ALPHA-TRIMMED MEAN RADIAL BASIS FUNCTIONS AND THEIR APPLICATION IN OBJECT MODELING

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1 ALPHA-TRIMMED MEAN RADIAL BASIS FUNCTIONS AND THEIR APPLICATION IN OBJECT MODELING Adrian G. Boq Ioannis Pitas Department University Thessaloniki of Informatis of Thessaloniki 54006, Greee ABSTRACT In this paper we use Radial Basis Funtion (RBF) networks for objet modeling in images. An objet is omposed from a set of overlapping ellipsoids and has assigned an output unit in the RBF network. Eah basis funtion an be geometrially represented by an ellipsoid. We introdue a new robust statistis based algorithm for training radial basis funtion networks. This algorithm relies on o-trimmed mean statistis. The use of the proposed algorithm in estimating ellipse parameters is analyzed. samples assigned to a basis funtion, a ertain number of them is eliminated from estimation. The number of data samples to be eliminated depends on the distribution [8, 91. W e analyze the estimation of the ellipse parameters when using o-trimmed mean statistis. The proposed algorithm is applied for modeling artifiial generated patterns and for segmenting a set of mirosope images. 2. THE RADIAL BASIS FUNCTION MODEL 1. INTRODUCTION Radial basis funtion neural network onsists of a two layer feed-forward struture employed for funtional approximation and lassifiation proposes. When used in pattern lassifiation an RBF network suessfully approximates the Bayesian lassifier [l, 21. In this ase, the underlying probability funtions are deomposed in a sum of kernel funtions with loalized support. The funtions, implemented by the hidden units, are usually hosen as Gaussian. The intersetion of an Gaussian with an hyperplane is geometrially an ellipsoid. Objets in images an be seen as omposed from a set of overlapping ellipti diss in 2-D [3], or ellipsoids in 3-D. A lassial RBF network training algorithm relies on the learning vetor quantization [l]. This learning algorithm represents the adaptive version of the moments approah, used for estimating the ellipse parameters [4]. However, the moments method is likely to provide biased estimates in the ase when the ellipses are overlapping or when they are embedded in noise. In [2, 5] marginal median and median of the absolute deviation [6] have been proposed as robust estimators for finding the RBF s hidden unit parameters. In this paper we analyze the o-trimmed mean RBF algorithm. The previous two algorithms are partiular ases of the new approah. a-trimmed mean has been extensively used in image filtering [6, 71. After ranking the data The RBF network has been employed in many appliations inluding urve modeling [l] or moving objet reognition [5]. Th e output unit models a omplex funtion omposed of a mixture of overlapping kernels. Eah kernel implements an unnormalized Gaussian funtion : &(X) = exp [-(X - pj) ET (X - &)I (1) where X represents the input feature vetor, and pj, Zj, the Gaussian ent,r vetor and ovariane matrix. The output represents a weighted sum of hidden units saled to the interval (0,l) by the sigmoidal funtion : yk(x) = 1 1 -I- exp [-Ef=lxkj4j(x)] where Xkj are output weights, L is the number of hidden units and k = 1,..., M are the output units. The parameters of the RBF network are found by training. A lassial algorithm for training RBF networks onsists of assigning a group of data samples to a basis funtion aording to the smallest Eulidean distane between every data sample and the basis funtion enter : IIX-~kII=~~~IIX-/ljII~ (3) The initial values of the enter vetors are taken randomly in the input range. From the data distribution, assigned to a basis funtion, the parameters of (2)

2 (1) are alulated. In [l] a variant of the learning vetor quantization, whih orresponds to lassial statistis assumptions, has been used. Marginal median and median of absolute deviation have been onsidered for alulating the hidden unit parameters in [2, 51. This algorithm has been alled median RBF (MRBF). 3. THE ALPHA-TRIMMED MEAN RBF TRAINING ALGORITHM We propose a new robust statistis based training algorithm. Let us order the data samples assigned to a speifi basis enter when using (3) and denote them aording to their rank as X(k) for k = 0,..., Nk, where Nk is the number of data samples assigned to the kth basis funtion. A general desription of many robust statistis algorithms onsists of assigning a weight to every data sample with respet to its rank and alulating the loation estimate as : Pk = Nk i= N, Wi X(i) wi i=o where Wi is the weight assigned to the data sample with respet to its rank. If Wi = 1 for i = 0,..., Nk then the loation is omputed by averaging. If Wi = 0 fori=o,..., F-l,i=%+l,..., NkandWi=l for i = C+ we obtain the marginal median estimator. In other robust statistis algorithms, Wi is replaed by a funtion whih dereases with respet to the distane of the ordered sample X(i) from the entral data sample X(s,). In this study we propose the a-trimmed mean algirithm [6] whih assigns Wi = 1 fori=&nk,..., Nk-&Nkin(d)andWi=Oforthe rest of data sample, where CY is the perentage of data samples to be eliminated from the estimation. This algorithm is alled a-trimmed mean RBF and estimates the enter of the basis funtion as : Nk-CVNI,. i=anh pk = Nk - 2Cdk x(i) The o-trimmed mean algorithm has been shown as a good hoie for the long and medium tail data distributions. This algorithm has been extensively used for image filtering [6? 71. The parameter Q is hosen aording to the data distribution. The following measure is used for esti- (4 (5) mating the tail of t,he data dist,ribution [8, 91 : U[O.5] - L[O.5] Q = U[O.O5] - L[O.O5] where WI, -WI re P resent the average of the upper and respetively lower p perentage of data samples. The number of data samples to be trimmed away relies diretly on the value of Q : - 1-Q a=2. When the distribution is long tailed, the amount of data samples to be trimmed is large, and when the distribution tail is short, the amount of data samples to be trimmed is small. For the seond order statistis parameters, we order the data samples aording to their Mahalanobis distane from the estimate fik : (6) L M(o) = ttltlf[(x - fij) et1 (X - fij)] 63) After ordering the data samples M(o) <... < M(N*), the estimate of the ovariane matrix is hosen as : C i=o (X(i),M - bj) (X(i),M - Dj) where X(~),M denotes the ith ordered data sample aording to the Mahalanobis distane (8). The ovariante matrix in (8) is initially estimated based on lassial seond order statistis, without trimming any data sample away from the given distribution, i.e. by onsidering am = 0 in (9). The number of data samples to be trimmed is alulated using : (9) (10) Both formulas used in (5,9) an be alulated from data histograms. For the output parameters we employ the bakpropagation algorithm as in [2]. The mixture of Gaussian funtions an approximate omplex funtions. If we drop the exp and the sign under the exponential in the expression (1) we obtain the equation of an ellipsoid. We an observe that for X E R2, if we interset the Gaussian funtion with a plane we obtain an ellipse. With ellipses we an model omplex 2-D objets [3]. The mean of the Gaussian funtion from (1) orresponds to the enter, the variane to the width and the ross-orrelation to the orientation of the ellipse.

3 4. ELLIPSE PARAMETERS ESTIMATION Let us onsider the equation of an ellipse in the analiti form : (X - p)!qx -/A) = 1 (11) where p denotes the enter vetor of the ellipse and E- its width and orientation. We denote the omponents of this matrix as : For the numerator from (13), similarly with the above derivation, we obtain : d&~;a~bd-dri z.,p J Z,,J J Y1nl YSUP x dydx = (12) The method based on moments for estimating the parameters of an ellipse is desribed in [4]. We onsider an ellipti dis as a uniform distributed statistis inside of an ellipti shape. We an alulate the ellipsis enter based on the first order moments : z.,p Y.-p J J J t,nj J Yld x dydx Jqbzl = ziy# p y y#yp dydx (13) where xinf, x sup, yinf, ysup denote the geometrial limits of the ellipse and the denominator denotes the area left after trunation. In order to evaluate the limits of integration we alulate the tangents to the ellipse, whih are parallel to the x axis [4] : (14) where v is the vetor normal to the ellipse and parallel with the I axis, and pl, is the x omponent of the enter vetor. After hoosing two vetors v1 = (1 0) and v;! = (-1 0) we derive the limits on the x axis. In the ase of trimming, let us denote with 2, the interval from the ellipse eliminated by a-trimming. The integration limits orresponding to the trunated ellipse are : I -1 xsup,xinf=pzf (x0-/&) (15) The integration limits on the y axis are derived from the equation (11) : ~(3: - pz) f,/b - (ab - *)(x - /.il,)2 Ysup, Yinf = /Jy- h (16) The trunated ellipse area an be expressed, after hanging the variable, and onsidering (15,16) : z,up YSUP J z,nj J Y:d dydx = The seond integral in (18) is zero and from (13,17,18) the result of the estimation is : Jwzl = /AT. (19) This result proves that the estimation of the ellipse enter by trimming is unbiased in the ase of perfet ellipses, V a E [O,;]. We evaluate the a-trimmed mean algorithm for estimating the parameters of the ellipse ovariane matrix (width and orientation). The equation of the ellipse, resulted after trimming QM of the data samples and after ranking them aording to the Mahalanobis distane with respet to the ellipse enter (8). is : (X - p) Irl(X -p) = 1 - (YM (20) The estimation of the ellipse width is given by normalized seond order moments [4] : E[a21 = ~:~~pjy:~:r(x - k)2 dydx = 1-y 1::: dydx (21) where xm,inf 7 XM,Jlrp, YM,inf, Y,M,~~~ are the extreme points of the ellipse after trimming. Following similar derivations as for (15) we obtain the integration limits on x : xm,sup,xm,inf = t% f lm=a (22) ab - 2 Similarly with (16) we derive the integration limits on y. We obtain the area of the ellipse in this ase as : 5M, up YM,#UP Lt., J dydx = Cl &itf - am)t (23) YM,*nI For the numerator in expression (21) we obtain : ZM,SUP YM,B~P (x - pz)2 dydx = L..., J YM,*nJ (l-r~)~b z2dddz = $ff$$24) J@=q -* a J

4 From (23) and (24) we derive the estimate variane : q$z] = (1 - am)* + 4(ab - 2) for the (25) Figures 2 (b)! (e). In Figures 2 (), (f) the result provided by r-trimmed mean RBF algorithm is displayed. The blood vessels are quite well segmented in the results provided by the r-trimmed mean RBF algorithm. We get a similar expression for the ross-orrelation. In the expression (25), i f we onsider am = 0 we obtain the same result as in [4] for estimating the normalized seond order moment of an ellipse. In order to orret this bias, when estimating ellipsis ovariane matrix we use the following expression, instead of (9) : ~j = N,-Sr~ NI, (x(i),m - bj) (X(i),M - i$) d=o (1 - am)(nk - 2aM Nk) 5. SIMULATION RESULTS (26) We have tested the o-trimmed mean RBF algorithm on artifiial generated patterns. We onsider three overlapping diss in a 512 x 512 image, without noise and with additive uniform distributed noise as shown in Figure 1 (a). We apply the lassial learning algorithm for RBF, MRBF [2] and a-trimmed mean RBF algorithm. The results are displayed in Figures 1 (b), () and (d), respetively. We onsider the oordinates of the white pixels as inputs of the network. The amount, of uniform distributed noise aounts for 4 % of the total number of pixels. The noise effet is onsidered as swithing the value of the pattern and bakground. As it an be observed in Figure 1, the initial pattern is deteriorated. The omparison measures are the average of the Eulidean distanes between the enters of the diss and those of the estimated model, and the total number of erroneous pixels in the reonstruted model with respet to the noise-free image. From Table 1 we an observe that a-trimmed mean RBF algorithm provides better results than the other two algorithms in modeling the objet. As it an be observed in Figure 1 the pattern orrupted by noise is better modeled by MRBF and a-trimmed mean RBF when ompared to the lassial statistis based approah. We have applied the proposed algorithm on a stak of 3-D mirosope images representing the inner struture of a tooth. In Figures 2 (a), (d) two frames of this stak of images are shown. The graylevel and the pixel oordinates have been used as inputs in the network. The graylevel aounts for the objet segmentation while the oordinates aount for the objet loalization. Both, the blood vessels and the bakground are modeled by RBF or a-trimmed mean RBF funtions. The result of the segmentation using the RBF, based on moving average and lassial variane is shown in 6. CONCLUSION In this study we introdue a new algorithm for estimating the parameters of hidden units in RBF networks. The proposed algorithm employs the a-trimmed mean statistis for alulating the hidden unit parameters. The number of data samples trimmed away in the proposed algorithm depends on the distribution. Ellipse parameter estimation using the a-trimmed mean RBF algorithm is analyzed. The proposed algorithm is applied in image modeling and segmentation. 131 T. Kimoto, Y. Yasuda, Shape desription and representation by ellipsoids, Signal Pro.: Image Comm., vol. 9, no. 3, pp , Mar [51 [71 7. REFERENCES L. Xu, A. Krzyzak, E. Oja: Rival penalized ompetitive learning for lustering analysis, RBF net, and urve detetion: IEEE Trans. on Neural Networks, vol. 4, no. 4, pp , A. G. Borg: I. Pitas, Median radial basis funtion neural network, IEEE Trans. on Neural Networks, vol. 7, no. 6, pp , R.M. Haralik, L.G. Shapiro. Computer and Robot Vision, vol. I, Reading, MA:Addison-Wesley, A. G. Borg, 1. Pitas, Moving sene segmentation using median radial basis funtion network, Pro. IEEE Inter. Symp. on Ciruits and Systems (IS- CAS 97), vol. I, Hong Kong, pp , I. Pitas, A. N. Venetsanopoulos. Nonlinear Digital Filters, Priniples and Appliations. Norwell, MA: Kluwer, S.R. Peterson, Y.-H. Lee, S.A. Kassam, Some statistial properties of Alpha-Trimmed mean and standard type M Filters, IEEE Trans. on Aoust., Speeh, Sig. Pro., vol. 36, no. 5, pp , P. Presott, Seletion of trimming proportions for robust adaptive trimmed means, J. Amerian Stat. Asso., , vol. 73, pp , D. M. Titteringt.on, Estimation of orrelation oeffiients by ellipsoidal trimming, Applied Statistis, no. 3, vol. 27, pp , 1978.

5 Noise-free Model ] Noisy Model (4 % noise) Algorithm Center Loation 1 Modeling 1 Center Loation 1 Modeling Estimation Error (%) Estimation Error (%) RBF MRBF o-trimmed RBF Table 1. Numerial omparison when modeling an artifiial pattern. (a) Artifiial generated (b) Modeling provided by () Modeling provided by (d) Modeling provided by image orrupted by noise. lassial RBF algorithm MRBF o-trimmed mean RBF. Figure 1. Comparative algorithm performane in modeling an artifiial pattern. Figure 2. Segmentation of blood vessels in a set of mirosope images representing inner struture of a tooth : (a), (d) ori g ina 1 mirosope images ; (b), ( e ) se g mentation provided by lassial RBF; (), (f) segmentation provided by o-trimmed mean algorithm.