On Incremental and Robust Subspace Learning

Transcription

1 On Incremental and Robust Subspace Learnng Yongmn L, L-Qun Xu, Jason Morphett and Rchard Jacobs Content and Codng Lab, BT Exact pp1 MLB3/7, Oron Buldng, Adastral Park, Ipswch, IP5 3RE, UK Emal: Abstract Prncpal Component Analyss (PCA) has been of great nterest n computer vson and pattern recognton. In partcular, ncrementally learnng a PCA model, whch s computatonally effcent for large scale problems as well as adaptable to reflect the varable state of a dynamc system, s an attractve research topc wth numerous applcatons such as adaptve background modellng and actve object recognton. In addton, the conventonal PCA, n the sense of least mean squared error mnmsaton, s susceptble to outlyng measurements. To address these two mportant ssues, we present a novel algorthm of ncremental PCA, and then extend t to robust PCA. Compared wth the prevous studes on robust PCA, our algorthm s computatonally more effcent. We demonstrate the performance of these algorthms wth expermental results on dynamc background modellng and mult-vew face modellng. Keywords Prncpal Component Analyss (PCA), ncremental PCA, robust PCA, background modellng, mult-vew face modellng 1 Introducton Prncpal Component Analyss (PCA), or the subspace method, has been extensvely nvestgated n the feld of computer vson and pattern recognton (Turk and Pentland, 1991; Murase and Nayar, 1994; Moghaddam and Pentland, 1997). One of the attractve characterstcs of PCA s that a hgh dmensonal vector can be represented by a small number of orthogonal bass vectors,.e. the Prncpal Components. The conventonal methods of PCA, 1

2 such as Sngular Value Decomposton (SVD) and egen-decomposton, perform n batch-mode wth a computatonal complexty of O(m 3 ) where m s the mnmum value between the data dmenson and the number of tranng examples. Undoubtedly these methods are computatonally expensve when dealng wth large scale problems where both the dmenson and the number of tranng examples are large. To address ths problem, many researchers have been workng on ncremental algorthms. Early work on ths topc ncludes (Gll et al., 1974; Bunch and Nelsen, 1978). Gu and Esenstat (Gu and Esenstat, 1994) developed a stable and fast algorthm for SVD whch performs n an ncremental way by appendng a new row to the prevous matrx. Chandrasekaran et al. (Chandrasekaran et al., 1997) presented an ncremental egenspace update algorthm usng SVD. Hall et al. (Hall et al., 1998) derved an egen-decomposton based ncremental algorthm. In ther extended work, a method for mergng and splttng egenspace models was developed (Hall et al., 2000). Lu and Chen (Lu and Chen, 2002) also ntroduced an ncremental algorthm for PCA model updatng and appled t to vdeo shot boundary detecton. In addton, the tradtonal PCA, n the sense of least mean squared error mnmsaton, s susceptble to outlyng measurements. To buld a PCA model whch s robust to outlers, Xu and Yulle (Xu and Yulle, 1995) treated an entre contamnated vector as an outler by ntroducng an addtonal bnary varable. Gabrel and Odoroff (Gabrel and Odoroff, 1983) addressed the general case where each element of a vector s assgned wth a dfferent weght. More recently, De la Torre and Black (De la Torre and Black, 2001) presented a method of robust subspace learnng based on robust M-estmaton. Brand (Brand, 2002) also desgned a fast ncremental SVD algorthm whch can deal wth mssng/untrusted data, however the mssng part must be known beforehand. One lmtaton of the prevous robust PCA methods s that they are usually computatonally ntensve because the optmsaton problem has to be computed teratvely 1, e.g. the self-organsng algorthms n (Xu and Yulle, 1995), the crss-cross regressons n (Gabrel and Odoroff, 1983) and the Expectaton Maxmsaton algorthm n (De la Torre and Black, 2001). Ths computatonal neffcency restrcts ther use n many applcatons, especally when real-tme performance s crucal. 1 It s mportant to dstngush an ncremental algorthm from an teratve algorthm. The former performs n the manner of prototype growng from tranng example 1,2,...to t, the current tranng example, whle the latter terates on each learnng step wth all the tranng examples 1,2,... and N untl a certan stop condton s satsfed. Therefore, for the PCA problem dscussed n ths paper, the complexty of algorthms n the order from the lowest to hghest s: ncremental, batch-mode and teratve algorthm. 2

3 To address the ssue of ncremental and robust PCA learnng, we present two novel algorthms n ths paper: an ncremental algorthm for PCA and an ncremental algorthm for robust PCA. In both algorthms, the PCA model updatng s performed drectly from the prevous egenvectors and a new observaton vector. The real-tme performance can be sgnfcantly mproved over the tradtonal batch-mode algorthm. Moreover, n the second algorthm, by ntroducng a smplfed robust analyss scheme, the PCA model s robust to outlyng measurements wthout addng much extra computaton (only flterng each element of a new observaton wth a weght whch can be returned from a look-up-table). The rest of the paper s organsed as follows. The new ncremental PCA algorthm s ntroduced n Secton 2. It s then extended to robust PCA n Secton 3 as a result of addng a scheme of robust analyss. Applcatons of usng the above algorthms for adaptve background modellng and multvew face modellng are descrbed n Secton 4 and 5 respectvely. Conclusons and dscussons are presented n Secton 6. 2 Incremental PCA Note that n ths context we use x to denote the mean-normalsed observaton vector,.e. x = x µ (1) where x s the orgnal vector and µ s the current mean vector. For a new x, f we assume the updatng weghts on the prevous PCA model and the current observaton vector are α and 1 α respectvely, the mean vector can be updated as µ new = αµ + (1 α)x = µ + (1 α)x (2) Construct p + 1 vectors from the prevous egenvectors and the current observaton vector y = αλ u, = 1, 2,..., p (3) y p+1 = 1 αx (4) where {u } and {λ } are the current egenvectors and egenvalues. The PCA updatng problem can then be approxmated as an egen-decomposton problem on the p + 1 vectors. An n (p + 1) matrx A can then be defned as A = [y 1, y 2,..., y p+1 ] (5) 3

4 Assume the covarance matrx C can be approxmated by the frst p sgnfcant egenvectors and ther correspondng egenvalues, C U np Λ pp U T np (6) where the columns of U np are egenvectors of C, and dagonal matrx Λ pp s comprsed of egenvalues of C. Wth a new observaton x, the new covarance matrx s expressed by C new = αc + (1 α)xx T = αu np Λ pp U T np + (1 α)xx T p αλ uu T + (1 α)xx T (7) =1 Substtutng (3), (4) and (5) nto (7) gves C new = AA T (8) Instead of the n n matrx C new, we egen-decompose a smaller (p+1) (p+1) matrx B, B = A T A (9) yeldng egenvectors {v new } and egenvalues {λ new } whch satsfy Bv new = λ new v new, = 1, 2,..., p + 1 (10) Left multplyng by A on both sdes and usng (9), we have AA T Av new = λ new Av new (11) Defnng u new and then usng (8) and (12) n (11) leads to = Av new (12) C new u new = λ new u new (13).e. u new s an egenvector of C new wth egenvalue λ new. 4

5 Algorthm 1 The ncremental algorthm of PCA 1: Construct the ntal PCA from the frst q(q p) observatons. 2: for all new observaton x do 3: Update the mean vector (2); 4: Compute y 1, y 2,..., y p from the prevous PCA (3); 5: Compute y p+1 (4); 6: Construct matrx A (5); 7: Compute matrx B (9); 8: Egen-decompose B to obtan egenvectors {v new {λ new }; 9: Compute new egenvectors {u new } (12). 10: end for } and egenvalues The algorthm s formally presented n Algorthm 1. It s mportant to note: 1. Incrementally learnng a PCA model s a well-studed subject (Gll et al., 1974; Bunch and Nelsen, 1978; Chandrasekaran et al., 1997; Hall et al., 1998; Hall et al., 2000; Lu and Chen, 2002; Brand, 2002). The man dfference between the algorthms, ncludng ths one, s how to express the covarance matrx ncrementally (e.g. Equaton (7)) and the formulaton of the algorthm. The accuracy of these algorthms s smlar because updatng s based on approxmatng the covarance wth the current p-ranked model. Also, the speed of these algorthms s smlar because they perform n a smlar way of egen-decomposton or SVD on the rank of (p + 1). Therefore, there s no need to compare the performance of these algorthms. However, we beleve the algorthm as presented n Algorthm 1 s concse and easy to be mplemented. Also, t s ready to be extended to the robust PCA whch wll be dscussed n the next secton. 2. The actual computaton for matrx B only occurs for the elements of the (p+1)th row or the (p+1)th column snce {u } are orthogonal unt vectors,.e. only the elements on the dagonal and the last row/column of B have non-zero values. 3. The update rate α determnes the weghts on the prevous nformaton and new nformaton. Lke most ncremental algorthms, t s applcaton-dependent and has to be chosen expermentally. Also, wth ths updatng scheme, the old nformaton stored n the model decays exponentally over tme. 5

6 3 Robust PCA Recall that PCA, n the sense of least squared reconstructon error, s susceptble to contamnated outlyng measurement. Several algorthms of robust PCA have been reported to solve ths problem, e.g. (Xu and Yulle, 1995; Gabrel and Odoroff, 1983; De la Torre and Black, 2001). However, the lmtaton of these algorthms s that they mostly perform n an teratve way whch s computatonally ntensve. The reason of havng to use an teratve algorthm for robust PCA s that one normally does not know whch part of a sample are lkely to be outlers. However, f a prototype model, whch does not need to be perfect, s avalable for a problem to be solved, t would be much easy to detect the outlers from the data. For example, we can easly pck up a cat mage as an outler from a set of human face mages because we know what the human faces look lke, and for the same reason we can also tell the whte blocks n Fgure 5 (the frst column) are outlyng measurements. Now f we assume that the updated PCA model at each step of an ncremental algorthm s good enough to functon as ths prototype model, then we can solve the problem of robust PCA ncrementally rather than teratvely. Based on ths dea, we develop the followng ncremental algorthm of robust PCA. 3.1 Robust PCA wth M-Estmaton We defne the resdual error of a new vector x by r = U np U T npx x (14) Note that the U np s defned as n (6) and, agan, x s mean-normalsed. We know that the conventonal non-robust PCA s the soluton of a least-squares problem 2 mn r 2 = (r j )2 (15) j Instead of sum-of-squares, the robust M-estmaton method (Huber, 1981) seeks to solve the followng problem va a robust functon ρ(r) mn ρ(r j ) (16) j 2 In ths context, we use subscrpt to denote the ndex of vectors, and superscrpt the ndex of ther elements. 6

7 Dfferentatng (16) by θ k, the parameters to be estmated,.e. the elements of U np, we have ψ(r j ) rj = 0, k = 1, 2,..., np (17) θ k j where ψ(t) = dρ(t)/dt s the nfluence functon. By ntroducng a weght functon w(t) = ψ(t) (18) t Equaton (17) can be wrtten as w(r j r j )rj = 0, k = 1, 2,..., np (19) θ k j whch can be regarded as the soluton of a new least-squares problem f w s fxed at each step of ncremental updatng mn w(r j )(rj )2 (20) j If we defne z j = w(r j )xj (21) then substtutng (14) and (21) nto (20) leads to a new egen-decomposton problem mn U np U T npz z 2 (22) It s mportant to note that w s a functon of the resdual error r j whch needs to be computed for each ndvdual tranng vector (subscrpt ) and each of ts elements (superscrpt j). The former mantans the adaptablty of the algorthm, whle the latter ensures that the algorthm s robust to every element of a vector. If we choose the robust functon as the Cauchy functon ρ(t) = c2 2 log(1 + ( t c )2 ) (23) where c controls the convexty of the functon, then we have the weght functon 1 w(t) = (24) 1 + (t/c) 2 7

8 Now t seems we arrve at a typcal teratve soluton to the problem of robust PCA: compute the resdual error wth the current PCA model (14), evaluate the weght functon w(r j ) (24), compute z (21), and egendecompose (22) to update the PCA model. Obvously an teratve algorthm lke ths would be computatonally expensve. In the rest of ths secton, we propose an ncremental algorthm to solve the problem. 3.2 Robust Parameter Updatng One mportant parameter needs to be determned before performng the algorthm: c n (23,24) whch controls the sharpness of the robust functon and hence determnes the lkelhood of a measurement beng an outler. In prevous studes, the parameters of a robust functon are usually computed at each step n an teratve robust algorthm (Huber, 1981; Hampel et al., 1986) or usng Medan Absolute Devaton method (De la Torre and Black, 2001). Both methods are computatonally expensve. Here we present an approxmate method to estmate the parameters of a robust functon. The frst step s to estmate σ j, the standard devaton of the jth element of the observaton vectors {x j }. Assumng that the current PCA model (ncludng ts egenvalues and egenvectors) s already a robust estmaton from an adaptve algorthm, we approxmate σ j wth σ j = max p =1 λ u j (25).e. the maxmal projecton of the current egenvectors on the jth dmenson (weghted by ther correspondng egenvalues). Ths s a reasonable approxmaton f we consder that PCA actually presents the dstrbuton of the orgnal tranng vectors wth a hyper-ellpse n a subspace of the orgnal space and thus the varaton n the orgnal dmensons can be approxmated by the projectons of the ellpse onto the orgnal space. The next step s to express c, the parameter of (23,24), wth c j = βσ j (26) where β s a fxed coeffcent, for example, β = s obtaned wth the 95% asymptotc effcency on the normal dstrbuton (Zhang, 1997). β can be set at a hgher value for fast model updatng, but at the rsk of acceptng outlers nto the model. To our knowledge, there are no ready solutons so far as to estmate the optmal value of coeffcent β. We use an example of background modellng to llustrate the performance of parameter estmaton descrbed above. A vdeo sequence of 200 frames s used n ths experment. The conventonal PCA s appled to the sequence to 8

9 obtan 10 egenvectors of the background mages. The varaton σ j computed usng the PCA model by Equaton (25) s shown n Fgure 1(a). We also compute the pxel varaton drectly over the whole sequence as shown n (b). Snce there s no foreground object appeared n ths sequence, we do not need to consder the nfluence of outlers. Therefore (b) can be regarded as the ground-truth pxel varaton of the background mage. For a quanttatve measurement, we compute the rato of σ j by Equaton (25) to ts groundtruth (subject to a fxed scalng factor for all pxels), and plot the hstogram n Fgure 1(c). It s noted that 1. the varaton computed usng the low-dmensonal PCA model s a good approxmaton of the ground-truth, wth most rato values close to 1 as shown n Fgure 1(c); 2. the pxels around mage edges, valleys and corners normally demonstrate large varaton, whle those n smooth areas have small varaton. 3.3 The Incremental Algorthm of Robust PCA By ncorporatng the process of robust analyss, we have the ncremental algorthm of robust PCA as lsted n Algorthm 2. The dfference from the non-robust algorthm (Algorthm 1) s that the robust analyss (lnes 3-6) has been added and x s replaced by z, the weghted vector, n lnes 7 and 9. For completeness of descrpton, we nclude the whole algorthm n Algorthm 2. It s mportant to note: 1. It s much faster than the conventonal batch-mode PCA algorthm for large scale problems, not to menton the teratve robust algorthm; 2. The model can be updated onlne over tme wth new observatons. Ths s especally mportant for modellng dynamc systems where the system state s varable. 3. The extra computaton over the non-robust algorthm (Algorthm 1) s only to flter a new observaton wth a weght functon. If the Cauchy functon s adopted, ths extra computaton s reasonably mld. However, even when more ntensve computaton lke exponental and logarthm nvolved n the weght functon w, a look-up-table can be bult for the weght tem w( ) n Equaton (21) whch can remarkably reduce the computaton. Note the look-up-table should be ndexed by r/c rather than r. 9

10 (a) (b) (c) Fgure 1: Standard devaton of ndvdual pxels σ j computed from (a) the low-dmensonal PCA model (approxmated) and (b) the whole mage sequence (ground-truth). All values are multpled by 20 for llustraton purpose. Large varaton s shown n dark ntensty. (c) Hstogram of the ratos of approxmated σ j to ts ground-truth value. Algorthm 2 The ncremental algorthm of robust PCA 1: Construct the ntal PCA from the frst q(q p) observatons. 2: for all new observaton x do 3: Estmate c j, the parameter of the robust functon, from the current PCA (25,26); 4: Compute the resdual error r (14); 5: Compute the weght w(r j ) for each element of x (24); 6: Compute z (21); 7: Update the mean vector (2), replacng x by z; 8: Compute y 1, y 2,..., y p from the prevous PCA (3); 9: Compute y p+1 (4), replacng x by z; 10: Construct matrx A (5); 11: Compute matrx B (9); 12: Egen-decompose B to obtan egenvectors {v new } and egenvalues {λ new }; 10 13: Compute new egenvectors {u new } (12). 14: end for

11 4 Robust Background Modellng Modellng background usng PCA was frstly proposed by Olver et al. (Olver et al., 2000). By performng PCA on a sample of N mages, the background can be represented by the mean mage and the frst p sgnfcant egenvectors. Once ths model s constructed, one projects an nput mage nto the p dmensonal PCA space and reconstruct t from the p dmensonal PCA vector. The foreground pxels can then be obtaned by computng the dfference between the nput mage and ts reconstructon. Although Olver et al. clamed that ths background model can be adapted over tme, t s computatonally ntensve to perform model updatng usng the conventonal PCA. Moreover, wthout a mechansm of robust analyss, the outlers or foreground objects may be absorbed nto the background model. Apparently ths s not what we expect. To address the two problems stated above, we extend PCA background model by ntroducng (1) the ncremental PCA algorthm descrbed n Secton 2 and (2) robust analyss of new observatons dscussed n Secton 3. We appled the algorthms ntroduced n the prevous sectons to an mage sequence n PET2001 datasets 3. Ths sequence was taken from a unversty ste wth a length of 3061 frames. There are manly two knds of actvtes happened n the sequence: (1) movng objects, e.g. pedestrans, bcycles and vehcles, and (2) new objects beng ntroduced nto or removed from the background. The parameters n the experments are: mage sze (grey-level), PCA dmenson p = 10, sze of ntal tranng set q = 20, update rate α = 0.95 and coeffcent β = Comparng to the Batch-mode Method In the frst experment, we compared the performance of our robust algorthm (Algorthm 2) wth the conventonal batch-mode PCA algorthm. It s nfeasble to run the conventonal batch-mode PCA algorthm on the same data snce they are too bg to be ft n the computer memory. We randomly selected 200 frames from the sequence to perform a conventonal batch-mode PCA. Then the traned PCA was used as a fxed background model. Sample results are llustrated n Fgure 2 (more results are avalable n the supplementary vdeo fle pets.mpg 4 ). It s noted that our algorthm successfully captured the background changes. An nterestng example s that, between the 1000th to 1500th frames (the 3 A benchmark database for vdeo survellance whch can be downloaded at 4 Avalable at yongmn/sctv2003/pets.mpg. 11

12 Fgure 2: Sample results of background modellng. From left to rght are the orgnal nput frame, reconstructon and the weghts computed by Equaton (24) (dark ntensty for low weght) of the robust algorthm, and the reconstructon and the absolute dfference mages (dark ntensty for large dfference) of the conventonal batch-mode algorthm. Results are shown for every 500 frames of the test sequence. 1st and 2nd rows n Fgure 2), a car entered nto the scene and became part of the background, and another background car left from the scene. The background changes are hghlghted by whte boxes n the fgure. The model was gradually updated to reflect the changes of the background. In 12

13 ths experment, the ncremental algorthm acheved a frame rate of 5 fps on a 1.5GHz Pentum IV computer (wth JPEG mage decodng and mage dsplayng). On the other hand, the fxed PCA model faled to capture the dynamc changes of the background. Most notceably are the ghost effect around the areas of the two cars n the reconstructed mages and the false foreground detecton. Fgure 3: The frst three egenvectors obtaned from the robust algorthm (upper row) and non-robust algorthm (lower row). The ntensty values have been normalsed to [0, 255] for llustraton purpose. 4.2 Comparng to the Non-Robust Method In the second experment, we compared the performance of the non-robust algorthm (Algorthm 1) and robust algorthm (Algorthm 2). After applyng both algorthms to the same sequence used above, we llustrate the frst three egenvectors of each PCA model n Fgure 3. It s noted that the non-robust algorthm unfortunately captured the varaton of outlers, most notceably the trace of pedestrans and cars on the walkway appearng n the mages of the egenvectors. Ths s exactly the lmtaton of conventonal PCA (n the sense of least squared error mnmsaton) as the outlers usually contrbute more to the overall squared error and thus devate the results from desred. On the other hand, the robust algorthm performed very well: the outlers have been successfully fltered out and the PCA modes generally reflect the 13

14 varaton of the background only,.e. greater values for hghly textured mage postons. (a) (b) Fgure 4: The frst dmenson of the PCA vector computed on the same sequence n Fgure 2 usng the robust algorthm (a) and non-robust algorthm (b). The mportance of applyng robust analyss can be further llustrated n Fgure 4 whch shows the values of the frst dmenson of the PCA vectors computed wth the two algorthms. A PCA vector s a p-vector obtaned by projectng a sample vector onto the p egenvectors of a PCA model. The frst dmenson of the PCA vector corresponds to the projecton to the most sgnfcant egenvector. It s observed that the non-robust algorthm presents a fluctuant result, especally when sgnfcant actvtes happened durng frames , whle the robust algorthm acheves a steady performance. Generally, we would expect that a background model (1) should not demonstrate abrupt changes when there are contnuous foreground actvtes nvolved, and (2) should evolve smoothly when new components beng ntroduced or old components beng removed. The results as shown n Fgure 4 depct that the robust algorthm performed well n terms of these crtera, whle the non-robust algorthm struggled to compensate for the large error from outlers by severely adjustng the values of model parameters. 14

15 Fgure 5: Sample results of mult-vew face modellng. From left to rght are: orgnal face mage, mean vectors and reconstructons of (1) vew-based egenface method, (2) Algorthm 2, (3) Algorthm 1, and (4) batch-mode PCA, respectvely. Results are shown for every 20 frames of the test sequence. 5 Mult-vew Face Modellng Modellng face across multple vews s a challengng problem. One of the dffcultes s that the rotaton n depth causes the non-lnear varaton to the 2D mage appearance. The well-known egenface method, whch has been successfully appled to frontal face detecton and recognton, can hardly provde a satsfactory soluton to ths problem as the mult-vew face mages are 15

16 largely out of algnment. One possble soluton to ths problem as presented n (Moghaddam and Pentland, 1997) s to buld a set of vew-based egenface models, however, the pose nformaton of the faces need to be known and the dvson of the vew space s often arbtrary and coarse. In the followng experments we compare the results of four methods: (1) vew-based egenface method (Moghaddam and Pentland, 1997), (2) Algorthm 2 (robust), (3) Algorthm 1 (non-robust), and (4) batch-mode PCA. The mage sequences were captured usng an electromagnetc trackng system whch provdes the poston of a face n an mage and the pose angles of the face. The mages are n sze of pxels and contan faces of about pxels. As face detecton s beyond the doman of ths work, we drectly used the cropped face mages by the poston nformaton provded by the trackng system. We also added unformly dstrbuted random nose to the data by generatng hgh-ntensty blocks wth sze of 4-8 pxels at varous mage postons. Note that the frst 20 frames do not contan generated nose n order to obtan a clean ntal model for the robust method. We wll dscuss ths ssue n the last secton. For method (1), we dvde the vew space nto fve segments: left profle, left, frontal, rght, and rght profle. So the pose nformaton s used addtonally for ths method. Fve vew-based PCA models are traned respectvely on these segments wth the uncontamnated data because we want to use the results of ths method as ground-truth for comparson. For methods (2) and (3), the algorthms perform ncrementally through the sequences. For method (4), the batch-mode PCA s traned from the whole sequence. The mages are scaled to pxels. The parameters for the robust method are the same as those n the prevous secton: p = 10, q = 20, α = 0.95 and β = 10. Fgure 5 shows the results of these methods (more results are avalable n the supplementary vdeo fle face.mpg 5 ). It s evdent that 1. the batch-mode method faled to capture the large varaton caused by pose change (most notceably s the ghost effect of the reconstructons; 2. although the vew-based method s traned from clean data and uses extra pose nformaton, the reconstructons are notceably blurry owng to the coarse segmentaton of vew space; 3. the non-robust algorthm corrupted quckly owng to the nfluence of the hgh-ntensty outlers; 5 Avalable at yongmn/sctv2003/face.mpg. 16

17 4. the proposed ncremental algorthm of robust PCA performed very well: the outlers have been fltered out and the model has been adapted wth respect to the vew change. 6 Conclusons PCA s a wdely appled technque n pattern recognton and computer vson. However, the conventonal batch-mode PCA suffers from two lmtatons: computatonally ntensve and susceptble to outlyng measurement. Unfortunately the two ssues have only been addressed separately n the prevous studes. In ths work, we developed a novel ncremental PCA algorthm, and extended t to robust PCA. The man contrbuton of ths paper s the ncremental algorthm for robust PCA. In the prevous work, the problem of robust PCA s mostly solved by teratve algorthms whch are computatonally expensve. The reason of havng to do so s that one does not know what part of a sample are outlers. However, the updated model at each step of an ncremental PCA algorthm can be used for outler detecton,.e. gven ths prototype model, one does not need to go through the expensve teratve process. Ths s the startng pont of our proposed algorthm. We have provded detaled dervaton of the algorthms. Moreover, we have dscussed several mplementaton ssues ncludng (1) approxmatng the standard devaton usng the prevous egenvectors and egenvalues, (2) selecton of robust functons, and (3) look-up-table for robust weght computng. These can be helpful to further mprove the performance. Furthermore, we appled the algorthms to the problems of dynamc background modellng and mult-vew face modellng. These two applcatons alone have ther own sgnfcance: the former extends the statc method of PCA background modellng to a dynamc and adaptve method by ntroducng an ncremental and robust model updatng scheme, and the latter makes t possble to model faces of large pose varaton wth a smple, adaptve, model. Nevertheless, we have experenced problems when the ntal PCA model contans sgnfcant outlers. Under these crcumstances, the assumpton (the prototype model s good enough for outler detecton) s broken, and the model would take long tme to recover. Although the model can recover more quckly by choosng a smaller update rate α, we argue that the update rate should be determned by applcatons rather than the robust analyss process. A possble soluton to ths problem s to learnng the ntal model usng the tradtonal robust methods. Owng to the small sze of ntal data, 17

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