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1 Te Varable Badwdt Mea Sft ad Data-Drve Scale Selecto Dor Comacu Vsvaata Rames Peter Meer Ima & Vsualzato Departmet Electrcal & Computer Eeer Departmet Semes Corporate Researc Ruters Uversty 755 Collee Road East, Prceto, NJ Brett Road, Pscataway, NJ 8855 Abstract We presettwo solutos for te scale selecto problem computer vso. Te rst oe s completely oparametrc ad s based o te te adaptve estmato of te ormalzed desty radet. Employ te sample pot estmator, we dee te Varable Badwdt Mea Sft, prove ts coverece, ad sow ts superorty over te ed badwdt procedure. Te secod tecque as a semparametrc ature ad mposes a local structure o te data to etract relable scale formato. Te local scale of te uderly desty s take as te badwdt wc mamzes te matude of te ormalzed mea sft vector. Bot estmators provde practcal tools for autoomous mae ad quas real-tme vdeo aalyss ad several eamples are sow to llustrate ter eectveess. Motvato for Varable Badwdt Te ecacy of Mea Sft aalyss as bee demostrated computer vso problems suc as track ad semetato [5, 6]. However, oe of te lmtatos of te mea sft procedure as deed tese papers s tat t volves te speccato of a scale parameter. Wle results obtaed appear satsfactory, we te local caracterstcs of te feature space ders scatly across data, t s dcult to d a optmal lobal badwdt for te mea sft procedure. I ts paper we address te ssue of locally adapt te badwdt. We also study a alteratve approac for data-drve scale selecto wcmposes a local structure o te data. Te proposed solutos are tested te framework of quas real-tme vdeo aalyss. We revew rst te trsc lmtatos of te ed badwdt desty estmato metods. Te, two of te most popular varable badwdt estmators, te balloo ad te sample pot, aretroduced ad ter advataes dscussed. We coclude te secto by sow tat, wt some precautos, te performace of te sample pot estmator s superor to bot ed badwdt ad balloo estmators.. Fed Badwdt Desty Estmato Te multvarate ed badwdt kerel desty estmate s deed by ^f() = d K ; : () were te d-dmesoal vectors f ::: represet a radom sample from some ukow desty f ad te kerel, K, s take to be a radally symmetrc, oeatve fucto cetered at zero ad terat to oe. Te termoloy ed badwdts due to te fact tat s eld costat across R d. As a result, te ed badwdt procedure () estmates te desty at eac pot by tak te averae of detcally scaled kerels cetered at eac of te data pots. For potwse estmato, te classcal measure of te closeess of te estmator ^f to ts taret value f s te mea squared error (MSE), equal to te sum of te varace ad squared bas MSE() = E = Var ^f() ; f() ^f() + Bas ^f() : () Us te multvarate form of te Taylor teorem, te bas ad te varace are appromated by [, p.97] ad Bas() (K)f() (3) Var() ; ;d R(K)f() (4) were (K) = R z K(z)dz ad R(K) =R K(z)dz are kerel depedet costats, z s te rst compoet of te vector z, ad s te Laplace operator. Te tradeo of bas versus varace ca be observed (3) ad (4). Te bas s proportoal to, wc meas tat smaller badwdts ve a less based estmator. However, decreas mples a crease te varace wcs proportoal to ; ;d. Tus for a ed badwdt estmator we sould coose tat aceves a optmal compromse betwee te bas ad varace over all R d,.e., mmzes te mea terated squared error (MISE) MISE() =E Z ^f() ; f() d : (5) Neverteless, te result badwdt formula (see [7, p.85], [, p.98]) s of lttle practcal use, sce t depeds o te Laplaca of te ukow desty be estmated. Te best of te curretly avalable data-drvemetods for badwdt selecto seems to be te plu- rule [5], wc was prove to be superor to least squares cross valdato ad based cross-valdato [], [6, / $. (C) IEEE

2 p.46]. A practcal oe dmesoal alortm based o ts metod s descrbed te Apped. For te multvarate case, see [, p.8]. Note tat tese data-drve badwdt selectors work well for multmodal data, ter oly assumpto be a certa smootess te uderly desty. However, te ed badwdt aects te estmato performace, by udersmoot te tals ad oversmoot te peaks of te desty. Te performace also decreases we te data ebts local scale varatos.. Balloo ad Sample Pot Estmators Accord to epresso (), te badwdt ca be vared two ways. Frst, by select a deret badwdt = () for eac estmato pot, oe ca dee te balloo desty estmator ^f () = () d K ; () : (6) I ts case, te estmate of f at s te averae of detcally scaled kerels cetered at eac datapot. Secod, by select a deret badwdt = ( ) for eac data pot we obta te sample pot desty estmator ^f () = ; ( ) K d ( ) : (7) for wc te estmate of f at s te averae of deretly scaled kerels cetered at eac data pot. Wle te balloo estmator as more tutve appeal, ts performace mprovemet over te ed badwdt estmator s scat. We te badwdt () scose as a fucto of te k-t earest ebor, te bas ad varace are stll proportoal to ad ; ;d, respectvely [8]. I addto, te balloo estmators usually fal to terate to oe. Te sample pot estmators, o te oter ad, are temselves destes, be o-eatve adterat- to oe. Ter most attractve property s tat a partcular coce of ( ) reduces cosderably te bas. Ideed, we ( )stake to be recprocal to te square root of f( ) = ( )= (8) f( ) te bas becomes proportoal to 4, wle te varace remas ucaed, proportoal to ; ;d [, 8]. I (8), represets a ed badwdt ad s a proportoalty costat. Sce f( ) s ukow t as to be estmated from te data. Te practcal approacsto use oe of te metods descrbed Secto. to d ad a tal estmate (called plot) off deoted by f. ~ Note tat by us f ~ stead of f (8), te ce propertes of te sample pot estmators (7) rema ucaed [8]. Varous autors [6, p.56], [7, p.] remarked tat te metod s sestve to te e detal of te plot estmate. Te oly provso tat sould be take s to boud te plot desty away from zero. Te al estmate (7) s owever ueced by te coce of te proportoalty costat, wc dvdes te rae of desty values to low ad destes. We te local desty s low,.e., f( ~ ) <, ( ) creases relatve to mply more smoot for te pot. For data pots tat verfy f( ~ ) >,te badwdt becomes arrower. A ood tal coce [7, p.] s to take as te eometrc mea of ~f( ) o :::. Our epermets ave sow tat for superor results, a certa deree of tu s requred for. Neverteless, te sample pot estmator proved to be almost all te tme muc better ta te ed badwdt estmator. Varable Badwdt Mea Sft We sow et tat start from te sample pot estmator (7) a adaptve estmator of te desty's ormalzed radet ca be deed. Te ew estmator, wc assocates to eac data pot a deretly scaled kerel, s te basc step for a teratve procedure tat we prove tocovere to a local mode of te uderly desty, we te kerel obeys some mld costrats. We called te ew procedure te Varable Badwdt Mea Sft. Due to ts ecellet statstcal propertes, we atcpate te etesve use of te adaptve estmator by vso applcatos tat requre mmal uma terveto.. Detos To smplfy otatos we proceed as [6] by troduc rst te prole of a kerel K as a fucto k : [ )! R suc tat K() = k(kk ). We also deote ( ) for all =:::. Te, te sample pot estmator (7) ca be wrtte as ^f K () =! k ; (9) d were te subscrpt K dcates tat te estmator s based o kerel K. A atural estmator of te radet off s te radet of ^f K () ^rf K () r ^f K () = = = " ; d+ d+ ; d+ ; ; k! ;!#! / $. (C) IEEE

3 6 4 were we deoted d+ d+ ; ; 3 ; 7 5 () () =;k () () ad assumed tat te dervatve of prole k ests for all [ ), ecept for a te set of pots. Te last bracket () represets te varable badwdt mea sft vector M v () d+ d+ ; ; ; () To see te scace of epresso (), we dee rst te kerel G as G() =C(kk ) (3) were C s a ormalzato costat tat forces G to terate to oe. Te, by employ (8), te term tat multples te mea sft vector () ca be wrtte as " were d+ ^f G () C ;!# = C ~ f( ) d " # ~ f( ) ; ~ f( ) ^f G () (4) (5) s oeatve ad terates to oe, represet a estmate of te desty of te data pots weted by te plot desty values f( ~ ). Fally, by us (), (), ad (4) t results tat M v () = P ; ~ ^rf K () f( ) =C ^f G () : (6) Equato (6) represets a eeralzato of equato (3) derved [6] for te ed badwdt mea sft. It sows tat te adaptve badwdt mea sft s a estmator of te ormalzed radet of te uderly desty. Te proportoalty costat, owever, depeds o te value of. We s creased, te orm of te mea sft vector also creases. O te oter ad, a small value for mples a small km v k. Due to ts eteral varablty of te mea sft orm, te coverece property ofa teratve procedure based o te varable badwdt mea sft s remarkable. Note also tat we s take equal to te artmetc mea of ~f( )o, te proportoalty costat becomes as ::: te ed badwdt case.. Propertes of te Adaptve Mea Sft Equato () sows a attractve beavor of te adaptve estmator. Te data pots ly lare desty reos aect a arrower eborood sce te kerel badwdt s smaller, but are ve a larer mportace, due to te wet = d+. By cotrast, te pots tat correspod to te tals of te uderly desty are smooted more ad receve a smaller wet. Te etreme pots (outlers) recevevery small wets, be tus automatcally dscarded. Recall tat te ed badwdt mea sft [5, 6] assocates te same kerel for eac datapot. Te most mportat property of te adaptve estmator s te coverece assocated wtts repettve computato. I oter words, f we dee te mea sft procedure recursvely as te evaluato of te mea sft vector M v () followed by te traslato of te kerel G by M v (), ts procedure leads to a statoary pot (zero radet) of te uderly desty. More speccally, we wll sow tat te pot ofcoverece represets a statoary pot of te sample pot estmator (9). Tus, te superor performace of te sample pot estmator traslates to superor performace for te adaptve mea sft. We deoteby y j te sequece of successve j= ::: locatos of te kerel G, were y j+ = d+ d+ y j ; y j ; j = ::: (7) s te weted mea at y j computed wt kerel G ad wets = d+,ady s te ceter of te tal kerel. Te desty estmates computed wt kerel K te pots (7) are o o ^f K = ^fk (j) ^fk (y j ) : (8) j= ::: j= ::: We sow Apped tat f te kerel K as a cove ad mootoc decreas prole ad te kerel G s deed accord to () ad (3), te sequeces (7) ad (8) are coveret. Ts meas tat te adaptve mea sft procedure talzed at a ve locato, coveres at a earby pot were te estmator (9) as zero radet. I addto, sce te modes of te desty are pots of zero radet, t results tat te coverece pot s a mode caddate. Te advatae of us te mea sft rater ta te drect computato of (9) followed by a searc for localmamastwofold. Frst, te overall computatoal complety of te mea sft s muc smaller ta tat of te drect metod. Te drect searc for mama requres a umber of desty fucto evaluatos tat creases epoetally wt te space dmeso. Secod, for may applcatos (see for eample [6]) weoly / $. (C) IEEE

4 eed to kow temode assocated wt a reduced set of data pots. I ts case, te mea sft procedure becomes a atural process tat follows te tral to te local mode. Te teratve procedure for mode detecto based o te varable badwdt mea sft s summarzed below. Varable Badwdt Mea Sft Alortm Gve te data pots f ::: :. Derve a ed badwdt ad a plot estmate ~f us te plu- rule (see Apped for te oe dmesoal plu- rule).. Compute lo = ; lo ~ f( ). 3. For eac datapot compute ts adaptve badwdt ( )= = ~ f( ) =. 4. Italze y wt te locato of terest ad compute teratvely (7) tll coverece. Te coverece pot s a pot of zero radet, ece, a mode caddate..3 Performace Comparso We compared te varable ad ed badwdt mea sft alortms for varous multmodal data sets tat ebted also scale varatos. Te ed badwdt procedure was ru wt a badwdt derved from te plu- rule ve Apped. Te plu- rule was developed for desty estmato [5] ad sce ere we are cocered wt desty radet estmato t s recommeded [, p.49] to use a larer badwdt to compesate for te eretly creased sestvty of te estmato process. We ave moded te plu- rule by alve te cotrbuto of te varace term. Ts cae was mataed for all te epermets preseted ts paper. Te costat of te adaptve procedure was kept as te eometrc mea of ~f( )o. ::: As oe ca see from Fures ad te ed badwdt mea sft resulted ood performace for te locatos were te local scale was te medum rae. However, te very arrow peaks were fused, wle te tals were broke to peces. O te oter ad, te adaptve alortm sowed superor performace, by coos a proper badwdt for eac data pot. 3 Semparametrc Scale Selecto 3. Motvato Te prevous two sectos followed purely oparametrc deas, sce o formal structure was assumed about te data. Imply oly a certa smootess of te uderly desty we used avalable alortms for Hstoram Value Number of Pots Number of Pots (a) (b) (c) Fure : A mture of data pots from eac N(5,), N(7,4), N(37,8), N(7,6), N(45,3). Te cotuous le s a scaled verso of te desty estmate. Te detected modes are marked proportoal to te umber of data pots tat covered to tem. (a) Hstoram of te data. (b) Varable Badwdt. (c) Fed Badwdt. scale selecto to derve a tal badwdt. Te crtero for badwdt selecto was a lobal measure (MISE), ece, aceved a optmal compromse betwee te terated squared bas ad te terated varace. Te, we moded ts badwdt for eac data pot, accord to te local desty. Te ma problem wt ts approacs tat for multdmesoal multmodal data, t s very dcult to determe te rt from te sample pots ad may of te practcal ssues are yet to be resolved [, p.8]. As a cosequece, most of practcal alortms use emprcal badwdt selecto rules tat are less depedet or eve depedet from te sample data. Ts mples a decrease ter performace we te put statstcs s ostatoary, as t appes most of / $. (C) IEEE

5 5 5 5 Hstoram Value Number of Pots Number of Pots (a) (b) (c) Fure : A mture of data pots from eac ep(3)+5, c(4)+5, loormal(,)+9, loormal(,)+9, 9-loormal(3,). Te cotuous le s a scaled verso of te desty estmate. Te detected modes are marked proportoal to te umber of data pots tat covered to tem. (a) Hstoram of te data. (b) Varable Badwdt. (c) Fed Badwdt. te tme vso tasks. 3. Normalzed Mea Sft Based Scale Selecto We propose ts secto a deret approac for badwdt selecto. Te dea s to mpose a local structure o te data by assum tat locally te uderly desty s spercal ormal wt ukow mea ad covarace matr = I. At a rst look, te task of d ad for eac data pot seems to be very dcult. To locally t a ormal to te multvarate data oe eeds a pror kowlede of te eborood sze wc te ukow parameters are to be estmated. If te estmato s performed for several eborood szes, a scale varat measure of te oodess of t s eeded. Fortuately, a smple soluto ests. It s based o te follow teorem, vald we te umber of avalable samples s lare. Teorem If te true desty f s ormal wt parameters ad = I, ad te ed badwdt mea sft s computed wt a spercal ormal kerel of badwdt, te, te badwdt ormalzed orm of te mea sft vector s mamzed we. Proof Recall tat te ed badwdt mea sft vector computed wt kerel G of badwdt ca be wrtte as M() = r ^fk () =C ^f G () : (9) Sce te true desty f s ormal wt covarace matr = I t follows tat te mea of ^fg (), E ^fg () ( + ) s also a ormal surface wt covarace ( + )I. Lkewse, by tak to accout () we ave E r ^f K () = r( + ). By assum tat te lare sample appromato s vald (see [8]) t results tat E r ^fk () plmm() = = =C E ^fg () =C = ; =C r( + ) ( + ) ( ; ) () + were plm deotes probablty lmt wt eld costat. Ts s equvalet to assum te sample sze sucetly lare to make te varaces of te meas relatvely small. Fally, te orm of te badwdt ormalzed mea sft s plmm() = k ; k () =C + a quatty tat as a uque postve mamum at =. Teorem leads to a very smple ad accurate scale selecto rule: te uderly desty as te local scale equal to te badwdt tat mamzes te orm of te ormalzed mea sft vector. We epect tat a smlar property olds te case of asotropc covarace matrces. 3.3 Scale Selecto Epermets Fure 3a sows a data set of sze =, draw from N(4,). Te badwdt ormalzed mea sft s represeted Fure 3b as a fucto of scale. Observe / $. (C) IEEE

6 Hstoram value (a) Norm of Mea Sft Vector Scale (b) Fure 3: Semparametrc scale selecto. (a) Iput data. N(,4), =. (b) Normalzed mea sft as a fucto of scale for te pots wt postve mea sft. Te upper curves correspod to te pots located far from te mea. Te curves are mamzed for =4. te accurate local scale dcato by te mama of te curves. Te same accurate results were obtaed for two ad tree dmesos. 4 Vdeo Data Aalyss A fudametal task vdeo data aalyss s to detect blobs represeted by collectos of pels tat are coeret spatal, rae, ad tme doma []. Te two dmesoal space of te lattce s kow as te spatal doma wle te ray level, color, spectral, or teture formato s represeted te rae doma. Based o te two ew estmators troduced Sectos ad 3 we preset et a autoomous tecque tat semet a vdeo frame to represetatve blobs detected te spatal ad color domas. Te tecque ca be aturally eteded to corporate tme formato, ts be oe of te subjects of our curret work. We selected te ortooal features I =(R + G + B)=3, I =(R;B)= ad I3 =(G;R;B)=4 from [] to represet te color formato. Due to te ortooalty of te features, te oe dmesoal plu- rule for badwdt selecto ca be appled depedetly for eac color coordate. As [5], te dea s to apply te mea sft procedure for te data pots te jot spatal-rae doma. Eac data pot becomes assocated to a pot of coverece wc represets te local mode of te desty a d = + 3 dmesoal space ( spatal compoets ad 3 color compoets). We employed a spercal kerel for te spatal doma ad a product kerel for te tree color compoets. Te ececy of te product kerel s kow to be very close to tat of spercal kerels [, p.4]. Due to te deret ature of te two spaces, te problem of badwdt selecto as bee treated dfferetly for eac space. A ed badwdt was rst derved for eac color compoet, based o te oe dmesoal plu- rule. Te, te plot desty as bee computed for eac pel, ad te adaptve color badwdts were determed accord to (8) for eac pel. Ts process as bee repeated for deret scales of te spatal kerel. Fally, te spatal scale as bee selected for eac pel accord to te semparametrc rule. As a result, eac pel receved a uque color badwdt for color ad a uque spatal badwdt. To obta te semeted mae, te adaptve mea sft procedure as bee appled te jot doma. Te blobs were deted as roups of pels tat ad te same coected coverece pots (see [5]). Te alortm s summarzed below. Adaptve Mea Sft Semetato Gve te mae pels f I I I3 :::, ad a rae of spatal scales r :::r S :. Derve,, 3, a ed badwdt for eac color feature.. For te spatal scale r, compute te adaptve badwdts ( r ), ( r ), 3 ( r ) ad determe te matude of te ormalzed mea sft vector M( r ). 3. Repeat Step. for te spatal scales r :::r S. 4. Select for eac pel a spatal scale r j accord to te semparametrc rule. Select also te color badwdts ( r j ), ( r j ), ad 3 ( r j ). 5. Ru te adaptve mea sft procedure, ad detfy te blobs as roups of pels av te same coected coverece pots. Altou te adaptve alortm as a creased complety, ts careful software mplemetato wt tree spatal scales (S=3) rus at about 8 frames/secod o a Dual Petum III at 9MHz for a vdeo frame sze of 34 pels. Fure4sows four eamples demostrat te semetato of color mae data wt very deret statstcs. Fure 5 sows te stablty of te alortm semet a color sequece obtaed by pa te camera. Te deted blobs were mataed very stable, altou te scee data caed radually alo wt te camera a. 5 Dscusso Te most attractve property of te tecques proposed ts paper s te automatc badwdt selecto bot color ad spatal doma. Te reaso we used two deret badwdt selecto tecques for te two spaces was ot arbtrary / $. (C) IEEE

7 Wle te color formato ca be collected across te mae, allow te computato of robust tal badwdt for color, te spatal propertes of te blobs vary drastcally across te mae, requr local decsos for spatal scale selecto. Te process deed by te mea sft tecque te color doma resembles blateral lter [9] (see also [3] for a dscusso o te lk betwee blateral lter, asotropc duso [], ad adaptve smoot [3]). Due to te wet of te data, te adaptve badwdt mea sft s more related to robust asotropc duso [4]. I te spatal doma, te mea sft s close to multscale tecques suc as [], ad te semparametrc scale selecto rule resembles prcple to tose developed [7, 9]. Te ucato of all tese deas s a terest subject for furter researc. Fure 5: Sequece of semeted maes used to test te stablty of our alortm. Frame sze: 3 4 pels. 5. ^ () =:357f ^SD (a)= ^TD (b) =7 5=7. 6. Solve te equato [R(K)=f (K) ^SD (^ ())] =5 ;=5 ; = were (K) ad R(K) are deed (3) ad (4), respectvely. Fure 4: Semetato eamples. Frame sze: 3 4 pels. APPENDIX Oe dmesoal plu- rule [5]. Compute ^ = Q 3 ; Q, te sample terquartle rae.. Compute a =:9^ ;=7, b =:9^ ;=9. X 3. ^TD (b) =;f(;) ; b ;7 v fb ; ( ; j ) j= were v s te st dervatve of te ormal kerel (see [] [p.77]). X 4. ^SD (a) =f(;) ; a ;5 v fa ; ( ; j ), j= were v s te fort dervatve of te ormal kerel. Coverece Proof for Varable Badwdt Mea Sft Sce s te te sequece ^f K s bouded, terefore, t s sucettosowtat ^f K s strctly mootoc creas,.e., f y j 6= y j+ te ^f K (j) < ^f K (j +), for all j = :::. By assum wtout loss of eeralty tat y j = we wrte ^f K (j +); ^f K (j) = = d " k y j+ ;! ; k Te covety of te prole k mples tat k( ) k( )+k ( )( ; )!# :(B.) (B.) for all [ ), 6=, ad sce k = ;, te equalty (B.) becomes k( ) ; k( ) ( )( ; ): (B.3) / $. (C) IEEE

8 Us ow (B.) ad (B.3) we ave ^f K (j +); ^f K (j) = = y> j+ d+ d+ ; ky j+k X! k k ;ky j+ ; k! y > j+ ;ky j+ k! ;! d+ d+ ad by employ (7) t results tat d+ (B.4)! : ^f K (j +); ^f K (j) X ky j+k (B.5) Sce k s mootoc decreas we ave ;k () () for all [ ). Te sum d+ s strctly postve, sce t was assumed to be ozero te deto of te mea sft vector (). Tus, as lo as y j+ 6= y j =, te rt term of (B.5) s strctly postve,.e., ^f K (j +); ^f K (j) >. Hece, te sequece ^f K s coveret. To sow te coverece of te sequece y j j= ::: we rewrte (B.5) but wtout assum tat y j =. After some alebra t results tat ^f K (j+); ^f K (j) X ky ;y j+ j k y j ; d+ (B.6) Sce ^f K (j +); ^f K (j) coveres to zero, (B.6) mples tat ky j+ ;y j k also coveres to zero,.e., y j! j= ::: s a Caucy sequece. But ay Caucy sequece s coveret te Eucldea space, terefore, y j j= ::: s coveret. Ackowledmet Peter Meer was supported by te NSF uder te rat IRI Refereces [] I.S. Abramso, \O Badwdt Varato Kerel Estmates - A Square Root Law," Te Aals of Statstcs, (4):7{3, 98. [] N. Auja, \A Trasform for Multscale Imae Semetato byiterated Ede ad Reo Detecto," IEEE Tras. Patter Aal. Mace Itell., 8:{35, 996. [3] D. Baras, \Blateral Flter ad Asotropc Duso: Towards a Ued Vewpot," Hewlett-Packard HPL--8(R.). Avalable at ttp: [4] M.J. Black, G. Sapro, D.H. Marmot, D. Heeer,, \Robust Asotropc Duso," Imae Process, 7(3):4{43, 998. [5] D. Comacu, P. Meer, \Mea Sft Aalyss ad Applcatos," IEEE It'l Cof. Comp. Vs., Kerkyra, Greece, 97{3, 999. [6] D. Comacu, V. Rames, P. Meer, \Real-Tme Track of No-Rd Objects us Mea Sft," IEEE Cof. Comp. Vs. Patt. Reco., Hlto Head, Sout Carola, Vol., 4{49,. [7] J. Elder, S.W. Zucker, \Local Scale Cotrol for Ede Detecto ad Blur Estmato," IEEE Tras. Patter Aal. Mace Itell., (7):699{76, 998. [8] P. Hall, T.C. Hu, J.S. Marro, \Improved Varable Wdow Kerel Estmates of Probablty Destes," Te Aals of Statstcs, 3():{, 995. [9] T. Ldeber, \Ede Detecto ad Rde Detecto wt Automatc Scale Selecto," It. J. Comp. Vso., 3():7{54, 998. [] Y. Ota, T. Kaade, T. Saka, \Color Iformato for Reo Semetato," Computer Grapcs ad Imae Process, 3:{4, 98. [] B. Park, J.S. Marro, \Comparso of Data- Drve Badwdt Selectors," J. Am. Statst. Assoc., 85(49):66{7, 99. [] P. Peroa, J. Malk, \Scale-Space ad Ede Detecto Us Asotropc Duso," IEEE Tras. Patter Aal. Mace Itell., (7):69{639, 99. [3] P. Sat-Marc, J.S. Ce, G.G. Medo, \Adaptve Smoot: A Geeral Tool for Early Vso", IEEE Tras. PAMI, 3(7):54-59, 99. [4] D.W. Scott, Multvarate Desty Estmato, New York: Wley, 99. [5] S.J. Seater, M.C. Joes, \A Relable Data-based Badwdt Selecto Metod for Kerel Desty Estmato," J. R. Statst. Soc. B, 53(3):683{69, 99. [6] J.S. Smoo, Smoot Metods Statstcs, New York: Sprer-Verla, 996. [7] B.W. Slverma, Desty Estmato for Statstcs ad Data Aalyss, New York: Capma ad Hall, 986. [8] T.M Stocker, \Smoot Bas Desty Dervatve Estmato," Amerca Stat. Assoc., 88(43):855{863, 993. [9] C. Tomas, R. Maduc, \Blateral Flter for Gray ad Color Imaes", It'l Cof. Comp. Vs., Bombay, Ida, 839{846, 998. [] M.P. Wad, M.C. Joes, Kerel Smoot, Lodo: Capma & Hall, 995. [] C. Wre, A. Azarbayeja, T. Darrell, A. Petlad, \Pder: Real-Tme Track of te Huma Body," IEEE Tras. Patter Aalyss Mace Itell., 9:78{ 785, / $. (C) IEEE