GloballyOptimal Greedy Algorithms for Tracking a Variable Number of Objects


 Sharlene Evans
 3 years ago
 Views:
Transcription
1 GloballyOpimal Greedy Algorihm for Tracking a Variable Number of Objec Hamed Piriavah Deva Ramanan Charle C. Fowlke Deparmen of Compuer Science, Univeriy of California, Irvine Abrac We analyze he compuaional problem of muliobjec racking in video equence. We formulae he problem uing a co funcion ha require eimaing he number of rack, a well a heir birh and deah ae. We how ha he global oluion can be obained wih a greedy algorihm ha equenially inaniae rack uing hore pah compuaion on a flow nework. Greedy algorihm allow one o embed preproceing ep, uch a nonmax uppreion, wihin he racking algorihm. Furhermore, we give a nearopimal algorihm baed on dynamic programming which run in ime linear in he number of objec and linear in he equence lengh. Our algorihm are fa, imple, and calable, allowing u o proce dene inpu daa. Thi reul in aeofhear performance. 62 rack deah rack deah d rack birh c b 62 rack birh rack birh 6 a e f Figure 1. We rea he problem of muliarge racking hrough a perpecive of paioemporal grouping, where boh a large number of group and heir paioemporal exen (e.g., he number of objec and heir rack birh and deah) mu be eimaed. We how he oupu of an efficien, linearime algorihm for olving hi compuaional problem on he ETHMS daae []. In hi video clip our mehod reurn hundred of correc rack, a eviden by he overlaid rack number. 1. Inroducion Our conribuion i grounded in a novel analyi of an ineger linear program (ILP) formulaion of muliobjec racking [14, 25, 3, 17, 2, 18]. Our work mo cloely follow he minco flow algorihm of [25]. We how ha one can exploi he pecial rucure of he racking problem by uing a greedy, ucceive horepah algorihm o reduce he bepreviou running ime of O(N 3 log2 N ) o O(KN log N ), where K i he unknown, opimal number of unique rack, and N i he lengh of he video equence. The inuiion behind he greedy approach em from hi urpriing fac (Fig.2): he opimal inerpreaion of a video wih k + 1 rack can be derived by a local modificaion o he oluion obained for k rack. Guided by hi inigh, we alo inroduce an approximae greedy algorihm whoe running ime cale linearly wih equence lengh (i.e., O(KN )), and i in pracice everal order of magniude faer wih no obervable lo in accuracy. Finally, our greedy algorihm allow for he embedding of variou preproceing or poproceing heuriic (uch a nonmaximum uppreion) ino he racking algorihm, which can boo performance. We conider he problem of racking a variable number of objec in a video equence. We approach hi ak a a paioemporal grouping problem, where all image region mu be labeled a background or a a deecion belonging o a paricular objec rack. From uch a grouping perpecive, one mu explicily eimae (a) he number of unique rack and (b) he paioemporal exen, including he ar/erminaion ime, of each rack (Fig.1). Approache o accomplihing he above ak ypically employ heuriic or expenive algorihm ha cale exponenially in he number of objec and/or uperlinearly in he lengh of he video. In hi paper, we ouline a family of muliobjec racking algorihm ha are: 1. Globally opimal (for common objecive funcion) 2. Locally greedy (and hence eay o implemen) 3. Scale linearly in he number of objec and (quai)linearly wih videolengh 1201
2 2. Relaed Work Claic formulaion of muliobjec racking focu on he daa aociaion problem of maching inance label wih emporal obervaion [11, 6, 7, 13]. Many approache aume manual iniializaion of rack and/or a fixed, known number of objec [14]. However, for many realworld racking problem, uch informaion i no available. A more general paioemporal grouping framework i required in which hee quaniie are auomaically eimaed from video daa. A popular approach o muliobjec racking i o run a lowlevel racker o obain rackle, and hen ich ogeher rackle uing variou graphbaed formalim or greedy heuriic [15, 22, 16, 1, 2]. Such graphbaed algorihm include flownework [25], linearprogramming formulaion [14], and maching algorihm [15]. One of he conribuion of hi paper i o how ha wih a paricular choice of lowlevel racker, and a paricular chedule of rack inaniaion, uch an algorihm can be globallyopimal. We rely on an increaingly common ILP formulaion of racking [14, 25, 3, 17, 2, 18]. Such approache reric he e of poible objec locaion o a finie e of candidae window on he pixel grid. Becaue andard linear programming (LP) relaxaion do no cale well, many algorihm proce a mall e of candidae, wih limied or no occluion modeling. Thi can produce broken rack, ofen requiring a econd merging age. Our calable algorihm i able o proce much larger problem and direcly produce aeofhear rack. Our work relie heavily on he minco flow nework inroduced for emporal daa aociaion in [25]. We compare our reul wih he minco olver ued in ha work [12], and verified ha our O(KN log N) algorihm produce idenical reul, and ha our approximae O(KN) algorihm produce nearidenical reul when properly uned. In concurren work, Berclaz e al. decribe a O(KN log N) algorihm for muliobjec racking in [4]. I i imilar in many repec wih ome difference: Our graph repreenaion ha a pair of node for each deecion. Thi allow u o explicily model objec dynamic hrough raniion co, and allow for a impler flowbaed analyi. In addiion, our algorihm inaniae rack in a greedy fahion, allowing for he inegraion of preproceing ep (e.g., nonmaxuppreion) ha improve accuracy. Finally, we alo decribe approximae O(KN) algorihm ha perform nearidenical in pracice. 3. Model We define an objecive funcion for muliobjec racking equivalen o ha of [25]. The objecive can be derived from a generaive perpecive by conidering a Hidden Markov 3 rack eimae 4 rack eimae x Figure 2. The inuiion behind our opimal greedy algorihm. Aume ha we are racking he x locaion of muliple objec over ime. On he lef, we how he opimal eimae of 3 objec rajecorie. Given he knowledge ha an addiional objec i preen, one may need o adju he exiing rack. We how ha one can do hi wih a horepah/minflow compuaion ha puhe flow from a ource o a erminal (middle). The oluion can revere flow along exiing rack o cu and pae egmen, producing he opimal 4rack eimae (righ). We furher peed up hi proce by approximaing uch edi uing fa dynamic programming algorihm. Model (HMM) whoe ae pace i he e of rue objec locaion a each frame, along wih a prior ha pecifie likely ae raniion (including birh and deah) and an obervaion likelihood ha generae dene image feaure for all objec and he background Independen rack We wrie x for a vecorvalued random variable ha repreen he locaion of a paricular objec, a given by a pixel poiion, cale, and frame number: x = (p, σ, ) x V (1) where V denoe he e of all paceime locaion in a video. Prior: We wrie a ingle rack a an ordered e of ae vecor T = (x 1,... x N ), ordered by increaing frame number. We wrie he collecion of rack a a e X = {T 1,... T K }. We aume ha rack behave independenly of each oher, and ha each follow a variablelengh Markov model: P (X) = P (T ) T X where P (T ) = P (x 1 ) ( N 1 n=1 ) P (x n+1 x n ) P (x N ) The dynamic model P (x n+1 x n ) encode a moohne prior for rack locaion. We wrie P (x 1 ) for he probabiliy of a rack aring a locaion x 1, and P (x N ) for he probabiliy of a rack raniioning ino a erminaion ae from locaion x N. If he probabiliy of erminaion i low, he above prior will end o favor longer, bu fewer rack o a o minimize he oal number of erminaion. If hee probabiliie are dependen on he paial coordinae of x, hey can model he fac ha rack end o erminae near image border or ar near enry poin uch a doorway. 1202
3 Likelihood: We wrie Y = {y i i V } for he e of feaure vecor oberved a all paceime locaion in a video. For example, hee could be he e of gradien hiogram feaure ha are cored by a lidingwindow objec deecor. We now decribe a likelihood model for generaing Y given he e of rack X. We make wo aumpion: 1) here exi a oneoone mapping beween a puaive objec ae x and paceime locaion index i and 2) rack do no overlap (T k T l = for k l). Togeher, boh imply ha a locaion can be claimed by a mo one rack. We wrie y x for he image feaure a locaion x; hee feaure are generaed from a foreground appearance model. Feaure vecor for unclaimed window are generaed from a background model: ( ) P (Y X) = P fg (y x ) P bg (y i ) (2) T X x T = Z T X x T l(y x ) where l(y x ) = P fg(y x ) P bg (y x ) and i V \X Z = i P bg (y i ) The likelihood i, up o a conan, only dependen on feaure of he window which are par of he e of rack. If we aume ha he foreground and background likelihood are Gauian deniie wih he ame covariance, P fg (y x ) = N(y x ; µ fg, Σ) and P bg (y x ) = N(y x ; µ bg, Σ), we can wrie he loglikelihoodraio a a linear funcion (log l(y x ) = w y x ), akin o a logiic regreion model derived from a clacondiional Gauian aumpion. Thi model provide a generaive moivaion for he linear emplae ha we ue a local deecor in our experimen Track inerdependence The above model i reaonable when he rack do no overlap or occlude each oher. However, in pracice we need o deal wih boh occluion and nonmaxima uppreion. Occluion: To model occluion, we allow rack o be compoed of ae vecor from nonconecuive frame e.g., we allow n and n+1 o differ by up o k frame. The dynamic model P (x n+1 x n ) for uch kframe kip capure he probabiliy of oberving he given kframe occluion. Nonmaxima uppreion: When we conider a dene e of locaion V, here will be muliple rack which core well bu correpond o he ame objec (e.g., a good rack hifed by one pixel will alo have a high probabiliy mach o he appearance model). A complee generaive model could accoun for hi by producing a cluer of image feaure around each rue objec locaion. Inference would explain away evidence and enforce excluion. In pracice, he ypical oluion i o apply nonmax uppreion (NMS) a a preproce o prune he e of candidae locaion V prior o muliobjec racking [1, 6, 14, 25]. In our experimen, we alo uilize NMS o prune he e V and a a heuriic for explaining away evidence. However, we how ha he NMS procedure can be naurally embedded wihin our ieraive algorihm (raher han a a preproce). By uppreing exra deecion around each rack a i i inanced, we allow for he poibiliy ha he prior can override he obervaion erm and elec a window which i no a local maxima. Thi allow he NMS procedure o exploi emporal coherence. The recen work of [2] make a imilar argumen and add an explici nonoverlapping conrain o heir ILP, which may acrifice racabiliy. We demonrae in Sec. 6 ha our imple and fa approach produce aeofhear reul. 4. MAP Inference The maximuim a poeriori (MAP) eimae of rack given he collecion of oberved feaure i: X = argmax P (X)P (Y X) (3) X = argmax P (T ) l(y x ) X T X x T (4) = argmax X log P (T ) + log l(y x ) (5) T X x T We drop he conan facor Z and ake logarihm of he objecive funcion o implify he expreion while preerving he MAP oluion. The above can be rewrien a an Ineger Linear Program: f = argmin C(f) (6) f wih C(f) = c i fi + c ij f ij + c i f i + c ifi i ij E i i (7).. f ij, f i, f i, f i {0, 1} and f i + j f ji = f i = f i + j f ij (8) where f i i a binary indicaor variable ha i 1 when paceime locaion i i included in ome rack. The auxiliary variable f ij along wih he econd conrain (8) enure ha a mo one rack claim locaion i, and ha muliple rack may no pli or merge. Wih a ligh abue of noaion, le u wrie x i for he puaive ae correponding o locaion i: c i = log P (x i ), c i = log P (x i ), () c ij = log P (x j x i ), c i = log l(y i ). 1203
4 frame 1 frame 2 frame 3 Figure 3. The nework model from [25] for hree conecuive frame of video. Each paceime locaion i V i repreened by a pair of node conneced by a red edge. Poible raniion beween locaion are modeled by blue edge. To allow rack o ar and end a any paioemporal poin in he video, each locaion i i conneced o boh a ar and erminaion node. All edge are direced and uni capaciy. The co are c i for red edge, c ij for blue edge and c i and c i for black edge. encode he rack ar, erminae, raniion, and obervaion likelihood repecively. We define he edge e E o pan he e of permiible ae raniion given by our dynamic model (Sec.3.1) Equivalence o nework flow To olve he above problem, we can relax he ineger conrain in (8) o linear box conrain (e.g., 0 f i 1) Thi relaxaion yield a uni capaciy nework flow problem whoe conrain marix i oally unimodular, implying ha opimal oluion o he relaxed problem will ill be inegral [1]. In paricular, aume ha we knew he number of rack in a video o be K. Le F K denoe he e of flow conervaion and uni capaciy conrain along wih he addiional conrain { F K = f ij, f i, fi, f i [0, 1], i f i = K, fi + j f ji = f i = fi + j f ij, i f i = K Minimizing C(f) ubjec o conrain F K i an inance of a minimum co flow problem [1, 25]. Such problem are imilar o maxflow problem (commonly ued in viion for olving graphcu problem [5]), excep ha edge in he flow nework are labeled wih a co a well a capaciy. The co of a flow i defined o be he um, over all edge, of he co of each edge muliplied by he flow hrough ha edge. Finding he MAP eimae of K rack correpond o finding a minimum co flow ha puhe K uni of flow from he ource o he ink. Figure 3 how an example flow nework conruced from he racking problem. Each paceime locaion i, or equivalenly puaive objec ae x i, correpond o a pair of node (u i, v i ) conneced by an edge of co c i. Each raniion beween ucceive window i repreened by an edge (v i, u j ) wih co c ij. Finally, node and are inroduced wih edge (, u i ) correponding o rack ar and edge (v i, ) for erminaion (wih co c i and c i repecively). All edge have uni capaciy. Puhing K uni of flow from o yield a e of K dijoin pah, each of which correpond o one of he opimal rack T X. 5. Finding minco flow Zhang e al. [25] decribe how o olve he above opimizaion problem in O(mn 2 log n) ime uing a puhrelabel mehod [12], where n i he number of node (e.g. deecion window) in he nework graph and m i he number of edge. Auming ha n and m cale linearly wih he number of frame N (reaonable given a fixed number of deecion per frame), he algorihm ake O(N 3 log N) o find K rack. Furhermore, he co of he opimal oluion, min f FK C(f) i convex in K [25] o one can ue a biecion earch over K (upperbounded by he number of deecion) o find he opimal number of rack wih a oal running ime O(N 3 log 2 N). In he following, we how ha one can olve he muliobjec racking problem in O(KN log N) by olving K +1 horepah problem. Thi coniderable reducion in complexiy i due o wo paricular properie of he nework in Fig.3: 1. All edge are uni capaciy. 2. The nework i a direced acyclic graph (DAG). The above condiion allow one o ue dynamic programming (DP) algorihm o compue hore pah. We decribe a novel DP algorihm ha i neceary o conruc a globallyopimal O(KN log N) algorihm. We alo how ha DP produce he opimal oluion for K = 1 in O(N) and highqualiy approximae oluion for K > 1 in O(KN). We begin by decribing he opimal O(KN log N) algorihm baed on ucceive hore pah (inroduced in Fig.2) Succeive Shorepah We now decribe a ucceive hore pah algorihm [1] for olving minco flow problem for DAG nework wih unicapaciy link. Given a graph G wih an inegral flow f, define he reidual graph G r (f) o be he ame a he original graph excep ha all edge ued in he flow f are revered in direcion and aigned negaive heir original co. We iniialize he algorihm by eing he flow f o be zero and hen ierae he following wo ep: 1. Find he minimumco pah γ from o in G r (f) 2. If oal co of he pah C(γ) i negaive, updae f by puhing uniflow along γ unil no negaive co pah can be found. Since each pah ha uni capaciy, each ieraion increae he oal flow by 1 and 1204
5 decreae he objecive by C(γ). The algorihm erminae afer K + 1 ieraion having found a minimum co flow. Puhing any furher flow from o will only increae he co. We refer he reader o [1] for a proof of he correcne of he algorihm bu give a brief ouline. We ay a flow f i F K feaible if i aifie he conrain e F K. A neceary and ufficien condiion for f o be a minimum co flow of ize K i ha i be F K feaible and ha here doe no exi a negaiveco direced cycle in G r (f). The ucceive horepah algorihm above ar wih a F 0 feaible flow and a each ieraion i yield a new flow which i F i  feaible. Furhermore, each ep of he algorihm modifie edge along a ingle pah and can be hown o no inroduce any negaive weigh cycle. Figure 4 how example ieraion of hi algorihm and he reuling equence of reidual graph. Noe ha he hore pah in he reidual nework may inance a new rack and/or edi previou rack by removing flow from hem (by puhing flow hrough he revere edge). In each ieraion, we need o find a hore pah. We would like o ue Dijkra algorihm o compue he hore pah in O(N log N), making he overall algorihm O(KN log N) where K i he opimal number of rack. Unforunaely, here are negaive edge co in our original nework, precluding he direc applicaion of Dijkra algorihm. Forunaely, one can conver any minco flow nework o an equivalen nework wih nonnegaive co [1]. Thi converion require compuing he horepah of every node from in he original graph G. For general graph wih negaive weigh, hi compuaion ake O(N 2 ) uing he BellmanFord algorihm [1]. For DAG, one can ue a O(N) dynamic programming algorihm, which we decribe below. The ucceive hore pah algorihm hu run in O(KN log N) operaion and reurn he global minima for our racking problem (Equaion 3) Dynamic Programming Soluion for K = 1 We now preen a O(N) dynamic programming (DP) algorihm for compuing he hore pah of every node o. We will alo how ha hi algorihm olve he min co flow problem for K = 1. Becaue each edge in he nework i of uni capaciy, he minimum co uni flow mu correpond o he hore pah from node o. Becaue he original nework graph i a DAG, one can conruc a parial ordering of node and ue DP o compue hore pah by weeping from he fir o la frame. Thi i imilar o DP algorihm for racking bu augmened o eimae boh he birh and deah ime of a rack. Aume ha node are ordered in ime, and le co(i) repreen he minimum co of a rack paing hrough node i. We iniialize co(i) for deecion in he fir frame o be co(i) = c i + c i. We can hen recurively compue he (a) (c) (e) Figure 4. Illuraion of ucceive hore pah algorihm. (a) The racking problem modeled a a graph a decribed in Fig.3. The algorihm hould end a given amoun of flow from ource node o he erminal. (b) One uni of flow f 1 i paed hrough he hore pah (in red) from ource o erminal. (c) The reidual graph G r(f 1) produced by eliminaing he hore pah and adding edge (in green) wih uni capaciy and negaive co wih he oppoie direcion. (d) The hore pah found in he reidual graph. In hi example, hi pah ue previouly added edge, puhing flow backward and ediing he previouly inanced rack. (e) Reidual graph afer paing wo uni of flow. A hi poin, no negaive co pah exi and o he algorihm erminae and reurn he wo rack highlighed in (f). Noe ha he algorihm ulimaely pli he rack inanced in he fir ep in order o produce he final opimal e of rack. In hi example only one pli happened in an ieraion, bu i i poible for a hore pah o ue edge from wo or more previouly inanced rack, bu i i very rare in pracice. Our dynamic programming algorihm canno reolve any pliing ince he reidual graph ha cycle, however he 2pa dynamic programming algorihm can reolve he iuaion when any new hore pah pli a mo one previouly inanced rack. co in ucceive frame a: co(i) = c i + min(π, c i ) where π = min c ij + co(j) j N(i) (10) where N(i) i he e of deecion from he previou k frame ha can raniion o deecion i. The co of he opimal ending a node i i hen co(i) + c i, and he overall hore pah i compued by aking a min over i. By caching he argmin value a each node, we can reconruc he hore pah o each node in a ingle backward weep. (b) (d) 5.3. Approximae DP oluion for K > 1 We now propoe a imple greedy algorihm o inance a variable, unknown number of dijoin, lowco rack. Sar wih he original neworkflow graph: (f) 1205
6 1. Find he hore pah from o uing DP If he co of he pah i negaive, remove node on he pah and repea. rack birh rack deah The above algorihm perform K + 1 ieraion of DP o dicover K rack he la inanced rack i ignored ince i increae he overall co. I running ime i O(KN ). A each ieraion, we have obained a feaible (bu no necearily minimum co) kuni flow. The ubopimaliy lie in he fac ha he above algorihm canno adju any previouly inanced rack baed on he demand o produce addiional rack. In ucceive age, i operae on a ube of he original graph raher han he reidual graph ued in he ucceive hore pah algorihm. Unforunaely dynamic programming can be direcly applied o he reidual graph Gr (f ) ince he reidual graph i no longer a DAG (Fig.4(c)). a rack deah c b rack deah d e d Figure 5. We how he reul of our algorihm, including eimaed rack birh and deah, on he Calech Pederian daae [8]. We how ypical reul on he ETHMS daae in Fig.1. men, hi decreaed compuaion ime by hree order of magniude. 6. Experimenal Reul 5.4. Approximae 2pa DP oluion for K > 1 Daae: Mo benchmark for muliobjec racking (e.g., PETS [24]) are deigned for aionary camera. We are inereed in moving camera applicaion, and o ue he Calech Pederian daae [8] and ETHMS daae [] o evaluae our algorihm. The Calech daae wa capured by a camera inalled on a moving car. I conain 71 video of roughly 1800 frame each, capured a 30 frame per econd. Since he ee conain heldou label, we evaluae ourelve uing all annoaed pederian on he raining e. The ETHMS daae conain fooage of a buy idewalk a een by a camera mouned on a child roller. Alhough hi daae conain boh lef and righ view o faciliae ereo, we ue only he lef view in our experimen. The daae conain four video of roughly 1000 frame each, capured a 14 fp. Boh daae include bounding box annoaion for people, while Calech alo provide rack ID. We manually annoaed ID on a porion of ETHMS. In order o compare our reul wih previou work, we ue he ame ETHMS video equence a [25] wih frame and ignore deecion maller han 24 pixel a hey did. Seup: We ran an ouofhebox prerained parbaed HOG pederian deecor [10] wih a conervaive NMS hrehold, generaing around 1000 deecion per frame of each video. We e he loglikelihood raio (he local co ci ) of each deecion o be he negaive core of he linear deecor (he diance from he deciion boundary of an SVM). We ue a boundedvelociy dynamic model: we define he raniion co cij o be 0, bu only connec candidae window acro conecuive frame ha paially overlap. We e birh and deah co (ci, ci ) o be 10. We experimened wih applying an addiional NMS ep wihin our greedy algorihm. We alo experimened wih occluion modeling by adding raniion which kip over k frame, wih k up o 10. We now decribe generalizaion of our DPbaed algorihm from 5.3 ha can alo inance new rack while performing mall edi of previouly inanced rack. We oberve ha mo of he ime he hore reidual pah doe no make large edi on previou rack. We ue he ame algorihm from Secion 5.1, bu perform an approximae horepah uing a 2pa DP algorihm raher han Dijkra algorihm. We perform a forward pa of DP a in (10), bu on Gr (f ) raher han G wih co(i) defined a he be forwardprogreing pah from he ource o node i (ignoring revered edge). We hen ue he co a iniial value for a backward pa aring from he la frame, defining N (i) o be he e of node conneced hrough revere edge o i. Afer hi pa, co(i) i he co of he be forward and backward progreing pah ending a i. One could add addiional pae, bu we find experimenally ha wo pae are ufficien for good performance while aving O(log N ) operaion over Dijkra approach Caching Our DP algorihm repeaedly perform compuaion on a erie of reduced or reidual graph. Much of hee compuaion can be cached. Conider he DP compuaion required for he algorihm from Secion 5.3. Once a rack i inanced, co(i) value for node whoe horepah inerec ha rack are no longer valid, and i i only hi mall number of node ha need o be reevaluaed in he nex ieraion. Thi e can be marked uing he following fac: any pah ha inerec a ome node mu hare he ame birh node. Each node can be labeled wih i birh node by propagaing a birh ID during meagepaing in DP. We hen only need o recompue co(i) for node ha have he ame birh node a a newly inanced rack. In our experi1206
7 Co DP (min a 444) 2 pa DP (min a 516) Succeive hore pah (min a 522) Co Number of rack Number of rack Figure 6. Co v. ieraion number for all hree algorihm on Calech daae. The ine how ha our 2pa DP algorihm produce rack whoe co i cloe o opimum while being order of magniude faer. Deecion Rae DP SSP HOG Deecion Rae Fale Poiive Per Frame DP SSP Fale Poiive Per Frame DP+NMS HOG Figure 7. Deecion rae veru FPPI on Calech daae [8] (lef) and ETHMS daae [] (righ). We compare our approximae 1 pa DP algorihm wih he opimal ucceive hore pah (SSP) algorihm and a HOGdeecor baeline. The DP perform a well a or even beer han he hore pah algorihm, while being order of magniude faer. We alo how ha by uppreing overlapping deecion afer each rack i inanced (DPNMS), we can furher improve performance. Scoring crieria: We ue deecion accuracy (a meaured by deecion rae and fale poiive per frame) a our primary evaluaion crieria, a i allow u o compare wih a wide body of relaed work on hee daae. To direcly core racker accuracy, variou oher crieria (uch a rack fragmenaion, ideniy wiching, ec.) have been propoed [21, 20, 25]. We alo ue rack ideniy o evaluae our algorihm below. Approximaion qualiy: We have decribed hree differen algorihm for olving he minimum co flow problem. Figure 6 how he flow co, i.e., he objecive funcion, veru ieraion number for all hree algorihm on he Calech daae. The DP algorihm follow he ucceive hore pah (SSP) algorihm for many ieraion bu evenually i i neceary o edi a previouly inanced rack (a in Figure 4) and he greedy DP algorihm begin o make ubopimal choice. However DP and SSP do no deviae much before reaching he minimum co and he 2pa DP which allow for a ingle edi follow SSP quie cloely. Thi figure ine how a cloe look a he co a he minimum. Since he 2pa algorihm can pli a mo one rack in each ieraion and i i very rare o ee wo pli a he ame ieraion, he co value for 2pa DP algorihm i very cloe o he opimum one. Raher han coring he co funcion, we can direcly compare algorihm uing rack accuracy. Figure 7 how deecion rae veru FPPI for he baeline deecor, DP, and SSP algorihm. Thee figure how ha DP and SSP are imilar in accuracy, wih DP performing even beer in ome cae. We upec he SSP algorihm produce (overly) hor rack becaue he 1 order Markov model enforce a geomeric diribuion over rack lengh. The approximae DP algorihm inadverenly produce longer rack (ha beer mach he ground ruh diribuion of lengh) becaue previouly inanced rack are never cu or edied. We henceforh evaluae our onepa DP algorihm in he ubequen experimen. We alo preen addiional diagnoic Lengh of % of window allowable occluion wih ID error Table 1. Evaluaing rack label error a a funcion of he lengh of he allowable occluion. We how reul for our DP algorihm applied o a porion of he ETHMS daae given ideal deeced window. Our DP algorihm cale linearly wih he lengh of allowable occluion. By allowing for longer occluion (common in hi daae), he % of window wih correc rack label ignificanly increae. experimen on he ETHMS daa, ince i conain on average more objec han Calech. Track ideniie: We evaluae rack ideniie on he ETHMS daae by uing our racker o compue rack label for groundruh bounding boxe. Thi i equivalen o running our racker on an ideal objec deecor wih zero mied deecion and fale poiive. Given a correpondence beween eimaed rack label and groundruh rack label, he miclaificaion rae i he fracion of bounding boxe wih incorrec label. We compue he correpondence ha minimize hi error by biparie maching [15]. We found occluion modeling o be crucial for mainaining rack ideniie. Our algorihm can repor rack wih kframe occluion by adding in raniion beween paceime window paced k frame apar. Our DP algorihm cale linearly wih k, and o we can readily model long 10frame occluion (Table 1). Thi grealy increae he accuracy of rack label on hi daa becaue uch occluion are common when nearby people pa he camera, occluding people furher away. Thi reul implie ha, given ideal local deecor, our racking algorihm produce rack ideniie wih 0% accuracy. NMSwihinheloop: In Figure 7, we ue he ETHMS daae o examine he effec of adding a NMS ep wihin 1207
8 Algorihm Deecion rae Fale poiive per frame [] ereo algorihm [25] algorihm [25] algorihm 2 wih occluion handling [23] woage algorihm wih occluion handling Our DP Our DP+NMS Table 2. Our algorihm performance compared o he previou aeofhear on he ETHMS daae. Pleae ee he ex for furher dicuion. our ieraive greedy algorihm. When applying [10] pederian deecor, we ue heir defaul NMS algorihm a a preproce o uppre deecion ha overlap oher highercoring deecion by ome hrehold. Afer inancing a rack during he DP algorihm, we uppre remaining window ha overlap he inanced rack uing a lower hrehold. Thi uppreion i more reliable han he iniial one becaue racked window are more likely o be rue poiive. Our reul ouperform all previouly publihed reul on hi daa (Table 2). Running ime: For he frame ETHMS daae, MATLAB LP olver doe no converge, he commercial mincoflow olver ued in [23] ake 5 econd, while our MATLAB DP code ake 0.5 econd. 7. Concluion We have decribed a family of efficien, greedy bu globally opimal algorihm for olving he problem of muliobjec racking, including eimaing he number of objec and heir rack birh and deah. Our algorihm are baed on a novel analyi of a minco flow framework for racking. Our greedy algorihm allow u o embed preproceing ep uch a NMS wihin our racking algorihm. Our calable algorihm alo allow u o proce large inpu equence and model long occluion, producing aeofhear reul on benchmark daae. Acknowledgemen: Funding for hi reearch wa provided by NSF Gran 0540 and , and ONR MURI Gran N Reference [1] R. Ahuja, T. Magnai, and J. Orlin. Nework flow: Theory, Algorihm, and Applicaion. Prenice Hall, [2] A. Andriyenko and K. Schindler. Globally opimal muliarge racking on a hexagonal laice. In ECCV, [3] J. Berclaz, F. Fleure, and P. Fua. Muliple objec racking uing flow linear programming. In Performance Evaluaion of Tracking and Surveillance (PETSWiner), 200 Twelfh IEEE Inernaional Workhop on, page 1 8. IEEE, [4] J. Berclaz, F. Fleure, E. Türeken, and P. Fua. Muliple Objec Tracking uing KShore Pah Opimizaion. IEEE Tranacion on PAMI, Acceped for publicaion in [5] Y. Boykov, O. Vekler, and R. Zabih. Fa approximae energy minimizaion via graph cu. IEEE PAMI, [6] Y. Cai, N. de Freia, and J. Lile. Robu viual racking for muliple arge. Lecure Noe in Compuer Science, 354:107, [7] W. Choi and S. Savaree. Muliple arge racking in world coordinae wih ingle, minimally calibraed camera. ECCV 2010, page 553 5, [8] P. Dollár, C. Wojek, B. Schiele, and P. Perona. Pederian deecion: A benchmark. In IEEE CVPR, June 200. [] A. E, B. Leibe, and L. Van Gool. Deph and appearance for mobile cene analyi. In ICCV, [10] P. Felzenzwalb, D. McAlleer, and D. Ramanan. A dicriminaively rained, mulicale, deformable par model. IEEE CVPR, [11] T. Formann, Y. BarShalom, and M. Scheffe. Sonar racking of muliple arge uing join probabiliic daa aociaion. IEEE Journal of Oceanic Engineering, 8(3):1 184, 1. [12] A. Goldberg. An efficien implemenaion of a caling minimumco flow algorihm. Journal of Algorihm, 22(1):1 2, 17. [13] M. Iard and J. MacCormick. Bramble: A bayeian mulipleblob racker. In ICCV, [14] H. Jiang, S. Fel, and J. Lile. A linear programming approach for muliple objec racking. In IEEE CVPR, [15] H. Kuhn, P. Haa, I. Ilya, G. Lohman, and V. Markl. The Hungarian mehod for he aignmen problem. Mahead, 23(3): , 13. [16] S. K. V. G. L. Leibe, B. Coupled deecion and rajecory eimaion for muliobjec racking. ICCV [17] Y. Ma, Q. Yu, and I. Cohen. Targe racking wih incomplee deecion. CVIU, 200. [18] S. Pellegrini, A. E, and L. V. Gool. Improving daa aociaion by join modeling of pederian rajecorie and grouping. In ECCV, [1] A. Perera, C. Sriniva, A. Hoog, G. Brookby, and W. Hu. Muliobjec racking hrough imulaneou long occluion and plimerge condiion. In IEEE CVPR, volume 1, [20] A. G. A. Perera, A. Hoog, C. Sriniva, G. Brookby, and W. Hu. Evaluaion of algorihm for racking muliple objec in video. In AIPR, page 35, [21] K. Smih, D. GaicaPerez, J. Odobez, and S. Ba. Evaluaing muliobjec racking. In CVPR Workhop. IEEE, [22] C. Sauffer. Eimaing racking ource and ink. In Proc. Even Mining Workhop. Cieeer. [23] J. Xing, H. Ai, and S. Lao. Muliobjec racking hrough occluion by local rackle filering and global rackle aociaion wih deecion repone. In IEEE CVPR, June 200. [24] D. Young and J. Ferryman. Pe meric: Online performance evaluaion ervice. In Join IEEE Inernaional Workhop on Viual Surveillance and Performance Evaluaion of Tracking and Surveillance (VSPETS), page 317 4, [25] L. Zhang, Y. Li, and R. Nevaia. Global daa aociaion for muliobjec racking uing nework flow. In CVPR,
2.4 Network flows. Many direct and indirect applications telecommunication transportation (public, freight, railway, air, ) logistics
.4 Nework flow Problem involving he diribuion of a given produc (e.g., waer, ga, daa, ) from a e of producion locaion o a e of uer o a o opimize a given objecive funcion (e.g., amoun of produc, co,...).
More informationTopic: Applications of Network Flow Date: 9/14/2007
CS787: Advanced Algorihm Scribe: Daniel Wong and Priyananda Shenoy Lecurer: Shuchi Chawla Topic: Applicaion of Nework Flow Dae: 9/4/2007 5. Inroducion and Recap In he la lecure, we analyzed he problem
More informationChapter 13. Network Flow III Applications. 13.1 Edge disjoint paths. 13.1.1 Edgedisjoint paths in a directed graphs
Chaper 13 Nework Flow III Applicaion CS 573: Algorihm, Fall 014 Ocober 9, 014 13.1 Edge dijoin pah 13.1.1 Edgedijoin pah in a direced graph 13.1.1.1 Edge dijoin pah queiong: graph (dir/undir)., : verice.
More information6.003 Homework #4 Solutions
6.3 Homewk #4 Soluion Problem. Laplace Tranfm Deermine he Laplace ranfm (including he region of convergence) of each of he following ignal: a. x () = e 2(3) u( 3) X = e 3 2 ROC: Re() > 2 X () = x ()e d
More informationOn the Connection Between MultipleUnicast Network Coding and SingleSource SingleSink Network Error Correction
On he Connecion Beween MulipleUnica ework Coding and SingleSource SingleSink ework Error Correcion Jörg Kliewer JIT Join work wih Wenao Huang and Michael Langberg ework Error Correcion Problem: Adverary
More informationHow Much Can Taxes Help Selfish Routing?
How Much Can Taxe Help Selfih Rouing? Tim Roughgarden (Cornell) Join wih Richard Cole (NYU) and Yevgeniy Dodi (NYU) Selfih Rouing a direced graph G = (V,E) a ource and a deinaion one uni of raffic from
More informationFortified financial forecasting models: nonlinear searching approaches
0 Inernaional Conference on Economic and inance Reearch IPEDR vol.4 (0 (0 IACSIT Pre, Singapore orified financial forecaing model: nonlinear earching approache Mohammad R. Hamidizadeh, Ph.D. Profeor,
More informationCHAPTER 11 NONPARAMETRIC REGRESSION WITH COMPLEX SURVEY DATA. R. L. Chambers Department of Social Statistics University of Southampton
CHAPTER 11 NONPARAMETRIC REGRESSION WITH COMPLEX SURVEY DATA R. L. Chamber Deparmen of Social Saiic Univeriy of Souhampon A.H. Dorfman Office of Survey Mehod Reearch Bureau of Labor Saiic M.Yu. Sverchkov
More informationOptimal Path Routing in Single and Multiple Clock Domain Systems
IEEE TRANSACTIONS ON COMPUTERAIDED DESIGN, TO APPEAR. 1 Opimal Pah Rouing in Single and Muliple Clock Domain Syem Soha Haoun, Senior Member, IEEE, Charle J. Alper, Senior Member, IEEE ) Abrac Shrinking
More informationPhysical Topology Discovery for Large MultiSubnet Networks
Phyical Topology Dicovery for Large MuliSubne Nework Yigal Bejerano, Yuri Breibar, Mino Garofalaki, Rajeev Raogi Bell Lab, Lucen Technologie 600 Mounain Ave., Murray Hill, NJ 07974. {bej,mino,raogi}@reearch.belllab.com
More informationRealtime Particle Filters
Realime Paricle Filers Cody Kwok Dieer Fox Marina Meilă Dep. of Compuer Science & Engineering, Dep. of Saisics Universiy of Washingon Seale, WA 9895 ckwok,fox @cs.washingon.edu, mmp@sa.washingon.edu Absrac
More informationSELFEVALUATION FOR VIDEO TRACKING SYSTEMS
SELFEVALUATION FOR VIDEO TRACKING SYSTEMS Hao Wu and Qinfen Zheng Cenre for Auomaion Research Dep. of Elecrical and Compuer Engineering Universiy of Maryland, College Park, MD20742 {wh2003, qinfen}@cfar.umd.edu
More informationHow has globalisation affected inflation dynamics in the United Kingdom?
292 Quarerly Bullein 2008 Q3 How ha globaliaion affeced inflaion dynamic in he Unied Kingdom? By Jennifer Greenlade and Sephen Millard of he Bank Srucural Economic Analyi Diviion and Chri Peacock of he
More informationState Machines: Brief Introduction to Sequencers Prof. Andrew J. Mason, Michigan State University
Inroducion ae Machines: Brief Inroducion o equencers Prof. Andrew J. Mason, Michigan ae Universiy A sae machine models behavior defined by a finie number of saes (unique configuraions), ransiions beween
More informationChapter 7. Response of FirstOrder RL and RC Circuits
Chaper 7. esponse of FirsOrder L and C Circuis 7.1. The Naural esponse of an L Circui 7.2. The Naural esponse of an C Circui 7.3. The ep esponse of L and C Circuis 7.4. A General oluion for ep and Naural
More informationA Comparative Study of Linear and Nonlinear Models for Aggregate Retail Sales Forecasting
A Comparaive Sudy of Linear and Nonlinear Model for Aggregae Reail Sale Forecaing G. Peer Zhang Deparmen of Managemen Georgia Sae Univeriy Alana GA 30066 (404) 6514065 Abrac: The purpoe of hi paper i
More informationDIFFERENTIAL EQUATIONS with TI89 ABDUL HASSEN and JAY SCHIFFMAN. A. Direction Fields and Graphs of Differential Equations
DIFFERENTIAL EQUATIONS wih TI89 ABDUL HASSEN and JAY SCHIFFMAN We will assume ha he reader is familiar wih he calculaor s keyboard and he basic operaions. In paricular we have assumed ha he reader knows
More informationThe Chase Problem (Part 2) David C. Arney
The Chae Problem Par David C. Arne Inroducion In he previou ecion, eniled The Chae Problem Par, we dicued a dicree model for a chaing cenario where one hing chae anoher. Some of he applicaion of hi kind
More informationRobust Bandwidth Allocation Strategies
Robu Bandwidh Allocaion Sraegie Oliver Heckmann, Jen Schmi, Ralf Seinmez Mulimedia Communicaion Lab (KOM), Darmad Univeriy of Technology Merckr. 25 D64283 Darmad Germany {Heckmann, Schmi, Seinmez}@kom.udarmad.de
More informationEmpirical heuristics for improving Intermittent Demand Forecasting
Empirical heuriic for improving Inermien Demand Forecaing Foio Peropoulo 1,*, Konanino Nikolopoulo 2, Georgio P. Spihouraki 1, Vailio Aimakopoulo 1 1 Forecaing & Sraegy Uni, School of Elecrical and Compuer
More informationAP Calculus AB 2010 Scoring Guidelines
AP Calculus AB 1 Scoring Guidelines The College Board The College Board is a noforprofi membership associaion whose mission is o connec sudens o college success and opporuniy. Founded in 1, he College
More informationAP Calculus BC 2010 Scoring Guidelines
AP Calculus BC Scoring Guidelines The College Board The College Board is a noforprofi membership associaion whose mission is o connec sudens o college success and opporuniy. Founded in, he College Board
More informationFourier Series Solution of the Heat Equation
Fourier Series Soluion of he Hea Equaion Physical Applicaion; he Hea Equaion In he early nineeenh cenury Joseph Fourier, a French scienis and mahemaician who had accompanied Napoleon on his Egypian campaign,
More informationHeat demand forecasting for concrete district heating system
Hea demand forecaing for concree diric heaing yem Bronilav Chramcov Abrac Thi paper preen he reul of an inveigaion of a model for horerm hea demand forecaing. Foreca of hi hea demand coure i ignifican
More informationFormulating CyberSecurity as Convex Optimization Problems
Formulaing CyberSecuriy a Convex Opimizaion Problem Kyriako G. Vamvoudaki, João P. Hepanha, Richard A. Kemmerer, and Giovanni Vigna Univeriy of California, Sana Barbara Abrac. Miioncenric cyberecuriy
More informationThe Transport Equation
The Transpor Equaion Consider a fluid, flowing wih velociy, V, in a hin sraigh ube whose cross secion will be denoed by A. Suppose he fluid conains a conaminan whose concenraion a posiion a ime will be
More information4.8 Exponential Growth and Decay; Newton s Law; Logistic Growth and Decay
324 CHAPTER 4 Exponenial and Logarihmic Funcions 4.8 Exponenial Growh and Decay; Newon s Law; Logisic Growh and Decay OBJECTIVES 1 Find Equaions of Populaions Tha Obey he Law of Uninhibied Growh 2 Find
More informationNanocubes for RealTime Exploration of Spatiotemporal Datasets
Nanocube for RealTime Exploraion of Spaioemporal Daae Lauro Lin, Jame T Kloowki, and arlo Scheidegger Fig 1 Example viualizaion of 210 million public geolocaed Twier po over he coure of a year The daa
More informationRepresenting Periodic Functions by Fourier Series. (a n cos nt + b n sin nt) n=1
Represening Periodic Funcions by Fourier Series 3. Inroducion In his Secion we show how a periodic funcion can be expressed as a series of sines and cosines. We begin by obaining some sandard inegrals
More informationUse SeDuMi to Solve LP, SDP and SCOP Problems: Remarks and Examples*
Use SeDuMi o Solve LP, SDP and SCOP Problems: Remarks and Examples* * his file was prepared by WuSheng Lu, Dep. of Elecrical and Compuer Engineering, Universiy of Vicoria, and i was revised on December,
More informationEquity Valuation Using Multiples. Jing Liu. Anderson Graduate School of Management. University of California at Los Angeles (310) 2065861
Equiy Valuaion Uing Muliple Jing Liu Anderon Graduae School of Managemen Univeriy of California a Lo Angele (310) 2065861 jing.liu@anderon.ucla.edu Doron Niim Columbia Univeriy Graduae School of Buine
More information17 Laplace transform. Solving linear ODE with piecewise continuous right hand sides
7 Laplace ransform. Solving linear ODE wih piecewise coninuous righ hand sides In his lecure I will show how o apply he Laplace ransform o he ODE Ly = f wih piecewise coninuous f. Definiion. A funcion
More informationTwoGroup Designs Independent samples ttest & paired samples ttest. Chapter 10
TwoGroup Deign Independen ample e & paired ample e Chaper 0 Previou e (Ch 7 and 8) Ze z M N e (oneample) M N M = andard error of he mean p. 989 Remember: = variance M = eimaed andard error p. 
More informationMTH6121 Introduction to Mathematical Finance Lesson 5
26 MTH6121 Inroducion o Mahemaical Finance Lesson 5 Conens 2.3 Brownian moion wih drif........................... 27 2.4 Geomeric Brownian moion........................... 28 2.5 Convergence of random
More informationOPTIMAL BATCH QUANTITY MODELS FOR A LEAN PRODUCTION SYSTEM WITH REWORK AND SCRAP. A Thesis
OTIMAL BATH UANTITY MOELS FOR A LEAN ROUTION SYSTEM WITH REWORK AN SRA A Thei Submied o he Graduae Faculy of he Louiiana Sae Univeriy and Agriculural and Mechanical ollege in parial fulfillmen of he requiremen
More informationNew Evidence on Mutual Fund Performance: A Comparison of Alternative Bootstrap Methods. David Blake* Tristan Caulfield** Christos Ioannidis*** and
New Evidence on Muual Fund Performance: A Comparion of Alernaive Boorap Mehod David Blake* Trian Caulfield** Chrio Ioannidi*** and Ian Tonk**** June 2014 Abrac Thi paper compare he wo boorap mehod of Koowki
More informationTSGRAN Working Group 1 (Radio Layer 1) meeting #3 Nynashamn, Sweden 22 nd 26 th March 1999
TSGRAN Working Group 1 (Radio Layer 1) meeing #3 Nynashamn, Sweden 22 nd 26 h March 1999 RAN TSGW1#3(99)196 Agenda Iem: 9.1 Source: Tile: Documen for: Moorola Macrodiversiy for he PRACH Discussion/Decision
More informationMathematics in Pharmacokinetics What and Why (A second attempt to make it clearer)
Mahemaics in Pharmacokineics Wha and Why (A second aemp o make i clearer) We have used equaions for concenraion () as a funcion of ime (). We will coninue o use hese equaions since he plasma concenraions
More informationChapter 2 Kinematics in One Dimension
Chaper Kinemaics in One Dimension Chaper DESCRIBING MOTION:KINEMATICS IN ONE DIMENSION PREVIEW Kinemaics is he sudy of how hings moe how far (disance and displacemen), how fas (speed and elociy), and how
More informationThe Role of the Scientific Method in Software Development. Robert Sedgewick Princeton University
The Role of he Scienific Mehod in Sofware Developmen Rober Sedgewick Princeon Univeriy The cienific mehod i neceary in algorihm deign and ofware developmen Scienific mehod creae a model decribing naural
More informationAP Calculus AB 2013 Scoring Guidelines
AP Calculus AB 1 Scoring Guidelines The College Board The College Board is a missiondriven noforprofi organizaion ha connecs sudens o college success and opporuniy. Founded in 19, he College Board was
More informationLecture III: Finish Discounted Value Formulation
Lecure III: Finish Discouned Value Formulaion I. Inernal Rae of Reurn A. Formally defined: Inernal Rae of Reurn is ha ineres rae which reduces he ne presen value of an invesmen o zero.. Finding he inernal
More informationSubsistence Consumption and Rising Saving Rate
Subience Conumpion and Riing Saving Rae Kenneh S. Lin a, HiuYun Lee b * a Deparmen of Economic, Naional Taiwan Univeriy, Taipei, 00, Taiwan. b Deparmen of Economic, Naional Chung Cheng Univeriy, ChiaYi,
More informationCrosssectional and longitudinal weighting in a rotational household panel: applications to EUSILC. Vijay Verma, Gianni Betti, Giulio Ghellini
Croecional and longiudinal eighing in a roaional houehold panel: applicaion o EUSILC Viay Verma, Gianni Bei, Giulio Ghellini Working Paper n. 67, December 006 CROSSSECTIONAL AND LONGITUDINAL WEIGHTING
More informationFormulating CyberSecurity as Convex Optimization Problems Æ
Formulaing CyberSecuriy a Convex Opimizaion Problem Æ Kyriako G. Vamvoudaki,João P. Hepanha, Richard A. Kemmerer 2, and Giovanni Vigna 2 Cener for Conrol, Dynamicalyem and Compuaion (CCDC), Univeriy
More informationMath 201 Lecture 12: CauchyEuler Equations
Mah 20 Lecure 2: CauchyEuler Equaions Feb., 202 Many examples here are aken from he exbook. The firs number in () refers o he problem number in he UA Cusom ediion, he second number in () refers o he problem
More informationAn approach for designing a surface pencil through a given geodesic curve
An approach for deigning a urface pencil hrough a given geodeic curve Gülnur SAFFAK ATALAY, Fama GÜLER, Ergin BAYRAM *, Emin KASAP Ondokuz Mayı Univeriy, Faculy of Ar and Science, Mahemaic Deparmen gulnur.affak@omu.edu.r,
More informationBilabel Propagation for Generic Multiple Object Tracking
Bilabel Propagaion for Generic Muliple Objec Tracking Wenhan Luo, TaeKyun Kim, Björn Senger 2, Xiaowei Zhao, Robero Cipolla 3 Imperial College London, 2 Toshiba Research Europe, 3 Universiy of Cambridge
More informationMultiresource Allocation Scheduling in Dynamic Environments
MANUFACTURING & SERVICE OPERATIONS MANAGEMENT Vol. 00, No. 0, Xxxxx 0000, pp. 000 000 in 15234614 ein 15265498 00 0000 0001 INFORMS doi 10.1287/xxxx.0000.0000 c 0000 INFORMS Mulireource Allocaion Scheduling
More informationCalculation of variable annuity market sensitivities using a pathwise methodology
cuing edge Variable annuiie Calculaion of variable annuiy marke eniiviie uing a pahwie mehodology Under radiional finie difference mehod, he calculaion of variable annuiy eniiviie can involve muliple Mone
More informationMaking a Faster Cryptanalytic TimeMemory TradeOff
Making a Faser Crypanalyic TimeMemory TradeOff Philippe Oechslin Laboraoire de Securié e de Crypographie (LASEC) Ecole Polyechnique Fédérale de Lausanne Faculé I&C, 1015 Lausanne, Swizerland philippe.oechslin@epfl.ch
More informationComplex Fourier Series. Adding these identities, and then dividing by 2, or subtracting them, and then dividing by 2i, will show that
Mah 344 May 4, Complex Fourier Series Par I: Inroducion The Fourier series represenaion for a funcion f of period P, f) = a + a k coskω) + b k sinkω), ω = π/p, ) can be expressed more simply using complex
More informationAppendix A: Area. 1 Find the radius of a circle that has circumference 12 inches.
Appendi A: Area workedou s o OddNumbered Eercises Do no read hese workedou s before aemping o do he eercises ourself. Oherwise ou ma mimic he echniques shown here wihou undersanding he ideas. Bes wa
More informationDividend taxation, share repurchases and the equity trap
Working Paper 2009:7 Deparmen of Economic Dividend axaion, hare repurchae and he equiy rap Tobia Lindhe and Jan Söderen Deparmen of Economic Working paper 2009:7 Uppala Univeriy May 2009 P.O. Box 53 ISSN
More informationSupply Chain Management Using Simulation Optimization By Miheer Kulkarni
Supply Chain Managemen Using Simulaion Opimizaion By Miheer Kulkarni This problem was inspired by he paper by Jung, Blau, Pekny, Reklaii and Eversdyk which deals wih supply chain managemen for he chemical
More informationOnline MultiClass LPBoost
Online MuliClass LPBoos Amir Saffari Marin Godec Thomas Pock Chrisian Leisner Hors Bischof Insiue for Compuer Graphics and Vision, Graz Universiy of Technology, Ausria {saffari,godec,pock,leisner,bischof}@icg.ugraz.a
More information4kq 2. D) south A) F B) 2F C) 4F D) 8F E) 16F
efore you begin: Use black pencil. Wrie and bubble your SU ID Number a boom lef. Fill bubbles fully and erase cleanly if you wish o change! 20 Quesions, each quesion is 10 poins. Each quesion has a mos
More informationMax Flow, Min Cut. Maximum Flow and Minimum Cut. Soviet Rail Network, 1955. Minimum Cut Problem
Maximum Flow and Minimum u Max Flow, Min u Max flow and min cu. Two very rich algorihmic problem. ornerone problem in combinaorial opimizaion. eauiful mahemaical dualiy. Minimum cu Maximum flow Maxflow
More informationsdomain Circuit Analysis
Domain ircui Analyi Operae direcly in he domain wih capacior, inducor and reior Key feaure lineariy i preered c decribed by ODE and heir Order equal number of plu number of Elemenbyelemen and ource ranformaion
More informationSignal Processing and Linear Systems I
Sanford Universiy Summer 214215 Signal Processing and Linear Sysems I Lecure 5: Time Domain Analysis of Coninuous Time Sysems June 3, 215 EE12A:Signal Processing and Linear Sysems I; Summer 1415, Gibbons
More informationANALYSIS AND COMPARISONS OF SOME SOLUTION CONCEPTS FOR STOCHASTIC PROGRAMMING PROBLEMS
ANALYSIS AND COMPARISONS OF SOME SOLUTION CONCEPTS FOR STOCHASTIC PROGRAMMING PROBLEMS R. Caballero, E. Cerdá, M. M. Muñoz and L. Rey () Deparmen of Applied Economics (Mahemaics), Universiy of Málaga,
More informationAutomatic measurement and detection of GSM interferences
Auomaic measuremen and deecion of GSM inerferences Poor speech qualiy and dropped calls in GSM neworks may be caused by inerferences as a resul of high raffic load. The radio nework analyzers from Rohde
More informationEconomics 140A Hypothesis Testing in Regression Models
Economics 140A Hypohesis Tesing in Regression Models While i is algebraically simple o work wih a populaion model wih a single varying regressor, mos populaion models have muliple varying regressors 1
More informationSAMPLE LESSON PLAN with Commentary from ReadingQuest.org
Lesson Plan: Energy Resources ubject: Earth cience Grade: 9 Purpose: students will learn about the energy resources, explore the differences between renewable and nonrenewable resources, evaluate the environmental
More informationDistributed Online Localization in Sensor Networks Using a Moving Target
Disribued Online Localizaion in Sensor Neworks Using a Moving Targe Aram Galsyan 1, Bhaskar Krishnamachari 2, Krisina Lerman 1, and Sundeep Paem 2 1 Informaion Sciences Insiue 2 Deparmen of Elecrical EngineeringSysems
More informationCHARGE AND DISCHARGE OF A CAPACITOR
REFERENCES RC Circuis: Elecrical Insrumens: Mos Inroducory Physics exs (e.g. A. Halliday and Resnick, Physics ; M. Sernheim and J. Kane, General Physics.) This Laboraory Manual: Commonly Used Insrumens:
More informationUnderstanding Sequential Circuit Timing
ENGIN112: Inroducion o Elecrical and Compuer Engineering Fall 2003 Prof. Russell Tessier Undersanding Sequenial Circui Timing Perhaps he wo mos disinguishing characerisics of a compuer are is processor
More informationThe Application of Multi Shifts and Break Windows in Employees Scheduling
The Applicaion of Muli Shifs and Brea Windows in Employees Scheduling Evy Herowai Indusrial Engineering Deparmen, Universiy of Surabaya, Indonesia Absrac. One mehod for increasing company s performance
More informationA Joint Probability Data Association Filter Algorithm for Multiple Robot Tracking Problems
A Join Probabiliy Daa Aociaion Filer Algorihm for Muliple Robo Tracing Problem Aliabar Gorji Daronolaei, Vahid Nazari, Mohammad Bagher Menhaj, and Saeed Shiry Amirabir Univeriy of Technology, Tehran, Iran.
More informationStock option grants have become an. Final Approval Copy. Valuation of Stock Option Grants Under Multiple Severance Risks GURUPDESH S.
Valuaion of Sock Opion Gran Under Muliple Severance Rik GURUPDESH S. PANDHER i an aian profeor in he deparmen of finance a DePaul Univeriy in Chicago, IL. gpandher@depaul.edu GURUPDESH S. PANDHER Execuive
More informationCapacitors and inductors
Capaciors and inducors We coninue wih our analysis of linear circuis by inroducing wo new passive and linear elemens: he capacior and he inducor. All he mehods developed so far for he analysis of linear
More information1 The basic circulation problem
2WO08: Graphs and Algorihms Lecure 4 Dae: 26/2/2012 Insrucor: Nikhil Bansal The Circulaion Problem Scribe: Tom Slenders 1 The basic circulaion problem We will consider he maxflow problem again, bu his
More informationSinglemachine Scheduling with Periodic Maintenance and both Preemptive and. Nonpreemptive jobs in Remanufacturing System 1
Absrac number: 050407 Singlemachine Scheduling wih Periodic Mainenance and boh Preempive and Nonpreempive jobs in Remanufacuring Sysem Liu Biyu hen Weida (School of Economics and Managemen Souheas Universiy
More information1. BACKGROUND 11 Traffic Flow Surveillance
AuoRecogniion of Vehicle Maneuvers Based on SpaioTemporal Clusering. BACKGROUND  Traffic Flow Surveillance Conduced wih kinds of beacons mouned a limied roadside poins wih Images from High Aliude Plaforms
More informationMorningstar Investor Return
Morningsar Invesor Reurn Morningsar Mehodology Paper Augus 31, 2010 2010 Morningsar, Inc. All righs reserved. The informaion in his documen is he propery of Morningsar, Inc. Reproducion or ranscripion
More informationModule 4. Singlephase AC circuits. Version 2 EE IIT, Kharagpur
Module 4 Singlephase A circuis ersion EE T, Kharagpur esson 5 Soluion of urren in A Series and Parallel ircuis ersion EE T, Kharagpur n he las lesson, wo poins were described:. How o solve for he impedance,
More informationRandom Walk in 1D. 3 possible paths x vs n. 5 For our random walk, we assume the probabilities p,q do not depend on time (n)  stationary
Random Walk in D Random walks appear in many cones: diffusion is a random walk process undersanding buffering, waiing imes, queuing more generally he heory of sochasic processes gambling choosing he bes
More informationAcceleration Lab Teacher s Guide
Acceleraion Lab Teacher s Guide Objecives:. Use graphs of disance vs. ime and velociy vs. ime o find acceleraion of a oy car.. Observe he relaionship beween he angle of an inclined plane and he acceleraion
More informationDuration and Convexity ( ) 20 = Bond B has a maturity of 5 years and also has a required rate of return of 10%. Its price is $613.
Graduae School of Business Adminisraion Universiy of Virginia UVAF38 Duraion and Convexiy he price of a bond is a funcion of he promised paymens and he marke required rae of reurn. Since he promised
More informationA Natural FeatureBased 3D Object Tracking Method for Wearable Augmented Reality
A Naural FeaureBased 3D Objec Tracking Mehod for Wearable Augmened Realiy Takashi Okuma Columbia Universiy / AIST Email: okuma@cs.columbia.edu Takeshi Kuraa Universiy of Washingon / AIST Email: kuraa@ieee.org
More informationAnswer, Key Homework 2 David McIntyre 45123 Mar 25, 2004 1
Answer, Key Homework 2 Daid McInyre 4123 Mar 2, 2004 1 This prinou should hae 1 quesions. Muliplechoice quesions may coninue on he ne column or page find all choices before making your selecion. The
More informationA Brief Introduction to the Consumption Based Asset Pricing Model (CCAPM)
A Brief Inroducion o he Consumpion Based Asse Pricing Model (CCAPM We have seen ha CAPM idenifies he risk of any securiy as he covariance beween he securiy's rae of reurn and he rae of reurn on he marke
More informationPart 1: White Noise and Moving Average Models
Chaper 3: Forecasing From Time Series Models Par 1: Whie Noise and Moving Average Models Saionariy In his chaper, we sudy models for saionary ime series. A ime series is saionary if is underlying saisical
More informationLong Term Spread Option Valuation and Hedging
Long Term Spread Opion Valuaion and Hedging M.A.H. Demper, Elena Medova and Ke Tang Cenre for Financial Reearch, Judge Buine School, Univeriy of Cambridge, Trumpingon Sree, Cambridge CB 1AG & Cambridge
More informationMaintaining MultiModality through Mixture Tracking
Mainaining MuliModaliy hrough Mixure Tracking Jaco Vermaak, Arnaud Douce Cambridge Universiy Engineering Deparmen Cambridge, CB2 1PZ, UK Parick Pérez Microsof Research Cambridge, CB3 0FB, UK Absrac In
More informationTwo Compartment Body Model and V d Terms by Jeff Stark
Two Comparmen Body Model and V d Terms by Jeff Sark In a onecomparmen model, we make wo imporan assumpions: (1) Linear pharmacokineics  By his, we mean ha eliminaion is firs order and ha pharmacokineic
More informationThe Role of Science and Mathematics in Software Development
The cienific mehod i eenial in applicaion of compuaion A peronal opinion formed on he bai of decade of experience a a The Role of Science and Mahemaic in Sofware Developmen CS educaor auhor algorihm deigner
More information1. The graph shows the variation with time t of the velocity v of an object.
1. he graph shows he variaion wih ime of he velociy v of an objec. v Which one of he following graphs bes represens he variaion wih ime of he acceleraion a of he objec? A. a B. a C. a D. a 2. A ball, iniially
More informationDISTRIBUTED DATA PARALLEL TECHNIQUES FOR CONTENTMATCHING INTRUSION DETECTION SYSTEMS. G. Chapman J. Cleese E. Idle
DISTRIBUTED DATA PARALLEL TECHNIQUES FOR CONTENTMATCHING INTRUSION DETECTION SYSTEMS G. Chapman J. Cleee E. Idle ABSTRACT Content matching i a neceary component of any ignaturebaed network Intruion Detection
More informationProcess Modeling for Object Oriented Analysis using BORM Object Behavioral Analysis.
Proce Modeling for Objec Oriened Analyi uing BORM Objec Behavioral Analyi. Roger P. Kno Ph.D., Compuer Science Dep, Loughborough Univeriy, U.K. r.p.kno@lboro.ac.uk 9RMW FKMerunka Ph.D., Dep. of Informaion
More informationYTM is positively related to default risk. YTM is positively related to liquidity risk. YTM is negatively related to special tax treatment.
. Two quesions for oday. A. Why do bonds wih he same ime o mauriy have differen YTM s? B. Why do bonds wih differen imes o mauriy have differen YTM s? 2. To answer he firs quesion les look a he risk srucure
More informationPROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE
Profi Tes Modelling in Life Assurance Using Spreadshees PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE Erik Alm Peer Millingon 2004 Profi Tes Modelling in Life Assurance Using Spreadshees
More informationOptimal Investment and Consumption Decision of Family with Life Insurance
Opimal Invesmen and Consumpion Decision of Family wih Life Insurance Minsuk Kwak 1 2 Yong Hyun Shin 3 U Jin Choi 4 6h World Congress of he Bachelier Finance Sociey Torono, Canada June 25, 2010 1 Speaker
More informationEconomics Honors Exam 2008 Solutions Question 5
Economics Honors Exam 2008 Soluions Quesion 5 (a) (2 poins) Oupu can be decomposed as Y = C + I + G. And we can solve for i by subsiuing in equaions given in he quesion, Y = C + I + G = c 0 + c Y D + I
More information9. Capacitor and Resistor Circuits
ElecronicsLab9.nb 1 9. Capacior and Resisor Circuis Inroducion hus far we have consider resisors in various combinaions wih a power supply or baery which provide a consan volage source or direc curren
More informationGraduate Macro Theory II: Notes on Neoclassical Growth Model
Graduae Macro Theory II: Noes on Neoclassical Growh Model Eric Sims Universiy of Nore Dame Spring 2011 1 Basic Neoclassical Growh Model The economy is populaed by a large number of infiniely lived agens.
More informationNewton s Laws of Motion
Newon s Laws of Moion MS4414 Theoreical Mechanics Firs Law velociy. In he absence of exernal forces, a body moves in a sraigh line wih consan F = 0 = v = cons. Khan Academy Newon I. Second Law body. The
More informationTrading Strategies for Sliding, Rollinghorizon, and Consol Bonds
Trading Sraegie for Sliding, Rollinghorizon, and Conol Bond MAREK RUTKOWSKI Iniue of Mahemaic, Poliechnika Warzawka, 661 Warzawa, Poland Abrac The ime evoluion of a liding bond i udied in dicree and
More informationHANDOUT 14. A.) Introduction: Many actions in life are reversible. * Examples: Simple One: a closed door can be opened and an open door can be closed.
Inverse Funcions Reference Angles Inverse Trig Problems Trig Indeniies HANDOUT 4 INVERSE FUNCTIONS KEY POINTS A.) Inroducion: Many acions in life are reversible. * Examples: Simple One: a closed door can
More informationA Universal Pricing Framework for Guaranteed Minimum Benefits in Variable Annuities *
A Universal Pricing Framework for Guaraneed Minimum Benefis in Variable Annuiies * Daniel Bauer Deparmen of Risk Managemen and Insurance, Georgia Sae Universiy 35 Broad Sree, Alana, GA 333, USA Phone:
More information