PARTICLE filter (PF) is a powerful tool for state tracking,

Transcription

1 1 Acceleraing Paricle Filer using Randomized Muliscale and Fas Mulipole Type Mehods Gil Shaba, Yaniv Shmueli, Ami Bermanis and Amir Averbuch Absrac Paricle filer is a powerful ool for sae racking using non-linear observaions. We presen a muliscale based mehod ha acceleraes he racking compuaion by paricle filers. Unlike he convenional way, which calculaes weighs over all paricles in each cycle of he algorihm, we sample a small subse from he source paricles using marix decomposiion mehods. Then, we apply a funcion exension algorihm ha uses a paricle subse o recover he densiy funcion for all he res of he paricles no included in he chosen subse. The compuaional effor is subsanial especially when muliple objecs are racked concurrenly. The proposed algorihm significanly reduces he compuaional load. By using he Fas Gaussian Transform, he complexiy of he paricle selecion sep is reduced o a linear ime in n and k, where n is he number of paricles and k is he number of paricles in he seleced subse. We demonsrae our mehod on boh simulaed and on real daa such as objec racking in video sequences. Index Terms paricle filer, muliscale mehods, nonlinear racking, fas mulipole mehod 1 INTRODUCTION PARTICLE filer (PF) is a powerful ool for sae racking, based on non-linear observaions, ha uses he Mone-Carlo approach [1]. PF implemens a recursive Bayesian filer where he probabiliy densiy funcion (PDF) is represened by a se of random samples (paricles) and no in is analyical form. The number of paricles conrols he approximaion accuracy. A large number of paricles will lead o a more accurae represenaion of he funcional form of he PDF. The paricles are propagaed and advanced under he conrol of he sysem dynamics and under arge measuremen model. In he Sequenial Imporance Resampling (SIR) version of he PF, paricles are resampled a each cycle by using imporance sampling on heir probabiliies ha are called weighs. The paricle wih he maximal likelihood is seleced o be he curren prediced sae of he arge. Unlike Kalman filers, PFs are no resriced by saionary linear-gaussian assumpions ha make hem more robus and suiable for a larger se of problems. Alhough he PF concep is fairly sraighforward, i becomes compuaionally expensive for pracical implemenaions, as a very large number of paricles are needed in order o accuraely approximae he PDF of he observed arge(s). The increase in compuaional power, in he las decades, have enabled o inroduce PF based soluions o real world problems. The role of PF acceleraion is o increase he number of propagaed paricles while mainaining he same compuaional cos. A large number of paricles represen he required disribuions more accuraely ha leads o beer resuls. Many sysems are geared o rack objecs ha are buried in huge daa-sreams such as video sequences, communicaion and elemeric daa. Predic- G. Shaba, Y. Shmueli, A. Bermanis and A. Averbuch are from Tel Aviv Universiy ing he nex objec sae or racking i has o be done in near real-ime especially by devices ha have limied compuaional resources such as embedded devices. In his work, we develop an improved PF algorihm ha avoids he need o compue he likelihood funcion (he weighs) for all he paricles. The paricle weigh compuaion is expensive in cases such as racking objecs in video sequences. Insead, he weighs are compued for only a subse of he paricles and heir values are used o esimae he weighs for he res of hem. To selec a represenaive se of paricles, we use he inerpolaive decomposiion (ID) mehod [2]. Then, he weighs are exended o he res of he paricles using a muliscale funcion exension (MSE) mehod [3], which is an applicaion of he Nysröm exension mehod ( [4], [5]) o paricles no in he seleced se. The exension is based on he similariies beween paricles ha are no in he seleced subse and he paricles in he seleced subse. The MSE uses radial Gaussian funcions wih varying scales o esimae he coefficiens of he Nysröm exension. The MSE mehod is shown in [3] o be boh accurae and numerically sable overcoming he deficiencies in he Nysröm mehod. Inerpolaive decomposiion (ID) is a mehod o approximae a marix by selecing a se of k independen columns ha consiue a basis. We use he ID mehod o selec paricles ha bes represen he PDF. To find his paricle subse we compue an affiniy marix A for he paricles and apply he ID algorihm o compue a k independen se of columns from A. These columns correspond o he mos relevan paricles. Once he paricle selecion process is compleed, he weighs are compued only for he seleced paricles. The weigh values are exended o he res of he paricles by he applicaion of MSE. The moivaion for using MSE, as our exension mehod, is based on he fac ha he PDF is generally a smooh funcion and ha he

2 2 MSE mehod is srongly relaed o a Gaussian process regression (GPR) [6], which is an exension mehod in he field of saisical inference. We use a randomized implemenaion of he ID algorihm, which is based on random projecions, o minimize he compuaional cos of selecing he mos relevan paricles [7]. To reduce he compuaional cos even furher, we propose a selecion algorihm ha is based on he Farhes Poin Sampling (FPS) [8] mehod combined wih densiy esimaion. We refer o our mehod, which combines he FPS algorihm wih densiy esimaor, as a weighed FPS (WFPS). The densiy esimaor was implemened using he Fas Mulipole Mehod (FMM) [9] ha asympoically has a lower compuaional cos. In our experimens, we were able o accelerae he weigh calculaion sep o be approximaely 10 imes faser compared o he sandard paricle filer. The running ime and racking error of he algorihm were compared o he sandard PF (sequenial imporance resampling) as well as o oher weigh inerpolaion mehods indicaing ha he proposed algorihm was able o mainain he same error rae wih a much shorer running ime. This paper is organized as follows: Secion 2 presens relaed work on PF acceleraion. Secion 3 describes he PF algorihm, explains is limiaions and how i can be used for objec racking in video sequences. Secion 4 describes he muliscale sampling and exension echniques wih is mahemaical ools such as he randomized inerpolaive decomposiion (ID). Secion 5 presens he full muliscale PF algorihm ha acceleraes he sandard PF. An addiional acceleraion is achieved by he algorihm in Secion 6 o overcome he muliscale PF scalabiliy boleneck. Experimenal resuls are presened in Secion 7. We compare beween he performances of he presened mehod and oher mehods in differen scenarios. 2 RELATED WORK PFs have been sudied in many works and used in differen applicaion domains such as compuer vision, roboics, arge racking and finance. Example applicaions include hand gesure-based inerface [10], realime racking of soccer players [11], mobile robo localizaion [12] and visual racking of human face [13]. The PF employs a sequenial Mone Carlo approach o solve recursive Bayesian filering problems. The Mone Carlo sampling mehod combined wih he Bayesian inference enables he PF o provide a soluion for nonlinear and non-gaussian problems. While PF can be robus o boh he inpu observaions disribuion and o he observaions noise level, is implemenaion is compuaionally expensive. Making i o work in real-ime (compuaionally efficien) has become a major challenge when objecs racking is done in high dimensional sae space, or when dealing wih muliple arges racking. Such insances require o use more paricles and hus he problem quickly becomes inracable. In addiion, due o he naure of he PF algorihm and is repeaed sampling mehod, he algorihm may suffer from sample degeneracy where mos of he paricles have negligible weighs. Over he las wo decades, differen variaions of he PF algorihm have emerged o overcome hese limiaions. Mehods such as Auxiliary PF [14], Gaussian Sum PF [15], Unscened PF [16] and Swarm Inelligence based PF [13] were developed in order o overcome hese limiaions by improving he underlining sampling echniques and by providing beer evaluaions o he proposal disribuion of he paricles in each sep of he algorihm. The problem of racking curves in a dense visual cluer is invesigaed in [17], where a mehod for learning he dynamical models using visual observaions and propagae a randomly generaed se over ime o achieve near real ime racking is inroduced. High dimensional models for racking people using Mone Carlo filering and hybrid hypohesis mehods are also sudied in [18], [19], [20], [21]. By using randomizaion mehods for improving he paricles re-sampling and applying specific assumpions o he dynamical models, efficien visual racking is achieved. Overcoming he uncerainy induced by occlusion, abrup moion or appearance changes while sill prevening sample impoverishmen problems is demonsraed in [22], [23]. Addiional mehods for esimaing he poserior densiies were suggesed in [24], [25], [26], [27]. These mehods propose effecive ways for represening he poserior densiy ha resul in using fewer paricles. The unscened PF (UPF), for example, defines poins ha capure sufficien disribuion saisics. Then, i propagaes each paricle and incorporaes he new observaion o produce a Gaussian esimaion of is proposal disribuion. By using he inpu observaions, he UPF can achieve more accurae proposal disribuion esimaion han wha he regular PF implemenaion achieves. In coninuous densiy propagaion [27], densiy approximaion and inerpolaion echniques are employed o represen he proposal disribuion efficienly. The densiy funcions are represened by Gaussian mixures where he number of componens, heir coefficiens and oher relaed saisical parameers are auomaically deermined by he algorihm. Similar sampling and approximaion mehods for acceleraing nonparameric belief propagaion (NBP) were also developed in [26]. The core of he NBP algorihm requires a repeaed sampling from producs of Gaussian mixure, which makes he algorihm compuaionally expensive. To accelerae he process, Gaussian mixure densiy approximaion using mode propagaion and kernel fiing is applied. The producs of he Gaussian mixure are approximaed accuraely by jus a few mode propagaions and kernel fiing seps. This significanly acceleraes he sampling mehod since i uses a fewer samples. The approach presened in his paper is similar in ha sense, bu unlike he above mehods we do no use fewer paricles o

3 3 accelerae he PF. Insead, we compue he weighs for only a paricle subse. This subse is seleced using marix decomposiion mehods and he remaining weighs are ieraively inerpolaed using muliscale exension mehods which are accurae and numerically sable. The advanage of compuing he weighs for only a subse of he paricles, enables generaing a large se of paricles and more freedom in choosing an appropriae subse for he esimaion of he disribuion, which leads o a more accurae esimaion and herefore beer performance. The challenge, which all hese mehods face, is how o incorporae a new observaion ino he esimaed proposal disribuion funcion of he paricles while mainaining a reasonable compuaion cos. Comprehensive uorials and surveys on differen PF versions and recen advances in PF mehods are given for example in [1], [28], [29]. 3 PARTICLE FILTER ALGORITHM PF is an on-line densiy esimaion echnique based on arge simulaion ha uses Mone Carlo mehods for solving a recursive Bayesian filering problem. PF is used o esimae he sae x i a ime i from noisy observaions y 1,..., y i. Dynamic sae space equaions are used for modeling and predicion of he arge sae. The basic idea behind he PF approach is o use a sufficienly large number of paricles. Each paricle is an independen random variable which represens a possible arge sae. For example, a sae can be locaion and velociy. In his case, each paricle represens a possible locaion and a possible velociy of he arge from a proposal disribuion. The sysem model is applied o he paricles in order o predic he nex sae. Then, each paricle is assigned a weigh, which represens is reliabiliy or he probabiliy ha i represens he real arge sae. The acual locaion (he oupu of he PF algorihm) is usually deermined as he maximum a poseriori probabiliy of he paricle s poserior disribuion. The algorihm robusness and accuracy are deermined by he number of paricipaing paricles. A large number of paricles is more likely o cover a wide sae subspace in he proximiy of he arge, as well as a beer approximaion of he sae disribuion funcion. However, he compuaional cos of such an improved racking is high since each paricle needs o be boh advanced in ime and weighed. This is repeaed in each cycle of he algorihm. A descripion of he PF algorihm flow is given by Algorihm 3.1. Assume ha p represens he proposal disribuion ha is used o predic he nex paricles saes. The opimal proposal disribuion is he arge s disribuion, which is given by p(x k x k 1, y k ). Since his compuaion is impracical, an esimaed disribuion, which is called he proposal disribuion, is used. In our case, his disribuion is compued by uilizing he evoluion of he sysem (a physical model) o he paricles. The weighs compuaion in Algorihm 3.1, sep 5, can be expensive in some cases. For example, when he Algorihm 3.1: Paricle Filer (SIR) Inpu: n number of paricles; x 0 iniial sae; y 1,..., y T curren observaions; q( ) proposal disribuion funcion; p( ) approximaed poserior disribuion funcion Oupu: x 1,..., x T esimaed observaions 1: Weighs iniializaion: w (i) 0 = 1 n, x(i) 0 p(x 0 ), i = 1,..., n. 2: for ime seps =1,...,T do 3: Resample n new paricles by heir disribuion deermined by he weighs w (i) 1. 4: Predicion: Apply he dynamic model o each paricle o esimae he nex sae using x 1 and y 1,..., y x (i) q(x x (i) 1, y 1,..., y ), i = 1,..., n. 5: Weighs calculaion: w (i) p(y x (i) )p(x (i) x (i) q(x x (i) 1, y 1,..., y ) 6: Weighs normalizaion: w (i) = w (i) n l=1 w(l) 7: x is se o be he paricle x (l) l = arg max w (i). 8: end for 1 i n 1 ), i = 1,..., n., i = 1,..., n. where PF algorihm is used for racking a arge in videos, i is common o use disances beween hisograms for weighing he measuremens. An RGB image wih 256 gray-levels for each pixel will have a hisogram of = 16, 777, 216 bins. Then, he disance beween wo hisograms h 1 and h 2 (boh vecors have lengh of size he number of bins) can be measured, for example, by he Bhaacharyya coefficien B = h T 1 h 2. (3.1) The color hisogram calculaion complexiy for a given paricle depends on he number of bins we use and he number of pixels we need o ierae, as each paricle poins o an image pach of he arge. Assume ha he number of bins is b and he number of paricles is n. The oal weigh calculaion complexiy in each cycle can be very expensive for large b and n values. On he oher hand, a large number of bins can help in improving he disance accuracy ha affecs he weighs esimaions. Despie he robusness of he PF algorihm, i suffers from several limiaions. Usually, each new observaion requires some preprocessing followed by a weigh calculaion of each paricle. Boh seps can be compuaionally expensive in applicaions such as compuer vision and roboics. In such problems, he observaion conains a

4 4 large amoun of daa. For example, when racking a arge wihin a video sequence, each measuremen consiss of an image frame ha may conain several millions of pixels (as in HD forma). In addiion, each paricle is assigned a weigh ha is based on some calculaion applied o a subse of he measured daa. This subse can be relaively large (for example, an image pach wih housands of pixels). As he number of paricles becomes large, he oal compuaional load can become exremely expensive. When dealing wih high dimensional paricles ha conain many parameers in each sae, he number of needed paricles increases exponenially o cover a region of ineres around he curren arge sae. However, using a large number of paricles is imporan o obain high diversiy for he paricles so hey will represen he soluion space adequaely. 4 MULTISCALE FUNCTION EXTENSION METHOD Given a se P n = {p 1, p 2,..., p n } of n paricles, we wan o esimae he values of heir weighs using a small subse P k P n of k paricles. Here, k n is a predefined number for which he weighs of P k are compued direcly. Formally, our goal is o inerpolae he weigh funcion w : P k R from P k o P n (as calculaed in Sep 5 in Algorihm 3.1). For ha purpose, we use he MSE mehod [3], which is a muliscale based algorihm. The MSE is an ieraive mehod. Each MSE ieraion conains wo phases: subsampling and exension. The firs phase is done by a special decomposiion, known as inerpolaive decomposiion (ID) [2], of an affiniies marix associaed wih P k. The second phase exends he funcion from P k o P n using he oupu from he firs (sampling) phase. The essenials of he MSE are described in Secions 4.1, 4.2 and in [3]. We use he following noaion: s denoes he scale parameer, s = 0, 1,..., ϵ s = 2 s ϵ 0 for some posiive number ϵ 0, and j-h singular value (in decreasing order) is denoed by σ j (A). w = (w 1, w 2,..., w k ) T are he values of he weigh funcion w on he paricles in P k, where w j is he weigh of p j. 4.1 Daa Subsampling Through ID of a Gaussian Marix Algorihm 4.1: Deerminisic inerpolaive decomposiion Inpu: An m n marix A and an ineger k, s.. k < min {m, n}. Oupu: An m k marix B, whose columns are a subse of A s columns, and a k n marix P s.. A BP 4k(n k) + 1σ k+1 (A) 1: Apply a pivoed QR algorihm o A (Algorihm in [30]), AP R = QR, where P R is an n n permuaion marix, Q is an m m orhogonal marix and R is an m n upper riangular marix, where he diagonal absolue values are decreasingly ordered. 2: Spli R and Q s.. [ ] R = R11 R 12, Q = [ ] Q 0 R 1 Q 2 22 where R 11 is k k, R 12 is k (n k), R 22 is (m k) (n k), Q 1 is m k and Q 2 is m (m k). 3: Define he m k marix 4: Define he k n marix P = [ I k B = Q 1 R 11 (4.4) R 1 11 R 12 ] P T R where I k is he k k ideniy marix. g (s) (r) = exp{ r 2 /ϵ s }. (4.1) For a fixed scale s, we define he funcions g (s) j : P n R, j = 1,..., n g (s) j (p) = g (s) (d(p j, p)) (4.2) o be a Gaussian of widh ϵ s cenered a p j. Here, d : P n P n R is a disance funcion defined among he paricles. For example, i can be he Euclidean disance beween heir coordinaes. Wihou loss of generaliy, we assume ha he chosen paricles are indexed such ha P k = {p 1,..., p k }. Le A (s) be he k k affiniies marix associaed wih P k, whose (i, j)-h enry is g (s) (d(p i, p j )). In oher words, A (s) (i, j) = g (s) (d(p i, p j )), i, j = 1,..., k. (4.3) Noe ha he j-h column of A (s) is he resricion of g (s) j o P k. Pk c is he complemenary se of P k in P n. The specral norm of a marix A is denoed by A and is Le s be a fixed scale. Our goal is o approximae w by a superposiion of he columns in he affiniy marix A (s), hen o exend w o p Pk c based on he affiniies beween p and he elemens of P k. Due o Bochner s heorem, as long as P k consiss of k disinc paricles, A (s), which is defined in Eq. 4.3, is sricly posiive definie. A firs sigh, we can solve he equaion A (s) c = w and by using he radialiy of g (s) (defined in Eq. 4.1), o exend w o p by w(p ) = k i=1 c ig (s) i (p ) (defined in Eq. 4.2), which is exac on P k. Tha is, w j = w(p j ), j = 1, 2,..., k. This mehod is known as he Nysröm exension [4], [5]. As proved in [3], he condiion number of A (s) is large for small values of s, namely A (s) is numerically singular. On he oher hand, a oo big s resuls in a shor disance inerpolaion. Moreover, even if we choose such s for which A (s) is numerically nonsingular and he inerpolaion is no for a oo shor disance, inerpolaion by a superposiion of ranslaed Gaussian of a fixed

5 5 widh will no necessarily fi he properies of w. In order o overcome he numerical singulariy of A (s), we apply he ID procedure o A (s). An ID of order k of an m n marix A consiss of an m k marix B whose columns consis of a subse of he columns of A, as well as a k n marix P, such ha a subse of he columns of P consiues a k k ideniy marix, and A BP in he sense ha A BP O(n, σ k+1 (A)). Usually, k is chosen o be he numerical rank of A up o a cerain accuracy δ > 0, i.e. k = #{j : σ j (A) δσ 1 (A)}. This selecion of k guaranees ha he columns of B consiue a well condiioned basis o he range of A, whose condiion number is of order δ. The deerminisic ID algorihm is described in Algorihm 4.1 whose complexiy is O(mn 2 ). Addiionally o Algorihm 4.1, here are randomized versions of he ID algorihm ha require less compuaional operaions. For example, Algorihm 4.2 is a random-projecions based algorihm [7]. I produces an ID for a general marix m n marix A and an ineger l < min{m, n}, s.. A BP 2 l mnσ l+1 (A). (4.5) The complexiy is lc A + lc A T + O(l 2 n log(n)), where C A is he cos of applying A o a vecor of lengh k, and C A T is he cos of applying A T o a vecor of lengh m. Algorihm 4.2 uses he deerminisic ID Algorihm 4.1 by applying i o a smaller marix han A. Each column of A (s), as defined in Eq. 4.3, corresponds o a single paricle in P k. The columns subse selecion from A (s) is equivalen o P k paricles ha are subsampled from he associaed P n paricles. 4.2 Muliscale Funcion Exension Algorihm Le A (s) B (s) P (s) be he ID of A (s), where B (s) is a k r marix, whose columns consiue a subse of he columns of A (s), and P (s) = {p s1,..., p sr } is is associaed sampled daase. The exension of he weigh funcion w from P k o Pk c is done by an orhogonal projecion of w on he columns space of B (s), and by exending he projeced funcion o Pk c in a similar manner o Nysröm exension mehod ha uses he radialiy of g (s). Algorihm 4.3, whose complexiy is O(kr 2 ), describes he single-scale exension algorihm. Since he columns of B (s) do no necessarily consiue a basis of R k, w (s) is no necessarily equal o w, namely he oupu of Algorihm 4.3 is no an inerpolan of w. This phenomenon is illusraed in Fig. 5.1 in [3]. In his case, we apply Algorihm 4.3 once again o he residual w w (s) wih a narrower Gaussian. This guaranees ha he nex-scale affiniies marix A (s+1) has a bigger numerical rank hen A (s). As a consequence, i guaranees a wider subspace o projec he residual on. The above is summarized in Algorihm 4.4 whose complexiy is O(k 3 ). Algorihm 4.2: Randomized inerpolaive decomposiion Inpu: An m n marix A and wo inegers l < k, s.. k < min{m, n} (for example, k = l + 8). Oupu: An m l marix B and an l n marix P ha saisfy Eq : Use a random number generaor o form a real k m marix G whose enries are i.i.d Gaussian random variables of zero mean and uni variance. Compue he k n produc marix W = GA. 2: Using Algorihm 4.1, form a k l marix S, whose columns consiue a subse of he columns of W and a real l n marix P, such ha SP W 2 4l(n l) + 1σ l+1 (W ). 3: From Sep 2, he columns of S consiue a subse of he columns of W. In oher words, here exiss a finie sequence i 1, i 2,..., i l of inegers such ha, for any j = 1, 2,..., l, he j-h column of S is he i j -h column of W. The corresponding columns of A are colleced ino a real m l marix B, such ha, for any j = 1, 2,..., l, he j-h column of B is he i j -h column of A. Then, he sampled daase is D s = {x i1, x i2,..., x il }. Algorihm 4.3: Single-scale exension Inpu: Scale parameer s, k r marix B (s), he associaed sampled daase P (s) = {p s1,..., p sk }, a new daa poin p Pk c, and he weigh funcion w : P k R o be exended. Oupu: The projecion w (s) = (w (s) 1, w(s) 2,... w(s) k )T of w = (w 1, w 2,..., w k ) T on B (s) and is exension w (s) o p. 1: Solve he leas squares problem min c w c R r B(s) 2 for c = (c 1, c 2,..., c r ) T. 2: Calculae he orhogonal projecion of w on he columns of B (s), w (s) = B (s) c. 3: Calculae he exension w (s) of w (s) o p using Eq. 4.2: w (s) r j=1 c j g (s) s j (p ). (4.6) 5 MULTISCALE PARTICLE FILTER (MSPF) In order o accelerae he PF algorihm when i runs on a large number of paricles, we apply paricles subsampling. We use he MSE mehod o compue he weighs for he res of he paricles. This will allow us o compue a relaively small number of paricle weighs in each cycle of he algorihm. This approach can be effecive if he paricle s weigh calculaion on all paricles is

6 6 Algorihm 4.4: Muliscale daa sampling and funcion exension Inpu: A daase of k paricles P k = {p 1,..., p k }, a posiive number ϵ 0, a new paricle p P c k, a weigh funcion w : P k R, o be exended (represened by w = (w 1, w 2,..., w k ) T where w j is he weigh of p j ), and an error parameer err 0. Oupu: An approximaion ŵ = (ŵ 1 ŵ 2... ŵ k ) T of w on P k ha saisfies w ŵ err and is exension ŵ o p. 1: Se he scale parameer s = 0, he approximaion o w, ŵ = 0, and he exension of ŵ o p, ŵ = 0. 2: while w ŵ > err do 3: Form he Gaussian affiniies marix A (s) (Eq. 4.3) on P k, wih ϵ s = 2 s ϵ 0. 4: Se r o be he numerical rank of A (s) (see Definiion 3.1 in [3]). 5: Apply Algorihm 4.1 o A (s) wih he parameer r o ge an k r marix B (s) and he associaed sampled daase P (s). 6: Apply Algorihm 4.3 o B (s), P (s), p, and w ŵ. We ge he approximaion ŵ (s) o w ŵ a scale s, and is exension ŵ (s) o p. 7: Accumulae he approximaions from sep 6: Se ŵ = ŵ + ŵ (s) and ŵ = ŵ + ŵ (s). 8: Se s = s : end while compuaionally expensive especially when he number of paricles is high. Algorihm 5.1 describes our modified PF algorihm ha suppors muliscale subsampling and exension. 5.1 Paricle Subsampling In each cycle in Algorihm 5.1, we firs resample a new se of k paricles from he se P n using heir weighs as he disribuion funcion. Once we apply he dynamic model o each paricle and advance i, new weighs have o be compued. Therefore, we firs selec a small subse from all he n paricles. The goal is o find a good se of represenaive paricle candidaes ha will capure he geomery and he aciviy of he source weigh funcion w : P n R. To idenify hese candidaes, we define a disance meric beween he n paricles using a weighed Euclidean disance beween each wo paricles viewed as vecors. Oher merics can be used as well. We selec he paricle candidaes using Algorihm 4.2 which is he randomized ID. We consruc an affiniy marix A (s) ha conains he affiniies d(p i, p j ) beween he paricles. The kernel, which is defined beween he paricles, is ( [A (s) d(pi, p j ) 2 ) ] ij = exp, i, j = 1,.., n. (5.1) ϵ s We calculae he affiniies beween all he paricles in P n such ha A (s) is an n n marix defined by Eq The number of candidaes we use is a mos k. The oupu from he randomized ID algorihm (Algorihm 4.2) will be he se P k of k paricles ha were seleced from P n. We compue direcly he weighs for he k seleced paricles. 5.2 Weigh Calculaion using Funcion Exension We obained a se of paricles P k wih heir calculaed weigh values. Nex, we coninue and compue he weighs for he res of he paricles ha are no included in P k. We compue he weigh value for each of he oher n k paricles by applying Algorihm 4.4 o he se P k ha has he firs k columns of he affiniy marix A (s). These columns conain he affiniies beween each pair of paricles in P k, he affiniies beween he paricles in P k and all he oher paricles. The oupu from Algorihm 4.4 is he weighs of he n k paricles ha were no seleced in he previous sep. This exension mehod allows us o skip a direc weigh compuaions for he remaining n k paricles. Therefore, we keep he enire se of paricles P n from which we resample a small se of paricles in he nex PF algorihm sep. This is especially beneficial when we canno compue he weighs for all paricles if he compuaion is oo expensive. Once he n k weighs are calculaed, we selec he paricle wih he maximum likelihood as he predicion resul and coninue o he nex algorihmic cycle. Weighs compuaion for k paricles and heir exension o he oher n k paricles reduce he number of operaions and acceleraes he PF as demonsraed in Secion 7. 6 ACCELERATING THE PARTICLE SAMPLING STEP The compuaional boleneck in Algorihm 5.1 lies in he selecion sep (Sep 5) where we sample a subse of size k from he source paricle se of size n. In his sep, he randomized ID algorihm requires O(kn 2 + k 2 nlogn) operaions (see Secion 5.3 in [7]). In addiion, he inpu kernel marix calculaion in he ID algorihm requires O(n 2 ) operaions. Therefore, Algorihm 5.1 does no scale well and he gained performance boos decreases as he number of paricles increases. To improve he performance of he sampling sep, a differen sampling mehod was developed. The mehod is based on a variaion of he Farhes Poin Sampling (FPS) algorihm [8], [31] combined wih radiional kernel densiy esimaion, which we refer o as Weighed-FPS (WFPS). The FPS algorihm begins by selecing a random daa poin and adding i ino he sampled se. Then, in each sep i adds he farhes daa poin from he sampled se, hus minimizing he disance beween he original daa poins and he sampled se. The resuled sampled se conains k daa poins which spans he original se. This sampling sep can be viewed as regular FPS selecion wih disance meric weighed by he paricles densiies o give prioriy

7 7 Algorihm 5.1: Muliscale PF (MSPF) Inpu: n number of paricles; x 0 iniial sae; y 1,..., y T curren observaions; q( ) proposal disribuion funcion; p( ) approximaed poserior disribuion funcion Oupu: x 1,..., x T esimaed observaions 1: Weighs iniializaion: w (i) 0 = 1 n, x(i) 0 p(x 0 ), i = 1,..., n. 2: for ime seps =1,...,T do 3: Resample n new paricles by heir disribuions deermined by he weighs w (i) 1. 4: Predicion: Apply he dynamic model q( ) o each paricle o esimae he nex sae using x 1 and y 1,..., y x (i) q(x (i) x (i) 1, y 1,..., y ), i = 1,..., n. 5: Selecion: Selec a subse of size k from he new paricles x (i) by compuing he affiniy marix A (s) (Eq. 5.1) and by using he ID Algorihm : Calculae he weighs of he k seleced paricles using w (i) p(y x (i) )p(x (i) x (i) q(x x (i) 1, y 1,..., y ) 1 ), i = 1,..., n. 7: Weigh exension: Calculae he weighs of he n k paricles using he MSE Algorihm : Weighs normalizaion: w (i) = w (i) n l=1 w(l) 9: x is se o be he paricle x (l) l = arg max w (i). 10: end for 1 i n, i = 1,..., n. where where {ϕ i } is a family of funcions ha correspond o a source funcion ϕ cenered around differen locaions x i, y j is a poin in a d-dimensional space and u i is a weigh. Direc evaluaion of he sum in Eq. 6.1 requires O(MN) operaions. In he FMM algorihm [9], we assume ha {ϕ i } can be expanded wih a mulipole series and wih a local series cenered a x and y, respecively, such ha ϕ(y) = p 1 n=0 b n (x )S n (y x ) + ϵ S (p) ϕ(y) = p 1 a n (y )R n (y y ) + ϵ R (p) n=0 (6.2) where S n and R n are he mulipole and he local basis funcions, respecively, x and y are he expansion ceners, {a n } and {b n } are he expansion coefficiens and ϵ S (p) and ϵ R (p) are he errors induced by runcaing he series afer p erms. Rewriing he sum in Eq. 6.1 using one of he expansions in 6.2 gives: v(y j ) = N u i ϕ i (y j ) i=1 = N i=1 p 1 u i n=0 c ni R n (y j y ), j = 1,..., M. (6.3) Here, c ni is he coefficien a n of ϕ i. By rearranging he expression in Eq. 6.3 we ge v(y j ) = p 1 n=0 = p 1 n=0 [ N ] u i c ni R n (y j y ) i=1 C n R n (y j y ). (6.4) The compuaion of Eq. 6.4 akes O(Mp+Np) operaions where p deermines he desired accuracy. The FMM can be used o compue he Gauss ransform efficienly and his is referred as he FGT. o dense areas. The sandard FPS algorihm is compued in O(klogn) operaions [32]. However, daa poins densiies calculaion, which uses a Gaussian kernel, akes O(n 2 ) operaions. In our implemenaion, we use he mulidimensional version of he Gauss Transform o calculae he densiy of each daa poin (in our case i is a paricle). We use he Fas Mulipole Mehod (FMM) as an efficien way o calculae he Gauss Transform ha is called he Fas Gauss Transform (FGT). This approach enables o reduce he compuaional cos of he paricle selecion sep from O(kn 2 + k 2 nlogn) o O(n + klogn) operaions. Secions 6.1 and 6.2 briefly describe he implemenaion of FMM and FGT, respecively. These descripions were adoped from [33]. 6.1 Fas Mulipole Mehod Assume ha we wan o evaluae he sum N v(y j ) = u i ϕ i (y j ), j = 1,..., M (6.1) i=1 6.2 Fas Gauss Transform (FGT) The FGT can be evaluaed by using he FMM direcly by choosing ϕ i (y) = e y xi 2 /h 2, and hen expanding he Gaussian using Hermie Polynomials. In one dimension, his yields e y x i 2 /h 2 = p 1 n=0 1 n! (x i x h ) n H n ( y x ) + ϵ(p) (6.5) h where H n are Hermie polynomials defined by H n (x) = ( 1) n e x2 d n dx (e x2 ). Exension o higher dimensions is n done by considering he mulivariae Gaussian funcion as a produc of univariae Gaussians where he series facorizaions are applied o each dimension, see [33]. Equipped wih a fas mehod o calculae he densiies, we presen in Secion 6.3 he full Weighed Farhes Poin Selecion (WFPS) algorihm for selecing he represenaives daa poins ha replaces he randomized ID Algorihm 4.2.

8 8 6.3 Weighed Farhes Poin Selecion (WFPS) Algorihm WFPS is a modificaion of he FPS Algorihm (described in Secion 6). In WFPS, he meric value is bigger for wo pairs of daa poins wih equal disance ha are locaed in a high densiy area han if hey are locaed in an area wih lower densiy. Therefore, he nex sampled daa poin in each sep in he FPS algorihm is seleced according o a densiy-weighed disance funcion. The idea of replacing he ID algorihm wih he WFPS is originaed from he empirical observaions ha choosing he mos linear independen columns of he affiniy marix by he ID is similar o choosing he acual daa poins based on disance and densiy. This can be viewed as selecing a daa poin which is he mos differen in disance erms. The densiy compuaion in he FPS algorihm is based on he observaion ha he ID algorihm favors columns of he affiniy marix ha correspond o disan daa poins wih high densiy. We found ha he WFPS algorihm yields an improved paricle selecion se in comparison o eiher randomly seleced or a regular FPS selecion. The densiy weighs cause he algorihm o prefer paricle selecions from dense areas. Similar augmenaion o he FPS algorihm is shown in [34]. Algorihm 6.1 describes he WFPS algorihm for selecing k daa poins from a se of n daa poins in R d. Algorihm 6.1: Weighed Farhes Poin Selecion (WFPS) Inpu: A se of daa poins X = {x 1,.., x n } in R d ; k he number seleced daa poins Oupu: k seleced daa poins S 1: Se w 1,..., w n o be he calculaed densiies of he daa poins in X using FGT (Eq. 6.4). 2: Se S = {x 1 }. 3: Se d s (x i ) = w i x i x 1 for all daa poins in X. 4: for sep=2,...,k do 5: Find he farhes daa poin in S: Condiion number Comparison beween he condiion number of he marix obained from choosing k rows from he kernel WFPS Deerminisic ID Randomized ID Random selecion Number of elemens seleced Fig Comparison beween he condiion number of a Gaussian kernel columns seleced by differen mehods. Condiion number Comparison beween he condiion number of he marix obained from choosing k rows from he kernel WFPS Deerminisic ID Randomized ID Random selecion Number of elemens seleced Fig Comparison beween he condiion number of a Gaussian kernel columns seleced by differen mehods (zoomed). s = arg max x X d s(x). 6: Add daa poin s o he se S. 7: Updae he disances of he daa poins in X: 8: end for d s (x i ) = min(d s (x i ), w i x i s ). Inuiively, he WFPS selecion mehod seems similar o he columns (paricles) chosen by he ID when i is applied o he affiniy marix. The reason for using ID is o find a numerically sable basis of columns (i.e. marix wih a small condiion number) o he affiniy marix. To verify ha he suggesed approach of WFPS provides a numerically sable columns (basis), we performed he following experimen: we generaed a normally disribued se of N poins (N = 1000), chose k poins ou of hem using FPS wih densiy esimaion, buil he N k affiniy marix and compued is condiion number. Then, we buil he N N affiniy marix and applied he ID o i (deerminisic and random), chose k columns from i and compued he condiion number. Nex, we compared he obained condiion number o he one obained using random sampling. Figs. 6.1 and 6.2 show ha he condiion number obained using FPS wih densiy esimaion, is close o his obained wih ID bu no as good. This provides a moivaion for he WFPS as a faser approach and a moivaion o he ID as a more accurae approach. When he selecion sep (Sep 5) in he deerminisic ID Algorihm 4.1 is replaced by he WFPS Algorihm 6.1 (where he inpu se o he WFPS is he paricle se P n ), we achieved even faser compuaional ime. The resuls are presened in Secion 7.4.

9 9 7 EXPERIMENTAL RESULTS The performance of he MSPF algorihm was evaluaed by preforming several experimens on objecs racking in boh synheic and real video sequences. The resuls were compared wih he resuls from oher racking mehods. We used a video sequence o rack a ball (Fig. 7.1), which moves in a non-linear way around a baskeball player. Each paricle is described by a vecor wih six coordinaes p = (x, y, v x, v y, w, h), which are he arge s locaion, speed in each axis, widh (w) and heigh (h), respecively. The arge s iniial sae p 0 is given as he inpu o he algorihm. Iniially, he algorihm exracs a color hisogram from a ile B T ha conains he arge. The arge s ile is a recangular defined by four poins B = {(x 1 2 w, y 1 2 h), (x w, y 1 2 h), (x 1 2 w, y h), (x w, y h)}. (7.1) This ile is used laer when i is compared o he oher color hisograms of he oher paricles. We also used he weighed Euclidean disance beween wo paricles as he disance beween he vecors ha represens hem such ha d(p (i), p (j) ) = p (i) p (j) 2. (7.2) This meric was used in he affiniy marix A (s) (Eq. 4.3) in Algorihm 4.4. In each cycle of he algorihm, we sampled he paricles and obained a new se of n paricles (see Sep 3 in Algorihm 5.1). The model equaions are applied o each paricle. In his case, i is done by adding he speed o he corresponding locaion coordinaes. Then, we perurbed each paricle by adding a Gaussian noise wih a sandard deviaion configured o each coordinae separaely. The sysem dynamics is formulaed as p = p 1 + n (7.3) where n is a random Gaussian noise vecor such ha n (i) N (0, σi 2), and σ2 i, i = 1,..., 6, represens he variance we assigned for his coordinae. For example, in a consan velociy, he variance of he velociy is zero, ha is σv 2 x = σv 2 y = 0. To calculae he weigh of each paricle p (i), i = 1,..., n, we process a ile, which is cenered a (x, y) wih widh w and heigh h defined in Eq. 7.1, by calculaing a color hisogram for all he pixels wihin he ile. Then, he hisogram is compared wih he hisogram from he original arge ile using he Bhaacharyya disance (Eq. 3.1). Therefore, he paricle s weigh is w (i) = h(b T ) T h(b i ), i = 1,..., n, (7.4) where h(b i ) is he color hisogram of ile B i. In he nex sep, we compue he weighs of he k paricles Fig A se of represenaive frames from a baskeball racking sequence. The objec is racked using he MSPF Algorihm 5.1 wih a direc compuaion of he weighs for 10% from he oal number of paricles. seleced by he randomized ID Algorihm 4.2. Their values are used for he weighs calculaion for he res of he paricles. This is done by applying he MSE wih he defined disance meric (Eq. 7.2). Once all he weighs were calculaed, hey are normalized and are used as he new disribuion values for he n paricles o be re-sampled in he nex phase. The resuls from he applicaion of he MSPF Algorihm 5.1 are applied o he baskeball sequence and hey are displayed in Fig The baskeball is being racked while he camera is moving and he background is changing consanly. The performance of he MSPF Algorihm 5.1 was esed wih differen n and k values. We found ha for mos racking asks, a small se of paricles (beween 5% o 10% from he oal number of paricles n) is sufficien as seen in Fig To measure he acceleraion of he MSPF Algorihm 5.1, he racking qualiy of each algorihm was esed along wih heir weigh compuaion ime. Each es was repeaed 10 imes and he average success racking rae was compued. The sandard PF (Algorihm 3.1) was esed wih a paricle range of paricles. The MSPF Algorihm was esed using paricles while a direc compuaion was done for 10% of he paricles. The resuls are shown in Fig We can see ha in he same compuaion ime, he racking success raio of MSPF Algorihm ouperforms he sandard PF (Algorihm 3.1). In addiion, he MSPF Algorihm 5.1 racking success graph has less jier han he sandard PF Algorihm. This is due o he fac ha he MSPF algorihm uses more paricles o cover he same sae space under similar compuaional cos hen he sandard PF Algorihm. For example, he sandard PF execuion ime wih 160

10 10 Percenage of frames wih successful racking Tracking Disance Error (RMSE) Muliscale(ID) muliscale(wfps) linear Cubic 10 Naive PF MuliScale PF Weigh Compuaion Time Fig Comparison beween he racking success rae for a given compuaional budge wih sandard PF (Alg. 3.1 which we refer o as he naive PF ) and MSPF (Alg. 5.1) Paricle Sampling Number Fig Comparison beween he RMSE for differen mehods: muliscale wih ID sampling (Alg. 4.2), muliscale wih WFPS sampling (Alg. 6.1), linear approximaion and cubic approximaion. paricles ook 80 seconds while achieving a 45% racking success rae. The MSPF achieved a 98% racking success rae when using 800 paricles where he weighs were compued for only 10% of hem while achieving he same execuion ime. 7.1 Comparison wih Oher Approximaion Mehods In order o compare beween he performance of he MSPF algorihm wih differen approximaion mehods, we esed he MSPF algorihm using differen approximaion mehods o calculae he paricle s weighs. For his comparison, we used a synheic movie. We generaed a video sequence by moving a colored disc over a sill image. The disc moved along a non-linear parameric funcion. This allows us o know he ground ruh of he arge a any frame. We applied he MSPF algorihm o he synheic video sequence several imes, each wih differen inerpolaion mehod. We compared he oal Roo Mean Square Error (RMSE) for each approximaion mehod measured on he disance beween he MSPF algorihm oupu and he real locaion of he arge. The MSPF Algorihm 5.1 achieved he lowes error rae even when we sampled beween 2%-5% paricles. When such subsampling rae was used, all he oher esed mehods fail (error grew). Nex, we compared beween he compuaional ime performing Algorihm 4.4 using differen sampling raes, by running he PF Algorihm 3.1. The weighs compuaion of all paricles ook 200 seconds (on average). From Fig. 7.4, we can see ha he MSPF Algorihm 5.1 achieved he lowes compuaional ime when he sampling rae was lower han 13% of he oal number of paricles. When he WFPS sampling was used, he compuaional ime was even beer. Overall, he MSPF Accumulaed Compuaion Time (Sec) None Muliscale(ID) Muliscale(WFPS) Paricle Sampling Facor Fig Compuaional ime of he MSPF wih differen sampling raes. The oal number of paricles is Algorihm 5.1 achieved he lowes compuaional ime while mainaining a low error rae. We repeaed he ess wih anoher video sequence where he disc locaion was se o simulae Brownian moion such ha he acceleraion was a random whie noise. The comparison beween he running ime and racking error rae showed similar resuls as in Figs. 7.3 and Muliple Targes Tracking The MSPF Algorihm 5.1 was esed on a video sequence ha conains muliple objecs. In such scenario, he racking can be achieved by using wo separae PF algorihms. Each PF uses a differen se of paricles and a separae

11 11 se of observaions. Here, each paricle describes a sae of a single arge. Anoher approach o rack muliple objecs is o creae a super-sae paricle, which describes he sae of all he objecs inside he video sequence. In his case, he number of fields inside he paricle vecor was n k where k is he number of arges and k is he number of parameers required o describe a single arge. In he laer scenario, he MSE Algorihm ouperformed he oher inerpolaion mehods since i works beer in high dimensions. The advanage of using he supersae paricle is by enabling o advance a paricle sae by dynamic model equaions ha ook ino accoun he sae of all he objecs wihin a paricle including dependencies beween objecs. Fig A seleced se of represenaive frames from he ennis game ha demonsraes he racking performance. The wo ennis players were racked by he applicaion of he MSPF Algorihm 5.1 wih a direc weighs compuaion for 10% from he oal number of paricles. In order o es he racking performance using he super-sae paricle, we racked wo ennis players in a video sequence. The players are represened by a single paricle wih 6 2 = 12 coordinaes, 6 for each player (locaion in x and y, velociy in x and y, widh and high). In each algorihmic cycle, he predicion sep advanced he paricles by he applicaion of he model equaions separaely o each coordinae. The weigh calculaion was done in each region separaely and hen muliplied he Bhaacharyya coefficien o obain a single weigh. Then, he exension sep was applied as before using he weighed Euclidean meric for each paricle ha has 12 coordinaes. By using Algorihm 5.1, we were able o rack boh arges successfully wih he lowes compuaional cos in comparison o oher exension mehods ha are based on sandard inerpolaion such as B- splines, cubics and neares neighbor. Fig. 7.5 displays he resuls from he applicaion of he MSPF Algorihm 5.1 o achieve muliple arges racking. We used 1500 paricles o rack boh players. In each sep of he algorihm, we calculaed he weighs for 150 seleced paricles and inerpolaed he weighs for he oher 1350 paricles by using he MSE Algorihm. The complee videos of he baskeball and ennis games racking can be viewed in our websie Comparison wih he EMD Measuremen Recenly, he Earh Moving Disance (EMD) [35] was used for paricles weigh compuaion since his paricle weigh fis deformable objecs [36]. The EMD compuaional cos is significanly higher han oher mehods such as color hisograms. The MSPF becomes effecive as he compuaional cos of he weighs increases. We esed Algorihm 5.1 wih he EMD meric o demonsrae how well he exension scheme fis i. Several runs were conduced on he Lemming sequence from he PROST daabase. Each run used several frames execued on i7-2630qm 2.9GHz processor. Weighs were calculaed for 10% from he oal number of paricles while he res of he paricles were esimaed using he MSE Algorihm. We verified ha he arge was no los during he racking procedure in each execuion when using MSPF algorihm. Table 7.1 shows he ime differences beween he sandard version of he PF algorihm ha uses he EMD meric (Algorihm 3.1) and our implemenaion ha uses he MSE mehod (Algorihm 4.2). For he laer, 10% of he paricles were sampled, and he MSE was applied o he oher 90% of he paricles. We can see ha he MSE algorihm reduces he PF oal compuaion ime. However, we can also observe ha when he number of paricles increases, he acceleraion becomes less significan, as seen in he second and hird columns. We analyze his scalabiliy issue in Secion Weighed FPS in he Selecion Sep Table 7.1 compares beween he running imes of he WFPS Algorihm 6.1 and he randomized ID as he selecion mehods. Each ime he MSPF Algorihm 5.1, which uses he EMD, was esed wih a differen paricle se sizes. Performance comparisons were done beween he following algorihms: sandard PF, PF wih MSE and ID selecion, PF wih MSE and WFPS selecion. Table 7.1 shows ha he acceleraion facor is high even when 10, 000 paricles are used. 1. hp://

12 12 TABLE 7.1 Comparison beween WFPS and ID acceleraion imes [sec], in he MSPF algorihm ha uses EMD. Sampling rae was 10% from he oal number of paricles. # of Time Time Time Acceleraion Paricles [No MSE] [MSE-ID] [MSE-WFPS] Facor Alg. 3.1 Alg. 4.2 Alg CONCLUSION In his work, several conribuions are presened. The PF compuaional ime was reduced by he applicaion of MSE mehod ha reduces he load of he paricle weigh calculaion. Therefore, i allows us o uilize more paricles wihin a given compuaional budge. This improves he PF performance. The modified PF algorihm was esed on real video sequences o successfully rack a single and muliple arges. In addiion, he performance of he PF was compared wih oher exension mehods and demonsraed ha he MSE managed o rack he arge wih much fewer paricles hen PF was using. These enhancemens can become effecive when muliple arges are racked in real-ime. ACKNOWLEDGMENT This research was parially suppored by he Israel Science Foundaion (Gran No. 1041/10), by he Israeli Minisry of Science & Technology (Grans No , ), by US - Israel Binaional Science Foundaion (BSF ) and by a Fellowship from Jyväskylä Universiy. REFERENCES [1] M. Arulampalam, S. Maskell, N. Gordon, and T. Clapp, A uorial on paricle filers for online nonlinear/non-gaussian bayesian racking, IEEE Transacions on Signal Processing, vol. 50, no. 2, pp , [2] H. Cheng, Z. Gimbuas, P. Marinsson, and V. Rokhlin, On he compression of low rank marices, SIAM Journal on Scienific Compuing, vol. 26, no. 4, pp , [3] A. Bermanis, A. Averbuch, and R. Coifman, Muliscale daa sampling and funcion exension, Applied and Compuaional Harmonic Analysis, vol. 34, pp , [4] C. Baker, The numerical reamen of inegral equaions. Clarendon press Oxford, 1977, vol. 13. [5] B. Flannery, W. Press, S. Teukolsky, and W. 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13 13 [34] Y. Eldar, M. Lindenbaum, M. Pora, and Y. Zeevi, The farhes poin sraegy for progressive image sampling, IEEE Transacions on Image Processing, vol. 6, no. 9, pp , [35] Y. Rubner, C. Tomasi, and L. Guibas, The earh mover s disance as a meric for image rerieval, Inernaional Journal of Compuer Vision, vol. 40, no. 2, pp , [36] S. Avidan, D. Levi, A. Bar-Hillel, and S. Oron, Locally orderless racking, in 2012 IEEE Conference on Compuer Vision and Paern Recogniion. IEEE, 2012, pp Gil Shaba is a PhD Suden in Tel Aviv Universiy. Gil was born in Tel Aviv, Israel. He received his B.Sc and M.Sc degrees in Elecrical Engineering from Tel-Aviv universiy, in 2002 and 2008, respecively. Prior o his PhD research, Gil was working in Applied Maerials as a compuer vision algorihm researcher. His research ineress include low rank approximaions, fas randomized algorihms, signal/image processing and machine learning. mehods. Yaniv Shmueli is a Ph.D. candidae a he Compuer Science School, Tel Aviv Universiy. He received his B.Sc. and M.Sc. degrees in Mahemaics and Compuer Science from Tel Aviv Universiy, Israel, in 1997 and 2003, respecively. Before saring his Ph.D. research, Yaniv worked a Kenshoo as R&D manager. Prior o ha, Yaniv led he cable sofware group a Texas Insrumens. His curren research ineress include dimensionaliy reducion, diffusion maps, machine learning, paricle filers and marix facorizaion Ami Bermanis received he B.Sc. degree in mahemaics and compuer science from he Technion-Israel Insiue of Technology, Haifa, Israel, in 2003 and he M.Sc. and Ph.D. degrees in applied mahemaics from Tel-Aviv Universiy, Tel-Aviv, Israel, in 2007 and 2012, respecively. He is a posdocoral fellow in he deparmen of compuer science, Tel Aviv Universiy, Tel-Aviv, Israel. His research ineress include mehods for Big Daa sampling and analysis. Amir Averbuch was born in Tel Aviv, Israel. He received he B.Sc and M.Sc degrees in Mahemaics from he Hebrew Universiy in Jerusalem, Israel in 1971 and 1975, respecively. He received he Ph.D degree in Compuer Science from Columbia Universiy, New York, in During and he served in he Israeli Defense Forces. In he was a Research Saff Member a IBM T.J. Wason Research Cener, Yorkown Heighs, in NY, Deparmen of Compuer Science. In 1987, he joined he School of Compuer Science, Tel Aviv Universiy, where he is now Professor of Compuer Science. His research ineress include applied harmonic analysis, big daa processing and analysis, waveles, signal/image processing, numerical compuaion and scienific compuing (fas algorihms).