BACKGROUND SUBTRACTION WITH EIGEN BACKGROUND METHODS USING MATLAB

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1 BACKGROUND SUBTRACTION WITH EIGEN BACKGROUND METHODS USING MATLAB 1 Ilmyat Sar 2 Nola Marna 1 Pusat Stud Komputas Matematka, Unverstas Gunadarma e-mal: lmyat@staff.gunadarma.ac.d 2 Pusat Stud Komputas Matematka, Unverstas Gunadarma e-mal: nola@staff.gunadarma.ac.d ABSTRACT Most background models are pxel-based and t s not robust to unstable background. One frame-based background model s egen background. In ths paper, we present egen background model and to support the dscusson of the theory, we perform some smulatons usng the software Matlab. The result of experment s to obtan good background we just use one egen vector correspondng one largest egen value. We obtan foreground mage wth add one parameter,. e threshold. The used threshold s 60. I. INTRODUCTION Detecton of foreground objects n vdeo often requres robust technques for background modelng. The basc dea of background modelng s to mantan an estmaton of the background mage whch should represent the scene wth no foreground objects and must be kept updated frame by frame so as to model both statc and dynamc pxels n the background. Movng objects can then be detected by a smple subtracton and threshold procedure. Many background modelng methods for background subtracton have been proposed. Most background models are pxel-based and very lttle attenton s gven to regon-based or frame-based methods. Typcal models, among many others, nclude Kalman flters [1], Gaussans [2], mxture of Gaussans (MoG) [3, 4] and kernel densty estmaton [5]. Despte beng capable of solvng many background mantenance ssues, pxel-based models are less effcent and effectve n handlng lght swtchng and tme-ofday problems [6]. Rather, frame-based background models may potentally handled such a problem more effcently. One frame-based background model s egen background.

2 In ths paper, we present egen background model and to support the dscusson of the theory, we perform some smulatons usng the software Matlab. The paper s organzed as follows. In secton II, we descrbe egen space model, effcent mplementaton of egen space model and background subtracton wth egen background whle results and concluson are presented n secton III and IV. II. EIGEN BACKGROUND In ths secton a descrpton of the egen background algorthm wll be gven. We wll start wth the defnton of egen space. The key element of ths method les n ts ablty of learnng the background model from unconstrant vdeo sequence, even when they contan movng foreground objects. Egen takes nto account neghborng statstcs. It thus has a more global defnton on background whch, hopefully, make t more robust to unstable backgrounds [7]. II.1 EIGEN SPACE MODELS We wll frst ntroduce the concept of egen value decomposton appled to the case of a sequence of mages. We follow manly the defntons of [8]. The followng notaton wll be used: talcs wth subscrpts to ndcate vectors and matrxes (A m,n s a matrx of m rows and n columns), bold letters wth subscrpts for mages ( B h,w s an mage wth heght h and wdth w). vector x, Gven an mage I, of sze h, w (heght, wdth) t can be rewrtten as a column I h,w x n,1, (1) where n = hw. Gven a sequence of N mages x 1 n,1, x 2 n,1,, x N n,1, the average mage s computed as: N x n,1 = 1 x N =1 n,1. (2) The covarance matrx of the sequence of mages s gven by: N C n,n = 1 (x N =1 n,1 x n,1)(x n,1. (3) x n,1) T The same covarance matrx can be obtaned, by rearrangng the sequence of mage n sngle matrx X n,n n whch each column contans one mage: 1 2 X n,n = [x n,1 x n,1 x N n,1 ]. (4) From ths C n,n can be obtaned by:

3 C n,n = 1 N (X n,n x n,1ii 1,N )(X n,n x n,1ii 1,N ) T, (5) n whch s a matrx n whch all elements are set to 1. By defnton a covarance matrx s real and symmetrc. Its egenvalue decomposton s guaranteed to exst then, and ts gven by: C n,n, U n,n = U n,n, A nn, (6) where U n,n are egenvectors, and A n,n s a dagonal matrx of egenvalues, ordered form the most sgnfcant one to the least relevant one. The egenvectors are orthonormal, so that U T n,n U n,n = I n,n, the dentty matrx. The memory requrement for such an algorthm are qute demandng. Gven a set of N mages of n pxels each, the covarance matrx obtaned, C n,n, wll have a sze of n 2. Let s as an example, consder an mage acqured wth a VGA camera, consdered nowadays a low resoluton devce. The amount of memory requred to store such a matrx s then: w = 640, h = 480 n = n 2 = 9.40e10. Moreover, the computaton of an egen value decomposton for a matrx of sze n 2 wth such large values s absolutely prohbtve. Clearly, the smplest soluton s to greatly reduce the adopted mage resoluton, wth the obvous sde effect of a lower qualty representaton of the scene. II.2 Effcent Implementaon of Egen Space Model Fortunately a much more effcent mplementaton s obtaned by smply observng the followng set of equatons [8,9]. Let s once agan rearrange a sequence of N mages as X n,n gven by equaton (4). We can then compute matrx as: If we defne: D N,N = 1 N (X n,n x n,1ii 1,N ) T (X n,n x n,1ii 1,N ). (7) A n,n = (X n,n x n,1ii 1,N ). (8) We can rewrte C n,n and D N,N as: C n,n = A n,n A T n,n. (9) D N,N = A T n,n A n,n. (10) As for, an egen value decomposton for D N,N exsts and t s gven by: D N,N V N,N = V N,N Λ N,N. (11)

4 1 Let s now take any of the egen vectors of D N,N, V N,1 wth ts respectve egen value δ. By defnton of egen value decomposton, the followng equatons hold: If we defne: A T n,n A n,n V N,1 = δ V N,1, (12) A n,n A T n,n A n,n V N,1 = δ A n,n V N,1, (13) (A n,n A T n,n )A n,n V N,1 = δ A n,n V N,1 (14) P n,1 = A n,n V N,1. (15) And usng equaton (9), we can then rewrte equaton (14) as: C n,n P n,1 = δ P n,1. (16) By defnton of egen decomposton, P n,1 s an egen vector of C n,n and δ s the respectve egen value. It s then clear that we can obtan the needed egen decomposton of matrx C n,n by solvng the decomposton of matrx D N,N. The advantage s that, n typcal applcatons, the number of observatons N s much smaller than the number of pxels n the mage n. We can thus wrte: N << n, P n,n = A n,n V N,N, C n,n P n,n = P n,n Λ N,N. Matrx P n,n s guaranteed to be orthogonal, but t s not orthonormal. We then need to normalze each vector. Two methods are possble: U n,1 = P n,1 / P n,1, (17) U n,1 =P n,1 / δ. (18) The second method (18), can be derved followng equaton n [7]. The advantage of usng (18) when compared to (17) s that t allows to save some computaton, because term does not need to be computed. Even f ths can appear as mnor detal, n a real tme mplementaton t can speed up the computaton. C n,n U n,n = U n,n Λ N,N. (19) Equaton (19) solves the same problem of equaton (6), reducng greatly the amount of computaton and memory needed, wth the only sde effect of returnng the N egen vectors out of the total avalable n, correspondng to the N most sgngcant egen values. Ths does not create problems though, snce the number of sgnfcant egen vectors, even for a scene wth many lghtng varatons s seldom hgher than few tens.

5 II.3 Background Subtracton wth Egen Background In ths secton we descrbe how to explot the decomposton of a sequence of mages to buld a background model and acheve a smple background subtracton algorthm. In the typcal scenaro, a background model s constructed usng a sequence of frames captured wth a vdeo camera, whle no object whch should be detected as foreground appears n the scene. These mages are arranged n matrx X n,n followng equaton (4) and the egen value decomposton s obtaned through equaton (26). In order to be robust to llumnaton changes takng place n the scene, the best opton s to select a sequence of mages n whch the expected varatons s mostly present. The model conssts thus of the followng matrces: x n,1 = average model mage Λ k,k = k largest egen values U n,k = correspondng k egen vectors. The detecton of foreground s obtaned usng the followng steps. Frst, gven a captured mage, rearranged as a vector x n,1, the reconstructon of such mage through the background model s : r k,1 = U T n,k (x n,1 x n,1), (20) and then reconstructed approxmaton mage (x n,1 ) as follows (x n,1 ) = U n,k r k,1 + x n,1. (21) Fnally, foreground movng pxels are detected by computng the dstance between the nput mage x n,1 and reconstructed one (x n,1 ) χ n,1 = { 1 0 f d 0 (x n,1, (x n,1 ) ) > τ, (22) otherwse where τ s a threshold and d 0 s defned as follows d 0 = x n,1 (x n,1 ). III. Expermental Results As an mportant part of ths paper, we also present the expermental results of our mplementaton. Our experments are employed on Matlab R2009a on 32-bt wndows system.

6 We frst analyse the effect of ncludng n the model mages contanng foreground, such as car movng of the street. Fgure 1 shows some of the frames that were used to compute the background model. Fgure 2 shows the obtaned average mage wth 1 largest egen vectors and Fgure 3 shows the background subtracton obtaned wth dfferent egen values. Fgure 1: Frames of the sequence that were used to compute the background model. In total 51 frames were used to compute the model. Fgure 2: Image and egen vectors, correspondng to1 largest egen values, of the background model obtaned wth the sequence shown n Fgure 1. Fgure 3: Top: form left to rght: the background mage use 5, 10, 20, and 30 largest egen value Bottom: from left to rght: background mage use 40, 45, 48, and 51 (all) largest egen value The background mage n Fgure 2 and 3, we reconstruct t wth formula n equaton (21). From Fgure 2 t s clear, we can get background from each sequence mage only use 1 largest egen value. Fgure 3 show the effect of egen value aganst the background mage n more detal. Growng number of the largest egen values are used, then t wll be more vsble foreground.

7 Based on secton 2.3, foreground mage we can obtan from equaton (22). Then to change graylevel mage that we get n equaton (22), we use threshold to get foreground mage of bnary mage. Fgure 4 show foreground mage wth dfferent egen value. Fgure 4: Foreground mage and egen vectors, correspondng to k largest egen values, of the background model obtaned wth the sequence shown n Fgure 1. Used threshold s 60. From Fgure 4, t s clear that the largest number of egen values can be used to obtan the good foreground mage s 40. In Fgure 4, used threshold s 60, because we have done smulaton and we obtan 60 s the best threshold value for all mages. Fgure 5 shows foreground mage reconstruct wth 1 largest egen value and dfferent threshold. Fgure 5. From left to rght. foreground mage reconstruct wth 1 largest egen value and dfferent threshold (τ). τ = 50, τ = 55, and τ = 70. IV. Concluson We presented the theory and mplementaton of background subtracton based on egen spaces. Effects of most parameters (number of k largest egen vector and threshold) choce have been llustrated, to provde the reader wth the nsghts needed to make an

8 nformed choce when selectng these parameters. The fnal choce s strongly dependent on the applcaton. From experment, t can be concluded that to obtan the background mage, we only use 1 egen vector correspondng 1 largest egen value. Foreground mage s obtaned by equaton (21) wth 1 egenvector correspondng 1 largest egen value and threshold equals 60. Ths threshold can be used for each sequence mage. References [1] Koller, D., Weber, J., Huang, T., Malk, J., Ogasawara, G., Rao, B., Russell, S.: Towards robust automatc traffc scene analyss n real-tme. In: Proceedngs of the Internatonal Conference on Pattern Recognton, Israel (1994) [2] Wren, C.R., Azarbayejan, A., Darrell, T., Pentland, A.P.: Pfnder: Real-tme trackng of the human body. IEEE Transactons on Pattern Analyss and Machne Intellgence 19(7) (1997) [3] Stauffer, C., Grmson,W.E.L.: Learnng patterns of actvty usng real-tme trackng. IEEE Transactons on Pattern Analyss andmachne Intellgence 22(8) (2000) [4] Lee, D.S.: Effectve gaussan mxture learnng for vdeo background subtracton.ieee Transactons on Pattern Analyss and Machne Intellgence 27(5) (2005) [5] Elgammal, A., Duraswam, R., Harwood, D., Davs, L.S.: Background and foreground modelng usng non-parametrc kernel densty estmaton for vsual survellance. Proceedngs of the IEEE 90(7) (2002) [6] Toyama, K., Krumm, J., Brumtt, B., Meyers, B.: Wallflower: Prncples and practce of background mantenance. In: ICCV (1). (1999) [7] Benezeth, Y., Jodon, P. M., Emle, B., Laurent, H., Rosenberger, C.: Comparatve study of background subtracton algorthms. Journal of Electronc Imagng (19). (2010). [8] Karaman, M., Goldmann, D., Yu, D., Skora, T.: Comparson of statc background segmentaton methods. Vsual Communcatons and Image Processng. (2005). [9] Turk, M., Pentland, A.: Egen for Recognton. Journal of Cogntve Neuroscence. (1991).