Generalized Principal Component Analysis (GPCA)

Transcription

1 Geeralized Pricipal Compoet Aalysis (GPCA) Reé Vidal Yi Ma Shakar Sastry Departmet of EECS, Uiversity of Califoria, Berkeley, CA ECE Departmet, Uiversity of Illiois, Urbaa, IL Abstract We propose a algebraic geometric approach to the problem of estimatig a mixture of liear subspaces from sample data poits, the so-called Geeralized Pricipal Compoet Aalysis (GPCA) problem. I the absece of oise, we show that GPCA is equivalet to factorig a homogeeous polyomial whose degree is the umber of subspaces ad whose factors (roots) represet ormal vectors to each subspace. We derive a formula for the umber of subspaces ad provide a aalytic solutio to the factorizatio problem usig liear algebraic techiques. The solutio is closed form if ad oly if 4. I the presece of oise, we cast GPCA as a costraied oliear least squares problem ad derive a optimal fuctio from which the subspaces ca be directly recovered usig stadard oliear optimizatio techiques. We apply GPCA to the motio segmetatio problem i computer visio, i.e. the problem of estimatig a mixture of motio models from 2-D imagery. 1. Itroductio Pricipal Compoet Aalysis (PCA) [4] refers to the problem of estimatig a liear subspace S R K of ukow dimesio k < K from N sample poits {x j S} N j=1. This problem shows up i a variety of applicatios i may fields, e.g., patter recogitio, data compressio, image aalysis, regressio, etc., ad ca be solved i a remarkably simple way from the sigular value decompositio (SVD) of the data matrix [x 1, x 2,...,x N ] R K N. Extesios of PCA iclude probabilistic PCA [9, 2], where the subspace is estimated i Maximum Likelihood sese usig a probabilistic geerative model, ad oliear PCA (NLPCA) [6], where the subspace is estimated after applyig a oliear embeddig to the data. I this paper, we cosider a alterative geeralizatio called Geeralized Pricipal Compoet Aalysis (GPCA), i which the sample poits {x j R K } N j=1 are draw from k-dimesioal liear subspaces of R K, {S i }, as illustrated i Figure 1 for =3,k =2,adK =3. I this case, the problem becomes oe of idetifyig each subspace without kowig which sample poits belog to which subspace. 1 Research supported by grats ONR N ad DARPA F C-3614, ad by UIUC ECE Departmet startup fud. S 3 x S 2 R 3 S 1 Figure 1: GPCA for =3, k =2ad K =3. Idetifyig three 2-dimesioal subspaces S 1,S 2,S 3 i R 3 from sample poits {x} draw from these subspaces. Geometric approaches to mixtures of pricipal compoets have bee proposed i the computer visio commuity i the cotext of 3-D motio segmetatio. The mai idea is to first segmet the data associated with each subspace, ad the apply stadard PCA to each group. I [5] (see also [1, 3]) it was show that if the pairwise itersectio of the subspaces is trivial, which implies that K k, oe ca use the SVD of all the data to build a similarity matrix from which the segmetatio ca be easily extracted. Whe the subspaces do itersect, the segmetatio of the data is usually obtaied i a ad-hoc fashio usig various clusterig algorithms, e.g., K-meas. A alterative algebraic solutio for the case of two plaes i R 3 was proposed i [7] i the cotext of 2-D segmetatio of trasparet motios. 2 See also [14] for 3-D segmetatio of two rigid motios. Probabilistic approaches to mixtures of pricipal compoets [8] assume that sample poits withi each subspace are draw from a ukow probability distributio. The membership of the data poits to each oe of the subspaces is modeled with a multiomial distributio whose parameters are referred to as the mixig proportios. The parameters of this mixture model are estimated i a Maximum Likelihood or Maximum a Posteriori framework as follows: oe first estimates the mixig proportios give a curret estimate for the subspaces, ad the estimates the subspaces give a curret estimate of the mixig proportios. This is usually doe i a iterative maer usig the Expectatio Maximizatio (EM) algorithm. Ufortuately, EM is 1 If the associatio betwee sample poits ad subspaces was kow, the the problem would reduce to stadard PCA applied to each subspace. 2 The authors thak Dr. David Fleet for poitig out this referece.

2 i geeral sesitive to iitializatio ad may ot coverge to the global optimum. Aother disadvatage is that it is hard to aalyze some theoretical questios such as the existece ad uiqueess of a solutio to the problem. Also there are may cases i which it is hard to solve the groupig problem correctly, yet it is possible to obtai a precise estimate of the subspaces. I such cases a direct estimatio of the subspaces (without groupig) seems more appropriate tha a estimatio based o icorrectly segmeted data Cotributios of this paper We propose a ovel algebraic geometric approach to mixtures of pricipal compoets, the so-called Geeralized Pricipal Compoet Aalysis (GPCA) problem. I the absece of oise, we cast GPCA i a algebraic geometric framework i which the umber of subspaces becomes the degree of a certai polyomial ad the ormals to each subspace become the factors (roots) of such a polyomial. We show that the umber of subspaces ca be obtaied from the rak of a certai matrix that depeds o the data. Give, the estimatio of the subspaces S i R K is essetially equivalet to a factorizatio problem i the space of homogeeous polyomials of degree i K variables. We prove that the factorizatio problem has a uique solutio which ca be obtaied from the roots of a polyomial of degree i oe variable ad from the solutio of K 2 liear systems i variables. Hece, the solutio is closed form whe 4. Ulike previous work, GPCA allows for arbitrary itersectios amog a arbitrary umber of differet subspaces ad does ot require previous kowledge of the segmetatio of the data or the umber of subspaces. I fact, the subspaces are estimated directly usig segmetatio idepedet costraits that are satisfied by all the poits, regardless of the subspace to which they belog. I the presece of oise, we cast GPCA as a costraied oliear least squares problem that miimizes the error betwee the oisy poits ad their projectios oto the subspaces. By covertig this costraied problem ito a ucostraied oe, we obtai a optimal fuctio from which the subspaces ca be directly recovered usig stadard oliear optimizatio techiques. We show that the optimal objective fuctio is just a ormalized versio of the algebraic error miimized by our aalytic solutio to GPCA. Although this meas that the aalytic solutio to GPCA may be sub-optimal i the presece of oise, we ca still use it as a global iitializer for our oliear algorithm or ay other iterative algorithm, such as K-meas or EM. Our solutio to GPCA ca be applied to various estimatio problems i which the data comes simultaeously from multiple (approximately) liear models. I this paper, we apply GPCA to the motio segmetatio problem i computer visio, i.e. the problem of estimatig a mixture of motio models from 2-D imagery. Applicatios to segmetatio of static ad dyamic textures are forthcomig [10]. 2 Problem formulatio ad aalysis I this paper, we cosider the followig geeralizatio of pricipal compoet aalysis (PCA). Problem 1 (Geeralized Pricipal Compoet Aalysis) Give a set of sample poits X = {x j R K } N j=1 draw from >1distict liear subspaces {S i R K } of dimesio k, 0 <k<k, idetify each subspace S i without kowig which sample poits belog to which subspace. By idetifyig the subspaces we mea the followig: 1. Idetify the umber of subspaces ad their dimesio; 2. Idetify a basis for each subspace S i (or for S i ); 3. Group or segmet the give N data poits ito the subspace(s) they belog to; I our aalysis of the GPCA problem, we will distiguish betwee the followig two cases: The geeral case of subspaces of ukow dimesio k, where 0 <k<k 1, ad the special case of hyperplaes of kow dimesio k = K 1. It turs out that the geeral case ca always be reduced to the special case, as log as all the subspaces {S i } have the same dimesio k (see [10] for the proof). This is because, from a geometric poit of view, the segmetatio of a sample set X draw from k-dimesioal subspaces of a space of dimesio K>kis preserved after projectig the sample set X oto a geeric 3 subspace P of dimesio k 1( K). A example is show i Figure 2, where two lies L 1 ad L 2 i R 3 are projected oto a plae P ot orthogoal to the plae cotaiig the lies. I geeral, there are various techical details ivolved i reducig the geeral case 0 <k<kto the special case k = K 1, e.g., how to determie the dimesio of the subspaces k ad how to choose the (k 1)-dimesioal subspace P. We refer the reader to [10] for all those details, ad cocetrate o the special but importat case k = K 1 from ow o. R 3 L 1 L 2 o l 1l2 Figure 2: Samples o two 1-dimesioal subspaces L 1,L 2 i R 3 projected oto a 2-dimesioal plae P. The membership of each sample is preserved through the projectio. 3 By geeric we mea except for a zero-measure set {P } of subspaces. P

3 3. A solutio to GPCA with k = K 1 I this sectio, we give the followig costructive solutio to the GPCA problem i the case k = K 1. Theorem 1 (GPCA with k = K 1) The GPCA problem with k = K 1 is algebraically equivalet to factorig a homogeeous polyomial of degree i K variables ito a product of polyomials of degree 1. The factorizatio problem ca be solved from the roots of a polyomial of degree i oe variable plus K 2 liear systems i variables. Thus GPCA with k =K 1 has a uique solutio, which ca be obtaied i closed form if ad oly if 4. We establish the equivalece betwee GPCA ad polyomial factorizatio i Sectio 3.1. We show how to estimate the umber of subspaces i Sectio 3.2 ad give a aalytic solutio to the factorizatio problem i Sectio 3.3. We summarize the overall algorithm i Sectio 3.4 ad preset a optimal algorithm i the presece of oise i Sectio GPCA ad polyomial factorizatio We otice that every (K 1)-dimesioal space S i R K ca be represeted by a ozero ormal vector b i R K as S i = {x R K : b T i x = b i1 x 1 b i2 x 2...b ik x K =0}. Sice the subspaces S i are all distict from each other, the ormal vectors {b i } are pairwise liearly idepedet. Imagie that we are give a poit x R K lyig o oe of the subspaces S i. Such a poit must satisfy the formula: (b T 1 x =0) (b T 2 x =0) (b T x =0), (1) which is equivalet to the followig homogeeous polyomial of degree i x with real coefficiets: p (x) = (b T i x) =0. (2) The problem of idetifyig each subspace S i is the equivalet to that of solvig for the vectors {b i } 1=1 from the oliear equatio (2). A stadard techique used i algebra to reder a oliear problem ito a liear oe is to fid a embeddig that lifts the problem ito a higherdimesioal space. Let R (K) =R [x 1,...,x K ] be the set of all homogeeous polyomials of degree i K variables. We otice that each R (K), ca be made ito a vector space uder the usual additio ad scalar multiplicatio. Furthermore, R (K) is geerated by the set of moomials x = x 1 1 x2 2 xk K, with 0 j, j =1,...,K, ad 1 2 K =. Sice there are a total of ( ) ( ) K 1 K 1 M = = (3) K 1 differet moomials, the dimesio of R (K) as a vector space is M. Therefore, we ca defie the followig embeddig (or liftig) from R K ito R M. Defiitio 1 (Veroese map) Give ad K, the Veroese map of degree, ν : R K R M, is defied as: ν : [x 1,...,x K ] T [...,x,...] T, (4) where x is a moomial of the form x 1 1 x2 2 xk K with chose i the degree-lexicographic order. With the so-defied Veroese map (also kow as the polyomial embeddig), equatio (2) becomes the followig liear expressio i the vector of coefficiets c R M : p (x) =ν (x) T c = c 1,..., K x 1 1 xk K =0, (5) where c R represets the coefficiet of the moomial x. Example 1 The case =2ad K =2correspods to segmetig two lies i R 2. These two lies are represeted by the polyomial p 2 (x) =(b 11 x 1 b 12 x 2 )(b 21 x 1 b 22 x 2 ).I this case the Veroese map is ν 2 (x) =[x 2 1,x 1 x 2,x 2 2] T ad the coefficiets are c =[b 11 b }{{ 21,b } 11 b 22 b 12 b 21,b }{{} 12 b 22 ] }{{} T. c 2,0 c 1,1 c 0,2 Give c, the slope of each lie ca be immediately computed from the roots of the polyomial c 2,0 w 2 c 1,1 w c 0,2 = Estimatio of the umber of subspaces Applyig equatio (5) to a give collectio of N M 1 sample poits {x j } N j=1 gives the followig system of liear equatios o the vector of coefficiets c ν (x 1 ) T L c =. ν (x 2 ) T. c =0 RN. (6) ν (x N ) T Sice the above liear system (6) depeds explicitly o the umber of subspaces, we caot estimate c directly without kowig i advace. It turs out that the estimatio of the umber of subspaces is very much related to the coditios uder which the solutio for c is uique (up to a scale factor), as stated by the followig propositio. Propositio 1 (Number of subspaces) Assume that a collectio of N M 1 sample poits {x j } N j=1 o differet (K 1)-dimesioal subspaces of R K is give. Let L i R N Mi be the matrix defied i (6), but computed with the Veroese map ν i (x) of degree i. If the sample poits are i geeral positio ad at least K 1 poits correspod to each subspace, the: >M i 1, i <, rak(l i ) = M i 1, i =, (7) <M i 1, i >. Therefore, the umber of subspaces is give by: =mi{i : rak(l i )=M i 1}. (8)

4 The ituitio behid Propositio 1 (see [10] for the proof) is that there is o polyomial of degree i<that is satisfied by all the data, hece rak(l i )=M i for i<. Coversely, there are multiple polyomials of degree i>, amely ay multiple of p (x), which are satisfied by all the data, hece rak(l i ) <M i 1 for i>. Thus the case i = is the oly oe i which system (6) has a uique solutio, amely the coefficiets c of the polyomial p (x). Remark 1 I the presece of oise, oe caot directly estimate from (8), because the matrix L i is always full rak. I practice we declare the rak of L i to be r if σ r1 /(σ 1 σ r ) <ɛ,whereσ k is the k-th sigular value of L i ad ɛ>0is a pre-specified threshold. We have foud this simple criterio to work well i our experimets. Remark 2 I the case of subspaces of arbitrary dimesio k, where0 <k<k, oe ca derive a rak coditio similar to (8) from which oe ca joitly estimate ad k. Such a rak coditio i GPCA is a atural geeralizatio of the rak coditio k = rak(l 1 ) i stadard PCA [10] Estimatio of the subspaces {S i } Propositio 1 ad the liear system i equatio (6) allow us to determie the umber of subspaces ad the vector of coefficiets c, respectively, from sample poits {x j } N. The rest of the problem becomes ow how to recover the ormal vectors {b i } from c. From (2) ad (5) we have p (x) = K c x = b ij x j. j=1 Therefore, recoverig {b i } from c is equivalet to factorig a give homogeeous polyomial p (x) R (K), ito distict polyomials i R 1 (K). 4 We ow preset a polyomial factorizatio algorithm that recovers the b i s from c. For simplicity, we will first preset a example with the case of two plaes i R 3, i.e. =2ad K =3, because it gives most of the ituitio about our geeral algorithm for arbitrary ad K. Example 2 Cosider the case =2ad K =3. The p 2 (x) =(b 11 x 1 b 12 x 2 b 13 x 3 )(b 21 x 1 b 22 x 2 b 23 x 3 ) =(b T 1 x)(b T 2 x) =[x 2 1,x 1 x 2,x 1 x 3,x 2 2,x 2 x 3,x 2 3]c = (b 11 b }{{ 21 )x } 2 1(b 11 b 22 b 12 b 21 )x }{{} 1 x 2 (b 11 b 23 b 13 b 21 )x }{{} 1 x 3 c 2,0,0 c 1,1,0 c 1,0,1 (b 12 b }{{ 22 )x } 2 2 (b 12 b 23 b 13 b 22 )x }{{} 2 x 3 (b 13 b 23 )x }{{} 2 3. c 0,2,0 c 0,1,1 c 0,0,2 4 Oe ca iterpret c as a vector represetatio of the symmetric part of the tesor b 1 b. I this case estimatig {b i } from c is equivalet to factorig such a symmetric tesor. The factorizatio algorithm we are about to preset ca be thought of as a SVD for symmetric tesors. We otice that the last three terms c 0,2,0 x 2 2 c 0,1,1x 2 x 3 c 0,0,2 x 2 3 correspod to a polyomial i x 2 ad x 3 oly, which is equal to the product of the last two terms of the origial factors, i.e. (b 12 x 2 b 13 x 3 )(b 22 x 2 b 23 x 3 ). After dividig by x 2 3 ad lettig w = x 2/x 3 we obtai q 2 (w). = c 0,2,0 w 2 c 0,1,1 w c 0,0,2 =(b 12 w b 13 )(b 22 w b 23 ). Sice c R 6 is kow, so is the secod order polyomial q 2 (w). Thus we ca obtai b13 b 12 ad b23 b 22 from the roots w 1 ad w 2 of q 2 (w). Sice b 1 ad b 2 are oly computable up to a scale factor, we ca actually divide c by c 0,2,0 (if ozero) ad set the last two etries of b 1 ad b 2 to be b 12 =1, b 13 = w 1, b 22 =1, ad b 23 = w 2. We are left with the computatio of the first etry of b 1 ad b 2. We otice that the coefficiets c 1,1,0 ad c 1,0,1 are liear fuctios of the ukows b 11 ad b 21. Therefore, if b 22 b 13 b 23 b 12 0, i.e. if w 1 w 2, the we ca obtai b 11 ad b 21 from the liear system [ ][ ] [ ] b22 b 12 b11 c1,1,0 =. (9) b 23 b 13 b 21 c 1,0,1 We coclude from the Example 2 that, except for the degeerate cases c 0,2,0 = b 12 b 22 =0or b 22 b 13 b 23 b 12 = 0, the factorizatio of a homogeeous polyomial of degree =2i K =3variables ca be doe as follows. 1. Solve for the last two etries of {b i } from the roots of a polyomial q (w) associated with the last 1 coefficiets of p (x). 2. Solve for the first K 2 etries of {b i } by solvig K 2 liear systems i variables. We ow show i Sectios ad how the above example ca be geeralized to arbitrary ad K, except for some degeerate cases. We aalyze such degeerate cases i Sectio ad briefly outlie how to hadle them. We summarize the overall algorithm i Sectio Solvig for the last 2 etries of each b i Cosider the last 1coefficiets of p (x): [c 0,...,0,,0,c 0,...,0, 1,1,..., c 0,...,0,0, ] T R 1, which defie the followig homogeeous polyomial of degree i the two variables x K 1 ad x K : c0,...,0,k 1, K x K 1 = (b ik 1 x K 1 b ik x K ). K 1 xk K Lettig w = x K 1 /x K, we obtai (b ik 1 x K 1 b ik x K )=0 (b ik 1 w b ik )=0.

5 Hece the roots of the polyomial q (w) =c 0,...,0,,0 w c 0,...,0, 1,1 w 1 c 0,...,0,0, are exactly w i = b ik /b ik 1, for all i =1,...,. Therefore, after dividig c by c 0,...,0,,0, we obtai the last two etries of each b i as: (b ik 1,b ik )=(1, w i ). (10) If b ik 1 =0for some i, the some of the leadig coefficiets of q (w) are zero ad we caot proceed as before, because q (w) has less tha roots. More specifically, assume that the first l coefficiets of q (w) are zero ad divide c by the (l 1)-st coefficiet. I this case, we ca choose (b ik 1,b ik )=(0, 1), fori =1,...,l, ad obtai {(b ik 1,b ik )} i= l1 from the l roots of q (w) usig equatio (10). Fially, if all the coefficiets of q (w) are zero, we set (b ik 1,b ik )=(0, 0), for all i =1,..., Solvig for the first K 2 etries of each b i We have demostrated how to obtai the last two etries of each b i from the roots of a polyomial of degree i oe variable. We are ow left with the computatio of the first K 2 etries of each b i. We assume that we have computed {b ij }, j = J 1,...K for some J, startig with J = K 2, ad show how to liearly solve for {b ij }.Asi Example 2, the key is to cosider the coefficiets of p (x) which are liear i x J. These coefficiets are of the form c 0,...,0,1,J1,..., K ad are liear i b ij. To see this, otice that the polyomial c 0,...,0,1,J1,..., K x J1 J1 xk K is equal to the partial derivative of p (x) with respect to x J evaluated at x 1 = x 2 = = x J =0. Sice ( ) ( i 1 ) (b T i x) = b ij (b T i x) (b T i x), x J l=1 l=i1 after evaluatig at x 1 = x 2 = = x J =0we obtai c0,...,0,1,j1,..., K x J1 J1 xk K = b ij gi J (x), (11) where i 1 gi J (x) = l=1 K j=j1 b lj x j l=i1 K j=j1 b lj x j (12) is a homogeeous polyomial of degree 1 i the last K J variables i x. LetVi J be the vector of coefficiets of the polyomial gi J(x). Notice that the vectors {V i J} are kow, because they are fuctios of the kow b ij s, for j J 1. Therefore we ca use equatio (11) to solve for the ukows {b ij } from the liear system b 1J c 0,...,0,1, 1,0,...,0 [ ] V J 1 V2 J V J b 2J. = c 0,...,0,1, 2,1,...,0.. (13) b J c 0,...,0,1,0,0,..., Degeerate cases I order for the liear system i (13) to have a uique solutio, the colum vectors {Vi J} (i the matrix o the left had side) must be liearly idepedet. We showed i [10] that this is ideed the case if ad oly if for all r s, 1 r, s, the vectors (b rj1,b rj2,...,b rk ) ad (b sj1,b sj2,...,b sk ) are pairwise liearly idepedet. This latter coditio is always satisfied, except for some degeerate cases described i Remark 3 below. I those degeerate cases, as log as the origial polyomial p (x) has distict factors, oe ca always perform a ivertible liear trasformatio o the data poits x x = T x, T R K K (14) that iduces a liear trasformatio o the vector of coefficiets c c = T c, T R M M, such that the ew vectors (b rj1,b rj2,...,b rk ) are pairwise liearly idepedet. We refer the reader to [10] for further details o the solutio of these degeerate cases. Remark 3 (Degeerate cases) There are essetially three cases i which the vectors (b rj1,b rj2,...,b rk ) are ot pairwise liearly idepedet: 1. The origial polyomial p (x) is such that the polyomial q (w) has repeated roots, e.g., p 3 (x) =(x 1 x 2 x 3 )(x 1 2x 2 2x 3 )(x 1 2x 2 x 3 ). 2. The polyomial q (w) associated with some factorable p (x), e.g., p (x) =(x 1 x 3 )x 3, has more tha oe zero leadig coefficiets. I this case we have (b i2,b i3 )=(0, 1) for more tha oe i. 3. The origial polyomial p (x) is ot factorable. This happes, for example, whe the vector of coefficiets c is corrupted with oise. I this case the polyomial q (w) may have complex roots, e.g., p (x) = x 2 1 x2 2 x 2x 3 x 2 3, ad oe could project these complex roots oto their real parts. This typically itroduces repeated real roots i the resultig polyomial, e.g., after projectio the above polyomial p (x) becomes x 2 1 x2 2 x 2x x GPCA algorithm for k = K 1 Algorithm 1 (GPCA algorithm for the case k = K 1) Give sample poits {x j } N j=1, fid the umber of subspaces ad their ormals {b i R K } as follows: 1. Apply the Veroese map of degree i, for i =1, 2,...,to the vectors {x j } N j=1 ad form the matrix L i R N Mi as i (6). Stop whe rak(l i )=M i 1 ad set the umber of subspaces to be the curret i. The solve for the vector of coefficiets c R M from the liear system L c =0ad ormalize so that c =1.

6 2. (a) Divide c by the first ozero coefficiet of q (w). (b) If the first l, 0 l, coefficiets of q (w) are equal to zero, set (b ik 1,b ik )=(0, 1) for i =1,...,l. Compute {(b ik 1,b ik )} i= l1 from the l roots of q (w) usig (10). (c) If all the coefficiets of q (w) are zero, set (b ik 1,b ik )=(0, 0), for i =1,...,. (d) If (b rk 1,b rk ) is parallel to (b sk 1,b sk ) for some r s, apply the trasformatio x x i (14) ad repeat 2(a), 2(b) ad 2(c) for the ew polyomial p (x ) to obtai {(b ik 1,b ik )}. 3. Give (b ik 1,b ik ), i =1,...,, solve for {b ij } from (13) for J = K 2,...,1. If a trasformatio T R K K was used i 2(d), the set b i = T T b i Optimal GPCA i the presece of oise I the previous sectio, we proposed a liear algorithm for estimatig a collectio of subspaces from sample data poits {x j } N j=1 lyig o those subspaces. I essece, Algorithm 1 solves for the ormal vectors {b i } from the set of oliear equatios (bt i x j )=0, j =1,...,N. From a optimizatio poit of view, Algorithm 1 gives a liear solutio to the oliear least squares problem ( N 2 (b T i x )) j (15) mi b 1,...,b S K 1 j=1 where S K 1 is the uit sphere i R K. I this sectio, we derive a optimal algorithm for recostructig the subspaces whe the sample data poits are corrupted with i.i.d. zero-mea Gaussia oise. We show that the optimal solutio ca be obtaied by miimizig a fuctio similar to the algebraic error i (15), but properly ormalized. Sice our derivatio is based o segmetatio idepedet costraits, we do ot eed to model the membership of each data poit with a probability distributio. Therefore, we do ot eed to iterate betwee model estimatio ad data segmetatio, as most iterative techiques do, e.g., K-meas ad EM. Istead, our approach elimiates the data segmetatio step algebraically ad solves the GPCA problem by directly optimizig over the ormals to each subspace. Let {x j } N j=1 be the give collectio of oisy data poits. We would like to fid a collectio of subspaces {S i } such that the correspodig oise free data poits { x j } N j=1 lie o those subspaces. That is, we would like to solve the costraied oliear least squares optimizatio problem N mi j=1 xj x j 2 (16) subject to (bt i xj )=0 j =1,...,N. By usig Lagrage multipliers λ j for each costrait, the above optimizatio problem is equivalet to miimizig the Lagragia fuctio ( N x j x j 2 λ j j=1 (b T i xj ) After takig partial derivatives w.r.t. x j we obtai 2( x j x j )λ j ). (17) b i (b T l xj )=0, (18) l i from which we ca solve for λ j /2 as bt i (x j x j ) l i (bt l x j ) = (bt i x j ) l i (bt l x j ) bi l i (bt l x j ) 2 bi l i (bt l x j ). (19) 2 Similarly, after premultiplyig (18) by ( x j x j ) T we get x j x j 2 = λj 2 (b T i x j ) (b T l xj ). (20) l i Replacig (19) ad (20) o the objective fuctio (16) gives Ẽ ({ x j }, {b i })= N j=1 ( ) 2 (bt i x j ) l i (bt l xj ) b i l i (bt l xj ) 2. (21) We ca obtai a objective fuctio o the ormal vectors oly by cosiderig first order statistics of c T ν (x j ). Sice this is equivalet to settig x j = x j i (21), we obtai the simplified objective fuctio ( N ) 2 (bt i x j ) E (b 1,...,b )= j=1 b i l i (bt l x j ) 2, (22) which is essetially the same as the algebraic error (15), but properly ormalized accordig to the chose oise model. By costructio, the error fuctio i (22) does ot deped o the segmetatio of the data, hece it ca be used to directly recover the subspace ormals {b i } from a set of N (K 1) data poits {x j } N j=1. Oe ca use Algorithm 1 to obtai a iitial estimate for ad {b i } ad the use stadard oliear optimizatio techiques to miimize (22). However, Algorithm 1 requires a much larger umber of poits N M 1, because it uses a overparameterized represetatio c R M of the ormal vectors. Remark 4 The optimal error i (21) has a very ituitive iterpretatio. If poit j belogs to group i, the b T i xj =0. Thus the cotributio of poit j to Ẽ reduces to ( b T i x j ) 2 l i (bt l x j ) ( 2 =(b l i (bt l x )) T i x j ) 2, (23) j which is the optimal fuctio to miimize for subspace i. Therefore, the optimal error Ẽ is just a clever algebraic way of writig a mixture of optimal fuctios for each subspace ito a sigle objective fuctio for all the subspaces.

7 4. Applicatios of GPCA I this sectio, we test GPCA o sythetic data ad preset various applicatios o 2-D ad 3-D motio segmetatio from 2-D imagery Experimets o sythetic data We first test GPCA (Algorithm 1) ad optimal GPCA (see Sectio 3.5) o sythetically geerated data. We radomly pick =2, 3, 4 collectios of N = 600 poits o k =2 dimesioal subspaces of R 3. Zero-mea Gaussia oise with stadard deviatio from 0% to 5% is added to the sample poits. We ru 1000 trials for each oise level. For each trial the error betwee the true (uit) ormals {b i } ad the estimates {ˆb i } is computed as error = 1 acos (b T ˆb ) i i (degrees). (24) Figure 3 plots the mea error as a fuctio of the oise level. I all the trials, the umber of subspaces was correctly estimated from equatio (8) as = 2, 3, 4. 5 Notice that the estimates of the algebraic algorithm (left) are withi 3.8, 8.5 ad 13.3 of the groud truth for =2, =3ad =4, respectively, while the estimates of the optimal algorithm (right) are withi 3.1, 6.4 ad 9.7 of the groud truth for =2, 3 ad 4, respectively. This is expected, because the algebraic algorithm uses a overparameterized represetatio c R M of the ormal vectors [b 1,..., b ] R K. Notice also that, as expected, the performace of both algorithms deteriorates as icreases Segmetatio of 2D traslatioal motios Cosider a image sequece whose 2-D motio field ca be modeled as a mixture of purely traslatioal motio models. That is, we assume that the optical flow u =[u, v, 1] T P 2 i a widow aroud every pixel ca take oe out of possible values {u i }, where the umber of models is ukow. Uder the Lambertia model, the optical flow u at pixel x = [x 1,x 2, 1] T P 2 is related to the partials of the image itesity y =[I x1,i x2,i t ] T R 3 at x by the well-kow brightess costacy costrait (BCC) y T u = I x1 ui x2 vi t =0. Thus the estimatio of multiple traslatioal motio models ca be casted as a GPCA problem with k =2ad K =3, i.e. the segmetatio of plaes i R 3. The optical flows {u i } correspod to the ormals to the plaes, ad the image partial derivatives {y j } N j=1 are the data poits. Furthermore, we iterpret the polyomial p (y) = (ut i y)=ũt ν (y) =0as the multibody brightess costacy costrait (MBCC) ad the vector of coefficiets ũ R M, where M =( 1)( 2)/2, as the multibody optical flow. 5 We used a threshold of ɛ = to compute the rak of L =2 =3 = =2 =3 = Figure 3: Error i the estimatio of the subspaces as a fuctio of oise for GPCA (left) ad optimal GPCA (right) Figure 4: Frames from the flower garde sequece (left) ad the image partials projected oto the I x1 -I t plae (right). (a) Tree (b) Houses (c) Grass Figure 5: Segmetatio of two frames from the flower garde sequece usig a mixture of three traslatioal models. Sice the MBCC icorporates multiple motio models, oe ca use a larger widow i the computatio of optical flow without havig the problem of itegratig image data across motio boudaries. Figures 4 ad 5 show the extreme situatio i which the whole image is used to estimate three traslatioal models for the flower garde sequece. Figure 4 shows two frames of the sequece ad the image partials for oe frame projected oto the I x1 -I t plae to facilitate visualizatio. We observe that the image partials lie approximately o three plaes through the origi, although the data is oisy ad cotais may outliers. We estimated three motio models by applyig Algorithm 1 to the image data, followed by the oliear algorithm described i Sectio 3.5. Figure 5 shows the segmetatio of the image pixels for two frames of the flower garde sequece accordig to the estimated motio models. Although we used a simple mixture of three traslatioal motios to model the 2-D motio field of the sequece, a good segmetatio of the tree, the houses ad the grass is obtaied. We did ot cluster pixels with low texture (y 0), e.g., pixels i the sky, sice they ca be assiged to either of the three models.

8 Remark 5 (Affie motio segmetatio) GPCA ca also be applied to the estimatio of a mixture of affie motio models {A i R 3 3 } from image data {(xj, y j )} N j=1. I this case the optical flow is modeled with the affie model u = A i x, thus the BCC becomes y T A i x =0. Affie motio segmetatio is the equivalet to estimatig {A i } from the multibody affie costrait (yt A i x)=0. This ca be doe by factorig this product of biliear forms. This problem ca be reduced to a collectio of GPCA problems with k = K 1=2as demostrated i [11] Segmetatio of liearly movig objects Cosider the problem of segmetig the 3-D motio of multiple objects udergoig a liear motio. That is, we assume that the scee ca be modeled as a mixture of purely traslatioal motio models, {e i R 3 }, where e i represets the epipole (traslatio) of object i relative to the camera betwee two cosecutive frames. Therefore, give the images x 1 P 2 ad x 2 P 2 of a poit i object i i the first ad secod frame, respectively, the rays x 1, x 2 ad e i must satisfy the well-kow epipolar costrait for liear motios x T 2 (e i x 1 )=0. Sice the epipolar costrait ca be coveietly rewritte as e T i (x 2 x 1 ) = 0, the segmetatio of liear motios is a GPCA problem with K =3ad k =2where the data poits are the epipolar lies l = x 2 x 1 R 3 ad the ormal vectors are the epipoles {e i }. Furthermore, we iterpret the polyomial p (l) = (et i l)=ẽt ν (l) =0as the multibody epipolar costrait ad the vector of coefficiets ẽ R M as the multibody epipole. We tested GPCA o a sequece with =2liearly movig objects. Figure 6(a) shows the first frame with N =92tracked features: 44 for the truck ad 48 for the car. Figure 6(b) plots the segmetatio of the features. There are o mismatches. The estimatio error for the epipoles was 3.3 for the truck ad 1.2 for the car. truck car (a) First frame (b) Feature segmetatio Figure 6: Segmetatio of =2liearly movig objects. Remark 6 (Multibody structure from motio (MSFM)) GPCA ca also be applied to the problem of estimatig a mixture of fudametal matrices {F i R 3 3 } from image pairs {(x j 1, xj 2 )}N j=1. I this case the epipolar costrait reads x T 2 F i x 1 =0ad the multibody epipolar costrait reads (xt 2 F i x 1 )=0. The MSFM problem is the equivalet to factorig this product of biliear forms. Such a problem ca be reduced to a collectio of GPCA problems with k = K 1=2as show i [13, 12]. 5. Discussio ad ope issues We have proposed a ovel geometric approach to the idetificatio of mixtures of subspaces (GPCA). We derived a formula for estimatig the umber of subspaces ad showed that GPCA is equivalet to estimatig ad factorig homogeeous polyomials. I the absece of oise, we preseted a aalytic solutio to the factorizatio problem based o liear algebraic techiques. I the presece of oise, we preseted oliear algorithm that miimizes the optimal error. We tested GPCA o sythetic data ad preseted various applicatios o 2-D ad 3-D motio segmetatio. Ope issues iclude a aalysis of the robustess of the polyomial factorizatio algorithm i the presece of oise. At preset the algorithm works well whe the umber ad dimesio of the subspaces is small, but the performace deteriorates as the umber of subspaces icreases. This is because the algorithm uses a overparameterized represetatio of the ormal vectors that eeds at least N M 1 poits, as opposed to the (K 1) poits eeded by the oliear algorithm. Whether it is possible to avoid workig i a space of dimesio M by usig somethig similar to the kerel trick i NLPCA [6], remais a ope questio. Refereces [1] T. Boult ad L. Brow. Factorizatio-based segmetatio of motios. I IEEE Workshop o Motio Uderstadig, pages , [2] M. Collis, S. Dasgupta, ad R. Schapire. A geeralizatio of pricipal compoet aalysis to the expoetial family. I Advaces i Neural Iformatio Processig Systems (NIPS), volume 14, [3] J. Costeira ad T. Kaade. A multibody factorizatio method for idepedetly movig objects. IJCV, 29(3): , [4] I. Jollife. Pricipal Compoet Aalysis. Spriger-Verlag, New York, [5] K. Kaatai. Motio segmetatio by subspace separatio ad model selectio. I ICCV, volume 2, pages , [6] B. Schölkopf, A. Smola, ad K.-R. Müller. Noliear compoet aalysis as a kerel eigevalue problem. Neural Computatio, 10(5): , [7] M. Shizawa ad K. Mase. A uified computatioal theory for motio trasparecy ad motio boudaries based o eigeeergy aalysis. I CVPR, pages , [8] M. Tippig ad C. Bishop. Mixtures of probabilistic pricipal compoet aalyzers. Neural Computatio, 11(2): , [9] M. Tippig ad C. Bishop. Probabilistic pricipal compoet aalysis. Joural of the Royal Statistical Society, 61(3): , [10] R. Vidal. Geeralized Pricipal Compoet Aalysis (GPCA). PhD thesis, Uiversity of Califoria at Berkeley, [11] R. Vidal ad S. Sastry. Segmetatio of dyamic scees from image itesities. I IEEE Workshop o Motio ad Video Computig, pages 44 49, [12] R. Vidal ad S. Sastry. Optimal segmetatio of dyamic scees from the multibody epipolar costrait. I CVPR, [13] R. Vidal, S. Soatto, Y. Ma, ad S. Sastry. Segmetatio of dyamic scees from the multibody fudametal matrix. I ECCV Workshop o Visual Modelig of Dyamic Scees, [14] L. Wolf ad A. Shashua. Two-body segmetatio from two perspective views. I CVPR, pages , 2001.