Dimensionality Reduction and Clustering on Statistical Manifolds

Transcription

1 Dmensonalt Reducton and Clusterng on Statstcal Manfolds Sang-Mook Lee and A. Lnn Abbott Vrgna Poltechnc Insttute and State Unverst Blacksburg, VA 24061, USA Phlp A. Araman U.S. Forest Servce, Southern Research Staton Blacksburg, VA 24060, USA Abstract Dmensonalt reducton and clusterng on statstcal manfolds s presented. Statstcal manfold [16] s a 2D Remannan manfold whch s statstcall defned b maps that transform a parameter doman onto a set of probablt denst functons. Prncpal component analss (PCA) based dmensonalt reducton s performed on the manfold, and therefore, estmaton of a mean and a varance of the set of probablt dstrbutons are needed. Frst, the probablt dstrbutons are transformed b an sometrc transform that maps the dstrbutons onto a surface of hper-sphere. The sphere constructs a Remannan manfold wth a smple geodesc dstance measure. Then, a Fréchet mean s estmated on the Remannan manfold to perform the PCA on a tangent plane to the mean. Expermental results show that clusterng on the Remannan space produce more accurate and stable classfcaton than the one on Eucldean space. 1. Introducton Texture segmentaton s of fundamental mportance n mage analss and pattern recognton tasks and have been studed extensvel [1,2,3,4]. Example approaches nclude transform methods [5,6], stochastc technques [7,8], and combned technques [9]. Also, curve evoluton technques are ganng n populart [10,11,12,13]. Most of the reported methods deal wth mage models that have two or more regons and assocated probablt denst functons. In [11,14], statstcs of mage regons are modeled wth parametrc methods, whle Km et al. [13] use Parzen s denst estmates as regon descrptors and then utlze an nformaton theoretc approach to mage segmentaton. A mxture of parametrc and nonparametrc methods has been proposed n [4], where dfferent technques are appled to dfferent feature spaces. Meanwhle, Sochen et al. [15] ntroduced a geometrc framework b whch mages and mage feature spaces are consdered as 2-dmensonal manfolds. Later, a statstcal manfold framework has been studed for texture mage segmentaton [16], whch s substantall dfferent from the work of [15] that constructs non-statstcal manfolds. However, there was a drawback n the statstcal manfold framework, whch ntroduces boundar offsets n some stuatons when creatng a metrc tensor map, consequentl decreasng effcenc n boundar detecton applcatons. The drawback has been overcome [17] b usng a dffuson scheme on statstcal manfolds and a substantal mprovement has made over the ntal work. The result s a more robust framework for localzng texture boundares. As a related work, a vector probablt dffuson scheme has been ntroduced b [18], where the probabltes are treated as a vector feld and then a mnmzaton problem s solved on the feld. In ths paper, we ntroduce a prncpal component analss (PCA) based dmensonalt reducton and texture clusterng scheme appled to the statstcal manfold framework. Statstcal manfold s defned as an embeddng map that assgns each mage coordnate wth a set of probablt denst functons (PDFs) of features. A multnomal dstrbuton representaton s used for the statstcal manfold to accommodate multmodal dstrbutons n dscrete spaces. Beng PDFs whose dscrete sum are unt, multnomal dstrbutons are confned on a hper-plane of n-smplex, whch s nce because t s possble to emplo lnear methods for dmensonalt reducton. Therefore, we frst use an ordnar PCA method to reduce dmensonalt of the multnomal dstrbuton. It s not requred for the reduced ones to have unt of summaton, and thus ther lneart s not guaranteed. On the other hand, an sometrc transformaton maps the multnomal dstrbutons nto ponts on a sphere, and a dstance between two ponts on the sphere are measured b the arc length of a great crcle connectng the ponts [19]. Applng a lnear method to ths manfold s not straghtforward snce a mean pont and ts tangent plane must be estmated on the sphercal manfold, whch leads to the use of specal projectons and to the estmaton of Fréchet mean [20]. In the next secton, we revew the defnton of statstcal manfolds and the dffuson scheme on statstcal

2 manfolds. Then, the dmensonalt reducton on statstcal manfolds and a clusterng method are dscussed n secton 3. Secton 4 presents some segmentaton results, and then secton 5 concludes the paper. 2. Statstcal Manfolds A Remannan manfold M p s an abstract surface of arbtrar dmenson p wth a proper choce of metrc. Then, an mage I(x) parameterzed n R 2, that s, x=(x,) R 2, s vewed as a 2-dmensonal Remannan manfold, M 2, embedded n R n wth a embeddng map (x, I(x)), where n=3 for ntenst mages and 5 for color mages. Smlarl, m-dmensonal feature spaces of an mage can be consdered as M 2 embedded n R m+2 [15] Statstcal embeddng Statstcall defned manfold has been ntroduced n [16], where each feature at a local coordnate x R 2 s represented b a set of PDFs rather than b determnstc values. Parametrc estmaton methods can be used for the feature statstcs, but n most cases the are not sutable to model multmodal dstrbutons. A Gaussan mxture model could be used for multmodal cases, but t bears hgh computatonal complext. Thus, here onl nonparametrc methods, such as smple normalzed hstogram or Gaussan kernel based estmaton are consdered. Accordngl, for an M-dmensonal feature space, the embeddng map becomes (x, f(θ 1 ;x),, f(θ M ;x)) called a statstcal embeddng. Means and varances of each feature can be used drectl as features, constructng ordnar non-statstcal mage or feature manfolds. Ths drectl leads to the work of [15], namel non-statstcal embeddng and non-statstcal manfold. Fgure 1 llustrates the dfference between non-statstcal and statstcal manfolds. Statstcal manfolds assocate each parameter locaton xp wth a set of PDFs of features whle non-statstcal manfolds map the parametrc space onto a set of scalar values of features. An example of the statstcal embeddng s depcted n Fgure 2. The PDFs n the frst column are estmated from a pont nsde the bod of the cheetah and the set on the second column from outsde the bod. It s clear that the PDFs for gra value, for nstance, for both ponts are dfferent: b-modal and mono-modal Manfold of multnomal dstrbutons In the statstcal embeddng descrbed above, PDF for a feature θ can be modeled wth a multnomal dstrbuton specfed b a parameterzaton z: M 2 (θ 1, θ 2 ) x x p =(x p, p ) x p =(x p, p ) x (a) Non-statstcal (b) Statstcal Fgure 1. Unlke non-statstcal manfolds whch map the parametrc space onto a set of scalar values of features, statstcal manfolds assocate each parameter locaton x p wth a set of PDFs of features. Fgure 2. Statstcal embeddng. The PDFs are estmated from nsde (frst column) and outsde (second) of the bod. p( θ, z) = δ( θ ) z, θ {1,2,... n+ 1}, z = 1, z > 0. (1) n 1 n 1 = 1 = 1 Then, a statstcal manfold S={p(θ, z)} can be dentfed as a n-smplex n R n+1 whose coordnate sstem s (z 1,, z n+1 ). That s, the multnomal dstrbutons are lad on the surface of the smplex (Fgure 3a). Meanwhle, Fsher s nformaton matrx defned as gj ( z) = Ez[ z j z ] = log f( θ, z) j log f( θ, z) dθ, (2) Θ provdes a geometr under a statstcal manfold, where = / z and Θ s a parameter space [21]. Alternatvel, the matrx can be represented as gj ( z) = 4 f( θ, z) j f( θ, z) dθ. (3) Θ Then an sometrc transformaton of p(θ, z), ξ = 2 z, = 1,..., n+ 1, (4) θ 1 (x p ) M 2 maps the n-smplex onto a surface of n+1-dmensonal sphere, S n, of radus 2. Ths leads to the fact that the Fsher nformaton corresponds to the nner product of tangent vectors to the sphere. Then, the nformaton dstance of two dstrbutons p and p s defned as d p p = (5) 1 n+ 1 (, ) 2cos ( = 1 zz) Orentaton Gra value Gabor feature θ 2 (x p ) whch s the arc length of a great crcle connectng the dstrbutons (Fgure 3b). Here, the length s a geodesc dstance. The geodesc dstance of (5) s used for a dffuson process on statstcal manfolds. f(θ 1 ;x p ) f(θ 2 ;x p )

3 z 3 z 1 z 2 (a) n-smplex Fgure 3. Multnomal dstrbutons as ponts on an n-smplex (a) and a sphere (b). ξ 1 (b) n+1-dmensonal sphere d(p,p ) ξ 2 lnes when the shape of PDFs graduall change over a boundar. The offset and the thckness of detected boundar depends on wndow sze used to estmate PDFs. Also, the use of K-L dvergence normall results n two thn lnes for texture boundares, whch s equall undesrable. Ths wll be shown n a later secton. Fgure 2f shows the result of a dffuson process on a statstcal manfold. Compare the locatons of the mark and the detected boundar. The dffuson corrects offsets falsel nduced n the prevous boundar detecton Dffuson on statstcal manfold A classcal ansotropc dffuson process proposed b Perona and Malk [22] can be used on the statstcal manfolds. Explctl, the dffuson equaton on statstcal manfold M 2 can be defned as f t = dv(c(d(f,f )) f) = c(d(f,f )) f + c f, (6) where dv s the dvergence operator, and and respectvel represent the gradent and Laplacan operators. We denote the dffused manfold as M 2. Dffuson on non-statstcal manfolds use an edge estmate θ as an argument of a monotoncall decreasng functon c( ) to acheve a conducton coeffcent. However, on statstcal manfold, the geodesc dstance s used for the conducton coeffcent snce edges on a statstcal manfold can be dentfed b the geodesc dstance. Then, followng the dscretzaton scheme n [22] and wth the choce of 2 2 cd ( ( pp, )) = exp( d( pp, ) / K ), (7) the dffuson process on a statstcal manfold becomes straghtforward and produces promsng results for further processng. Fgure 4 shows a sgnfcant dfference between statstcal and non-statstcal manfolds usng a snthetc mage (2a) generated from two known PDFs (2b) of the same mean and varance of ntenst. To dentf the texture boundar, a metrc tensor map, defned as a determnant of metrc tensor, s calculated. The metrc tensor for statstcal manfold s based on PDF dssmlart measures such as Kullback-Lebler (K-L) dvergence and wll be defned later n secton 2.4. Tensor maps based on non-statstcal manfolds, whch take onl parametrc nformaton of a PDF nto account, faled to locate the desred texture boundar (2c, 2d), unless pror knowledge of the PDFs are provded. In contrast, the one on statstcal manfolds results n a successful localzaton of the texture boundar (2e). One drawback of usng PDF dssmlart measures s that the nduce offsets from true boundares (the ellow mark), or sometmes produce thck (d) (e) (f) Fgure 4. Statstcal manfolds can be used to separate regons that have the same mean and varance but look dfferent. The test mage (a) was generated from two dfferent PDFs, f 1 and f 2 n (b). A technque based on non-statstcal manfolds faled to accuratel locate the boundar of the two regons (c,d), but statstcal manfold framework successfull dentfes the texture boundar (e). In addton, dffuson on the statstcal manfold produces a better result (f) Metrc tensor A metrc tensor matrx contans nformaton related to the geometrc structure of a manfold and s used to measure dstances on manfolds. The determnant of the metrc tensor matrx s a good ndcator used for edge detecton n varous mage processng applcatons. The calculaton of metrc tensor matrx requres partal dervatves wth respect to the parametrc doman. Lee et al. [16] used K-L dvergence as an approxmaton of the partal dervatves of PDF f(x) at locaton x=(x, ). That s, where (a) f ( x) KLx ( f ) x = KL( f ( x), f ( x+ δ x) ) = 1 2 kl( f( x), f( x+ δx)) + kl( f( x+ δx), f( x)) ( δ ) (8) kl( f( x), f( x+ δx)) = f( x)log f( x)/ f( x+ x ) (9) f 1 f 2 (b) (c)

4 Then, the metrc tensor matrx for a statstcal manfold s defned as 1+ KLxKLx KLxKL τ ( x ) = KLxKL 1 KLKL, (10) + and ts determnant measures statstcal dssmlart of nearb features on manfolds. The determnant of τ(x) s much larger than unt when the manfold has a hgh statstcal gradent, whle the value s close to unt at locatons where the manfold s statstcall statonar. 3. Clusterng on statstcal manfolds Clusterng a set of multnomal dstrbutons ma be carred out straghtforward b applng a smple k-means algorthm wth an approprate dstance measure. Also, kernel based methods could be used to mplctl handle the dstance measure. However, due to the curse of hgh dmenson and ts nstablt nduced n clusterng results, benefts are often acqured when a dmenson of nput space s reduced. Fgure 5 shows an example of nstablt n clusterng when used full rank feature space. The result on the left s produced b applng an ordnar k-means clusterng algorthm to the PDFs n statstcal manfold framework. Not to menton a computatonal complext, clusterng result s not trustworth. In contrast, reduced features space wth three prncpal components of PCA scheme produces more accurate clusterng results (on the rght n fgure 5) compared to the full feature space. In ths secton, we nvestgate a PCA based dmenson reducton technque for a set of multnomal dstrbutons. The dstrbutons are frst transformed b (4) onto a surface of n+1 dmensonal sphere. The surface s a Remannan sub-manfold n R n+1 and s denoted as S n here. mean and the sample data set. However, our experments show better clusterng results when a Fréchet mean s used. Fréchet mean mnmzes the sum of squared dstance along geodescs on Remannan manfolds and unquel exsts when the dstrbutons are lmted to a suffcentl small part of the manfold [23,24]. Multnomal dstrbutons mapped onto S n satsf ths condton snce all coordnates are postve. A gradent descent algorthm establshed b [23] and expressed dfferentl b others [24,25,26] s llustrated n Fgure 6. Frst, the ponts ξ S n for =1, N (Fgure 6a) n are projected onto a tangent plane T S at t t b an nverse projecton (Fgure 6b), 1 v exp ( ξ ), (11) = t where exp (v) projects a pont v on a tangent plane at onto a sphere. An expectaton s calculated on the tangent plane and projected back onto the sphere b a projecton exp (Fgure 6c), + 1 = exp ( E[ v ]). (12) t t Explct expressons for these projectons are gven as [27], 1 2 1/2 exp ( ξ) = [1 ( ξ) ] ( ξ ( ξ) )arcos( ξ). (13) n exp ( v) = cos( v ) + sn( v ) v/ v ( v T S, v 0) Wth a proper selecton of startng pont, the algorthm converges wthn a few teratons. Fgure 7 shows the dfference of the Fréchet and the arthmetc means for the mage shown n Fgure 2a. v (a) E[v ] Fgure 5. An example shows nstablt of clusterng wth a large nput dmenson Fréchet means Gven a set of multnomal dstrbutons, L={p 1, p 2,, p N }, p S={p(θ, z), =1, N, a mean dstrbuton of the set can be obtaned b ndependentl computng the arthmetc mean of each coordnate. Then, the arthmetc mean mnmzes the sum of squared dstance between the x t Tangent plane v t t+1 t E[v ] (b) (c) Fgure 6. Iteratve method to estmate Fréchet mean on a sphere. Ponts (a) on the sphere are projected on a tangent plane (b) at an ntal pont of mean estmate. An expectaton s calculated and projected back onto the sphere (c). Iterate the procedure untl the mean converges.

5 Fréchet Arthmetc Eucldean space decreases as the number of prncpal components ncrease. Fnall, n Fgure 11, the clusterng method s tested wth varous natural mages presented n [29]. For most cases, color nformaton alone s used to construct feature PDFs. Fgure 7. Fréchet mean dstrbuton vs. arthmetc mean dstrbuton. The are dfferent n general Dmensonalt reducton Let ŷ S n be a Fréchet mean estmated b the teraton method descrbed n the prevous secton, and let V={v }, = 1,, N, be the projectons at the mean. Then, the set V n spans the entre T ˆS when N»n, and an ordnar sngular value decomposton can be appled to extract egenspace of the tangent plane. Ths smpl leads to a PCA based dmensonalt reducton of the tangent plane, and the clusterng can be appled to ths reduced space. The shape of the reduced space s arbtrar and nonlnear, and accordngl, a nonlnear technque such as LLE (locall lnear embeddng) [28] could be a possble choce for the clusterng. However, t s unrealstc n terms of speed to use algorthms of complext of O(N 2 ) when N s large. So, n ths paper, a smple k-means method s used for clusterng at the cost of msclassfcaton. Fgure 8 shows the case. The object s labeled as green n Fgure 8a, but some of background s msclassfed as objects. Among the msclassfcaton, the ones ndcated wth ellpses should belong to background. These partcular areas correspond to the porton ndcated n Fgure 8b, whch can be correctl classfed wth LLE method. 4. Results Several textured or mxed mages are tested for clusterng. PDFs are estmated wth 32 bns and wth dfferent features for dfferent mages. Then, the are dffused wth 20 teratons. Each teraton requres less than 2 seconds for mages of sze on a 2.8GHz Pentum 4 machne wth Matlab TM mplementaton. The top row of the fgure 9 shows the tensor maps on manfolds M 2 2 (left) and M (rght) for the test mage n fgure 2. The dffuson process removes offsets and merges two thn lnes across boundares. Clusterng results on both manfolds are depcted rght below the tensor maps. As expected, more smooth and accurate clusterng 2 s acheved on M. Next experment compares stablt of clusterng n between Eucldean R n+1 (frst column) and S n (second column) spaces. From top to bottom n Fgure 10, 3, 5, and 7 prncpal components are used for clusterng, respectvel. It s evdent that the S n space produces more accurate results than Eucldean space. Also, clusterng stablt n (a) Fgure 8. A case of msclassfcaton. Fgure 9. Intenst values are used as features. The dffuson process generates smooth and strong boundar on statstcal manfolds, consequentl producng accurate clusterng results. Fgure 10. Clusterng results and stablt comparson n between Eucldean and S n spaces. (b)

6 Fgure 11. Natural mage segmentaton wth the clusterng on statstcal manfolds. Expect the frst sample whch uses ntenst value as feature, color nformaton n RGB alone s used as feature. 5. Concluson A clusterng scheme on statstcall defned manfolds s presented. The algorthm handles textured or mxed mages wth one framework. Our experment on natural mage segmentaton shows promsng results n classfng textured or mxed mages, even f we used onl one feature, that s, color nformaton. The framework could be more powerful f multple feature spaces are consdered. References [1] R.M. Haralck, Statstcal and structural approaches to texture, IEEE Proceedngs, vol. 67, no. 5, pp , [2] A. Jan and F. Farrokhna, Unsupervsed texture segmentaton usng Gabor flters, Pattern Recognton, 24, pp , [3] G. Hale and B.S. Manjunath, Rotaton nvarant texture classfcaton usng a complete space-frequenc model, IEEE Trans. on Image Processng, vol. 8, no. 2, pp , [4] M. Rousson, T. Brox, and R. Derche, Actve unsupervsed texture segmentaton on a dffuson based feature space, Proc. of IEEE Conf. on Computer Vson and Pattern Recognton, vol. 2, pp. II , June [5] A. Lane and J. Fan, Texture classfcaton b wavelet packet sgnatures, IEEE Trans. on Pattern Analss and Machne Intellgence, vol. 15, pp , [6] M. Unser, Texture classfcaton and segmentaton usng wavelet frames, IEEE Trans. on Image Processng, vol. 4, pp , November [7] C. Bouman and M. Shapro, A multscale random feld model for Baesan mage segmentaton, IEEE Trans. on Image Processng, vol. 3, pp , [8] B. Manjunath and R. Chellappa, Unsupervsed texture segmentaton usng Markov random felds models, IEEE Trans. on Pattern Analss and Machne Intellgence, vol. 13, pp , Ma [9] L. Lepsto, I. Kunttu, J. Auto, and A. Vsa, Classfcaton of non-homogenous texture mages b combnng classfers, Proc. of IEEE Intl. Conf. on Image Processng, vol. 1, pp. I , [10] G. Sapro, Vector (self) snakes: a geometrc framework for color, texture, and multscale mage segmentaton, Proc. of IEEE Intl. Conf. on Image Processng, vol. 1, pp , September [11] N. Paragos and R. Derche, Geodesc actve regons for supervsed texture segmentaton, Intl. Journal of Computer Vson, vol. 46, no. 3, pp , [12] T.F. Chan and L.A. Vese, Actve contours wthout edges, IEEE Trans. on Image Processng, vol. 10, no. 2, pp , [13] J. Km, J.W. Fsher III, A. Yezz, Jr., M. Cetn, and A.S. Wllsk, Nonparametrc methods for mage segmentaton usng nformaton theor and curve evoluton, Proc. of IEEE Intl. Conf. on Image Processng, vol. 3, pp , [14] M Rousson and R. Derche, A varatonal framework for actve and adaptve segmentaton of vector valued mages, Proc. of Workshop on Moton and Vdeo Computng, pp , [15] N. Sochen, R. Kmmel and R. Mallad, A general framework for low level vson, IEEE Trans. on Image Processng, vol. 7, no. 3, pp , [16] S. Lee, A.L. Abbott, N.A. Clark and P.A. Araman, Actve contours on statstcal manfolds and texture segmentaton, Proc. of IEEE Intl. Conf. on Image Processng, vol. 3, pp , 2005.

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