E^/E^ _fc=i ' k=l where I denotes the index of the allowable path Te in the constraint set, and Ki is the total number of moves in 3V The monotonicity conditions in Eq. 6.58 hold as

For instance, in Sakoe-Chiba local continuity constraints given in Figure 6.23, I = 1,2,3, and Kt = 2,1,2, respectively for ?l,72,73 resulting in Qmax = 2 and Qmin = 1/2. Normally, Qmax = 1/Qmin. The values of Qmax and Qmm for different types of local continuity constraints are given in Table 6.4.

The boundaries of the allowable regions (global path constraints) can be defined using the values of Qmax and Qmjn as


y max

Eq. 6.73 defines the range of the points that can be reached using a legal path based on a local constraint from the beginning point (1,1). Likewise, Eq. 6.74 specifies the range of points that have a legal path to the ending point (N, M) defined by Itakura [252] (Itakura constraints). Figure 6.27 shows the effects of the global constraints on the optimal search region defined by the parallelogram (Itakura constraints) in the (N, M) grid. An additional global path constraint has been proposed by Sakoe and Chiba [533] as

where K0 denotes the maximum allowable absolute temporal difference between the two variable trajectories at any given sampling instance. This constraint further decreases the search range as well as the potential misalignments by trimming off the edges of the parallelogram in the grid.

Table 6.4. Allowable local path specifications and associated Qmax and Qmin values for different types of local continuity constraints given in Figure 6.24 [484]


Allowable paths

q max


3*2 — (1,1) - (0, 1)


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