled nput observatons and L L-flter the homogeneous ones. The spatal mpulse detectors ere derved from mpulse detectors [8] [9] hch detect mpulses or mage detals n all mage pxels of nput observatons. The spatal medan order statstcs detector (SMD) [4] belongs to the most popular SID. The decson rule of SMD s gven by IF med { x } x Threshold () THE D = ELSE D = 0

3 48 R. HUDEC MIXED OISE SUPPRESSIO I COLOR IMAGES BY SIGAL-DEPEDET LMS L-FILTERS If the dfference beteen the -th nput sample magntude and the medan one s more than the threshold value the nput sample s mared as the detected mpulse. 3.2 Adaptve Multchannel Sgnal-Dependent LMS L-Flter Adaptve sgnal-dependent LMS P L-flter (SD LMS P L-F) hch s shon n Fg. utlzes to LMS P L-flters to process homogeneous or detaled mage areas. Adaptve sgnal-dependent LMS C L-flter (SD LMS C L-F) s shon n Fg.2. SD LMS C L-flter s based on assumpton that flter coeffcents of partal flters can be adapted jontly by suppressng the correlated nose (color pxel s degraded n each channel). Moreover mpulse detecton probablty at the same poston n every component s hgh. SD LMS C L-F s based on the same prncple as SD LMS P L-F and can employ any modfcaton of SID. The usage of partal flters s only dfference. The output of SD LMS C L-flter s gven by y LLMSR y LLMSG L F f D Level < = y LLMSB y = y H LMSR y H LMSG L F f D Level = y H LMSB (3) The flter s advantageous n the smaller number of flter coeffcents (54 for 3 3 square flter ndo) n smple updatng and ndependent shape of flter mas of partal flters. The flter does not explot correlaton among channels hch s ts man dsadvantage. Fg. 2. The adaptve sgnal-dependent LMSC L-flter. Input color data n -th observatons X are dvded to the color components. Snce the detector desgned for grayscale mages s used as SID the mpulse detecton has to be done for one of color components. Consderng SID decson the processed -th nput observaton serves for adaptng one of partal flters. Thus H LMS P L-F s the partal flter that processes detaled nput observatons and L LMS P L-F processes homogeneous nput observatons. The output of SD LMS P L-F s gven as a combnaton of the partal L-flter outputs and t can be expressed as y y = y P L LMS P H LMS f f = D < Level (2) = D Level SD LMS P L-F s advantageous n employng correlaton among color channels and ther jont processng. On the other hand hgher computaton complexty (operaton th matrces) and hgher number of flter coeffcents (62 for 3 3 square flter ndo) are dsadvantages of SD LMS P L-F. Moreover the shape of the flter ndo can be changed for the hole partal LMS P L-flter only. 3.3 Adaptve Component-se Sgnal-Dependent LMS L-Flter 4. Expermental Results For experments reference color mages Lena and Mandrll ere used (Fg. 4a d). They ere corrupted by a mxed nose (Fg. 4b e) consstng of addtve Gaussan hte nose th the standard devaton σ = 20 and a correlated mpulsve nose th the probablty p = 0% (degraded pxel at the same poston n each channel). Ths mxed nose as denotes as G20CI0. Flterng results ere evaluated by MAE (Mean Absolute Error) MSE (Mean Square Error) R (ose Reducton) MAER (Mean Absolute Error Reducton) and CD (Color Dfference) crtera respectvely [] [5]. R and MAER crtera are n db scale. We used the SMD detector as SID n experments [4]. Results of adaptve L-flters ere compared th VMF L2 flter (Vector Medan Flter th L2 norm) and some are shon n the Table [6]. SD processng mproved suppresson of the mxed nose n comparson th conventonal component-se or multchannel flters. The above-mentoned mpulse detector for grayscale statc mages as used for mpulse detecton n each color channel. Best results ere acheved by detecton n R channel. Input parameter for SMD detector as determned to Level = and ts optmal threshold as obtaned expermentally. Detaled R and CD dependences from threshold are shon n Fg. 3. Flterng nosed Lena mage (less composte mage) better results ere acheved compared to SMD SD R LMS P L-flter. For Mandrll (more composte mage) the SMD SD R LMS C L-flter s preferable. Optmal thresholds for SMD detector Level = for both mages are shon n the Tab. 2.

4 RADIOEGIEERIG VOL. 2 O. 3 SEPTEMBER Flter Lena MAE MSE R MAER CD osed VMF L LMS P L-F LMS C L-F SMDSD R LMS P L-F SMDSD R LMS C L-F c) Mandrll osed VMF L LMS P L-F LMS C L-F SMDSD R LMS P L-F SMDSD R LMS C L-F d) Tab.. The flter performance ndces acheved for color mages Lena and Mandrll corrupted by G20CI0 nose. Flter Lena Mandrll SCOSD2SD R LMS P L-F SCOSD2SD R LMS C L-F Tab. 2. Optmal thresholds for SMD detector (Level= G20CI0 nose). In case of a dfferent defnton of the optmal threshold for R and CD crterons the optmal value s determned by R parameter (R s derved by mnmzng the MSE crteron functon of LMS algorthm). a) Fg. 3. R and CD dependences from threshold a-b) Lena fltered by SMD SD R LMS P L-flter c-d) Mandrll fltered by SMDSD R LMS C L-flter. In the R G B sense the orgnal mage of Lena contans around 47% 26% 27% and mage of Mandrll 36% 34% 30% of the total color poer. Obvously the R -channel s domnant and therefore the best results ere acheved n ths color channel (but they are not robust). For robustness n adjustng flters thresholds from 60 to 90 have to be chosen by detecton n G or B channels. Generally the detecton has to be done n a non-domnant channel. Tab. 2 shos that a hgher value of threshold ll be used for fltraton of detals n mages and vce versa. Snce percepton of detals and mpulses s the same subjectve evaluaton of fltered mages founds better detaled mages. Fltered color mages are shon n Fg. 4c f. Improvement of a smoothng parameter by SD L-flters s vsble n the face detal of Lena (Fg. 5). Adaptaton step of LMS algorthm as µ = 0-7. b) 5. Concluson In the paper to adaptve sgnal-dependent LMS L- flters ere descrbed: the adaptve SD component-se one and SD multchannel one. Both flters employ spatal mpulse detectors desgned for grayscale mages. The descrbed adaptve SD flters ere desgned for a mxed nose removed from color mages. Moreover the paper contans optmal thresholds for dfferent composte mages and recommendatons for sgnal-dependent processng of color mages by adaptve L-flters. Due to ther hgh computatonal complexty they can be used n offlne applcatons.

5 50 R. HUDEC MIXED OISE SUPPRESSIO I COLOR IMAGES BY SIGAL-DEPEDET LMS L-FILTERS a) b) c) d) e) f) Fg. 4. Color mages. (a) Orgnal Lena (b) Lena corrupted by mxed G20CI0 nose (c) Lena fltered by SMD SD R LMS P L-flter (d) Orgnal Mandrll (e) Mandrll corrupted by mxed G20CI0 nose (f) Mandrll fltered by SMD SD R LMS C L-flter. The adaptve SD L-flters ere proven that ther nose-suppresson characterstcs are able to reduce ell the mxed nose n color mages. Explotng vector detecton of mpulses and detals can mprove the fltraton results n the future modfcatons. a) b) Fg. 5. Fltered mages of Lena s face by (a) VMF L2 (b) SMD SD R LMS P L-flter. References [] KOTROPOULOS C. PITAS I. Adaptve LMS L-flters for nose suppresson n mage. IEEE Transactons on Image Processng. 996 vol. 5 no. 2 p [2] KOCUR D. HUDEC R. MARCHEVSKÝ S. Suppresson of mxed nose n the smlar mages by usng adaptve LMS L-flters. Radoengneerng vol. 9 no. 4 p [3] KOTROPOULOS C. PITAS I. Adaptve multchannel margnal L- flters. SPIE Optcal Engneerng. 999 vol. 38 no. 4 p [4] HUDEC R. MARCHEVSKÝ S. Extenson of mpulse detectors to spatal dmenson and ther utlsaton as stch n the LMS L-SD flter. Radoengneerng. 200 vol. 0 no. p. -5. [5] FORD A. ROBERTS A. Colour space conversons p. -3. [6] ASTOLA J. HAAVISTO P. EUVO Y. Vector medan flters. Proceedngs of the IEEE. 990 vol. 78 no. 4 p [7] KOTROPOULOS C. PITAS I. GABRAI M. Multchannel adaptve L-flters n color mage processng. In Proceedngs of the IEEE Internatonal Conference on Image Processng ICIP ' Lausanne (Stzerland) p [8] LUKÁČ R. STUPÁK Cs. A Class of mpulse detectors controlled by a threshold. In Proceedngs of the Internatonal Conference Informaton and Algorthms Prešov: Unversty of Prešov p [9] MOUCHA V. MARCHEVSKÝ S. LUKÁČ R. STUPÁK Cs. Číslcová fltráca obrazových sgnálov. Košce: Vydavateľstvo Vojensej letecej aademe M.R. Štefana v Košcach 2000.