Discontinuity Preserving Noise Removal Method based on Anisotropic Diffusion for Band Pass Signals

Transcription

1 isconini Preservin Noise emoval Mehod based on Anisoropic iffsion for Band Pass Sinals Sasan Mahmoodi School of lecronic and omper Science Universi of Sohampon Sohampon U sm3@ecs.soon.ac.k Absrac A nonlinear disconini-preservin mehod for noise removal for band pass sinals sch as sinals modlaed wih Binar Phase-Shif ein BPS modlaion is proposed in his paper. This mehod is inspired b he anisoropic diffsion alorihm o remove noise and preserve disconiniies in band pass sinals modlaed wih a sinle freqenc. I is demonsraed here ha nonlinear noise removal mehod for a real valed band pass sinal reqires a solion for a nonlinear parial differenial eqaion which is of forh order in space and second order in ime. The resls presened in his work show beer performance in nonlinear noise removal for real valed band pass sinals in comparison wih he previos work in he lierare ewords Anisoropic iffsion; Band Pass Sinals; Noise emoval; arrier Sinal; isconini Preservaion; I. INTOUTION Anisoropic diffsion for ede-preservin noise removal in imaes is iniiall p forward b Perona and Malik PM in heir seminal work [1]. Geri e al. eneralize he nonlinear noise removal mehod of PM o 3 volmeric MI imaes []. A color anisoropic diffsion is also developed b Sapiro e al. [3] o remove he noise and preserve edes in color imaes. A robs esimaion mehod o erac a piecewise smooh imae from an oriinal nois imae is sesed b Black e al. [4]. rvare-preservin parial differenial eqaions Ps are emploed b Tshmperie [5] o propose a fas anisoropic smoohin alorihm for he noise removal of mli-valed imaes. Lo e. al [6] propose a echniqe o smooh nois imaes b eploiin an ede-srenh srae o improve he preservaion of he deails of he smoohed imae. Finall a nonlinear filerin scheme based on a linear parial differenial eqaion of he pe of he Hea eqaion is proposed b Mahmoodi [7] o remove noise in band pass sinals b preservin disconiniies and modlain carrier sinal. The mehod proposed in [7] however can onl be sed for comple valed sinals. In order for his mehod o be sed for real valed sinals Hilber ransform is emploed in [7] o prodce comple valed sinals o of heir real valed ones. A new parial differenial eqaion is herefore reqired for noise removal in real valed sinals. This is no a rivial problem. Or conribion in his paper is ha a hiher order parial differenial eqaion based on he one invesiaed in [7] is proposed o perform he noise removal of real valed band pass sinals wih no reqiremen o se Hilber ransform o prodce he imainar par of a real valed sinal. This paper is srcred as follows. The heor is presened in secion II. Secion III deals wih implemenaion isses. Nmerical resls are hen presened in secion IV. onclsions are finall drawn in secion V. II. THOY The noise removal mehod proposed b Mahmoodi [7] for band pass sinals b preservin disconiniies is based on a linear parial differenial eqaion of he followin form: 1 wih iniial condiion where : : > and are he smoohed sinal oriinal nois sinal consan vales respecivel and 1. In eqaion 1 i is assmed ha he carrier sinal has freqenc and and are comple valed sinals. qaion 1 is associaed wih one carrier sinal wih freqenc. The nonlinear noise removal proposed in [7] can herefore be onl sed for comple valed sinals havin real and imainar pars. Sinals in real life however are real valed. Sch a sinal reqires a real valed carrier sinal. The specrm of a real valed carrier sinal herefore consiss of wo freqenc componens in freqenc domain in freqencies and. The anisoropic diffsion eqaion associaed wih freqenc is wrien as: 3 An iniial condiion similar o is also sed for his eqaion. qaions 1 and 3 can be rewrien as: MMSP 13 Sep. 3-Oc. 13 Pla Sardinia Ial /13/$ I MMSP13

2 5 To desin a noise removal echniqe for real valed sinals eqaions 4 and 5 need o be combined. In his paper we propose he followin eqaion combinin eqaions 4 and 5: Or 6 We assme he followin iniial condiions for eqaion 6: 7 8 I is noed ha in eqaion 6 : : are real valed sinals. B akin Laplace ransform wih respec o ime and Forier ransform wih respec o space from boh sides of eqaion 6 i becomes clear ha he reslin ransfer fncion conains wo cenral freqencies and. This is indeed similar o facorin mehod in he solion of wave parial differenial eqaion where wave eqaion is facored ino wo mliplin erms: one responsible for he wave ravelin in he lef direcion and he oher one in chare of he wave ravelin in he rih direcion [11]. I is also imporan o noice ha addiion and/or sbracion of eqaions 4 and 5 does no prodce he desired resls and herefore i does no lead o a ssem capable of dealin wih a real valed sinal conainin wo carrier freqencies a and. Before we se eqaion 6 o propose a nonlinear filerin alorihm o remove noise from real valed band pass sinals and preserve disconiniies and he carrier sinal le s invesiae linear eqaion 6. The propaaor of eqaion 6 is a real valed Gabor-like filer shown in fire 1 for consan vales of 1 5. and 15. Fire 1: The propaaor of eqaion 6 Zero mean Gassian noise is added o he noiseless sinal shown in fire -a o prodce a nois sinal depiced in fire -b wih SN.154. qaion 6 is applied o he nois sinal of fire -b wih 1 5. and 15 o prodce he smoohed sinal shown in fire -c. The hea eqaion wih 1 and 15 whose propaaor is a Gassian filer has also been applied o he nois sinal of fire -b o obain he smoohed sinal shown in fire - d. As shown in his fire he carrier sinal has been filered o and he smoohed sinal is compleel disored b he hea eqaion. In he ne secion eqaion 6 wih iniial condiions 7 and 8 are discreized and sed nonlinearl o remove he noise and preserve disconiniies and he real valed carrier sinal wih freqenc. III. IMPLMNTATION ISSUS We emplo an eplici finie difference scheme for he discreizaion of eqaion 6. A cenral finie difference approach is sed for spaial discreizaion and a forward finie difference scheme is emploed for emporal discreizaion. For he discreizaion prposes eqaion 6 can be wrien in he followin form: 9 B sin an eplici discreizaion scheme eqaion 9 can herefore be discreized as follows: L 1 where L Fncion can be chosen as one of he followin fncions: ep q 11 MMSP13 363

3 Or a d Fire : Linear filerin of eqaion 6 a a noiseless sinal wih a carrier freqenc of. 5 b Nois sinal prodced b conaminain he noiseless sinal of fire -a wih a zero mean Gassian noise wih SN.154 c he sinal smoohed b applin eqaion 6 wih 1. 5 and 15 o he nois sinal of fire -b d he sinal smoohed b applin he hea eqaion wih 1 and 15 o he nois sinal of fire -b b c q where q is a parameer deermined b sers. An disconini in sinal componens wih freqencies and wold prodce a local maima in erms sch as and in eqaion 9. The erms and L herefore approach o zero o preserve he disconini in case a local maimm is deeced in and. Throho his paper is se o ni. Time is reaed virall and corresponds o ieraions in he alorihm. The clidian disance beween wo consecive smoohed sinals is sed as a soppin crierion. If his disance is less han a hreshold he alorihm sops and he alorihm is said o have convered. Parameer is sed o ads he speed of he converence. Lare vales of ma increase he speed of he converence and ma also lead o he insabili of he alorihm as i is epeced in an eplici finie difference srae. Lower vales for on he oher hand aranees he alorihm s converence in he epense of lower speeds of he converence. Finall parameer q shold be seleced accordin o he amon of disconiniies in he oriinal noiseless sinals. Over-smoohin in he smoohed sinal ma occr if larer vales for q are chosen b he ser. This oversmoohin is demonsraed in he smoohed sinal wih some smoohed no preserved disconiniies. Smaller vales for q on he oher hand in a ver nois environmen ma lead o he failre of noise removal and preservin some disconiniies associaed wih noise raher han he oriinal noiseless sinal. Generall i is bes o choose lower vales for q in he presence of moderae noise o preserve disconiniies. We also noe ha o avoid he effecs associaed wih he ncerain principle [8] i is alwas assmed ha here is a leas one fll ccle of he carrier sinal beween wo consecive disconiniies. IV. NUMIAL SULTS In his secion we appl he nonlinear band pass filer derived in eqaion 1 on nois band pass sinals conainin disconiniies. Fire 3-a shows a noiseless band pass sinal wih freqenc. 5 conainin disconiniies. This is an eample of a diial sinal modlaed b sin Binar Phase- Shif ein BPS modlaion [9]. Zero mean Gassian noise is added o his noiseless sinal o resl in he nois sinal wih SN 1.49 shown in fire 3-b. Or nonlinear 364 MMSP13

4 a a b b c c Fire 3: Noise removal in a band pass sinal BPS modlaed sinal conainin disconiniies a Oriinal noiseless sinal wih freqenc. 5 wih disconiniies a locaions 1 and 3 b Nois sinal conaminaed wih zero mean Gassian noise wih SN1.49 c he sinal smoohed wih or mehod proposed here for q1 and.5 d he sinal smoohed wih a nonlinear low pass filer based on anisoropic eqaion [7] wih q1 and.5 d Fire 4: Noise removal of a band sinal conainin disconiniies a an oriinal noiseless band pass sinal wih.5 conainin a disconini a he locaion of b Nois sinal conaminaed wih a zero mean Gassian noise wih he sandard deviaion of. c The sinal smoohed b he mehod proposed in [7] wih q1 and. 1 d he sinal smoohed b or filerin mehod proposed here wih q1 and. 1. d 365 MMSP13

5 filer proposed here wih q 1 and. 5 is applied o he nois sinal of fire 3-b o obain he smoohed sinal shown in fire 3-c. As can be seen from his fire Noise is removed and carrier sinal as well as disconiniies are preserved. I akes or nonlinear filerin alorihm onl 1.6 seconds o convere o he solion shown in fire 3-c in a 64-bi Malab version 7.11 rnnin on a P worksaion wih a PU wih freqenc.67 GHz. A nonlinear low pass filer based on anisoropic diffsion eqaion [7] wih q 1 and.5 is also applied o he nois sinal of fire 3-b o obain he smoohed sinal shown in fire 3-d. As can be seen from fire 3-d he smoohed sinal has lower amplide owin o low pass filerin and also he disconiniies are considerabl smoohed. In he ne eperimen we demonsrae ha or alorihm proposed in his paper is compeiive in comparison wih a nonlinear band pass filer reqirin o compe he imainar par of a real sinal b eploiin he Hilber ransform [7]. In he nonlinear filerin scheme proposed here however here is no need o compe he Hilber ransform of he real valed sinal i.e. or parial differenial eqaion is applied o onl real valed sinals. Fire 4-a shows a noiseless band pass sinal wih a disconini a he locaion of and.5. Zero mean Gassian noise wih a sandard deviaion of. is hen added o he noiseless sinal o prodce he nois sinal of fire 4-b. The band pass nonlinear filerin process proposed in [7] wih q1 and. 1is applied o he nois sinal of fire 4-b o obain he smoohed sinal of fire 4-c. The nonlinear filerin echniqe proposed in his paper wih q1 and. 1 is also applied o he nois sinal of fire 4-b o compe he smoohed sinal shown in fire 4-d. As can be seen from fire 4 or noise removal alorihm based on a new parial differenial eqaion proposed here prodces compeiive resls in comparison wih he work presened in [7] wih no frher reqiremen and hassle of he Hilber ransform compaion needed o compe he imainar par of real valed sinals. FNS [1] P. Perona J. Malik Scale-Space and de eecion sin Anisoropic iffsion I Transacions on Paern econiion and Machine Inellience Vol. 1 No. 7 pp [] G. Geri O. bler. ikinis F. Jolesz Nonlinear Anisoropic Filerin of MI aa I Transacions on Medical Imain Vol. 7 No. 11 pp [3] G. Sapiro.L. inach Anisoropic iffsion of Mli-Valed Imaes wih Applicaions o olor Filerin I Transacions on Imae processin Vol. 5 No. 11 pp [4] M.J. Black G. Sapiro.H. Marimon. Heeer obs Anisoropic iffsion I Transacions on Imae Processin Vol. 7 No. 3 pp [5]. Tschmperie Fas Anisoropic Smoohin of Mli-Valed Imaes sin rvare-preservin Ps Inernaional Jornal of omper Vision Vol. 68 No. 1 pp [6] H. Lo L. Zh H. in opled Anisoropic iffsion for Imae Selecive Smoohin Sinal Processin Vol. 86 No. 1 pp [7] S. Mahmoodi Anisoropic iffsion for Noise emoval of Band Pass Sinals Sinal Processin Vol. 91 No.5 pp [8] V. Harvin B. Joricke The Uncerain Principle in Harmonic Analsis Spriner-Verla [9] H. Sern S. Mahmod ommnicaion Ssems Pearson Prenice Hall 4. [1] L. Hanzo S.X.Nq T. eller W.T. Webb Qadrare Amplide Modlaion: From Basics o Adapive Trellis-oded Trbo-qalised and Space-Time oded OFM MA and M-MA Ssems Wile- Blackwell 4. [11] G.B. Arfken H.J. Weber Mahemaical Mehods for Phsiciss Academic Press he 7 h diion 1. V. ONLUSION A nonlinear noise removal alorihm based on a new formlaion for band pass sinals o preserve disconiniies is presened in his paper. The filerin alorihm proposed here demonsraes sperior performance over he nonlinear low pass filer based on anisoropic diffsion. In conras wih he previos work of he noise removal for band pass sinals presened in [7] he alorihm presened here has he advanae ha he Hilber ransform is no reqired o prodce comple valed sinals needed for he mehod proposed in [7]. Or alorihm direcl is applied o he real valed sinals wih eqivalen resls obained from he alorihm presened in [7]. Frhermore he mahemaical framework presened in his paper paves he wa for noise removal alorihms siable for band pass sinals conainin carrier sinals wih more han one freqenc mli-freqenc carrier sinals sch as sinals modlaed b sin Orhoonal Freqenc-ivision Mliplein OFM modlaion scheme [1]. 366 MMSP13