3D Periodic Human Motion Reconstruction from 2D Motion Sequences
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1 3D Peiodic Human Motion Reconstuction fom D Motion Sequences Zonghua Zhang and Nikolaus F. Toje BioMotionLab, Depatment of Psychology Queen s Univesity, Canada zhang, toje@psyc.queensu.ca Abstact In this epot, we pesent and evaluate a method of econstucting thee-dimensional (3D) peiodic human motion fom two-dimensional (D) motion sequences. Based on a Fouie decomposition of a taining set of 3D data, we constuct a linea, mophable epesentation. Using this epesentation a lowdimensional linea model is leaned by means of Pinciple Component Analysis (PCA). Twodimensional test data ae now pojected onto this model and the esulting 3D econstuctions ae evaluated. We pesent two diffeent simulations. In the fist expeiment, we assume the D pojection matix to be known. In the second expeiment, the hoizontal viewpoint is unknown and is being ecoveed fom the data.. Intoduction Human motion contains a wealth of infomation about the actions, intentions, emotions, and pesonality taits of a peson and plays an impotant ole in many application aeas, such as suveillance, compute games, films, and biomechanics. The human visual system is so sensitive to human motion pattens that in a visual scene, one can quickly and efficiently ecognize human behavios and lean many aspects of human motion. What ou visual system seems to have achieved effotlessly so fa is still a challenging poblem fo atificial vision systems, although the computeized analysis of human motion is gaining moe and moe inteest. Many eseaches have conducted studies on human motion analysis, and geneal suveys can be found in ecent eview papes [,]. Most of the existing eseach [37] has been focused on the poblem of tacking and ecognizing human motion though motion sequences, while less attention has been paid to the poblem of econstucting 3D human motion fom D motion sequences. 3D motion econstuction is simila to 3D shape econstuction fom a sequence of images, but it is widely believed that 3D motion econstuction is much hade than the latte because it equies not only 3D postues deived fom D motion sequences but also tempoal coespondence between the model and D motion sequences. Theefoe, 3D motion econstuction fom D motion sequences is still a challenging task. The impotance of 3D motion econstuction stems fom applications such as suveillance and monitoing, human body animation, and automatic ecognition of human activities fom video databases. Unlike D motion, which is highly view-dependent, 3D human motion can povide obust ecognition and suveillance. Howeve, multiple cameas ae usually needed to get 3D motion infomation. Fo the pupose of ealizing a vitual acto and multimedia application, even the most skilled modeles and animatos ae not able to accuately epoduce the espective motions fom image sequences. Human motion analysis is oughly divided into two categoies: model-based methods and model-fee methods [8]. In the model-based methods, a human model is used to epesent the obseved subjects, while in the model-fee methods, the motion infomation is diectly utilized in a sequence of images to ecognize o analyze human motion. The use of a human model seems to be the pefeed tend. Model-based methods equie coespondence-based epesentations, which esult in a sepaation of the oveall infomation into ange-specific infomation and domain-specific infomation [9,0]. In the case of biological motion data, ange specific infomation efes to the state of the acto at a given time in tems of cuent position o of a numbe of featue points. Domain specific infomation, on the othe hand, efes to when a given position occus. In the model of Kakadiais [], motion estimation of human movement is obtained fom multiple cameas. Bowden et al [] mentioned the econstuction of 3D postue fom monocula image sequences. Just like econstucting 3D face shape fom a single image based on a mophable model [3], Bowden et al econstucted the 3D
2 stuctue of a subject fom a single view of its outline. In fact, only sepaate 3D poses can be econstucted without using tempoal coespondence. Recently, Toje [9] developed a famewok that tansfomed biological motion data into a linea epesentation. Lineaizing the data by using a PCA-based epesentation, he constucted a sex classifie with a easonable classification pefomance. Giese et al [4] pesented a leaning-based appoach fo the epesentation of complex motion pattens based on linea combinations of pototypical motion sequences. Both authos [9,4] pointed out that spatio-tempoal coespondences between image sequences ae the key issue fo the development of efficient models that pefom well on ecognition and classification tasks. The esulting models have been temed mophable models since linea combinations of existing data esult in motions which epesent membes of the same motion class as the oiginal motions. Locomotion pattens such as walking and unning ae univesal yet complex human activities. Locomotion involves a high level of inteaction between the cental nevous system and vaious muscles to allow an individual to keep the body upight and balanced while moving aound in an odely and stable manne. The solutions to the poblem of geneating a stable gait depend on masses and dimensions of the paticula body and its pats and ae theefoe highly individualized. Gait pattens can in fact seve biometic puposes and have been used in the context of ecognition and tacking tasks [3,4]. In this epot, we will exploe the poblem of econstucting 3D human walking fom D motion sequences. Human walking is a special class of motion chaacteized not only by abundant similaities but also by vaiations along many diffeent lines. Given a set of walking data epesented in tems of a mophable model, the similaities ae epesented by the aveage motion patten, while the vaiations ae expessed in tems of the covaiance matix. Based on the set of walking data, pincipal component analysis (PCA) can be used to find a low dimensional, othonomal basis system that would efficiently span a motion space in which individual walking pattens ae appoximated in tems of a linea model. We offe a epesentation that is able to captue the edundancy in gait pattens in an efficient way. The vision poblem is not the pupose of this pape and hee we make two assumptions that facilitate and focus the poblem at this point. Fistly, we assume that the image pocessing poblem of tacking featue points in the D image plane is solved. We epesent human motion in tems of tajectoies of a seies of makes located at the main joints of the body. Figue illustates the locations of these makes. Secondly, D motion sequences ae assumed to be othogaphic pojections of 3D motion. Both assumptions ae not citical to the geneal idea outlined in this pape. The tacking poblem can be teated completely independently and in fact thee ae many laboatoies woking on this poblem [57]. Figue. Human motion epesentation by joints Establishing spatio-tempoal coespondences is a peequisite fo constucting a linea mophable model. Hee, ou input data consist of the motion tajectoies of discete make points so that spatial coespondence is solved in a canonical way and not an issue. Since we use low-dimensional Fouie expansions to appoximate ou data, establishing tempoal coespondence is educed to adjusting fequency and phase of the Fouie components. We systematically develop a theoy fo ecoveing 3D walking data fom D motion sequences. Section biefly descibes data acquisition with a make-based motion captue system. The details of the linea model and the 3D motion econstuction ae given in Section 3. We econstuct 3D walking data and evaluate the poposed method in Section 4. Finally, some conclusions and possible futue extensions ae given in Section 5.. Walking data acquisition Eighty paticipants seved as subjects to acquie walking data. A set of 4 eto-eflective makes was attached to thei bodies. Paticipants wee equested to
3 walk on a teadmill and 3D motion data wee ecoded using a motion captue system (Vicon; Oxfod Metics, Oxfod, UK) equipped with 9 CCD high-speed cameas. The system tacked the 3D tajectoies of the makes with a sub-millimete spatial esolution and a sampling ate of 0 Hz. Fom the tajectoies of these makes we computed the location of vitual makes positioned at the majo joints accoding to a standad biomechanical model (BodyBuilde, Oxfod Metics). Each postue is descibed in tems of 5 makes positioned at the majo joints of the body as illustated in Fig.. 3. Motion econstuction Using the obtained walking data, we fist constuct a linea mophable model by PCA fom Fouie epesentations of human examples, and then econstuct the missing motion data fom the linea model. 3. Linea model Human walking can be egaded as a time seies of postues p ( : t =,, ; T, =,,, R, whee R is the numbe of walkes and T is the numbe of sampled postues fo walke. Because each joint has thee coodinates, the epesentation of a postue p ( including 5 joints is a 45-dimensional vecto. Human walking can be efficiently descibed in tems of low-ode Fouie seies [5]. A compact epesentation fo a paticula walke, which we call hee Fouie epesentation, utilizes the aveage postue p, 0 the chaacteistic postues of the fundamental fequency ( p, q ) and second hamonic ( p, q ) of a discete Fouie expansion, and the fundamental fequency ( ω ) itself to chaacteize a paticula walke: p( = p0 + p sin( ω + q cos( ω () + p sin(ω + q cos(ω Since the aveage postue and each of the chaacteistic postues ae 45-dimensional vectos and the fequency is a scala, the dimensionality of a Fouie epesentation at this stage is 45 *5 + = 6. The fundamental fequency ω is used to obtain 3D peiodic human motion fom Fouie epesentation. Fo each specific walking patten p (, using Eq.(), we can get a Fouie epesentation w : w ( p p, q, p, q, ω ) = () 0,,,,,, The advantage of Fouie epesentation is that it povides the possibility to successfully apply linea opeations and to detemine tempoal coespondence. Tempoal infomation is included in the fequency and phase. Afte computing the fequency and phase of a walking seies by Fouie analysis, we can epesent the walking seies with zeo phase and fequencyindependent chaacteistic postues as shown in Eq. (). Consequently, tempoal coespondence is obtained by scaling and tanslating the time axis. Evey Fouie epesentation can be teated as a point in a 6-dimensional linea space. We applied PCA to all the Fouie epesentations in ode to lean the main motion vaiation and to futhe educe dimensionality. All the Fouie epesentations ae concatenated into a matix W with each column containing one walke w, concatenating p 0,, p,, q,, p,, q, and ω in tun. Computing PCA on the matix W esults in a decomposition of each paamete set w into an aveage value w and a seies of othogonal eigen-walkes e n : w = w + k, e (3) n n w = ( / R) w denotes the aveage value of all the R columns and k n, is the coefficient fo walke. A linea model is spanned by the fist N eigen-walkes e en, which epesent the pincipal vaiation. The exact dimensionality N of the model depends on the equied accuacy of the appoximation but will geneally be much smalle than R. A Fouie epesentation of walke w can now be epesented as a linea combination of eigen-walkes by using the obtained coefficients k n,. 3. Reconstuction We denote a single D motion sequence as pˆ ( : t =,,, T, which is epesented in tems of its discete Fouie components: ( pˆ, pˆ, qˆ, pˆ, ˆ,ω) ˆ 0 q n w = (4) ˆ 0 The aveage postue p, and the chaacteistic postues p ˆ, qˆ, ˆp and ˆq only contain D infomation. These postues and fequency ae concatenated into a column vecto ŵ with 30 *5 + = 5 enties. Reconstucting the full 3D motion means to find the ight solution w of the equation: w ˆ = Cw (5) 3
4 6 5 Hee C : R R is the pojection matix and at this point we assume that it is known. In some cases with an unknown viewpoint, the pojection matix is a function of the view angle. In the following we only fomulate the econstuction with a known viewpoint and those cases with an unknown viewpoint can be fomulated by the same pocedue. Equation (5) is an unde-detemined equation system and can only be solved if additional constaints can be fomulated. This is done in tems of the linea model outlined in the pevious section. Substituting Eq. (3) into Eq. (5), we get: w N ˆ = C + w k e n n (6) n= Denoting w ˆ = Cw and e ˆn = Cen, Eq. (6) can be ewitten as: a known viewpoint. The missing dimensions wee econstucted and evaluated at one degee inteval fom the left (negative) to the fontal and finally to the ight (positive) pofile viewpoints. In the second expeiment, we test whethe we can etieve 3D motion even without knowing the hoizontal viewpoint fom which a walke is seen. The D motion sequence was the othogaphic pojection of a 3D walke onto a vetical plane fom an unknown viewpoint. In both expeiments, we used a leave-one-out coss validation pocedue: One walke is excluded fom the data base when ceating the linea model and is late used fo testing. This pocedue is epeated fo evey walke. 4.. Linea model constuction N ˆ w ˆ = k e ˆ n n (7) w n= o, using matix notation, as: wˆ wˆ = EK ˆ (8) This is an ove-detemined linea equation system. We appoximate a solution accoding to a least-squae citeion using the pseudo-invese: ˆ T K = ( E Eˆ) ˆ T E ( wˆ w ˆ ) (9) Afte the coefficients K ae estimated, the missing infomation w = ( p 0, p, q, p, q ) of Fouie epesentation is synthesized in tems of the espective linea combination of eigen-walkes e en. The econstucted Fouie epesentation w is the combination of the D Fouie epesentation ŵ and the econstucted w : w = [ wˆ, w ] = ([ pˆ ],[ ˆ, ],[ ˆ, 0, p0 p p q q ], [ pˆ, ],[ ˆ, (0) p q q ],ω ) 3D peiodic human motion is obtained fom the econstucted Fouie epesentation w by using Eq. (). 4. Expeiments Using walking data acquied with a make-based motion captue system, we conducted two diffeent expeiments. In the fist expeiment we assume that the viewpoint fom which the D test walke is seen is known. The D motion sequence was the othogaphic pojection of one 3D walke onto a vetical plane fom Figue. The cumulative vaiance coveed by the pincipal components computed acoss 80 subjects Fouie epesentations wee applied sepaately to all walking data. On aveage, the fundamental fequency coveed 9.9% of the total vaiance and the second hamonic accounted fo anothe 6.0%, which means that the fist two hamonics explain 97.9% of the oveall postual vaiance of a walke. The sets of all the Fouie epesentations wee now submitted to a PCA. The fist 8 pincipal components accounted fo 95% vaiance of all the Fouie epesentations, so they wee chosen as eigen-walkes in the linea model (Fig. ). 4.. Motion econstuction Pojecting the Fouie epesentation of a D motion sequence onto the linea model can lead to a 3D Fouie epesentation. 3D walking data wee econstucted at one degee intevals fom the left pofile-view to the ight pofile-view. Figue 3 4
5 illustates the oiginal and the econstucted motion sequences fo one subject fom thee viewing angles (0, 30 and 90 degees), coesponding to the top, middle and bottom ows. In ode to bette visualize the econstucted motion, we displayed the esults fom the diections othogonal to missing diections. This figue shows five equal inteval postues in one motion cycle. The dashed lines and the solid lines ae the oiginal and the econstucted motion sequences, espectively. The esults showed that the oiginal and the econstucted walking sequences wee vey simila in the coesponding postues. Figue 3. Oiginal and econstucted walking data fom thee diffeent viewing angles. In the expeiments, the D motion sequence is the othogaphic pojection of a 3D walke onto a vetical plane, which means that the actual motion in the missing dimension is known befoehand. Theefoe, the econstuction method can be evaluated by compaing the oiginal data with the econstucted data. In ode to give a quantified evaluation, an absolute mean squae econstuction eo is defined fo each joint: T j j Eabs ( j) = ( pm ( p ( ) () T t= whee p j m( ) and j p ( ) ae the oiginal and econstucted data fo the j th ( j =,,, 5 ) joint in the missing dimension. Fom diffeent viewpoints, human walking will have diffeent vaiances, so a elative mean squae econstuction eo is also defined fo each joint: E el T j j ( j) = ( pm( p ( ) Tσ m t= () whee σ m is the aveage oveall vaiance of the 5 joints in the missing dimension. The econstuction eo of one walke is the aveage value of the econstuction eos fo the 5 joints: 5 E = E( j) (3) 5 j= In Eq. (3), E denotes the absolute econstuction eos o the elative econstuction eos of each joint. The aveage econstuction eos among all the walkes fo a cetain viewpoint wee calculated and Fig. 4 shows these absolute aveage eos and elative aveage eos fom diffeent viewpoints (-9090 degees). As can be seen fom Fig. 4(a), thee ae diffeent absolute econstuction eos in the negative and positive oblique viewpoints, which implies that human walking is slightly asymmetic. The vey small asymmety of vaiances in Fig. 4(b) also confims this point. Gait symmety and asymmety is a popula topic of eseach [6] since human gait intechanges the left and ight legs and the opposite side of ams with a phase shift of half a peiod. Both legs pefom the same motion but out of phase fom each othe by half a peiod. In most cases, these motions ae opeated in space and time, satisfying the ules of spatial and tempoal symmety. Theefoe, it is easonable to assume that nomal gait is symmetic. In fact, in ablebodied gait, thee ae diffeences in the ight and left limbs, so gait asymmety was also addessed in the liteatue [6]. Ou expeimental esults povide evidence that the actions of fou limbs ae sometimes asymmetic duing human walking. Human body expends a specific amount of enegy fo locomotion and it has been suggested that asymmetic walkes should expend moe enegy than symmetic walkes. Ou econstuction method can analyze amputee s gait to get symmetic atificial limbs to save enegy. Figue 4(c) illustates that the fontal and the oblique viewpoints have small elative econstuction eos, while the pofile viewpoints have geat elative econstuction eos. The poposed method can etieve most vaiances in the fontal and oblique viewpoints (nea fontal) by the D pojection of the linea model along these viewpoints. The geat eos in the pofile and oblique viewpoints (nea pofile) imply that fewe vaiances ae coveed by the D pojection of the linea model along these viewpoints. The diffeent elative econstuction eos imply that human walking has moe divese actions in the fontal and oblique viewpoints (nea fontal), while the pofile and oblique viewpoints (nea pofile) contain moe simila actions. 5
6 4.3. Viewpoint econstuction Using 3D walking examples to geneate D motion sequences and to build up the linea model, we can econstuct and evaluate the viewpoint of a D walke. Assuming that a D walking sequence emains constant walking diection and the viewpoint has only otation aound the vetical axis (that is, hoizontal viewpoin, we can estimate the view angle using the otated aveage postue w (α ) and the otated eigenwalkes E (α ). Equation (8) can be epesented as: ( α, K ) = ag min wˆ wˆ ( α) Eˆ( α) K (4) α, K Figue 5. Illustation of the aveage econstuction view angles fom diffeent viewpoints of degees fo 80 walkes Figue 4. Illustation of the aveage econstuction eos fom diffeent viewpoints of degees fo 80 walkes. (a) Absolute econstuction eos. (b) Vaiance. (c) Relative econstuction eos. The imum solution of this minimization poblem can be found by solving a nonlinea ove-detemined system and the view angle is α. As such, the missing data is the linea combination of eigen-walkes by using the imum coefficient K. As in the fist expeiment, a leave-one-out pocedue was applied to all the walkes and the aveage econstuction angles fom diffeent viewpoints ae illustated in Fig. 5. The solid line is the aveage econstucted angles and the dashed lines ae the aveage standad deviations. The expeimental esults show that we can pecisely obtain the viewpoints by ou poposed linea model. It is possible to identify and ecognize human gait fom D image sequences of diffeent viewpoints by using the obtained view angle and missing data. The poo econstuction in pofile views and the good econstuction in fontal views fit with data on human pefomance in biological motion ecognition tasks [9,7]. 6
7 5. Conclusion and futue wok We have investigated and evaluated the poblem of econstucting 3D peiodic human motion fom D motion sequences. A linea model, which is based on linea combination of eigen-walkes, was constucted fom Fouie epesentations of the examples by using PCA. Fouie epesentation of a paticula subject is a compact desciption of human walking, so not only is tempoal coespondence detemined by the fequency and phase, but also computational expenditue and stoage space equiement ae saved. Expeiments and quantified evaluations wee made on walking data. The esults and quantified eo analysis explained that the poposed method can easonably econstuct 3D human walking data fom D motion sequences. We also veified that the linea model can estimate paametes like the view angle of walking diection, fom D motion sequences with constant walking diection. It is possible to povide a view-independent gait identification and ecognition method fom monocula motion sequences. Anothe application of the poposed model is to use it as a pio fo videobased tacking. In ou study, we assume that the positions of the joints ae known. In pactical applications, ecognizing and tacking joints is a challenging task. Human motion is non-igid and pivotal otation about some joints, so the linea model is limited to captue global vaiations in human activities. In ou futue wok, nonlinea motion models will be incopoated into the econstuction method and nonpeiodic human motion econstuction will also be studied by dynamic pogamming to detemine tempoal coespondence. 7. Refeences [] J. Aggawal and Q. Cai. Human Motion Analysis: A Review. Compute Vision and Image Undestanding, 73(3): , 999. [] L. Wang, W. Hu and T. Tan. Recent Developments in Human Motion Analysis. Patten Recognition, 36(3): , 003. [3] D. Cunado, M. Nixon and J. Cate. Automatic Extaction and Desciption of Human Gait Models fo Recognition Puposes. Compute Vision and Image Undestanding, 90(): -4, 003. [4] L. Wang, T. Tan, H. Ning and W. Hu. Silhouette Analysis-based Gait Recognition fo Human Identification. IEEE Tansactions on Patten Analysis and Machine Intelligence, 5(): , 003. [5] A. Jepson, D. Fleet, and T. El-Maaghi. Robust Online Appeaance Models fo Visual Tacking. IEEE Tansactions on Patten Analysis and Machine Intelligence, 5(0): 96-3, 003. [6] H. Ning, T. Tan, L. Wang and W. Hu. Kinematics- Based Tacking of Human Walking in Monocula Video Sequences. Image and Vision Computing, (5): 49-44, 004. [7] R. Utasun and P. Fua. 3D Tacking fo Gait Chaacteization and Recognition. In the 6 th Intenational Confeence on Automatic Face and Gestue Recognition, 004, South Koea [8] T. Moeslund and E. Ganum. A Suvey of Compute Vision-Based Human Motion Captue. Compute Vision and Image Undestanding, 8(3): 3-68, 00. [9] N. F. Toje. Decomposing Biological Motion: A Famewok fo Analysis and Synthesis of Human Gait Pattens. Jounal of Vision, (5): , 00. [0] J. Ramsay and B. Silveman. Functional Data Analysis. Spinge-Velag, 997. [] I. Kakadiais and D. Metaxas. Model-Based Estimation of 3D Human Motion. IEEE Tansactions on Patten Analysis and Machine Intelligence, (): , 000. [] R. Bowden, T. Mitchell and M. Sahadi. Non-linea Statistical Models fo the 3D Reconstuction of Human Pose and Motion fom Monocula Image Sequences. Image and Vision Computing, 8(0): , 000. [3] V. Blanz and T. Vette. Face Recognition Based on Fitting a 3D Mophable Model. IEEE Tansactions on Patten Analysis and Machine Intelligence, 5(9): , 003. [4] M. Giese and T, Poggio. Mophable Models fo the Analysis and Synthesis of Complex Motion Pattens. Intenational Jounal of Compute Vision, 38(): 59-73, 000. [5] M. Unuma, K. Anjyo and R. Takeuchi. Fouie Pinciples fo Emotion-based Human Figue Animation. Compute Gaphics Poceedings of SIGGRAPH 95, 9-96, 995. [6] H. Sadeghi, P. Allad, F. Pince, H. Labelle. Symmety and Limb Dominance in Able-Bodied Gait: a Review. Gait and Postue, (): 34-45, 000. [7] G. Mahte and L. Mudoch. Gende Discimination in Biological Motion Displays Based on Dynamic Cues. Poceedings of the Royal Society of London. Seies B: Biological Sciences, 58, 73-79,
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