Recursive Implementation of Anisotropic Filters

Transcription

1 Rcursiv Implmtatio of Aisotropic Filtrs Zu Yu Dpartmt of Computr Scic, Uivrsit of Tas at Austi Abstract Gaussia filtr is widl usd for imag smoothig but it is wll kow that this tp of filtrs blur th imag faturs.g., dgs. Two tsios of Gaussia filtrs will b discussd i this surv. O is th aisotropic filtrig bilatral filtrig or PDE-basd aisotropic diffusio for fatur-prsrvig smoothig ad th othr is th rcursiv implmtatio of various filtrs that ca largl rduc th computatioal tim i crtai coditios. W will also discuss th combiatio of aisotropic filtrig ad rcursiv implmtatio for imag smoothig such that th aisotropic smoothig could b do i a fast wa.. Itroductio Nois is commol s i ma tps of imags for ampl, biomdical imags ad rmot ssig imags. It is i a grat dmad to smooth th imags bfor othr tasks could b coductd. Gaussia lowpass filtrig is kow to b a fficit ad simpl wa for imag smoothig. Howvr, it is wll kow that Gaussia filtrig blurs th imag dgs whil smoothig imag ois. Th raso is that Gaussia filtrs ar isotropic i th ss that all surroudig pils affct th ctr pil i a similar fashio rgardlss thir itsit variatios. Hc, th dgs ad th ois ar tratd i th sam wa, which ilds ois rductio as wll as dg blurrig. A ampl of such ffct ca b s from Fig. b. To rmd th problm of traditioal Gaussia filtrig, popl hav proposd lots of mthods, trig to achiv th goal of fatur-prsrvig smoothig. All of ths mthods ar calld aisotropic filtrig ad ca b groupd ito two catgoris. O is calld Bilatral Filtrig [7,, 3], which is a straightforward tsio of Gaussia filtrig. Th othr is a PDE-basd tchiqu, calld aisotropic hat diffusio [4, 8]. W shall s mor dtails o th various tchiqus o ths topics i th followig sctios. I Fig. c,

2 w show a ampl of bilatral filtrig o th sam ois imag. W ca clarl s th diffrc btw th isotropic filtrig ad th aisotropic filtrig. Obviousl th lattr o givs bttr rsults. Aothr importat issu rgardig imag filtrig is th implmtatio of th various filtrs. Gaussia low-pass filtrig ca b implmtd b dirct covolutio, which is furthr acclratd b FFT. Aisotropic filtrs, howvr, ar grall much mor tim-cosumig although som authors studid fast algorithms for bilatral filtrig []. Aothr fast wa for implmtig Gaussia filtrig is b rcursiv schm [5, 6]. Rcursiv implmtatio rquirs a costat ad small umbr of MADDs multiplicatios ad additios rgardlss th siz of th ighborhood big cosidrd. Hc, th tim complit of rcursiv implmtatio of Gaussia filtrig is quit small compard to othr squtial algorithms of Gaussia filtrig [5]. Howvr, th rcursiv schm was origiall dsigd for isotropic Gaussia filtrig ad littl work has b do o combiig th rcursiv schm ad th aisotropic filtrig. I this projct our goal is to plor th possibilit of applig th rcursiv implmtatio tchiqu o th aisotropic filtrig. This surv is orgaizd as follows. W shall bgi i t sctio b rviwig various tchiqus for aisotropic imag smoothig. Th i Sctio 3, w shall giv a brif dscriptio of various rcursiv tchiqus that hav b s i litraturs. I Sctio 4 w will s som prvious work that combis th rcursiv tchiqus ad aisotropic filtrig ad our proposd approach will b brifl dscribd i Sctio 5. Fiall w giv our coclusio i Sctio 6. Origial Nois Imag Isotropic Filtrig Aisotropic Filtrig Figur Eampl of isotropic Gaussia filtrig ad aisotropic filtrig o a mdical imag.

3 . Aisotropic Filtrs Aisotropic filtrig is grall rprstd i two diffrt was. O is b bilatral filtrig [7] ad th othr is b PDE-basd aisotropic hat diffusio [4]. I th followig w will dscrib both was o b o ad latr o th clos rlatioship btw ths approachs will b show. Th bilatral filtrig [7] is a straightforward tsio of Gaussia low-pass filtrig. As w kow, th Gaussia filtrig is dfid b a fuctio as follows: g σ,, πσ whr σ is a giv valu, kow as stadard dviatio. It is obvious that this fuctio is isotropic with rspct to th ctr. Thrfor, a dirct us of this fuctio o th ois imags will rsult i blurrd dgs sic this fuctio ol cosidrs th spatial iformatio without cosidrig th imag iformatio aroud th ctr. Th basic ida of bilatral filtrig is to add a additioal trm to th wightig fuctio i such that th imag iformatio is tak ito accout: g', f, f 0,0 σ d σ c, whr σ d ad σ c ar prst paramtrs. Th ampl i th followig figur, rproducd from [], shows th diffrc btw th Gaussia filtrig show i b ad th bilatral filtrig show i d. a b c d Figur Illustratio of bilatral filtrig []. I a w show a ampl of ois imag ad th spatial krl Gaussia fuctio g, is show i b. Th imag-rlatd wightig fuctio scod trm i g, is show i c. Th combid wightig fuctio g, i d shows a aisotropic proprt.

4 Rctl a fast implmtatio of bilatral filtrig was proposd b Durad t al []. Th acclratd th bilatral filtrig b usig a picwis-liar approimatio i th itsit domai ad appropriat subsamplig. Elad [3] discussd th bilatral filtrig from a liar algbra poit of viw ad poitd out som possibl was to improv it. Aothr commol s tchiqu to raliz aisotropic filtrig is dfid b hat diffusio quatios. Th first modl about th aisotropic diffusio is wll kow as Proa-Malik modl as s i [4]. Proa- Malik modl is dfid b a o-liar PDE hat diffusio as follows [4]: u t div g u u 3 whr g. is a positiv o-icrasig fuctio which supprsss diffusio aroud imag dgs whr th orm of th gradit is high. A commo choic for g. is giv b g s K s 4 whr K is a prst costat. Strictl spakig, howvr, Proa-Malik modl is ot a aisotropic diffusio modl sic th wightig fuctio g. is just a scalar fuctio, which dos ot idicat th aisotropic diffusio aroud a poit. A tru aisotropic diffusio modl was discussd b J. Wickrt [8], who dfid th diffusio PDE as follows: u t div D u u 5 whr D. is dfid as a tsor of u. This matri givs a dirctio whr th diffusio is prfrrd ad aothr dirctio whr th diffusio is supprssd. Similar to th rlatioship btw Gaussia filtrig ad liar hat diffusio, thr is also a clos rlatioship btw bilatral filtrig ad PDE-basd aisotropic hat diffusio as discussd i []. Both bilatral filtrig ad aisotropic hat diffusio ar quit tim-cosumig to implmt. I th followig sctios w will s o of th stratgis to rduc th computatioal tim of both tps of filtrs b usig rcursiv implmtatio.

5 3. Rcursiv Implmtatio Th rcursiv implmtatio of svral tps of filtrs had b discussd i litraturs. I [,, 3], Drich studid th rcursiv implmtatio of svral tps of filtrs with potial wightig fuctios. I [3], Th author discussd two tps of filtrs. O is calld scod ordr rcursiv filtr dfid b:, 6 k S whr is a prst costat ad k is chos as k 7 such that. S Assum th iput sigal ad output sigal ar ad, rspctivl. Th th rcursiv ralizatio of th filtrig with covolutio mask dfid b 6 is drivd b th followig causal ad ati-causal squcs:, 8 k k It is clar that th abov rcursiv implmtatio of such filtr rquirs 8 multiplicatios ad 7 additios pr output pil. This fact idicats th computatioal advatag of rcursiv implmtatio of filtrs. I [3], th authors also discussd th rcursiv ralizatios of th first ordr filtr ad th drivativs of both first ordr ad scod ordr filtrs. Th claimd that thir rcursiv implmtatio of filtrs is much computatioall fastr tha th o-rcursiv implmtatio if o paralllism is cosidrd. Howvr, as poitd out i [5], all th rcursiv implmtatios s i [,, 3] ar basd o o- Gaussia filtrig, that is, th ar all basd o filtrs dfid b potiall wightig fuctios. Thrfor, th do ot hav th imprssiv proprtis that Gaussia filtrs hav. For ampl, th

6 potiall dfid filtrs ar ot isotropic i D, ot circularl smmtric. I [5], th authors proposd a rcursiv implmtatio of th Gaussia filtr. Thir approach is basd o a ratioal approimatio of th Gaussia fuctio giv b: t / g t 4 π a a t a t a t ε t 9 whr a , a , a ad a Th rror ε t is provd to b limitd to ε t < Accordig to th ratioal approimatio of th Gaussia fuctio, o ca driv th followig rcursiv implmtatio of Gaussia filtrig: forward : backward w B B w bw b b w b b b 0 0 b w 3 b whr B b b b b ad b, b b ar costats drivd from th stadard dviatio σ of th 3 / 0, 3 Gaussia filtr. It was claimd i [5] that th abov rcursiv implmtatio of th Gaussia filtr ol rquirs 6 MADDs multiplicatios ad additios pr output pil. It is fastr tha th rcursiv schm s i Eq. 8 ad furthrmor, it givs a mor isotropic circularl smmtric impuls rspos [5]. Now w giv a short summar o th tim complit of som of th abov algorithms. Assum th siz of th imag big cosidrd is N N ad th siz of th mask widow is M M. Algorithms Tim Complit Gaussia low-pass filtrig dirct covolutio O N M Bilatral filtrig [7] O N M Proa-Malik modl [4] O N K, whr K is umbr of itratios Rcursiv implmtatio of potial filtr [3] Rcursiv implmtatio of Gaussia filtr [5] 8 MADDs multiplicatios ad additios pr pil 6 MADDs multiplicatios ad additios pr pil Not: Accordig to [5], th rcursiv implmtatio s i 0 is fastr tha FFT-basd Gaussia lowpass filtrig.

7 4. Rcursiv Implmtatio of Aisotropic Filtrs Rcursiv implmtatio of aisotropic filtrs is th mai goal of this projct. As far as I kow, thr hav b svral paprs dalig with this issu. Th first papr is th o b Alvarz t al. [9, 0], who discussd th rcursiv ralizatio of th oliar vrsio of th potial filtrs as s i Eq. 6. Th ida is that, istad of usig costat paramtr, w ca cosidr a varig paramtr o th orm of th drivativ of th iput sigal., which dpds h t, t whr h. is a o-dcrasig fuctio with h 0 ε > 0. For simplicit, th authors i [9, 0] chos h. as a liar fuctio: h s ε M s, whr M is a positiv costat. W ow show som simulatio rsults that ar rproducd from [9]. I Fig. 3, w show th rsults b th approach s i [9, 0]. W ca s that as M gos smallr ad smallr, th smoothd imags bcom mor ad mor blurrd. I Fig. 4, w show th simulatio rsults b Proa-Malik modl [4]. From all ths imags, w ca s that th rcursiv implmtatio ca grat rsults as good as thos w s i Proa- Malik modl but th rcursiv implmtatio clarl rquirs much lss computatioal tim. I [5, 6], th authors discussd th rcursiv implmtatio of th act Gaussia filtr. Th D Gaussia fuctio th cosidrd could b aisotropic i th ss that th pricipal ais of th Gaussia fuctio could b i a oritatio ad th associatd dviatios σ u ad σ v could hav a aspct Iput imag M 0.5 M 0. M 0.05 Figur 3 Th simulatio rsults b th approach s i [9, 0]. Rproducd from [9]

8 Iput imag K 0 K 0 K 50 Figur 4 Th simulatio rsults b th Proa-Malik approach s i [4]. Rproducd from [9] ratio. Howvr, th assumd th Gaussia fuctio to b uiforml applid i vrwhr of th imag. Thrfor, thir mthod ca ol b usd to smooth or dtct th lis with a spcific oritatio. This assumptio is crtail ivalid for imag smoothig with arbitraril oritd faturs. 5. Proposd Approach W hav s th two tsios of Gaussia filtrig. O is aisotropic filtrig for highr smoothig qualit ad th othr is rcursiv implmtatio of filtrs for lss computatioal tim. Th goal of this projct is to combi ths two tsios togthr such that a fast implmtatio of aisotropic filtrs could b possibl. Although w hav s som prvious work o this issu, it is still itrstig to go furthr i this dirctio. Our ida is to dsig a aisotropic vrsio of th rcursiv filtrs as discussd i [5, 6]. Our mthod will b quit similar to th o usd i [9, 0] ad th ida is also b rplacig th costat propagatio paramtrs with varig paramtrs. Howvr, sic th filtrs that w ar goig to work o ar diffrt from that s i [9, 0], th dtaild approach is pctd to b diffrt ad th rsultig impuls rspos as wll as th primtal rsults should also b diffrt. 6. Coclusio I this rport w rviwd som prvious work o two tsios of th classic Gaussia filtrs. O is th aisotropic filtrig ad th othr is rcursiv implmtatio of filtrs. W also saw som work that combid ths two tsios togthr, ildig a rcursiv implmtatio of aisotropic filtrs. W

9 propos a approach that is rlatd to th rcursiv implmtatio of th Gaussia filtr [5, 6] ad th rcursiv implmtatio of aisotropic potial filtr [9, 0]. Mor dtails will b availabl soo. Rfrcs [] R. Drich, Optimal dg dtctio usig rcursiv filtrig, Procdig of st Itratioal Cofrc o Computr Visio, Lodo, Ju 8-, 987. [] R. Drich, Usig Ca's critria to driv a rcursivl implmtd optimal dg dtctor, Itratioal Joural of Computr Visio, vol., o., pp , Ma 987. [3] R. Drich, Fast algorithms for low-lvl visio, IEEE Tras. Pattr Aalsis ad Machi Itlligc, vol., o., pp 78-87, 990. [4] P. Proa, J. Malik, Scal-spac ad dg dtctio usig aisotropic diffusio, IEEE Tras. o Pattr Aalsis ad Machi Itlligc, vol., o. 7, pp , 990. [5] I.T. Youg, L.J. Vlit, Rcursiv implmtatio of th Gaussia filtr, Sigal Procssig, vol. 44, pp. 39-5, 995. [6] L.J. Vlit, I.T. Youg ad P.W. Vrbk, Rcursiv Gaussia drivativ filtrs, Procdigs of th 4th Itratioal Cofrc o Pattr Rcogitio, vol., pp , Brisba, Australia, 6-0 August 998. [7] C. Tomasi ad R. Maduchi, Bilatral filtrig for gra ad color imags, I Proc. IEEE It. Cof. o Computr Visio, pp , 998. [8] J. Wickrt, Aisotropic Diffusio I Imag Procssig, ECMI Sris, Tubr, Stuttgart, ISBN , 998. [9] L. Alvarz, R. Drich ad F. Sataa, "Rcursivit ad PDE's i Imag Procssig", Uivrsidad d Las Palmas d G.C. Tchical Rport availabl oli at: Octobr 999. [0] L. Alvarz, R. Drich ad F. Sataa, Rcursivit ad PDE's i imag procssig, Proc. Itratioal Cofrc o Pattr Rcogitio, vol., pp. 4-48, Barcloa, Spai, Sptmbr, 000. [] D. Barash, A fudamtal rlatioship btw bilatral filtrig, adaptiv smoothig ad th oliar diffusio quatio, IEEE Tras. Pattr Aalsis ad Machi Itlligc, vol. 4, o. 6, pp , Ju 00. [] F. Durad ad J. Dors, Fast bilatral filtrig for th displa of high-damic-rag imags, I Proc. ACM Cofrc o Computr Graphics SIGGRAPH, pp , 00. [3] M. Elad, O th bilatral filtr ad was to improv it, IEEE Trasactios O Imag Procssig, vol., o. 0, pp. 4-5, Octobr 00. [4] I.T. Youg, L.J.va Vlit ad M.va Gikl, Rcursiv Gabor filtrig, IEEE Tras. Sigal Procssig, vol. 50, No., Novmbr 00. [5] J.M. Gusbrok, A.W.M. Smuldrs, ad J. va d Wijr. Fast aisotropic Gauss filtrig. Proc. 7th Europa Cofrc o Computr Visio A. Hd, G. Sparr, M. Nils, ad P. Johas, ditors, volum, pags 99-. Sprigr Vrlag LNCS 350, Cophag, Dmark, 00. [6] J.M. Gusbrok, A.W.M. Smuldrs, ad J. va d Wijr. Fast aisotropic Gauss filtrig. IEEE Tras. Imag Procssig, accptd, 003.