N 1 ky y k2x g yj xi 2 fk j 1fk j d2 j 1 j figure

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Unformatted text preview: 3697, 5440, 6081, and 6681 are shown (left-right, topdown). nhd+2 i=1 i=1 xi nh i=1 h (A.6) ^ ^ Since fK (j + 1) ; fK (j ) converges to zero, (A.6) implies that kyj+1 ; yj k also converges to zero, i.e., yj j=1 2::: is a Cauchy sequence. This completes the proof, since any Cauchy sequence is convergent in the Euclidean space. p ^^ Proof that the distance d(^ q) = 1 ; (p q) is a p^ metric i The proof is based on the properties of the Bhattacharyya coe cient (17). According to the Jensen's inequality 8, p.25] we have v um m m sq Xp X ^u uX q = 1 ^^ (p q) = pu qu = pu p t ^u ^^ ^^ u u=1 u=1 u=1 (A.7) 2 2y>+1 xi ; kyj+1 k2 = nh1+2 j d # n xi 2 ; ky k2 X g xi 2 j +1 h h i=1 (A.4) 7 ^ ^ with equality i p = q. Therefore, d(^ q) = p^ p ^^ ^ 1 ; (p q) exists for all discrete distributions p and ^ ^^ q, is positive, symmetric, and is equal to zero i p = q. The triangle inequality can be proven as follows. ^^ Let us consider the discrete distributions p, q, and ^, r and de ne the associated m-dimensional points p = ;p ;p p^ p^ p1 : : : pm > , q = q1 : : : qm > , and r = ^ ^ ;p p^ r1 : : : rm > on the unit hypersphere, centered at ^ the origin. By taking into account the geometric interpretation of the Bhattacharyya coe cient, the triangle inequality d(^ ^) + d(^ ^) d(^ q) pr qr p^ (A.8) is equivalent to q q q 1 ; cos( p r )+ 1 ; cos( q r ) 1 ; cos( p q ): (A.9) If we x the points p and q , and the angle between p and r , the left side of inequality (A.9) is minimized when the vectors p , q , and r lie in the same plane. Thus, the inequality (A.9) can be reduced to a 2dimensional problem that can be easily demonstrated by employing the half-angle sinus formula and a few trigonometric manipulations. 11] A. Djouadi, O. Snorrason, F.D. Garber, \The Quality of Training-Sample Estimates of the Bhattacharyya Coe cient," IEEE Trans. Pattern Analysis Machine Intell., 12:92{97, 1990. 12] A. Eleftheriadis, A. Jacquin, \Automatic Face Location Detection and Tracking for Mo...
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This note was uploaded on 11/27/2011 for the course MATH 3484 taught by Professor Staff during the Fall '10 term at University of Central Florida.

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