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1 IEEE CVPR 000 RealTime Trackin of NonRiid Objects usin Mean Sift Dorin Comaniciu Visvanatan Rames Peter Meer Imain & Visualization Department Electrical & Computer Enineerin Department Siemens Corporate Researc Ruters University 755 Collee Road East, Princeton, NJ Brett Road, Piscataway, NJ Abstract A new metod for realtime trackin of nonriid objects seen from a movin camera isproposed. Te central computational module is based on te mean sift iterations and nds te most probable taret position in te current frame. Te dissimilarity between te taret model (its color distribution) and te taret candidates is expressed by a metric derived from te Battacaryya coecient. Te teoretical analysis of te approac sows tat it relates to te Bayesian framework wile providin a practical, fast and ecient solution. Te capability of te tracker to andle in realtime partial occlusions, sinicant clutter, and taret scale variations, is demonstrated for several imae sequences. Introduction Te ecient trackin of visual features in complex environments is a callenin task for te vision community. Realtime applications suc as surveillance and monitorin [0], perceptual user interfaces [4], smart rooms [6, 8], and video compression [] all require te ability to track movin objects. Te computational complexity of te tracker is critical for most applications, only a small percentae of a system resources bein allocated for trackin, wile te rest is assined to preprocessin staes or to ilevel tasks suc as reconition, trajectory interpretation, and reasonin [4]. Tis paper presents a new approac to te realtime trackin of nonriid objects based on visual features suc as color and/or texture, wose statistical distributions caracterize te object of interest. Te proposed trackin is appropriate for a lare variety of objects wit dierent color/texture patterns, bein robust to partial occlusions, clutter, rotation in dept, and canes in camera position. It is a natural application to motion analysis of te mean sift procedure introduced earlier [6, 7]. Te mean sift iterations are employed to nd te taret candidate tat is te most similar to a iven taret model, wit te similarity bein expressed by a metric based on te Battacaryya coecient. Various test sequences sowed te superior trackin performance, obtained wit low computational complexity. Te paper is oranized as follows. Section presents and extends te mean sift property. Section 3 introduces te metric derived from te Battacaryya coef cient. Te trackin aloritm is developed and analyzed in Section 4. Experiments and comparisons are iven in Section 5, and te discussions are in Section 6. Mean Sift Analysis We dene next te sample mean sift, introduce te iterative mean sift procedure, and present a new teorem sowin te converence for kernels wit convex and monotonic proles. For applications of te mean sift property in low level vision (lterin, sementation) see [6].. Sample Mean Sift Given a set f :::n of n points in te d dimensional space R d, te multivariate kernel density estimate wit kernel K(x) and window radius (bandwidt), computed in te point s iven by ^f(x) = n d K x ; xi : () Te minimization of te averae lobal error between te estimate and te true density yields te multivariate Epanecnikovkernel [5, p.39] K E (x) = c; d (d + )( ;kxk ) if kxk < 0 oterwise () were c d is te volume of te unit ddimensional spere. Anoter commonly used kernel is te multivariate normal K N (x) =() ;d= exp ; kxk : (3) Let us introduce te prole of a kernel K as a function k : [0 )! R suc tat K(x) = k(kxk ). For example, accordin to () te Epanecnikov prole is k E (x) = c; d (d + )( ; x) if x< 0 oterwise (4) and from (3) te normal prole is iven by k N (x) =() ;d= exp ; x : (5) Employin te prole notation we can write te density estimate () as ^f K (x) = n d k x ;! : (6) We denote (x) =;k 0 (x) (7) assumin tat te derivative of k exists for all x [0 ), except for a nite set of points. A kernel G can be dened as G(x) =C(kxk ) (8)
2 were C is a normalization constant. Ten, by takin te estimate of te density radient as te radient of te density estimate we ave ^rf K (x) r^f K (x)= n d+ = n d+ " ( ; x) x ;!# 4 x ; x ; x (x ; ) k 0 i! = x; x; n d+ 3 ;x! 5 (9) can be assumed to be were x; nonzero. Note tat te derivative of te Epanecnikov prole is te uniform prole, wile te derivative ofte normal prole remains a normal. Te last bracket in (9) contains te sample mean sift vector M G (x) and te density estimate at x ^f G (x) x i x; x; C n d ; x (0)! x ; () computed wit kernel G. Usin now (0) and (), (9) becomes ^rf K (x) = ^fg (x) =C M G(x) () from were it follows tat M G (x) = ^rf K (x) =C ^f G (x) : (3) Expression (3) sows tat te sample mean sift vector obtained wit kernel G is an estimate of te normalized density radient obtained wit kernel K. Tis is a more eneral formulation of te property rst remarked by Fukunaa [5, p.535]. j= :::. A Sucient Converence Condition Te mean sift procedure is dened recursively by computin te mean sift vector M G (x) and translatin te centerofkernel G by M G (x). Let us denote by y j te sequence of successive locations of te kernel G, were y j+ = x i y j ; y j ; j = ::: (4) is te weited mean at y j computed wit kernel G and y is te center of te initial kernel. Te density estimates computed wit kernel K in te points (4) are n o n o ^f K = ^fk (j) ^fk (y j ) : (5) j= ::: j= ::: Tese densities are only implicitly dened to obtain ^rf K. However we need tem to prove te converence of te sequences (4) and (5). Teorem If te kernel K as a convex and monotonic decreasin prole and te kernel G is dened accordin to (7) and (8), te sequences (4) and (5) are converent. Te Teorem eneralizes te converence sown in [6], were K was te Epanecnikov kernel, and G te uniform kernel. Its proof is iven in te Appendix. Note tat Teorem is also valid wen we associate to eac datapoint a positive weit w i. 3 Battacaryya Coecient Based Metric for Taret Localization Te task of ndin te taret location in te current frame is formulated as follows. Te feature z representin te color and/or texture of te taret model is assumed to ave a density function q z, wile te taret candidate centered at location y as te feature distributed accordin to p z (y). Te problem is ten to nd te discrete location y wose associated density p z (y) is te most similar to te taret density q z. To dene te similarity measure we takeinto account tat te probability of classication error in statistical ypotesis testin is directly related to te similarity of te two distributions. Te larer te probability of error, te more similar te distributions. Terefore, (contrary to te ypotesis testin), we formulate te taret location estimation problem as te derivation of te estimate tat maximizes te Bayes error associated wit te model and candidate distributions. For te moment, we assume tat te taret as equal prior probability to be presentatany location y in te neiborood of te previously estimated location. An entity closely related to te Bayes error is te Battacaryya coecient, wose eneral form is de ned by [9] Z ppz (y) [p(y) q]= (y)q z dz : (6) Properties of te Battacaryya coecient suc as its relation to te Fiser measure of information, quality of te sample estimate, and explicit forms for various distributions are iven in [, 9]. Our interest in expression (6) is, owever, motivated by its near optimality iven by te relationsip to te Bayes error. Indeed, let us denote by and two sets of parameters for te distributions p and q and by =( p q ) a set of prior probabilities. If te value of (6) is smaller for te set tan for te set, it
3 can be proved [9] tat, tere exists a set of priors for wic te error probability for te set is less tan te error probability for te set. In addition, startin from (6) upper and lower error bounds can be derived for te probability of error. Te derivation of te Battacaryya coecient from sample data involves te estimation of te densities p and q, for wic we employ te istoram formulation. Altou not te best nonparametric density estimate [5], te istoram satises te low computational cost imposed by realtime processin. We estimate te discrete density ^q P m = f^q u :::m (wit ^q u = ) from te mbin istoram of P te taret model, wile m ^p(y) =f^p u (y) :::m (wit ^p u = ) is estimated at a iven location y from te mbin istoram of te taret candidate. Hence, te sample estimate of te Battacaryya coecient isiven by ^(y) [^p(y) ^q] = p^pu (y)^q u : (7) Te eometric interpretation of (7) is te cosine of te anle between te mdimensional, unit vectors ;p^p ::: p^p m > and ;p^q ::: p^q m >. Usin now (7) te distance between two distributions can be dened as d(y) = p ; [^p(y) ^q] : (8) Te statistical measure (8) is well suited for te task of taret localization since:. It is nearly optimal, due to its link to te Bayes error. Note tat te widely used istoram intersection tecnique [6] as no suc teoretical foundation.. It imposes a metric structure (see Appendix). Te Battacaryya distance [5, p.99] or Kullback diverence [8, p.8] are not metrics since tey violate at least one of te distance axioms. 3. Usin discrete densities, (8) is invariant to te scale of te taret (up to quantization eects). Historam intersection is scale variant [6]. 4. Bein valid for arbitrary distributions, te distance (8) is superior to te Fiser linear discriminant, wic yields useful results only for distributions tat are separated by te meandierence [5, p.3]. Similar measures were already used in computer vision. Te Cerno and Battacaryya bounds ave been employed in [0] to determine te eectiveness of ede detectors. Te Kullback diverence as been used in [7] for ndin te pose of an object in an imae. Te next section sows ow to minimize (8) as a function of y in te neiborood of a iven location, by exploitin te mean sift iterations. Only te distribution of te object colors will be considered, altou te texture distribution can be interated into te same framework. 4 Trackin Aloritm We assume in te sequel te support of two modules wic sould provide (a) detection and localization in te initial frame of te objects to track (tarets) [, 3], and (b) periodic analysis of eac object to account for possible updates of te taret models due to sinicant canes in color []. 4. Color Representation Taret Model Let fx? i :::n be te pixel locations of te taret model, centered at 0. We dene a function b : R! f :::m wic associates to te pixel at location x? i te index b(x? i ) of te istoram bin correspondin to te color of tat pixel. Te probability of te color u in te taret model is derived by employin a convex and monotonic decreasin kernel prole k wic assins a smaller weit to te locations tat are farter from te center of te taret. Te weitin increases te robustness of te estimation, since te periperal pixels are te least reliable, bein often aected by occlusions (clutter) or backround. Te radius of te kernel prole is taken equal to one, by assumin tat te eneric coordinates x and y are normalized wit x and y, respectively. Hence, we can write ^q u = C k(kx? i k ) [b(x? i ) ; u] (9) were is te Kronecker delta function. Te normalization constant C is derived by imposin te condition P m ^q u =, from were C = k(kx? i k ) (0) since te summation of delta functions for u =:::m is equal to one. Taret Candidates Let f :::n be te pixel locations of te taret candidate, centered at y in te current frame. Usin te same kernel prole k, but wit radius, te probability of te color u in te taret candidate is iven by X ^p u (y) =C n k! y ; [b(x i ) ; u] () were C is te normalization constant. Te radius of te kernel prole determines te number of pixels (i.e., te scale) of te P taret candidate. By imposin te m condition tat ^p u =we obtain C = y;xi k(k k ) : () Note tat C does not depend on y, since te pixel locations are oranized in a reular lattice, y bein one of te lattice nodes. Terefore, C can be precalculated for a iven kernel and dierent values of. 3
4 4. Distance Minimization Accordin to Section 3, te most probable location y of te taret in te current frame is obtained by minimizin te distance (8), wic is equivalent tomaxi mizin te Battacaryya coecient ^(y). Te searc for te new taret location in te current frame starts at te estimated location ^y 0 of te taret in te previous frame. Tus, te color probabilities f^p u (^y 0 ) :::m of te taret candidate at location ^y 0 in te current frame ave to be computed rst. Usin Taylor expansion around te values ^p u (^y 0 ), te Battacaryya coecient (7)isapproximated as (after some manipulations) [^p(y) ^q] (^y 0 )^q u + ^q u ^p u (y) ^p p^pu u (^y 0 ) (3) were it is assumed tat te taret candidate f^p u (y) :::m does not cane drastically from te initial f^p u (^y 0 ) :::m, and tat ^p u (^y 0 ) > 0 for all u = :::m. Introducin now () in (3) we obtain [^p(y) ^q] were w i = p^pu (^y 0 )^q u + C [b( ) ; u] Xn s w i k s! y ; (4) ^q u ^p u (^y 0 ) : (5) Tus, to minimize te distance (8), te second term in equation (4) as to be maximized, te rst term bein independent of y. Te second term represents te density estimate computed wit kernel prole k at y in te current frame, wit te data bein weited by w i (5). Te maximization can be eciently acieved based on te mean sift iterations, usin te followin aloritm. Battacaryya Coecient [^p(y) ^q] Maximization Given te distribution f^q u :::m of te taret model and te estimated location ^y 0 of te taret in te previous frame:. Initialize te location of te taret in te current frame wit ^y 0, compute te distribution f^p u (^y 0 ) :::m,andevaluate [^p(^y 0 ) ^q] = P m p^pu (^y 0 )^q u :. Derive teweits fw i :::n accordin to (5). 3. Based on te mean sift vector, derive te new location of te taret (4) ^y = x iw i w i ^y0 ; ^y0 ; : (6) Update f^p u (^y ) :::m,andevaluate [^p(^y ) ^q] = P m p^pu (^y )^q u : 4. Wile [^p(^y ) ^q] <[^p(^y 0 ) ^q] Do ^y (^y 0 + ^y ). 5. If k^y ; ^y 0 k < Stop. Oterwise Set ^y 0 ^y and o to Step. Te proposed optimization employs te mean sift vector in Step 3 to increase te value of te approximated Battacaryya coecient expressed by (4). Since tis operation does not necessarily increase te value of [^p(y) ^q], te test included in Step 4 is needed to validate te new location of te taret. However, practical experiments (trackin dierent objects, for lon periods of time)sowed tat te Battacaryya coecient computed at te location dened by equation (6) was almost always larer tan te coecient correspondin to ^y 0. Less tan 0:% of te performed maximizations yielded cases were te Step 4 iterations were necessary. Te termination tresold used in Step 5 is derived by constrainin te vectors representin ^y 0 and ^y to be witin te same pixel in imae coordinates. Te trackin consists in runnin for eac frame te optimization aloritm described above. Tus, iven te taret model, te new location of te taret in te current frame minimizes te distance (8) in te neiborood of te previous location estimate. 4.3 Scale Adaptation Te scale adaptation sceme exploits te property of te distance (8) to be invariant to canes in te object scale. We simply modify te radius of te kernel prole wit a certain fraction (we used 0%), let te trackin aloritm to convere aain, and coose te radius yieldin te larest decrease in te distance (8). An IIR lter is used to derive te new radius based on te current measurements and old radius. 5 Experiments Te proposed metod as been applied to te task of trackin a football player marked by a anddrawn ellipsoidal reion (rst imae of Fiure ). Te sequence as 54 frames of pixels eac and te initial normalization constants (determined from te size of te taret model) were ( x y ) = (7 53). Te Epanecnikov prole (4) as been used for istoram computation, terefore, te mean sift iterations were computed wit te uniform prole. Te taret istoram as been derived in te RGB space wit bins. Te aloritm runs comfortably at 30 fps on a 600 MHz PC, Java implementation. Te trackin results are presented in Fiure. Te mean sift based tracker proved to be robust to partial occlusion, clutter, distractors (frame 40 in Fiure ), 4
5 and camera motion. Since no motion model as been assumed, te tracker adapted well to te nonstationary caracter of te player's movements, wic alternates abruptly between slow and fast action. In addition, te intense blurrin present in some frames and due to te camera motion, did not inuence te tracker performance (frame 50 in Fiure ). Te same eect, owever, can larely perturb contour based trackers Mean Sift Iterations Frame Index Fiure : Te number of mean sift iterations function of te frame index for te Football sequence. Te mean number of iterations is 4:9 per frame. Te number of mean sift iterations necessary for eac frame (one scale) in te Football sequence is sown in Fiure. One can identify two central peaks, correspondin to te movement ofte player to te center of te imae and back to te left side. Te last and larest peak is due to te fast movement from te left to te rit side. 0.9 Fiure : Football sequence: Trackin te player no. 75 wit initial window of7 53 pixels. Te frames 30, 75, 05, 40, and 50 are sown. Battacaryya Coefficient Initial location Converence point 0 0 Y Fiure 3: Values of te Battacaryya coecient correspondin to te marked reion (8 8 pixels) in frame 05 from Fiure. Te surface is asymmetric, due to te player colors tat are similar to te taret. Four mean sift iterations were necessary for te aloritm to convere from te initial location (circle). To demonstrate te eciency of our approac, Fiure 3 presents te surface obtained by computin te Battacaryya coecient for te rectanle marked in Fiure, frame 05. Te taret model (te selected elliptical reion in frame 30) as been compared wit te taret candidates obtained by sweepin te elliptical reion in frame 05 inside te rectanle. Wile most of te trackin approaces based on reions [3, 4,] 0 X
6 must perform an exaustive searc in te rectanle to nd te maximum, our aloritm convered in four iterations as sown in Fiure 3. Note tat since te basin of attraction of te mode covers te entire window, te correct location of te taret would ave been reaced also from farter initial points. An optimized computation of te exaustive searc of te mode [3] as a muc larer aritmetic complexity, dependin on te cosen searc area. Te new metod as been applied to track people on subway platforms. Te camera bein xed, additional eometric constraints and also backround subtraction can be exploited to improve te trackin process. Te followin sequences, owever, ave been processed wit te aloritm uncaned. A rst example is sown in Fiure 4, demonstratin te capability of te tracker to adapt to scale canes. Te sequence as 87 frames of pixels eac and te initial normalization constants were ( x y )= (3 37). Fiure 5 presents six frames from a minute sequence sowin te trackin of a person from te moment se enters te subway platform till se ets on te train ( 3600 frames). Te trackin performance is remarkable, takin into accounttelow quality ofte processed sequence, due to te compression artifacts. A torou evaluation of te tracker, owever, is subject to our current work. Te minimum value of te distance (8) for eac frame is sown in Fiure 6. Te compression noise determined te distance to increase from 0 (perfect matc) to a stationary value of about 0:3. Sinicant deviations from tis value correspond to occlusions enerated by oter persons or rotations in dept of te taret. Te lare distance increase at te end sinals te complete occlusion of te taret. 6 Discussion By exploitin te spatial radient of te statistical measure (8) te new metod acieves realtime trackin performance, wile eectively rejectin backround clutter and partial occlusions. Note tat te same tecnique can be employed to derive te measurement vector for optimal prediction scemes suc as te (Extended) Kalman lter [, p.56, 06], or multiple ypotesis trackin approaces [5, 9, 7, 8]. In return, te prediction can determine te priors (denin te presence of te taret in a iven neiborood) assumed equal in tis paper. Tis connection is owever beyond te scope of tis paper. A patent application as been led coverin te trackin aloritm toeter wit te Kalman extension and various applications [9]. We nally observe tat te idea of centroid computation is also employed in []. Te mean sift was used for trackin uman faces [4], by projectin te Fiure 4: Subway sequence: Te frames 500, 59, 600, 633, and 686 are sown (leftrit, topdown). istoram of a face model onto te incomin frame. However, te direct projection of te model istoram onto te new frame can introduce a lare bias in te estimated location of te taret, and te resultin measure is scale variant. Gradient based reion trackin as been formulated in [] by minimizin te enery of te deformable reion, but no realtime claims were made. APPENDIX ProofofTeorem Since n is nite te sequence ^f K is bounded, terefore, it is sucienttosow tat ^f K is strictly monotonic increasin, i.e., if y j 6= y j+ ten ^f K (j) < ^f K (j +), for all j = :::. By assumin witout loss of enerality tat y j = 0 we can write ^f K (j +); ^f K (j) = = n d " k! y ; x j+ i ; k # :(A.) 6
7 d Frame Index Fiure 6: Te detected minimum value of distance d function of te frame index for te minute Subway sequence. Te peaks in te rap correspond to occlusions or rotations in dept of te taret. For example, te peak of value d 0:6 corresponds to te partial occlusion in frame 3697, sown in Fiure 5. At te end of te sequence, te person bein tracked ets on te train, wic produces a complete occlusion. Fiure 5: Subway sequence: Te frames 340, 356, 3697, 5440, 608, and 668 are sown (leftrit, topdown). Te convexity of te prole k implies tat k(x ) k(x )+k 0 (x )(x ; x ) (A.) for all x x [0 ), x 6= x, and since k 0 = ;, te inequality (A.) becomes k(x ) ; k(x ) (x )(x ; x ): Usin now (A.) and (A.3) we obtain ^f K (j +); ^f K (j) n d+ = n d+ " y > j+ kxi k ;ky j+ ; k (A.3) y > j+ ;ky j+ k = n d+ X n ;ky j+ k # (A.4) and by employin (4) it results tat ^f K (j +); ^f K (j) n d+ ky j+k : (A.5) Since k is monotonic decreasin we P ave ;k 0 (x) n (x) 0 for all x [0 ). Te sum is strictly positive, since it was assumed to be nonzero in te denition of te mean sift vector (0). Tus, as lon as y j+ 6= y j = 0, te rit term of (A.5) is strictly positive, i.e., ^f K (j +); ^f K (j) > 0. Consequently, te sequence ^f K is converent. Toprove te converence of te sequence y j j= ::: we rewrite (A.5) but witout assumin tat y j = 0. After some alebra we ave ^f K (j+); ^f K (j) n ky j+;y d+ j k y j;! (A.6) Since ^f K (j + ) ; ^f K (j) converes to zero, (A.6) implies tat ky j+ ; y j k also converes to zero, i.e., yj is a Caucy sequence. Tis completes te j= ::: proof, since any Caucy sequence is converentinte Euclidean space. p Proof tat te distance d(^p ^q) = ; (^p ^q) is a metric Te proof is based on te properties of te Battacaryya coecient (7). Accordin to te Jensen's inequality [8, p.5] we ave (^p ^q)= p^pu^q u = s ^q u ^p u ^p u vu u t X m ^q u = (A.7) 7
8 wit p equality i ^p = ^q. Terefore, d(^p ^q) = ; (^p ^q) exists for all discrete distributions ^p and ^q, is positive, symmetric, and is equal to zero i ^p = ^q. Te trianle inequality can be proven as follows. Let us consider te discrete distributions ^p, ^q, and ^r, and dene te associated mdimensional points p = ;p^p ::: p^p > m, q = ;p^q ::: p^q > m, and r = ;p^r ::: p^r > m on te unit yperspere, centered at te oriin. By takin into account te eometric interpretation of te Battacaryya coecient, te trianle inequality d(^p ^r)+d(^q ^r) d(^p ^q) (A.8) is equivalent to q ; cos( p r )+ q ; cos( q r ) q ; cos( p q ): (A.9) If we xte points p and q, and te anle between p and r, te left side of inequality (A.9) is minimized wen te vectors p, q, and r lie in te same plane. Tus, te inequality (A.9) can be reduced to a  dimensional problem tat can be easily demonstrated by employin te alfanle sinus formula and a few trionometric manipulations. 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