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

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

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.

Ask a homework question - tutors are online