A COMPARISON AND EVALUATION OF MOTION INDEXING ...

etween the two sequences using a dynamic programming algorithm and the following
optimization criterion:
cij = dij + min (ci−1,j−1, ci,j−1, ci−1,j) ,
where cij is the minimal cost between the subsequences (a1, a2, · · · , ai) and (b1, b2, · · · , bj).
A warping path is a path through the matching cost matrix from the element c11 to
element cnm consisting of those cij elements that have contributed to the distance in
cnm.
The global warp cost, GC(a, b), of the two sequences is defined as shown below:
GC(a, b) = 1/p
p
wi,
where wi are the elements belonging to the warping path and p is the number of
elements in the path. The warping path is typically subject to several constraints:
1. Boundary conditions: The path starts in the left bottom corner and ends in the
right top corner. This means that the warping path starts and finishes inthe
i=1
diagonally opposite corner cells of the cost matrix.
2. Continuity: The path advances gradually, step by step. The indices i and j
increase by a maximum of 1 unit at each step. This restricts the possible steps
in the warping path to adjacent cells including diagonally adjacent cells. Given
wk = (i, j), then wk−1 = (i ′ , j ′ ), where (i − i ′ ) ≤ 1 and (j − j ′ ) ≤ 1
3. Monotonicity: Both indices i and j used for traversing through sequences never
decrease. This forces the points in the path W to be monotonically spaced in
time. Given wk = (i, j), then wk−1 = (i ′ , j ′ ), where i − i ′ 0 and j − j ′ 0.
Dynamic Time Warping (DTW) finds the best global match or alignment between
the two sequences by time warping them optimally. The Figure 4.1 shows the result
of DTW on two sequences of a pick box from floor action.
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