Particle Filtering for Geometric Active Contours with Application to Tracking Moving and Deforming Objects

Transcription

1 Paricle Filering for Geomeric Acive Conours wih Applicaion o Tracking Moving and Deforming Objecs Yogesh Rahi Namraa Vaswani Allen Tannenbaum Anhony Yezzi Georgia Insiue of Technology School of Elecrical and Compuer Engineering Alana, GA, USA {yogesh.rahi,annenba}@bme.gaech.edu,{namraa,ayezzi}@ece.gaech.edu Absrac Geomeric acive conours are formulaed in a manner which is parameerizaion independen. As such, hey are amenable o represenaion as he zero level se of he graph of a higher dimensional funcion. This represenaion is able o deal wih singulariies and changes in opology of he conour. I has been used very successfully in saic images for segmenaion and regisraion problems where he conour (represened as an implici curve) is evolved unil i minimizes an image based energy funcional. Bu racking involves esimaing he global moion of he objec and is local deformaions as a funcion of ime. Some aemps have been made o use geomeric acive conours for racking, bu mos of hese minimize he energy a each frame and do no uilize he emporal coherency of he moion or he deformaion. On he oher hand, racking algorihms using Kalman filers or Paricle filers have been proposed for finie dimensional represenaions of shape. Bu hese are dependen on he chosen paramerizaion and canno handle changes in curve opology. In he presen work, we formulae a paricle filering algorihm in he geomeric acive conour framework ha can be used for racking moving and deforming objecs. 1 Inroducion The problem of racking moving and deforming objecs has been a opic of subsanial research in he field of Compuer Vision; see [1, 30] and he references herein. In his paper, we propose a scheme which combines he advanages of paricle filering and geomeric acive conours realized via level se models for dynamic racking. In order o appreciae his mehodology, we briefly review some previous relaed work. Firs of all, a number of differen represenaions of shape have been proposed in lieraure ogeher wih algorihms for racking using such represenaions. In paricular, he noion of shape has been found o be very useful in his enerprise. For example, he shape of a se of N discree poins (called landmarks) in R M is defined [12, 8] as he equivalence class of R MN under he Euclidean similariy group in R M. The dynamics of he similariy group defines he global moion while he dynamics of he equivalence class defines he deformaion. In [34], he auhors define a prior dynamical model on he deformaion and on he similariy group parameers. A paricle filer [9] is hen used o rack he deformaion and he global moion over ime. The possible parameerizaions of shape are of course very imporan. We should noe ha various finie dimensional parameerizaions of coninuous curves have been proposed, perhaps mos prominenly he B-spline represenaion used for a snake model as in [30]. Isard and Blake (see [1] and references herein) apply he B-spline represenaion for conours of objecs and propose he Condensaion algorihm [10] which reas he affine group parameers as he sae vecor, learns a prior dynamical model for hem and uses a paricle filer [9] o esimae hem from he noisy observaions. Since his approach only racks he affine parameers i canno handle local deformaions of he deforming objec (see e.g., he fish example in Secion 5.1). One possible soluion proposed in [37], is o use deformable emplaes o model prior shapes allowing for many deformaion modes of shapes. Some oher approaches o racking are given in [4, 29, 15]. Anoher approach for represening conours is via he level se echnique [20, 27] which is an implici represenaion of conours. For segmening a shape using level ses, an iniial guess of he conour is deformed unil i minimizes an image-based energy funcional. Differen energy funcionals which uilize differen feaures of he image have been

2 used in lieraure; see e.g. [13, 17, 3, 14, 16, 2]. Some previous work on racking using level se mehods is given in [38, 19, 21, 36, 11]. In [19], he auhors use dynamic acive conours for racking. The sae space is defined by he conour s posiion and he deformaion velociy. Explici observaions of he conour and he conour deformaion velociy are incorporaed ino he PDE using an error injecion echnique. The predicion sep in his approach is obained using he principle of leas acion which uses he curren image (and hence he prediced conour velociy is correlaed wih he observaion). This can be a problem if here are some oulier observaions (such as occlusions) and hence hey define a separae occlusion handling mehod. The work in his paper exends he ideas presened in [36, 11]. More precisely, in [36], he auhors propose a definiion for moion and shape deformaion for a deforming objec. Moion is parameerized by a finie dimensional group acion (e.g. Euclidean or Affine) while shape deformaion is he oal deformaion of he objec conour (infinie dimensional group) modulo he finie dimensional moion group. This is called deformoion. Tracking is hen defined as a rajecory on he finie dimensional moion group. This approach relies only on he observed images for racking and does no use any prior informaion on he dynamics of he group acion or of he deformaion. As a resul i fails if here is an oulier observaion or if here is occlusion. To address his problem, [11] proposes a generic local observer o incorporae prior informaion abou he sysem dynamics in he deformoion framework. They impose a consan velociy prior on he group acion and a zero velociy prior on he conour. The observed value of he group acion and he conour is obained by a join minimizaion of he energy. This is linearly combined wih he value prediced by he sysem dynamics using an observer. This approach suffers from wo problems. Firs, as in [36], hey mus perform a join minimizaion over he group acion and he conour a each ime sep which is compuaionally very inensive. Second, for nonlinear sysems such as he one used in [11], here is no sysemaic way o choose he observer o guaranee sabiliy. The presen paper addresses he above limiaions. We formalize he incorporaion of a prior sysem model along wih an observaion model. A paricle filer is used o esimae he condiional probabiliy disribuion of he group acion and he conour a ime, condiioned on all observaions up o ime. Oher approaches closely relaed o our work are given in [30, 23]. Here he auhors use a Kalman filer in conjuncion wih acive conours o rack nonrigid objecs. The Kalman filer was used for predicing possible movemens of he objec, while he acive conours allowed for racking deformaions in he objec. This paper is organized as follows: In he nex secion we discuss he paricle filer and level se mehod. In Secion 3 we describe he sae space model and Secion 4 discusses he algorihm in deail. Experimenal resuls are given in Secion 5. Limiaions and fuure work are discussed in Secion 6. 2 Preliminaries In his secion, we review some basic noions from he heory of level se evoluions and paricle filering which we will need in he sequel. 2.1 Paricle Filering Le X R n be a sae vecor evolving according o he following difference equaion: X +1 = f (X ) + u where u is i.i.d. random noise wih known probabiliy disribuion funcion (pdf), p u,. A discree imes, observaions Y R p become available. These measuremens are relaed o he sae vecor via he observaion equaion: Y = h (X ) + v where v is measuremen noise wih known pdf p v,. I is assumed ha he iniial sae disribuion denoed by π 0 (dx), he sae ransiion kernel denoed by K (X, X +1 ) = p u, (X +1 f (X )) and he observaion likelihood given he sae, denoed by g (Y X ) = p v, (Y h (X )), are known. The paricle filer (PF) [9, 7] is a sequenial Mone Carlo mehod which produces a each ime, a cloud of N paricles, {X (i) } N i=1, whose empirical measure closely follows π (dx Y 0: ), he poserior disribuion of he sae given pas observaions (denoed by π (dx) in he res of he paper). The PF was firs inroduced in [9] as he Bayesian Boosrap filer and is firs applicaion o racking in compuer vision was he Condensaion algorihm [10]. The algorihm sars wih sampling N imes from he iniial sae disribuion π 0 (dx) o approximae i by π0 N (dx) = 1 N N i=1 δ (dx) and hen implemens he X (i) 0 Bayes recursion a each ime sep. Now, he disribuion of X 1 given observaions upo ime 1 can be approximaed by π 1 1 N (dx) = 1 N N i=1 δ (dx). The predicion sep samples he new sae from he X (i) 1 disribu- X (i) ion K 1 (X (i) 1,.). The empirical disribuion of his new cloud of paricles, π 1 N (dx) = 1 N N i=1 δ (dx) is an X(i) approximaion o he condiional probabiliy disribuion of X given observaions upo ime 1 (predicion disribuion). In he updae sep, each paricle is weighed in proporion o he likelihood of he observaion a ime, Y, i.e. w (i) = (i) X ) g (Y N i=1 g (Y X (i) ).

3 1 N N i=1 w(i) π N (dx) = δ (dx) is hen an esimae of X(i) π (filering disribuion). We resample N imes wih replacemen from π N (dx) o obain he empirical esimae π N (dx) = 1 N N i=1 δ (dx). Noe ha boh π X (i) N and πn approximae π bu he resampling sep is used because i increases he sampling efficiency by eliminaing samples wih very low weighs. 2.2 Curve Evoluion Using Level Ses Geomeric acive conours evolving according o edge based and/or region based energy flow are very commonly used for image segmenaion. In hese mehods, saring from an iniial esimae, he curve deforms under he influence of various forces unil i fis he objec boundaries. The curve evoluion equaion is obained by reducing an energy E image as fas as possible, i.e., by doing a gradien descen on E image. In general, E image may depend on a combinaion of image based feaures and exernal consrains (smoohness, shape ec) [17, 3]. The level se mehods of Osher and Sehian [20, 28] offer a naural and numerically robus implemenaion of such curve evoluion equaions. Level ses have he advanage of being parameer independen (i.e. hey are implici represenaion of he curve) and can handle opological changes naurally. We now briefly go over he level se represenaion of a given curve evoluion equaion. Le C(p, τ) : S 1 [0, θ) R 2 be a family of curves saisfying he following evoluion equaion: C E image = C = βn (1) τ where, τ is an arificial ime-marching parameer. The basic idea of he level se approach is o embed he conour as he level se of a graph Φ : R 2 R and hen evolve he graph so ha his level se moves according o he prescribed flow. In his manner, he level se may develop singulariies and change opology while Φ iself remains smooh and mainains he form of a graph. Formulaing he correc evoluion of Φ amouns o solving Φ(C, τ) C τ + Φ τ = 0. (2) so ha he level se of ineres mainains a consan value as he graph, Φ evolves. Choosing he zero level se of Φ o define C and choosing Φ o be negaive inside C and posiive ouside C, allows us o wrie N = Φ/ Φ. So he level se implemenaion of (1) becomes: Φ τ = β Φ (3) Finally, given an iniial curve, one mus generae an iniial level se funcion. A well known scheme [28] is o use a signed disance funcion. 3 The Sae Space Model Le A denoe he 6-dimensional affine parameer vecor (see equaion 8) and µ denoe he conour (represened as he zero level se of Φ) a ime. We propose o use he affine parameers and he conour as he sae, i.e. X = [A, µ ] and rea he image a ime as he observaion, i.e. Y = Image(). The predicion sep for X consiss of: 1. predicing he local deformaions in he shape of he objec 2. predicing he affine moion of he objec. The affine moion predicion in (2) is obained from he sae dynamics for A, which is given by a firs or second order (consan velociy or acceleraion) auoregressive (AR) model on he affine parameers (see Secion 4.1). Since he curve is infinie-dimensional, i is difficul o have a predicion model for local shape deformaion. Hence, we assume ha: predicion for local shape deformaion a ime = local shape deformaion a ime 1. This can be realized by doing a gradien descen on he image energy E image (any image dependen energy, for e.g. equaion (9)) a ime 1: C = C 1 = f L CE(µ 1, Y 1 ). (4) Thus, he predicion for X depends on X 1 = [A 1, µ 1 ] and he observaion a 1, Y 1. In equaion (4), fce L (µ, Y ) is given by L ieraions of gradien descen: µ k = µ k 1 α k µ E image (µ k 1, Y ), k = 1, 2, 3,.., L wih f L CE(µ, Y ) = µ L, µ 0 = µ The choice of L depends on he paricular PDE used for doing curve evoluion. In our experimens, we found ha L = 4 was a good choice. If µ 1 is evolved unil convergence, one would reach a local minimum of he energy E image. Bu his is no desirable since he local minimum would be independen of all saring conours in is domain of aracion and would only depend on he observaion, Y 1. Thus he sae a ime would loose is dependence on he sae a ime 1 and his may cause loss of rack in cases where he observaion is bad. Bu if µ 1 is evolved only a fixed number of imes, i will deviae he conour only a lile (in a direcion which reduces he energy E image as fas as possible) so ha paricles wih sae closer o he rue sae will have smaller energy han oher paricles and hese will ge propagaed during he resampling sep. Now, he probabiliy of observaion Y = Image() given sae X = [A, µ ] can be defined as p(y X ) = e E image(µ,y ) (5)

4 4 The Algorihm Based on he descripion above, he proposed algorihm can be wrien as follows: 1. Predicion Sep: Perform L seps of curve evoluion for each sample as follows: 1 C (i) = C (i) 1 = f CE(µ L (i) 1, Y 1) Generae samples {Ã(i) Thus we have, µ (i) } N i=1 using: Ã (i) = f AR (A (i) 1, u(i) 1 ) µ (i) π(a, µ Y 1: 1 ) 2. Updae Sep: = Ã(i) (C (i) 1 ) N 1 N δ Ã (i) i=1 (a) Weigh each sample by, µ (i) w (i) e E image( µ (i),y ) = N j=1 e E image( µ (j),y ) Thus we have π(a, µ Y 1: ) N i=1 w (i) δã(i), µ (i) (A, µ ) (A, µ ) (b) Resample from he above disribuion o generae N paricles {A (i), µ (i) } disribued according o π(a, µ Y 1: ), i.e. π(a, µ Y 1: ) N 1 N δ A (i) i=1 3. Go back o he predicion sep for + 1.,µ (i) (A, µ ) Noe : Scaling E image by a consan facor will affec he resampling sep. This scaling facor will decide how much one russ he sysem model versus he observaion model. We discuss he deails of he above algorihm in he following subsecions 2. 1 To find he bes conour a ime 1, find he MAP esimae, i.e., find he paricle wih he maximum probabiliy and evolve only his conour unil E image is minimized or unil a user defined crieria is saisfied. This is he bes esimae of he posiion and shape of he objec a ime 1. 2 Noe ha he above algorihm differs from he sandard paricle filer in ha he predicion sep is a funcion of he previous sae and also he previous observaion. Remark 1 Noe ha, one could include he curve evoluion equaion in he updae sep once he observaion a ime is available. However, his will change he sae X based on he observaion Y. Thus, he exising convergence resuls [7] of he paricle filering esimae of he poserior π N o he rue poserior π as N canno be applied. We are working on sudying how his modificaion migh change he convergence resuls, if a all. 4.1 The AR model In he above algorihm f AR could be any suiable predicion funcion which can model he dynamics of moion of he moving objec. Raher han conjuring up a model ha is merely plausible, one can learn he dynamics of moion from a raining se. This can be done using an auoregressive (AR) model. Below, we describe he second-order AR process in which he affine parameers a a given ime depend on wo previous ime-seps: A +1 A = B 1 (A A) + B 2 (A 1 S) + B 0 w +1 (6) where A is he N x dimensional affine parameer vecor (8), B 1, B 2, B 0 are N x N x marices learned a priori, w +1 is a vecor of N x independen random N(0,1) variables and A is he seady sae mean of he model. We refer he ineresed reader o [1] for furher deails on how o learn hese parameer marices and he advanages of using he secondorder model (AR-2) versus he firs-order model (AR-1) Learning Affine Moion Many approaches [35, 18] have been repored in he lieraure for finding he affine parameers ha relae one image o he oher. Mos of hese mehods require a se of feaure poins o be known before one can find he affine parameers ha relae hem. In [22] he auhor proposes a mehod which does no require feaure poins o be known, insead only he source and arge images are required. The affine ransformaion ha relaes he curve C() and C( 1) is given by: C(x, y, ) = C(m 1 x+m 2 y +m 5, m 3 x+m 4 y +m 6, 1) where, m i are he affine parameers. In order o esimae hese parameers, he following quadraic error is o be minimized: E( m) = [C(x, y, ) x,y ω C(m 1 x + m 2 y + m 5, m 3 x + m 4 y + m 6, 1)] 2 which is linearized and hen minimized o give [ ] 1 [ ] m = d d T d k x,y ω x,y ω (7)

5 where he scalar k and he vecors d, m are given as 3 : k = C + xc x + yc y d T = (xc x yc x xc y yc y C x C y ) m = (m 1 m 2 m 3 m 4 m 5 m 6 ) T (8) Derivaion deails are available in [22]. Once he affine parameer vecor m is known for he raining se, he AR model parameer marices can be learned as given in [1]. 4.2 The Model of Chan and Vese Many mehods [5, 38, 24, 31, 14] which incorporae geomeric and/or phoomeric (color, exure, inensiy) informaion have been shown o segmen images robusly in presence of noise and cluer. In he predicion sep above, f CE could be any edge based or region based (or a combinaion of boh) curve evoluion equaion. In our numerical experimens we have used he Mumford-Shah funcional [17] as modelled by Chan and Vese [3] o obain he curve evoluion equaion, which we describe briefly. We seek o minimize he following energy: E image = E cv (c 1, c 2, Φ) = (f c 1 ) 2 H(Φ)dx dy + (f c 2 ) 2 (1 H(Φ)) dx dy + ν H(Φ) dx dy where c 1 and c 2 are defined as: f(x, y)h(φ)dx dy f(x, y)(1 H(Φ))dx dy c 1 =, c 2 = H(Φ)dx dy (1 H(Φ))dx dy H(Φ) is he Heaviside funcion defined as: { 1 Φ 0, H(Φ) = 0 else (9) (10) f(x, y) is he image and Φ is he level se funcion as defined in Secion 2.2 before. The Euler-Lagrange equaion for his funcional can be implemened by he following gradien descen [3, 17]: [ ( ) ] Φ Φ = δ ɛ(φ) ν div (f c 1 ) 2 + (f c 2 ) 2 Φ (11) where, ɛ δ ɛ (s) = π(ɛ 2 + s 2 ) 3 Noe: he subscrips in his equaion denoe parial derivaives 4.3 Dealing wih Muliple Objecs In principle, he Condensaion filer [1] could be used for racking muliple objecs. The poserior disribuion will be muli-modal wih each mode corresponding o one objec. However, in pracice i is very likely ha a peak corresponding o he dominan likelihood value will increasingly dominae over all oher peaks when he esimaion progresses over ime. In oher words, a dominan peak is esablished if some objecs obain larger likelihood values more frequenly. So, if he poserior is propagaed wih fixed number of samples, evenually, all samples will be around he dominan peak. This problem becomes more pronounced in cases where he objecs being racked do no have similar phoomeric or geomeric properies. We deal wih his issue as given in [33] by firs finding he clusers wihin he sae densiy o consruc a Voronoi essalaion [25] and hen resampling wihin each Voronoi cell separaely as follows: 1. Every sep, build an imporance funcion which resuls in equal number of samples being aken in each Voronoi cell 2. Every N seps, rescale he weighs in each cell so ha he peak weigh is 1. Oher soluions proposed by [26, 29, 15] could also be used in ackling his problem of sample impoverishmen. 4.4 Coping wih Occlusions Many acive conour models [14, 24, 6] which use shape informaion have been repored in he lieraure. Prior shape knowledge is necessary when dealing wih occlusions. In paricular, in [38], he auhors incorporae shape energy in he curve evoluion equaion o deal wih occlusions. Any such energy erm can be used in he proposed model o deal wih occlusions. In numerical experimens we have deal wih his issue in a slighly differen way by incorporaing he shape informaion in he updae sep, (see algorihm sep 2) insead of he predicion sep, i.e. we calculae he weigh for each paricle using he following: w i +1 = λ 1 e Ei cv N j=1 e Ej cv + λ 2 (1 Edissimilariy i N ) j=1 Ej dissimilariy (12) where λ 1 + λ 2 = 1 and E dissimilariy is he dissimilariy measure d 2 (Φ, Φ i ) as given in [6] by, d 2 (Φ, Φ i ) = (Φ Φ i ) 2 h(φ ) + h(φ i ) 2 wih h(φ) = H(Φ) H(Φ) dx dy dx dy,

6 where Φ and Φ i are he level se funcions of a emplae shape and he curren conour shape respecively and H(Φ) is he Heaviside funcion as defined before in (10). The dissimilariy measure gives an esimae of how differen any wo given shapes (in paricular, heir corresponding level ses) are. So, higher values of E dissimilariy indicae more dissimilariy in shape. Using his sraegy, paricles which are closer o he emplae shape are more likely o be chosen han paricles wih occluded shapes (i.e., shapes which include he occlusion). 5 Experimens In his secion we describe some experimens performed o es he proposed racking algorihm. Resuls of applying he proposed mehod on hree image sequences are given below. The model of Chan and Vese [3], as described earlier, was used for curve evoluion. Level se implemenaion was done using narrow band evoluion [28]. Learning [1] was performed on images wihou he background cluer, i.e. on he oulines of he objec. 5.1 Fish Sequence In he fish video, he shape of he fish undergoes sudden deformaion as he fish urns or ges parially occluded (see Figure 3, Frames 167, 181). This local shape deformaion canno be modelled using an affine moion model. Hence, such moion is difficul o rack using he sandard Condensaion filer [1]. As can been seen in he images, (Figure 3) he proposed mehod can robusly rack nonrigid deformaions in he shape of he fish. Noe ha, no shape informaion eiher in curve evoluion or in he weighing sep was used in racking his sequence, i.e. we did no use he dissimilariy erm specified in Secion 4.4. For his es sequence, an AR-1 model [1] was used for affine moion predicion. 5.2 Car Sequence In his sequence, he car is occluded as i passes hrough he lamp pos. I is unclear if he sandard Condensaion algorihm will be able o rack he car all he way, since he shape of he car (including he shadow) undergoes a change which is no affine. Noice ha he shadow of he car moves in a non-linear way from he side o he fron of he car. On he oher hand, rying o rack such a sequence using geomeric acive conours (for example, (11)) wihou any shape energy gives very poor resuls as shown in Figure 1. However, using he proposed mehod and a weighing sraegy as described in Secion 4.4 he car can be successfully racked (Figure 2). Noe ha we used equaion (11) for he curve evoluion which does no conain any shape erm. A second-order auoregressive model was used for f AR. 5.3 Couple Sequence The walking couple sequence demonsraes muliple objec racking. In general, racking such a sequence by he sandard Condensaion mehod [1] can give erroneous resuls when he couple come very close o each oher or ouch each oher, since he measuremens made for he person on he righ can be inerpreed by he algorihm as coming from he lef. One soluion has been proposed in [29]. Our mehod naurally avoids his problem since i uses region based energy E cv (9) and weighing as given in Secion 4.4 o find he observaion probabiliies. To rack muliple objecs, we used he mehod described in Secion 4.3. Since he number of frames in he video is less (abou 22) no dynamical moion model was learn. This video demonsraes he fac ha, he proposed algorihm can rack robusly (see Figure 4) even when he learn model is compleely absen. 6 Limiaions and Fuure Work In his paper, we proposed a paricle filering algorihm for geomeric acive conours which can be used for racking moving and deforming objecs. The proposed mehod can deal wih parial occlusions and can rack robusly even in he absence of a learn model. The above framework has several limiaions which we inend o overcome in our fuure work. Firs, we have o include some kind of shape informaion when we rack objecs which undergo major occlusions. This resrics our abiliy o rack highly deformable objecs in such siuaions. Secondly, he algorihm migh perform poorly if he objec being racked is compleely occluded for many frames. Also, in our curren framework he predicion sep for he conour is deerminisic. We use his model because adding noise o an infinie dimensional represenaion of he conour is no easy. Noneheless in [32], he auhors have performed PCA on a se of signed disance funcions of raining shapes o obain principal direcions of variaion of he signed disance funcion for a class of shapes. We can adop a similar idea and add noise in he principal variaion direcions. This approach can also provide a shape prior. References [1] A. Blake and M. Isard, ediors. Acive Conours. Springer, [2] V. Caselles, F. Cae, T. Coll, and F. Dibos. A geomeric model for acive conours in image processing. Numerische Mahemaik, 66:1 31, 1993.

7 (a) Frame 35 (b) Frame 47 (c) Frame 59 Figure 1. Tracking using equaion (11) wihou paricle filer (a) Frame 16 (b) Frame 39 (c) Frame 48 (d) Frame 64 Figure 2. Car Sequence: Number of paricles = 50 (a) Frame 34 (b) Frame 167 (c) Frame 181 (d) Frame 215 Figure 3. Fish Sequence: Number of paricles = 25 (a) Frame 2 (b) Frame 9 (c) Frame 15 (d) Frame 18 Figure 4. Couple Sequence: Number of paricles = 100

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