Diffusions and Confusions in Signal and Image Processing

Transcription

1 Journal of Mahemacal Imagng and Vson 4: , 200 c 200 Kluwer Academc Publshers. Manufacured n The Neherlands. Dffusons and Confusons n Sgnal and Image Processng N. SOCHEN Deparmen of Appled Mahemacs, School of Mahemacal Scences, Tel Avv Unversy, Tel Avv 69978, Israel sochen@mah.au.ac.l R. KIMMEL AND A.M. BRUCKSTEIN Deparmen of Compuer Scence, Technon-Israel Insue of Technology, Technon Cy, Hafa 32000, Israel ron@cs.echnon.ac.l freddy@cs.echnon.ac.l Absrac. In hs paper we lnk, hrough smple examples, beween hree basc approaches for sgnal and mage denosng and segmenaon: ) PDE axomacs, 2) energy mnmzaon and 3) adapve flerng. We show he relaon beween PDE s ha are derved from a maser energy funconal,.e. he Polyakov harmonc acon, and non-lnear flers of robus sascs. Ths relaon gves a smple and nuve way of undersandng geomerc dfferenal flers lke he Belram flow. The relaon beween PDE s and flers s medaed hrough he shor me kernel. Keywords: Ansoropc dffuson, selecve smoohng, geomerc flerng. Inroducon Averagng s a sandard procedure for smoohng nosy daa and summarzng nformaon, bu can have raher dangerous and msleadng resuls. Oulers, even f rare by defnon, can dsor he resuls consderably f hey are gven smlar weghs o ypcal daa. Such concerns led o he developmen of so-called robus esmaon procedures n sascal daa analyss, procedures ha daa-adapvely deermne he nfluence each daa pon wll have on he resuls. Only recenly were such deas and mehods mpored o sgnal and mage processng and analyss [3, 6,, 6]. The applcaon of he robus sascs deas n sgnal and mage analyss lead o he nroducon of varous non-lnear flers. To fx deas and ge a perspecve on he serous problems ha mus be addressed we shall consder below a seres of smple examples. A seemngly dfferen approach o denosng and segmenaon s based on geomerc properes of sgnals. The flerng s done, n hs approach, by solvng a non-lnear Paral Dfferenal Equaon PDE). The dervaon of he PDE s based eher on axoms and requremens, such as nvarance, separably [, 5, 7] ec., or as a by produc of a mnmzaon process for an energy funconal [5, 8, 2]. In hs paper we dscuss, hrough smple examples, he nmae connecon beween he above-menoned sgnal and mage processng mehodologes: ) PDE axomacs, 2) energy mnmzaon and 3) adapve smoohng flers. We show he relaon beween PDE s ha are derved from a maser energy funconal and non-lnear flers. Ths relaon gves a smple and nuve way of undersandng he Belram flow, and connecs beween geomerc dfferenal flers and classcal lnear and non-lnear flers. We show ha he Belram flow, whch resuls from he mnmzaon process of he Polyakov acon, s relaed o non-lnear flers of specal ype upon choosng a L γ nduced merc and afer dscrezaon of he

2 96 Sochen, Kmmel and Brucksen correspondng paral dfferenal equaon. A dfferen non-lnear fler s consruced va a shor me analyss of he same PDE. The shor me approach s dfferen snce we analyze and evenually dscreze he soluon of he dfferenal equaon. I s also more general snce we rea a varey of flows by no specfyng he explc form of he merc a he ouse. In hs way we have a clear nuve undersandng of he adapve averagng as a Gaussan wegh funcon on he manfold ha s defned by he daa. The flows dffer n he geomery arbued o he manfold hrough dfferen choces of he merc. The paper s organzed n an ncreasng order of echncaly. In Secon 2 we dscuss smple examples ha make he basc dea clear. In Secon 3 we deal wh he more echncal consderaons of he shor me analyss and show how dfferen approaches are relaed o, or specal cases of, hs framework. We consruc he approprae nonlnear fler for he challengng problem of he averagng of consraned feaures n Secon 4. Image denosng s dscussed and demonsraed n Secon 5 and our conclusons are summarzed n Secon 5. Few of he more echncal compuaons can be found n he Appendx. 2. Averagng Daa for Smoohng and Cluserng To make our presenaon as smple as possble we frs lm our dscusson o ses of scalar and vecors. One dmensonal sgnals are analyzed nex, and few remarks on hgher dmenson sgnals, e.g. mages, can be found n he las subsecon. 2.. Averagng Scalar Varables Values n IR) Suppose we are gven a se of real numbers x, x 2,...x N } and we would lke o provde someone wh a ypcal represenave value ha somehow descrbes hese numbers. A naural choce would be her average,.e. x = = ) x ) N Whle for ses lke 2.83, 3.7, 3.22, 2.97, 3.05} hs s a reasonable choce, clearly s no for a se of values comprsng he number 0 0 and a housand values n he nerval ɛ, + ɛ) for ɛ = 0.0. In hs second case, a ypcal number n he se s around and 0 0 may be regarded as an ouler o be eher dscarded or gven specal consderaon. To deal wh such suaons we could propose he followng process of smoohng he nal se of values o produce new ses ha more clearly exhb he nner srucure of he values n hs se. For each =, 2,...,N do x new = α)x old + j= j where 0 < α <. In marx form X new w x old, x old ) j x old j, 2) α w x, x 2 ) w x, x N ) w 2 x 2, x ) α w 2 x 2, x N ) = Xold, w N x N, x ) w N x N, x 2 ) α 3) where X T = x, x 2,...,x N ). The weghs may be chosen so ha w x, x j ) = α, j= j and w x, x j ) reflecs how much nfluence x j has on x based, say, on how far or how dfferen x j s from x. We furher assume ha he dsance s always posve and symmerc. For example, we could choose wh w x, x j ) = β = α j= j β + x j x γ j, + x j x γ Such a choce wll make nearby pons more nfluenal on he dsplacemen of x oward s new locaon. If, however, we shall choose w x, x j ) = α for all N, he dynamcs of updae becomes x new = α)x old + α N j. x old j, 4)

3 Dffusons and Confusons 97 Fgure. Random pons on IR propagae va adapve averagng wh γ = 0,,.2, 2 lef o rgh). and all pons converge owards he mean of he nal locaon. Indeed we wll have n hs case α α α... N N X new α α = α N N X old, 5) } α N α } B and he crculan marx here B s dagonalzed by he Fourer Transform. I s easy o see ha repeaed applcaon of he updae rule 5) wll yeld asympocally he vecor N [,,...,], [x, x 2,...,x N ],.. where V, W s he scalar produc of vecors. Here he lnear analyss was applcable whle n he general case s que dffcul o say wha wll be he exac dynamcs of he se of pons. However, f he se of pons would comprse wo subses of values, one clusered around and he oher around 0 0 hen he correspondng non-lnear dynamcs wll have he wo clusers convergng o he wo cenrods averages) snce pons from he oher se wll be deweghed by abou. In fac hs would precsely be he +0 0 behavor f we choose he wegh funcon wx, y) o be wx, y) = χdx, y) > Th })β, where dx, y) s he dsance beween x and y. Here χ s he ndcaor funcon for he predcae n he curly brackes and he hreshold Th can be chosen as 0,000. Therefore, we have seen ha we can devse an averagng procedure va he wegh or dsance +dx,x j ) γ funcons ha wll accomplsh he followng: If he daa appears n well-defned separaed) clusers of pons wh dsances beween clusers of Th or more, he resul of erave applcaon of he smoohng process wll be a se of cenrods ha are ypcal values n fac he averages) of each cluser. The same ype of effec s acheved whou he need of a hreshold parameer wh he weghs w j x, x j ) = β, for γ some posve consan. Ths s very nce snce we have a daa-adapve smoohng process ha provdes an effecve cluserng of he daa pons no a varable sze se of ypcal values. See Fg Averagng Vecors Values n IR d ) Gven N pons n IR d we consruc a smoohng process ha moves each pon owards he cenrod of s cluser see Fg. 2). Le X =, 2,...,N be N vecors where X = x, x 2,...,x d )T. We average each pon n s local neghborhood, gvng only small wegh o far away pons ha may belong o a dfferen cluser. Specfcally we choose X new = α)x old + j= j w X, X j )X old j. 6) Ths smoohng process acs on each componen as follows x a ) new = α) x a ) old + j= j w X, X j ) x a ) old j a =,...,d. 7) Ths averagng process s smlar, for each componen, o he scalar process Eq. 2) wh he noable dfference ha he weghs depend on he dsance beween he pons n IR d. Ths means ha he averages of he

4 98 Sochen, Kmmel and Brucksen Fgure 2. Random vecors pons n IR 2 ) averagng process wh γ = 0.5,.2, 2 lef o rgh). componens are no ndependen. We ake for example he Eucldean dsance beween pons: dx, X j ) = d ) x a 2, x a j a= and choose a wegh w as a funcon of hs dsance wh w X, X j ) = β = α j= j β + dx, X j ) γ j + dx, X j ) γ Such a choce wll make nearby pons more nfluenal on he dsplacemen of x. The dsance ha nvolve all he componens of a pon provdes he couplng n Eq. 7) beween he averagng processes of he dfferen componens. See Fg Averagng Pons on a Crcle Values n S ) Suppose nex ha we have a se of values on he un crcle S, encoded say as complex numbers va e θ j } where θ j [0, 2π). We would lke o devse a process of averagng over he values on S ha can effecvely summarze for us he nal se of values. I s clear ha we canno smply average he complex numbers drecly. Such an operaon wll mos lkely resul n a. value no on he un crcle. An alernave s o average he angles θ j } as numbers beween [0, 2π], however, we mmedaely realze ha θ = ɛ and θ 2 = 2π ɛ wll yeld an average of π, far away from boh ɛ, and 2π ɛ on he crcle. The proper averagng n hs case should obvously be θ av = 0, and ceranly no π. The problem here arses from he jump n our mappng of S no he represenaon space of he angles. Clearly he average of θ and θ 2 should be he angle ha bsecs he shores arc ha connecs θ o θ 2. Suppose we have N pons on he un crcle and we wsh o devse a process of averagng ha wll yeld a beer represenaon of hese pons e.g. va cluserng hem). We shall map hese pons P θ ), P 2 θ 2 ) P N θ N ) no mulple mages on a real lne see Fg. 3) exhbng he fac ha P j corresponds o θ j +k 2π}, k = 0, ±, ±2... Hence, we have a perodc confguraon of pons on R ha exhbs maps) he sngle angle parameer. Now, he angular) dsance beween wo pons on IR can be smply read as a dsance on IR : dp, P 2 ) beng defned as he closes dsance on IR beween he mulple represenaons of P and P 2. Ths corresponds o always measurng he dsance on S by he smalles arc connecng P and P 2 as s naural). The averagng smoohng) process for hese pons can now be defned as follows: For all P look a he pons Fgure 3. Averagng S values.

5 Dffusons and Confusons 99 n s symmerc 2π-neghborhood,.e. consder he nerval P π, P + π ). Then consder all pons n hs neghborhood as Q j s. Now compue P nex = α)p + α j wdp, Q j ))Q j Clearly hs process wll move P a b oward he weghed average of he nfluenal pars Q j }, wh he nfluence funcon defned o depend on he dsance beween P and Q j. I s clear from he above defned process ha perodcy s here preserved, hence we shall always ge a correc represenaon of N pons on a crcle, and he varous possbles for choosng nfluence funcons w) wll lead o a varey of ypes of cluserng processes. I s also clear ha, alhough he averagng process defned above works on nfnely many represenaons of each pon, s praccal mplemenaon can easly be done by an updang process of only N pons. I was smply a concepual advanage o look a he problem on IR raher han work drecly on S. Noe ha he dsance defned hs way s he geodesc dsance on he crcle Adapvely Smoohng a Dscree Tme Seres Suppose ha we are gven a dscree me seres X = x, x 2,...,x N ) T. Here x = x ) and hs fac makes hs seres dfferen from he seres of scalars we dscussed above. Ses of hs ype appear ofen n almos every feld of research rangng from Engneerng and Physcs hrough Bology and Psychology o Economy and Socology. Our adapve averagng s based, as above, on he dsance funcon dx, x j ): x n+ = α)x n + j w d x n, x n j )) x n j, where he superscrp n s he eraons ndex. In order o go from hs prncple o a specfc algorhm we have o specfy he smoohng funcon w and more mporan we have o decde wha s he approprae dsance beween generc wo measuremens x and x j. We may choose a smoohng funcon w x, x j ) = β + dx, x j ) γ j. 8) where γ defnes he degree of smoohng waned. The dsance dx, x j ) could be j or could be α j +β x x j or could even be dsance measured on he sgnal curve. For wo adjacen measuremens x and x + he dsance on he sgnal s d 2 x, x + ) = x + x 2. Noe ha he ndex ndcaes ha hs dsance s beween wo adjacen pons. The dsance beween generc pons s he sum of he dsances beween he pars of adjacen pons ha connec he gven generc pons. Assumng, wh no loss of generaly, ha j > reads j dx, x j ) = d k x k, x k+ ). k= Ths expresson s smply an approxmaon of he lengh of he curve whch connecs x and x j Adapvely Smoohng an Image An mage s a wo-dmensonal analog of he dscree me seres. Denoe he gray value a a pxel I j = I x, y j ) and we assume equal spacng n he x and y drecons. Denosng he mage wll clearly necessae a smoohng process. The specal naure of mages demands he average o be aken over pxels ha resemble he cener pxel I j. Ths average should be aken over projecons of he same objec and be ndependen of pxels ha descrbe dfferen objecs or ha are smply n a regon far from he pxel of neres. The smoohng sep can herefore be aken as I n+ j where, agan = α)ij n + w j d I n j, Ikl)) n I n kl. kl j w j I j, I kl ) = β + di j, I kl ) j) kl). 9) However, here we represen he mage as a wodmensonal surface whch s he dscrezed graph of I x, y). The dsance s measured on he mage surface as follows. For any pxel he dsance o pxels n he closes neghborhood,.e. 3 3 sencl, s calculaed as d 2 near I j, I kl ) = k 2 + j l 2 + I j, I kl 2.

6 200 Sochen, Kmmel and Brucksen For furher away pons he dsance s defned as sum of dsances of near neghbors along he shores pah on he manfold ha connecs he wo pons. Recenly, he fas marchng mehod on curved domans algorhm was developed n [3] can compue hese geodesc dsances very effecvely. See, Fg. 4 for adapve smoohng wh varous γ values. 3. Shor Tme Kernel for Non-Lnear Dfferenal Averagng The soluon of he lnear dffuson equaon s gven as a convoluon wh a Gaussan. The Gaussan funcon wh a me dependen varance s called he kernel of he dfferenal equaon. For a non-lnear dffuson he suaon s dfferen and a global n me) kernel does no exs. If, however, we are neresed n he evoluon over a shor me span, more can be sad. A kernel can be consruced for he shor me evoluon of he non-lnear dffuson equaon [9]. We demonsrae hs and clarfy he deas ha le a he bass of any of he geomercal evoluon equaons n he res of hs secon. 3.. One-Dmensonal Sgnal Le us frs consder a one dmensonal sgnal Y p). The sgnal can be hough of as a curve embedded n IR 2, as X p), X 2 p)) = p, Y p)) see Fg. 5. The equaon we analyze s he geomerc hea equaon Y p) = Y ss p), 0) where s s an arclengh defned n erms of a general merc by he relaon ds 2 = gp) dp 2, and for any funcon f we denoe by f s he dervave f/ s = gp) f/ p. Ths s he graden descen equaon for he Polyakov acon S[Y, g] = dp gg X ) 2 ) p + X 2 2 ) p = ds Xs 2 + Y s 2 ). If we furher assume ha he merc on he curve s nduced from he amben Eucldean IR 2 space hen gp) = + Yp 2 and gs) = X s 2 + Y s 2 = from he defnon of he arclengh s. The Polyakov funconal has a very smple geomerc meanng n hs case: S[Y ] = ds Xs 2 + Y s 2 ) = ds = lengh. Noe however ha ha oher mercs are possble, e.g. he L γ -norm gp) = ) X γ ) Y γ ) 2 γ +. p p Noe ha hese funcons are mercs snce hey ransform properly under a general reparameerzaon p p p). In our canoncal coordnae sysem p, Y p)) he merc adm he form gp) = ) Y γ ) 2 γ +. p Use of hese mercs for he defnon of a dsance leads, upon dscrezaon, o he smoohng funcon w) of he prevous secon. From an axomac pon of vew we may also wsh o consruc a curve flow whch s nvaran under dfferen groups of ransformaons. Invarance under he Eucldean group ha ncludes ranslaons and roaons leads o he Eucldean nduced merc, and resuls n a curvaure flow projeced on he Y axs. Ths projecon acon preserves edges longer along he flow, enhancng hs way he boundary beween dfferen regons n he mage as wll be seen below. Dfferen requremens such as nvarance under dfferen groups of ransformaons) lead o a dfferen form for he merc and o a dfferen flow. We analyze all hese possbles a he same me by no specfyng any parcular form of he merc and leavng as a free parameer of he framework. The flow s he one-dmensonal analog of he Belram flow Y = g p gg p Y ) g Y ) where g s he merc on he curve. Anoher way o wre hs equaon s Y = D p p Y = p + A )g p Y = g 2 p Y + A py ) 2)

7 Dffusons and Confusons 20 Fgure 4. Image adapve averagng wh γ = 0,,.5, 2 op down), wh a cross-secon ha shows he edge preservng propery.

8 202 Sochen, Kmmel and Brucksen Fgure 5. The sgnal as a curve. where D p = p + A s he covaran dervave, A = 2 g p g s he connecon and A = A g p g = 2 g p g. Ths equaon can descrbe a varey of curve evoluon dynamcs upon dfferen choces of he merc g. In more complcaed suaons we encouner he same form of Eq. 2) where he connecon A depends on he merc of he embeddng space as well. In flows wh g dependng on p Y Eq. 2) s non-lnear, and prevens he exsence of kernels for long me nervals. If however g hen Eq. ) becomes he usual hea equaon wh he well known Gaussan smoohng kernel. Upon dscrezaon Y p) Y hs equaon assumes he followng form Y n+ = j where n s he eraon ndex and W n s a marx whose enres depend on he values of Y n and p Y n. For one eraon only we can hnk abou he updae rule as f W n s a fxed funcon of p descrbng he underlyng curve whch s fxed durng hs one shor me updae. Afer Y s updaed he merc s updaed and hen agan s fxed durng he nex updae of Y. The couplng beween he merc or connecon), ha descrbes he geomery, and he feaure of neres Y, prevens a global fundamenal soluon o exs, ye we can fnd such soluon for each me sep n whch he geomery s fxed. In order o derve he shor me kernel of hese equaons we use he followng ansaz, known n physcs as he WKB approxmaon, Y p, + ɛ) = W n j Y n j K p, p ; ɛ)y p, ) dp, 3) where he kernel s assumed o be of he form K p, p ; ) = Hp, p, ) exp p, }. 4) We ake, whou loss of generaly, Hp, p, ) = consan, see Appendx A for he deals. Ths form s a generalzaon of he Gaussan kernel soluon of he lnear dffuson equaon. The funcon depends on he dffuson ensor only whle H depends on he deals of all he erms of he equaon. The valdy of hs approxmaon procedure can be found n [9] for example. Noe ha upon dscrezaon of he curve p = ah, where h = L/N, L s he lengh of he curve, and N s he number of segmens, hs equaon akes he form of a sysem of lnear equaons X new a = b= K ab X old b. 5) Ths has he same form as he one descrbed n he prevous secon when we dscussed averagng sequences of scalars and vecors. Inserng Eq. 3) n Eq. ) we see ha he kernel K p, p ; ) as a funcon of p sasfes he same equaon as X. As a power seres n we ge p, p )) ) + O K p, p ; ) 2 = 2 g p) p p, p )) 2 +O )) K p, p ; ). 6) For shor mes only he mos sngular par s domnan, namely p) = g p) p p)) 2, 7) where by abuse of noaons p) s a shorhand for p, p ), and wh boundary condon p ) = 0. 8) Ths equaon can be rewren as an algebrac dfferenal sysem of equaons Fp,, Z) = Z 2 gp) = 0 Z = p. The algebrac equaon s solved o yeld 9) Z = g. 20)

9 Dffusons and Confusons 203 Fgure 6. Gaussan on he curve. The resulng dfferenal equaon s solved by separaon of varables ha yelds he soluon d = gp) dp, 2) ) p) = p 2 gd p = p 2 ds). 22) 4 4 p Ths equaon has a smple nerpreaon. The kernel s a Gaussan on he curve,.e. he convoluon s performed on he sgnal see Fg. 6). Dsances from a pon should be measured on he sgnal self and no on he grd ha happens o be used n order o descrbe. Ths way, pons on one sde of a sgnfcan jump are farher away from he pons on he oher sde of he jump and herefore have small nfluence on he average here. Ths explans he edge preservng naure of he geomerc hea equaon and he fler ha emerges ou of. In Fg. 7 a one dmensonal edge s convolved wh a Gaussan kernel once wh suppor along he x axs, and once, nrnscally defned,.e., wh a suppor of he sgnal self. The varance n boh cases was defned o yeld smlar resuls along he fla areas. We see ha he kernel defned on he sgnal beer preserves he bmodal naure of he daa One-Dmensonal Curve Embedded n IR n Equaon ) can be easly generalzed o an arbrary one-dmensonal curve embedded n IR n by smply applyng he geomerc dffuson equaon componenwse: X = g p gg p X ) g X p =, 2,...,n. 23) Fgure 7. Smoohng a nosy sgnal green sold lne) once wh a Gaussan along he p axs red dashed lne) and wh he kernel defned on he sgnal self blue doed lne). For a beer vew see hp:// ron/pub.hml Ths equaon can descrbe a varey of curve evoluon dynamcs upon dfferen choces of he merc g. Noe ha he merc nvolves all componens leadng o a sysem of coupled equaons. In order o derve he shor me kernel of hese equaons we use he followng ansaz X p, + ɛ) = K p, p ; ɛ)x p, ) dp 24) where he kernel form, lke n Eq. 4) s gven by kp, p ; ) = C exp p, }. 25) Now, C s a consan and s he same as n he prevous subsecon,.e., p) = 4 p p 2 ) g dp ) = p 2 ds 4 p = 4 sp, 2. 26) sp, p ) s he dsance on he curve beween pon p and he pon p Affne Invaran Averagng Once we have a formulaon for he kernel n erms of an arc-lengh hrough he relaon ds 2 = g dp 2, we

10 204 Sochen, Kmmel and Brucksen can envson he consrucon of group nvaran kernels by mposng nvarance properes on he merc. The smples example s he specal affne nvaran merc of a curve,.e. sp 0, p ) = See e.g. [, 5, 9]. p 3.4. Relaon o Oher Works p 0 X p, X pp /3 dp. 27) The Blaeral Fler. The shor me kernel can be approxmaed n a small wndow by approxmang he negral p0 +dp p 0 g dp = gp) dp. We can use he Eucldean nduced merc,.e., Then, ) f 2 g = +. p 4 = gp) dp) 2 = = dp 2 + df 2, + ) ) f 2 dp 2 p whch s exacly he blaeral fler ha was recenly proposed by Tomas and Manduch, [2, 0, 23] The TV Dgal Fler. Recenly Chan, Osher and Shen [7] analyzed non-lnear fler of he followng form U new = h αβ Uβ old, β α where he sum s over neghbors of α ncludng α self). The weghs h αβ are of he form w αβ h αβ = λ + γ α w αγ for α β. The dagonal wegh s λ h αα = λ + γ α w. αγ These flers are consruced by drec dscrezaon of he dfferenal operaor ha resuls n a graden descen mnmzaon of he TV funconal. In pracce s a regularzed funconal ha s mnmzed leadng o a drec relaon o he Belram flow whch s based on he Eucldean nduced merc see [2] for deals). Wrng he dscrezed dfferenal operaor as a marx whose enres depend on he sgnal appear also n [24, 25]. The shor me approach s dfferen snce we analyze and evenually dscreze he soluon of he dfferenal equaon. I s also more general snce we rea a varey of flows by no specfyng he explc form of he merc a he ouse. In hs way we have a clear nuve undersandng of he adapve averagng as a Gaussan wegh funcon on he manfold ha s defned by he daa. The flows dffer n he geomery arbued o he manfold hrough dfferen choces of he merc Relaon o Projecve Averagng. In [4] an adapve non-lnear and projecve nvaran averagng process was nroduced. I was demonsraed ha s equvalen o a projecve nvaran PDE ha generaes he curve flow. 4. Averagng Consraned Feaures One ofen encouner, n some sgnal and mage processng asks, a more complex feaure-space sgnal. Examples of such feaures are color, exure, curvaure, dervave vecor feld, orenaon vecor feld ec. The feaure space may have non-rval geomery represened by a merc [8, 2, 4, 20, 22]. Regularzaon or smoohng of he feaure space s frequenly done wh a non-lnear dffuson sysem of equaons. In hs secon we derve he shor me kernel for he dffuson of an orenaon defned over a one-dmensonal curve. 4.. Orenaon Feld over a Sgnal In he case of averagng an orenaon feld, we defne he orenaon a each pon p, Y p)) as he angle beween he feaure vecor and he p axs. By defnon, he vecor feld has a un magnude and hs propery should be preserved durng he smoohng flow. The vecor feld s a map f : C S, and he geomerc smoohng flow may here oo be derved as a graden descen flow from he Polyakov acon defned as f p) = g f p) + Ɣ p f ) 2 g, 28)

11 Dffusons and Confusons 205 see [4], where Ɣ = h f h, 29) and h s he merc on S. Thus, for example, f he componens of he orenaon vecor feld are f and e such ha f 2 + e 2 =, hen he merc on he pach ha s descrbed by f s calculaed as follows e ds 2 = df 2 + de 2 = df 2 + f df = + f 2 ) df 2 = f 2 From he defnon we ge ds 2 = hdf 2, h = f 2. ) 2 f 2 df2. The calculaon of Ɣ s sraghforward and gves Ɣ = h f h = f 2 ) f f 2 ) f = f = fh. 2 Le us wre Eq. 28) n a more suggesve form: f = p + ) 2 g p g + Ɣ p f g p f. 30) We smplfy hs expresson by nong ha Fnally we ge Ɣ p f = 2 h f h p f = 2 h p h. f = p + A )g p f = g p + A) p f where A = 2 g p g + 2 h p h and A = 2 g p g + 2 h p h. Repeang he analyss of he prevous subsecons we derve he soluon of he form K p, p ; ) = Hp, p, ) exp p, }. 3) and develop hp, p ; ) n a Taylor seres Hp, p ; ) = n H n p, p ). n=0 The leadng erm s he same as n he non-consraned dffuson and herefore p) = 4 p Here g s he nduced merc p 2 gdp ) = p 2 ds). 32) 4 p g = + Y p ) 2 + h p f ) 2 and whle n he prevous case H = consan here he suaon s dfferen and a careful analyss see Appendx) gves H = Ch /4 + O) 33) where C s a consan and h s he merc on S Orenaon Feld over a Curve Embedded n IR n The above analyss apples o a consraned vecor feld defned over a curve embedded n IR n. The map s f : C S and he PDE assocaed wh s f p) = g f p) + Ɣ p f ) 2 g 34) The only dfference s n he srucure of he nner merc g. g = = X p ) 2 + h p f ) 2 ) 2 ) X 2 f p + f 2 X X p. The shor me kernel s he same as n he above subsecon. I s mporan o undersand ha hese equaons are of he same form for any coordnae sysem we choose up o ransformaon of coordnaes of course). We need a leas wo chars o cover S bu we can also adop he deas developed n Secon 2 and consruc for each pon he coordnae sysem where he orenaon s he furhes pon from he sngulary.

12 206 Sochen, Kmmel and Brucksen Fgure 8. In each couple he lef s he orgnal mage, whle he rgh s he flered one processed wh he Belram approxmaed by he blaeral fler. The convoluon kernel s e ds2 = e dx2 +dy di 2 )/6, normalzed, wh wndow sze of Image Denosng Resuls The shor me kernel for mages s concepually a sraforward generalzaon of he D case. I s gven as a weghed Gaussan on he mage manfold see [20] for deales): K p, p ) = C e dp, 2 + O 0 ). Here p and p are wo pons on he mage manfold and dp, p ) s he dsance beween hem.e. he lengh of he shores geodesc on he manfold beween hese pons. The resuls for small wndow are gven below n Fg Summary and Conclusons We brefly explored here he relaon beween PDE based flers, classcal sgnal processng lnear flers, and non-lnear flers. I was shown ha he shor me kernels of he Belram flow may be consdered as approxmaons for known lnear and non-lnear flers. The dscusson covers Gaussan flers, he non-lnear blaeral flers, and furher non-rval robus and daaadapve flers. Our approach yelds a unfed and comprehensve vew on he relaon beween hese seemngly unrelaed se of ools. Appendx We analyze n hs appendx he me srucure of a general WKB approxmaon. The Kernel obeys he equaon K = g p + A) p K 35) and we defne a WKB kernel n he form K p, p ; ) = Hp, p, ) exp p, }. 36)

13 Dffusons and Confusons 207 and develop Hp, p ; ) n a Taylor seres Hp, p ; ) = n H n p, p ) n=0 Inserng hs ansaz n he dffuson Eq. 35) we ge on one hand K exp p, } = 2 Hp, p, ) + n=0 exp n ) n 3 2 Hn p, p ) 2 p, }. On he oher hand we fnd p K = Hp, p ; ) p exp p, } 3/2 + p H exp p, } /2 and 2 p K = Hp, p ; ) 2 p 3/2 2 p H) p ) 3/2 + Hp, p ; ) p ) 2 5/2 + 2 p Hp, p ; ) /2 exp p, } exp p, } exp p, exp p, }. By comparson of he coeffcens of he power seres we ge from he leadng erm and from he second erm H 0 = g p ) 2 H 0 H 2 H 0 = g H p ) 2 2 p ) p H 0 ) h 0 p + A) p ). 37) The coeffcen of H s he equaon for he leadng erm and hus assumed o be sasfed. For he res, we know from he soluon of he frs equaon ha p = g } and 2 g) + g p = 2 g = ψ 2 g pg + g 2. Usng hese denes we ge = 2 ψ g pg + g g 2 g g 2 H 0 = H 0 2 p + A p ) + 2 p ) p H 0 ) ) = H 0 2 g pg + 2 g + A g + 2 g p H 0 ) from whch we fnally oban p H g p g + ) 2 A H 0 = 0. The form of A when he feaure space s no consraned s A = g p g = 2 g p g. We conclude ha n hs case H 0 = consan. In he consraned case A = 2 g p g + 2 h p h. The coeffcen H 0 s n hs case H 0 = Ch /4 where C s a consan of negraon. Acknowledgmens Smulang dscussons wh Mchael Elad of Ne2Wreless, Doron Shaked and Danny Barash of HP Labs. on blaeral flers and he Belram flow, are graefully acknowledged. We hank Zeev Schuss for dscussons and for ponng us o very valuable references. References. L. Alvarez, F. Guchard, P.L. Lons, and J.M. Morel, Axoms and fundamenal equaons of mage processng, Arch. Raonal Mechancs, Vol. 23, 993.

14 208 Sochen, Kmmel and Brucksen 2. D. Barash, Blaeral flerng and ansoropc dffuson: owards a unfed vewpon, HP Labs., Techncal Repor HPL , M. Black, G. Sapro, D. Marmon, and D. Heeger, Robus ansoropc dffuson, IEEE Trans. on Image Processng, Vol. 7, No. 3, pp , A.M. Brucksen and D. Shaked, On projecve nvaran smoohng and evoluons of planar curves and polygons, Journal of Mahemacal Imagng and Vson, Vol. 7, pp , E. Calab, P.J. Olver, C. Shakban, A. Tannenbaum, and S. Haker, Dfferenal and numercally nvaran sgnaure curves appled o objec recognon, In. J. Compuer Vson, Vol. 26, pp , D. Comancu, and P. Meer, Mean shf analyss and applcaons, n Proc. of he 7h IEEE In. Conf. on Compuer Vson CA, USA, 999, Vol. 2, pp F.C. Chan, S. Osher, and J. Shen, The dgal TV fler and nonlnear denosng, UCLA-Techncal Repor, T. Chan and J. Shen, Varaonal resoraon of non-fla mage feaures: Models and algorhms, Techncal Repor, Mah- UCLA, J.K. Cohen, F.G. Hagn, and J.B. Keller, Shor me asympoc expansons of soluons of parabolc equaons, Journal of Mahemacal Analyss and Applcaons, Vol. 38, pp. 82 9, M. Elad and D. Shaked, Personal communcaon. In HP Labs Israel, D. Geman and G. Reynolds, IEEE Trans. on PAMI, Vol. 4, pp , R. Kmmel, R. Mallad, and N. Sochen, Images as embedded maps and mnmal surfaces: Moves, color, exure, and volumerc medcal mages, Inernaonal Journal of Compuer Vson, Vol. 39, No. 2, pp. 29, R. Kmmel and J.A. Sehan, Compung geodesc pahs on manfolds, Proceedngs of Naonal Academy of Scences, USA, Vol. 95, No. 5, pp , R. Kmmel and N. Sochen, Orenaon dffuson or how o comb a porcupne?, Journal of Vsual Communcaon and Image Represenaon, D. Mumford and J. Shah, Boundary deecon by mnmzng funconals, n Proc. of CVPR, Compuer Vson and Paern Recognon, San Francsco, P. Perona, Orenaon dffusons, IEEE Trans. on Image Processng, Vol. 7, No. 3, pp , P. Perona and J. Malk, Scale-space and edge deecon usng ansoropc dffuson, IEEE-PAMI, Vol. 2, pp , L. Rudn, S. Osher, and E. Faem, Nonlnear oal varaon based nose removal algorhms, Physca D, Vol. 60, pp , S. Sapro and A. Tannenbaum, Affne nvaran scale space, In. Journal of Compuer Vson, Vol., No., pp , N. Sochen, Sochasc processes n vson, I: From Langevn o Belram, CC Pub #285 June 999, Technon, Israel. 2. N. Sochen, R. Kmmel, and R. Mallad, A geomercal framework for low level vson, IEEE Trans. on Image Processng, Vol. 7, No. 3, pp , B. Tang, G. Sapro, and V. Caselles, Drecon dffuson, n Inernaonal Conference on Compuer Vson, C. Tomas and R. Manduch, Blaeral flerng for gray and color mages, n Proc. of he IEEE Inernaonal Conference on Compuer Vson, 998, pp R. Vogel, and M.E. Oman, Ierave mehods for oal varaon denosng, SIAM. J. Sc. Sas. Compu., Vol. 7, No., pp , J. Wecker, Ansoropc Dffuson n Image Processng, Teubner: Sugar, 998. Nr Sochen, fnshed hs B.Sc. sudes n Physcs 986 and hs M.Sc. sudes n heorecal physcs 988 boh n he Unversy of Tel-Avv. Receved hs PhD n Theorecal physcs 992 from he Unversé de Pars-Sud whle conducng hs research n he Servce de Physque Théorque a he Cenre d Eude Nucleare a Saclay, France. He connued wh a one year research n he Ecole Normale Supereure n Pars on he Haue Eude Scenfque fellowshp, and a hree years NSF fellowshp n he physcs deparmen of he Unversy of Calforna a Berkeley. I s n Berkeley ha hs neress shfed from quanum feld heores and negrable models, relaed o hgh-energy physcs and srng heory, o compuer vson and mage processng. He spen one year n he Physcs Deparmen a he Unversy of Tel-Avv and wo years n he Faculy of Elecrcal Engneerng n he Technon-Israel Insue of Technology. Currenly he s a senor lecurer n he Deparmen of Appled Mahemacs, Unversy of Tel-Avv. He s also a member of he Ollendorf Cener n he Technon. Hs man research neress are he applcaons of dfferenal geomery and sascal physcs n mage processng and compuaonal vson. Ron Kmmel receved hs B.Sc. wh honors) n compuer engneerng n 986, he M.S. degree n 993 n elecrcal engneerng, and he D.Sc. degree n 995 from he Technon Israel Insue of Technology. Durng he years he served as an R&D offcer n he Israel Ar force. Durng he years he has been a posdocoral fellow a Lawrence Berkeley Naonal Laboraory, and he Mahemacs

15 Dffusons and Confusons 209 Deparmen, Unversy of Calforna, Berkeley. Snce 998, he has been a faculy member of he Compuer Scence Deparmen a he Technon, Israel. Hs research neress are n compuaonal mehods and her applcaons ncludng opcs n: Dfferenal geomery, numercal analyss, non-lnear mage processng, geomerc mehods n compuer vson, and numercal geomery mehods n compuer aded desgn, roboc navgaon, and compuer graphcs. Dr. Kmmel was awarded he Alon Fellowshp, he HTI Posdocoral Fellowshp, and he Wolf, Guwrh, Ollendorff, and Jury fellowshps. He has been a consulan of HP research Lab n mage processng and analyss durng he years Snce 2000, he s a long erm consulan of Jgam research group. Technon Israel Insue of Technology and hs Ph.D. n Elecrcal Engneerng from Sanford Unversy, Sanford, CA, USA. He was he recpen of he Hershel-Rch Innovaon award wh Y. Pnuel) for he DgDURER, a dgal engravng sysem, 993. The paper Why he Ans-Trals Look so Sragh and Nce seleced as one of he op sores n mahemacs for 993 by Dscover Magazne, and revewed n Scence and The New Scens. Hs oher awards have ncluded he Paern Recognon Socey Ousandng Conrbuon Honorable Menon wh Y. Eldar and N. Krya), 992 and The Henry Taub award n Compuer Scence, 990. He s currenly a Faculy member n he Deparmen of Compuer Scence a Technon Israel Insue of Technology. He prevously held faculy posons n Elecrcal Engneerng a Technon and was a Vsng Professor a Sanford Unversy n He has been a permanen vsor a Bell Laboraores, almos every year snce 987. Alfred M. Brucksen holds The Technon Ollendorff Char n Scence. He receved hs B.S. and M.S. n Elecrcal Engneerng from