Image Decomposition of Partly Noisy Images

Transcription

1 Avalable ole at wwwscecedrectcom Proceda Egeerg 9 () 6 66 Iteratoal Workshop o Iformato ad Electrocs Egeerg (IWIEE) Image Decomposto of Partly Nosy Images Ruhua u ab** Ruzh Ja a ad yu Su a a School of Mathematcs ad Statstcs Chogqg Uversty of Techology Cha b Isttute of Automato Chese Academy of Sceces Bejg Cha Abstract I ths paper we frstly propose a ew varatoal model for partly osy mage decomposto by troducg a texture detectg fucto Secodly we prove the exstece ad uqueess of mmal solutos of our proposed model Thrdly we gve our algorthm ad wrte dow the pseudo-codes of the specfc steps Fally we show some umercal expermets ad the smulato results show that our model ca effectvely decompose cartoo ad texture for mages wth Gaussa ose Publshed by Elsever td Ope access uder CC BY-NC-ND lcese Keywords: mage decomposto texture cartoo texture detectg fucto Itroducto The mage decomposto s a mportat work mage processg Gve f be a mage cotaed texture e f = u v where u s cartoo part ad v s oscllatg cotaed texture May PDEmethods have bee offered for ths kd mage decomposto such as AABC model ROF model ad other models[-] I ths paper we maly dscuss a decomposto method for osy mages I 99 Rud[8] et al preseted the ROF model to remove the ose The model s m E ( u) = Du u f As we kow the model ca also be used for mage decomposto I 9 [5] et al desged a varatoal deosg model for partly textured mages by troducg a texture detectg fucto the ROF model The model s to mmze the followg eergy fuctoal * *Correspodece author:lruh@sohucom Publshed by Elsever td do:6/jproeg6 Ope access uder CC BY-NC-ND lcese

2 Ruhua u et al / Proceda Egeerg 9 () m E ( u) = u dx ( g)( u f ) dx where g s the texture detectg fucto accordg to the dervatves of the osy mage wth texture et al use the total varato(tv) flow to remove the ose that s f t = dv ( f f ) Ad the they offered to exted the structure tesor by usg the frst ad secod order dervatves to extract the texture feature a osy mage The sx feature chaels are as follows u = fx u = fx u 3 = fx f x u 4 = fxx u 5 = fxx u 6 = fxx Sce the ose f s amplfed by takg dervatves they aga used the TV flow to deose each part u e u t = dv( u u ) ad gave a detectg fucto as g( x x) = ( kλ ( x x) ) where k s a postve costat ad Λ ( x x) s the largest egevalue of the geometrc matrx 6 6 ( u ) ( ) ( ) x u u x x = = 6 6 ( u ) ( u ) ( u ) x x x = = From the above costructo we ca see that each compoet of u s ear zero such that g goes to oe the homogeeous regos; at least oe compoet of u s very large such that g goes to zero the regos wth texture or ose Models metoed above are maly amed at ose-free mage decomposto ad partly textured mage deosg Hece the mage decomposto task s challegg ad mportat for a osy mage et f be decomposed to a sum of three compoets e f = u v w such that u represets the cartoo compoet of f whch s the geometrc or structural compoet of f v s the texture compoet ad w s the ose compoet I 4 Vese[] et al used two steps for osy mage decomposto Frstly they removed ose Secodly they decomposed deosy mage to cartoo ad texture parts As we kow the results of mage decomposto are a drect relatoshp wth deosy mages At preset there are o well methods to decompose texture ad ose The rest of the paper s orgazed as follows I secto we costruct a ovel decomposto model ad wrte out the executo algorthm I secto 3 we show the exstece of mmal solutos of our proposed model I secto 4 we show some expermetal results I secto 5 we gve some coclusos ad dscussos Our Proposed Model ad Algorthm I ths secto we develop a ew model for osy mage decomposto spred by ROF model ad the texture detectg fucto[5] We are to mmze the followg eergy fuctoal m E ( u v) g Du μ = ( g) v dx ( f u v) dx where ad μ are two postve costats u BV( ) s cartoo part of f v ( ) s texture part ad w ( ) s resdue compoet or ose e w= f u v We call that the frst term s the regularzato term o cartoo part u the secod term s test term o texture compoet v ad the thrd term s fdelty term Accordg to the defto of texture detecto fucto for g ( ) we ca see that the frst ad thrd terms play more roles whe g teds to oe the cartoo doma ad the secod ad thrd terms take the more mportat part whe g goes to zero the texture or ose rego I order to solve the mmal solutos of our proposed model we ca frst compute the Euler- agrage equatos Ad the we use the steepest descet method to fd the evolvemet equatos There are as followg u τ = dv g u u μ u v f () ( ) ( )

3 64 Ruhua u et al / Proceda Egeerg 9 () 6 66 v τ = ( gv ) μ( f u v) () I ths paper we follow the alteratg teratve algorthm to mplemet the above evoluto equatos the alteratg mmzato algorthm s rug the order of () () () () We wll employ pseudo-codes to wrte dow the specfc steps the followg Frstly we compute the texture detectg fucto g () Deosg a osy mage f by TV flow () Italze u = f τ = τ (Default ) For (Default ) ( ( ( ))) ( ) ( ) ( ) ( u u τ dv u u ) ε Ed We ca get deosg result ued I If the mage s a ose-free mage we just take = 3 () Deosg the sx chaels usg TV flow Italze τ = τ (Default ) u = I u = I u = I I u = I u = I u = I ( ) For (Default ) u u dv( u ( u )) x x 3 x x 4 xx 5 xx 6 xx ( ) ( ) ( ) ( ) τ ε Ed (3) Computg the detectg fucto g Secodly we decompose the osy mage by the evoluto equatos Italze u () = z v () = u () 55* g τ = τ τ = π (Default ) k = k (Default 5 for osy mage 5 7 for ose-free mage) μ = μ (Default ) = For (Default ) τ ( ( ε )) μτ ( ) ( ) ( ) ( ) ( ) v v τ ( g) v μτ ( f u v ) ( ) ( ) ( ) ( ) ( ) ( ) u u dv g u u u v f ( ) Ed After rug we ca receve the results of mage decomposto by our model whch clude cartoo part u ued texture part v ved ad resdual part or ose w f u v 3 Exstece of Mmal Solutos I ths secto we prove exstece of mmal solutos of our eergy fuctoal model I order to facltate the proof we ca wrte dow the above model as follow μ m E ( u v w) = g Du ( g) v dx w dx u BV ( ) v ( ) w ( ) (3) where w= f u v Theorem et f ( ) there s a uque mmal soluto ( uvw ˆ ˆ ˆ) BV( ) ( ) ( ) the above eergy fuctoal model (3) Proof: (Exstece) et {( u v w ) } be a mmzg sequece for the fuctoal (3) there s a geeralzed costat M such that μ E( u v w ) = g Du ( g) v dx w dx M Accordgly to g Du M ( g) v dx M w M g let g = g () we Sce () where have g = gdx g = gdx v M s ebesgue measure Thus Du M Therefore we get two sequeces { v } { w } whch are uformly bouded ( ) So there exst two subsequeces { v } { w } ad two fuctos wˆ v ˆ ˆ Due to ( ) ( ) w w weak bouded ad strog covergece ( ) such that ˆ v v weak we have the above two sequeces whch are uformly ( )

4 Ruhua u et al / Proceda Egeerg 9 () Accordg to the above results we ear u = f v w f v w M Rewrtg Pocare-Wrtger equalty bouded ( ) ( ) ad ( ) u u Cu we ga the sequece{ u } whch s uformly BV BV So there exst a subsequece { u } ad û BV( ) such that u uˆ BV w Because to lower sem-cotuty for BV( ) ad ( ) we ca derve μ Euvw ( ˆ ˆ ˆ) lmf Eu ( v w) lmf gdu ( gv ) dx w dx (Uqueess) We ca easly draw that the eergy fuctoal Euvw ( ) s covex o u v w respectvely Next we wll prove the strct covexty of the eergy fuctoal (3) If there are a costat t () ad two vectors ( u v w) = such that E t u v w t u v w = te u v w t E u v w ( ( ) ( )( ) ) ( ) ( ) ( ) that s E( t( u v w) ( t)( u v w) ) μ μ = t g Du ( g ) vdx w ( t) g Du ( g ) vdx w g ( t Du ( t ) Du μ ) ( ( ) ) ( )( ( ) ) t w t w = g tv t v dx Comparg the secod ad thrd terms of the frst equato wth those of the thrd equato respectvely there are w = w v = v Ad because f = u v w u v w = we ca see u = u Therefore the mmzg eergy fuctoal Euvw ( ) s strctly covex o u v w respectvely I the ed we ca receve that there s a uque mmal soluto 4 Numercal Expermets I ths secto we show some umercal results by our proposed model the followg I our expermets we take the space step h = tme step τ = τ = teratos (a) (b) (c) (d) (e) (f) Fg Decomposto of the osy ogo mage (a) Nose-free ogo mage (b) Nosy ogo mage (c) Cartoo part of our proposed model (d) Texture of our proposed model (e) Resdual part or ose of our proposed model (f) Texture detectg fucto (a) (b) (c) (d) (e) (f) Fg Decomposto of the osy Badge mage (a) Nose-free Badge mage (b) Nosy Badge mage (c)cartoo part of our proposed model (d) Texture of our proposed model (e) Resdual part or ose of our proposed model (f) Texture detectg fucto

5 66 Ruhua u et al / Proceda Egeerg 9 () 6 66 Fg ad Fg show the decomposto results of the ogo mage ad Badge mage wth Gaussa ose By observg the cartoo parts of Fg-(c) we ca see that the cartoo parts are decomposed well form the osy ogo ad Badge mages I the same way we ca kow that the texture parts are largely separated from the osy ogo ad Badge mages by aalyzg Fg(d) ad Fg(d) but they cota a part of ose at the same tme Texture detecto fuctos Fg-(f) bascally detfy the texture parts of osy mages but also clude a fracto of the edge of ose 5 Cocluso ad Dscussos I ths paper we have proposed a ew model for partly osy mage decomposto by troducg a texture detectg fucto Smultaeously we also prove the exstece ad uqueess of solutos of the mmal fuctoal (3) From the umercal smulatos we ca see that our proposed model ot oly processes well for ose-free mages but also deals wth mages wth Gaussa ose However we ca ot perfect decompose texture ad ose for osy mages Further study wll focus o these problems such as osy mages wth large varace multplcatve osy mages the exstece of the evoluto equatos etc Ackowledgemets Ths s partly supported by NSFC of Cha (9739) the Chogqg Mucpal Scece ad Techology(CSTC) foudato of Cha(CSTCJJA433 CSTC BB3) Chogqg CMEC foudato of Cha(KJ88)CQUT foudato (ZQ3) Refereces []G Aubert ad J Aujol A varatoal approach to remove multplcatve ose SIAM Joural o Appled Mathematcs 68(4) 8 pp []J Aujol G Aubert Blac ad A Chambolle Image decomposto to a bouded varato compoet ad a oscllatg compoet Joural of Mathematcal Imagg ad Vso () 5 pp7-88 [3]S Chao ad D Tsa A mproved asotropc dffuso model for detal ad edges preservg smoothg Patter Recogto etters 3(3) pp -3 [4]G Glboa N Oche ad Y Zeev Varatoal deosg of partly textured mages by spatally varyg costrats IEEE Trasactos o Image Processg 5(8) 6 pp 8-89 [5]F C She C- She ad G Zhag Varatoal deosg of partly textured mages Joural Vsual Commucato ad Image Represetato (4) 9 pp 93-3 [6] Y Meyer Oscllatg patters mage processg ad some olear evoluto equatos The Ffteeth Dea Jacquele B ews Memoral ectures Uversty ectures Seres [7]M Nkolova A varatoal approach to remove outlers ad mpulse ose Joural of Math Imagg ad Vso (-) 4 pp 99- [8] Rud S Osher ad E Fatem Nolear total varato based ose removal algorthms Physca D 6 99 pp [9] Vese ad S Osher Modelg textures wth total varato mmzato ad oscllatg patters mage processg Joural of Scetfc Computg 9(-3) 3 pp [] Vese ad S Osher Image deosg ad decomposto wth total varato mmzato ad oscllatory fuctos Joural of Mathematcal Imagg ad Vso (/) 4 pp 7-8