Mean shift

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Unformatted text preview: n shift was used for tracking human faces 4], by projecting the APPENDIX Proof of Theorem 1 ^ Since n is nite the sequence fK is bounded, therefore, it is su cient to show that f^K is strictly monotonic ^ increasing, i.e., if yj 6= yj+1 then f^K (j ) < fK (j + 1), for all j = 1 2 : : :. By assuming without loss of generality that yj = 0 we can write f^K (j + 1) ; f^K (j ) = ! # n" 1 X k yj+1 ; xi 2 ; k xi 2 (A.1) = : nhd i=1 6 h h 0.7 0.6 0.5 d 0.4 0.3 0.2 0.1 0 3000 3500 4000 4500 5000 5500 Frame Index 6000 6500 7000 Figure 6: The detected minimum value of distance d function of the frame index for the 2 minute Subway2 sequence. The peaks in the graph correspond to occlusions or rotations in depth of the target. For example, the peak of value d 0:6 corresponds to the partial occlusion in frame 3697, shown in Figure 5. At the end of the sequence, the person being tracked gets on the train, which produces a complete occlusion. and by employing (14) it results that n 1 ky k2 X g f^K (j + 1) ; f^K (j ) j +1 d+2 nh The convexity of the pro le k implies that k(x2 ) k(x1 ) + k0 (x1 )(x2 ; x1 ) (A.2) for all x1 x2 2 0 1), x1 6= x2 , and since k0 = ;g, the inequality (A.2) becomes k(x2 ) ; k(x1 ) g(x1 )(x1 ; x2 ): (A.3) Using now (A.1) and (A.3) we obtain f^K (j + 1) ; f^K (j ) n 1 X g xi 2 kx k2 ; ky ; x k2 1 = nhd+2 " 2y>+1 j n X i=1 n X i=1 i h g xi h xi g j +1 h 2 : (A.5) Since k is monotonic decreasing we have ;k0(x) P g(x) 0 for all x 2 0 1). The sum n=1 g xi 2 i h is strictly positive, since it was assumed to be nonzero in the de nition of the mean shift vector (10). Thus, as long as yj+1 6= yj = 0, the right term of (A.5) is strictly ^ ^ positive, i.e., fK (j + 1) ; fK (j ) > 0. Consequently, the ^K is convergent. sequence f To prove the convergence of the sequence yj j=1 2::: we rewrite (A.5) but without assuming that yj = 0. After some algebra we have ! n 1 ky ;y k2X g yj ;xi 2 f^K (j +1);f^K (j ) d+2 j +1 j Figure 5: Subway2 sequence: The frames 3140, 3516,...
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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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