Smoothing of Ultrasound Images with the p-lag FIR Structures

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1 Recet Researches Telecommucatos, Iformatcs, Electrocs ad Sgal Processg Smoothg of Ultrasoud Images wth the -lag FIR Structures L. J. Morales-Medoza, Y. S. Shmaly, R. F. Vázquez-Bautsta ad O. G. Ibarra-Mazao FIEC, Veracruz Uversty Av. Veustao Carraza, col. Revolucó, Poza Rca Ver. Méxco. DICIS, Guaauato Uversty Ctra. Salamaca-Valle Km. 3.5, comudad Palo Blaco, Salamaca Gto. Méxco. Abstract: A aalyss s gve of the secfc ots of the ose ower ga of the -lag meda hybrd FIR flters alcatos to ultrasoud mage rocessg. A mortat feature of such a ga s that, at some cross ots, t coverges to the reduced degree oe. The effect of smoothg s evaluated terms of the sgal-to-ose rato ad root mea square error metrcs. We show that mage ehacg has the best aearace f the FIR smoother degree ad are both otmzed. Key-Words: FIR smoothg, Ultrasoud, Hybrd meda structure, Power ga. Itroducto The roblem of savg a shar edge wth a smultaeous ehacg the mage s tycal for ultrasoud alcatos. Ultrasoud magg s a techque that s wdely used a varety of clcal alcatos. Due to the blur ad tycally o Gaussa ose, a org ultrasoud mage has a oor resoluto. The roblem of savg a shar edge wth a smultaeous ehacg the mage s tycal for mage rocessg. The bgger varety of the flters bomedcal mage rocessg, secfcally ultrasoud, are owadays a tool very owerful to the motorg of Fetus, study of the Heart, Kdey, Lver, Cysts, Tssue ad huma carotd artery dsease ad so fort. A overall aorama of methods of olear flterg followg the meda strategy has bee gve by Ptas ad Veetsaooulos [] alog wth the mortat modfcatos for a large class of olear flters emloyg the order statstcs. The algorthm ssues for the flter desg have bee dscussed []. I [3], the fte mulse resose (FIR) meda hybrd flters (MHF) strategy has bee roosed wth alcatos to mage rocessg. A mortat ste ahead has bee made [4], where the FIR MHF structures were desged. I the sequel, the MHF structures have extesvely bee vestgated, develoed, ad used by may authors. Bascally, hybrd FIR structures ca be desged usg dfferet tyes of estmators. Amog ossble solutos, the olyomal estmators occuy a secal lace, sce the olyomal models ofte well formalze a osteror kowledge about udergog rocesses. Relevat sgals are tycally rereseted wth degree olyomals to ft a varety of ractcal eeds. Examles of alcatos of olyomal structures ca be foud sgal rocessg [5], tmescales ad clock sychrozato [6], mage rocessg [7], ultrasoud mage rocessg [4], etc. The olyomal estmators sutable for such structures ca be obtaed from the geerc form of the -ste redctve ubased FIR flter roosed [8]. Such estmators usually rocess data o fte horzos of N ots that tycally obtas a ce restorato. Polyomal Image Model A two-dmesoal mage s ofte rereseted as a k c k r matrx M {µ, }. To rovde two dmesoal flterg, the matrx ca be wrtte the form of a row-ordered vector or a colum-ordered vector, resectvely ISBN:

2 Recet Researches Telecommucatos, Iformatcs, Electrocs ad Sgal Processg x x r c [ µ µ µ ] T,, kr kc, µ kc, kr [ µ µ µ ] T, kc,, k µ r kc, kr, (). () The flterg rocedure s the ofte aled twce, frst to () ad the to (), or vce versa. To rereset a two-dmesoal electroc mage wth () ad (), oe may also substtute each of the vectors wth the dscrete tme-varat determstc sgal x that, tur, ca be modeled o a horzo of some N ots state sace. 3 Sgal Model ad Problem Formulato A electroc mage x ca be substtuted wth the dscrete tme-varat determstc sgal. Followg [8], such a sgal ca be rereseted state sace by the state ad observato equatos as follows, resectvely x, (3) N + A x N+ y Cx + v, (4) where x [x x x K ] s the K vector of the states, y s the measuremet reresetg a electroc mage, v s the measuremet ose, the K measuremet matrx s C [ 0 0], ad K K tragular matrx A s secfed wth ( τ )... ( K )!( τ ) τ... ( K )!( τ )... ( K 3)!( τ ) K τ K 0 K 3 A 0 0. (5) The oseless model (3) roects ahead from N + to wth the fte degree Taylor seres exaso as x τ ( N + ) K q xl+ q q 0 q! q. (6) If we troduce the l-degree olyomal flter ga h l (N, ), that s -shft deedet, the the estmate of the electroc mage x ca be obtaed o a averagg horzo of N ots by the covoluto N + xˆ ( N, ) y, (7) where > 0 stads for redctve FIR flterg, 0 for FIR flterg, ad < 0 for smoothg FIR flterg. It has bee show [8] that, order for the estmate to be ubased, the ga h l (N, ) must satsfy the followg codtos: N + N + ( N ),, (8) ( N ) u, 0, u [, l]. (9) It s also kow from Kalma-Bucy flter theory that the order of the otmal (ad so ubased) flter s the same as that of the system. That meas that, for the K-state model, the ga ca be rereseted wth the l-degree olyomal such that l l h ( N, ) a ( N, ), (0) 0 l where, a l (N, ) are the olyomal coeffcets ad l must be chose such that l K. The coeffcets for (0) have bee foud [8] the form of a l M( + ) ( N, ) ( N, ) ( ), () D( N, ) where a short (l + ) (l + ) symmetrc matrx D(N, ) s gve by d0 d dl d d dl+ D, () dl dl+ d l D s the determat of (), ad M (+) s the mor of (). I accordace wth [8], the comoet of () ca be determed usg the Beroull olyomals B (x) as: d r+. (3) r r+ ( N ) [ B ( N+ ) B ( ) ] r, + 4 Low Degree Polyomal Gas I FIR flterg, a estmate s obtaed va the dscrete covoluto aled to measuremet. That ca be doe f to rereset the state sace model o a averagg terval of some N ots. Referrg to (0), the low-degree olyomal gas ca thus be defed wth 3 h ( ) a ( ), (4) 0 ISBN:

3 Recet Researches Telecommucatos, Iformatcs, Electrocs ad Sgal Processg Below, we derve ad vestgate the relevat uque gas for the uform, lear, quadratc ad cubc models coverg a overwhelmg maorty of ractcal eeds. A) Uform model A model that s uform over a averagg horzo of N ots s the smlest oe. The relevat sgal s characterzed wth oe state ad the flter ga s rereseted, by (7), wth the 0-degree olyomal as N N + h0 ( ) (5) 0 otherwse The ose ower ga (NPG) of ths flter s - varat, g 0 () /N. Because ths ga s assocated wth smle averagg, t s also otmal for a commo task: reducg radom ose whle retag a shar ste resose. B) Lear Model For the lear redcto, l, the ga s ram ( ) a ( ) a ( ) h 0 + havg the coeffcets ad, (6) ( N )( N ) + ( N + ) a 0( ), (7) N ( N + ) ( N ) ( N ) 6 a ( ). (8) N C) Quadratc ad Cubc Models For the quadratc ad cubc models, the gas of the ubased FIR flters become, resectvely ( ) a ( ) ( ) ( ) 0 + a a ( ) a ( ) ( ) ( ) 03 + a3 a3 a ( ) 3 h +, (9) h (0) 33 where the coeffcets are defed [8]. I the Fg., we sketch (5), (6), (9) ad (0) for N 3 ad 0. 5 Cross Pot of the NPG Nose the FIR estmator ca be evaluated terms of the ose ower ga (NPG) defed [8] as l N + g ( N, ) ( N, ). () It turs out that the reduced order gas ca be foud at some cross ots secfed [9, 0, ] for dfferet gas (uform, ram, quadratc ad cubc), as N, () N N +, (3) N N (4) N 3 + 5( 3N 7), 0 (5) N 36 5( 3N 7). 0 (6) As a examle, these ots are show Fg. for N 3 h 3() h () h () h 0() Fg. : The uque FIR flter gas h 0 (), h (), h () ad h 3 () for N 3 wth 0. 3 Fg. : Cross ots betwee the low-degree olyomal gas for N ISBN:

4 Recet Researches Telecommucatos, Iformatcs, Electrocs ad Sgal Processg 6 FMH Structure The block dagram of the basc FIR meda hybrd (FMH) structure was develoed [4] to maxmze the SNR the row ad colum vectors. Here, the ut sgal y s fltered wth two FIR flters. The forward FIR flter (FIRFW) comutes the ots o a horzo to the left from the ot. I tur, the backward FIR flters (FIRBW) rocesses data o the same legth horzo lyg to the rght from. The estmates ca thus be formed for < 0 as, resectvely, N + ( ) FW xˆ ( ) y, (6) N + ( ) BW xˆ ( ) y. (7) + The outut sgal xˆ ( ) s obtaed usg the olear oerator called the meda. I the FMH structure, the ut y ad the oututs of the FIR flters, xˆ BW ( ) ad xˆ FW ( ), lay the role of etres. Followg the meda flter strategy, the outut xˆ ( ) becomes equal to the termedate value that s stated by the oerator [ ] BW FW ( ) MED xˆ ( ), y, xˆ ( ) xˆ. (8) Note that the best flterg result ca be obtaed f oe sets roerly the smoother lag or redcto ste the FIR flters. Because the basc structure show [4], s commoly uable to obta ce mage ehacg, that s owg to a small umber of the etres, the more sohstcated FMH structures exlotg dfferet would rovde better erformace. Gve the dscrete mage mag x (, ), where [, P] ad [, Q], ad the relevat ehaced ad recostructed mag xˆ (, ), the SNR ca be estmated ( db) by SNR db P Q [ x(, ) ] 0 log0 P Q [ ( ) ( )], (9) x, xˆ, I tur, the RMSE ca be estmated wth RMSE PQ P Q [ x(, ) xˆ (, ) ] Fg. 3: Orgal ultrasoud mage test. (30) 7 Smulatos For further vestgato, we chose a real ultrasoud mage of xels showed Fg. 3. The cture has bee formed uder the codtos dscussed [] ad [3]. Further, the mage was cotamated wth both addtve whte Gaussa ad seckle ose comoets as show Fg. 4. The smulato codtos were take as follows: N 3, the -lag was allowed to be at the cross ots of 5, 4, 6, 3 7 ad 36 3 (see Fg. ), ad the ose varace was set as σ 0.. To rovde the mage ehacg, we emloyed the meda structures fttg (9). Image ehacemet ca quattatvely be evaluated usg the SNR ad RMSE metrcs. Fg. 4: Nosy Image wth σ 0.. ISBN:

5 Recet Researches Telecommucatos, Iformatcs, Electrocs ad Sgal Processg TABLE I. Quattatve evaluato for N 3. l RMSE SNR(dB) Fg. 5: Ehaced mage wth l, N 3 ad 5. Fg. 6: Ehaced mage wth l, N 3 ad max(, ) 6. The results are sketched Fg 5 Fg. 7 ad we evaluate the ehacemets terms of SNR (30) ad RMSE (3) as show Table at each of the cross ots (Fg. ). As ca be see, rovded N 3, the best ehacg s acheved wth the quadratc ga of degree l that s the reduced order cubc ga, l 3, at the aforemetoed cross ots. 8. Coclusos I ths aer, we vestgated effect of the secfc cross ots featured to the -lag ubased smoothg FIR flters wth low-degree olyomals gas o ehacg ultrasoc mages emloyg the FMH structures. The results are llustrated Fg. 5 Fg. 7 rovg that smoothg FIR flterg works more effcetly FMH structures tha the redctve oe [4]. Table I gves us the relevat estmated values of the SNRs ad RMSEs. As ca be cocluded, the crease the smoothg FIR flter degree does ot oblgatorly lead to better deosg ad error reducto. O the other had, better ehacg s acheved wth the lags, allowg for reducg the smoother degree. That meas that both l ad must be otmzed FMH structures order to maxmze SNR ad mmze RMSE for each of the artcular mages. Ackowledgmet The frst author would lke to thak at PROMEP ad the FIEC Veracruzaa Uversty for the suort of ths vestgato. Refereces Fg. 7: Ehaced mage wth l 3, N 3 ad max( 3,, 36 ) 3. [] I. Ptas ad A. Veetsaooulos, Nolear Dgtal Flters Prcles ad Alcatos, Kluwer Academc Publshers, 990. [] N. Kaloutsds ad S. Theodords, Adatve System Idetfcato ad Sgal Processg Algorthms, Pretce Hall, 993. [3] J. Astola ad P. Kuosmae, Fudametals of Nolear Dgtal Flters, CRC Press, 997. ISBN:

6 Recet Researches Telecommucatos, Iformatcs, Electrocs ad Sgal Processg [4] P. Heoe ad A. Neuvo, FIR-meda hybrd flter wth redctve FIR substructures, IEEE Tras. o Acoustc, Seech, ad Sgal Processg, vol. 36, o. 6, , Jue , Housto, TX. USA, Arl 30 May, 009. [5] Dumtrescu B., Postve Trgoometrc Polyomals ad Sgal Processg Alcatos, Srger, Dordrecht, 007. [6] Y. S. Shmaly, A ubased FIR flter for TIE model of a local clock alcatos to GPS-based tmekeeg, IEEE Tras. o Ultrasoc, Ferroelectrcs ad Frequecy Cotrol, vol. 53, o. 5, , May 006. [7] T. Bose, F. Meyer, ad M. Q. Che, Dgtal Sgal ad Image Processg, J. Wley, New York, 004. [8] Y. S. Shmaly, A ubased -ste redctve FIR flter for a class of ose-free dscrete tme models wth deedetly observed states, Sgal, Image & Vdeo Processg, vol. 3, o.,. 7-35, Ju [9] L. J. Morales-Medoza, Y. S. Shmaly ad O. G. Ibarra-Mazao, Ehacg Ultrasoud Images usg Hybrd FIR Structures, Image Process./Ch. 6, Eds., , ITech: Vea, December 009. [0] Y. S. Shmaly ad L. J. Morales-Medoza, FIR Smoothg of Dscrete-tme Polyomal Sgals Sace State, IEEE Tras. o Sgal Processg, vol. 58, o. 5, , May 00. [] L. J. Morales-Medoza, Y. S. Shmaly, ad S. Perez-Caceres, A aalyss of cross ots the low-degree olyomal gas of -lag ubased smoothg FIR flters Proc. IEEE Medterraea Electrotechcal Coferece (MELECON), , Valletta, Malta, Arl 6-8, 00. [] K. Sgh & N. Malhotra, Ste-by-Ste Ultrasoud Obstetrcs, Mc-Graw Hll, 004. [3] D. Leve, Ultrasoud Clcs, Elsever, Bosto USA, 007. [4] L. J. Morales-Medoza, Y. Shmaly, O. G. Ibarra-Mazao, A Aalyss of Hybrd FIR structures Alcatos to Ultrasoud Image Processg st Iteratoal Coferece o Comutatoal ad Iformato Scece (WSEAS),. ISBN: