1 4.7 TIME ALIGNMENT AND NORMALIZATION Linear time normalization:

2 4.7 TIME ALIGNMENT AND NORMALIZATION

3 m(k): a nonnegative (path) weighting factor M ϕ : a (path) normalizing factor

4 4.7 TIME ALIGNMENT AND NORMALIZATION

Dynamic Programming-Basic Considerations We are interested in obtaining the optimal sequence of moves and the associated minimum cost from any point i to any other point j in As many points as necessary. For every pair of points (i,j) We define to be a nonnegative cost that represents the cost of moving directly from the ith point to the jth point in one step. According to Bellman: An optimal policy has the property that, whatever the initial state and decisions are, the remaining decisions must constitute an optimal policy with regard to the state resulting from the first decision

Dynamic Programming-Basic Considerations To determine the minimum cost path between points i and j, the following dynamic program is used:

Dynamic Programming-Basic Considerations A recursion allowing the optimal path search to be conducted incrementally

Example: Example: Shortest path from A to E in M=3 moves Shortest path from A to E in M=3 moves 8 A C E B D

A BCD E ∞ AB C D E 2 4 ∞ 8 M=1M=2M=3

∞ 6 ∞ A BCD E 3 AB C D E 2 4 ∞ 8 ∞ ∞ 3 6 M=1M=2M=3

∞ 5 ∞ A BCD E ∞ AB C D E 2 4 ∞ 8 ∞ ∞ 3 6 M=1M=2M=3 ∞ ∞ ∞ 7

Dynamic Programming-Basic Considerations

Dynamic Programming-Basic Considerations 1- Initialization 2- Recursion

Dynamic Programming-Basic Considerations 3- Termination 4- Path backtracking

15 Time-Normalization Constraints Endpoint constraints Endpoint constraints Monotonicity constraints Monotonicity constraints Local continuity constraints Local continuity constraints Global path constraints Global path constraints Slope weighting Slope weighting

Endpoint Constraints

Monotonicity Conditions

Local Continuity Constraints We define a path P as a sequence of moves, each specified by a pair of coordinate increments,

Local Continuity Constraints For a path that begins at (1,1), which point we designate k=1, we normally set (as if the path originates from (0,0)) and have:

Local Continuity Constraints

Global Path Constraints For each type of local constraints, the allowable regions can be Defined using the following two parameters: Normally, Q max =1/Q min

22 Values of and for different types of paths 2ITAKURA 3VII VI 3V 2IV 2III 2II 0I TYPE

Global Path Constraints We can define the global path constraints as follows: The first equation specifies the range of the points in the (i x, i y ) plane that can be reached from the beginning point (1,1) via the allowable path according to the local constraints. The second equation specifies the range of points that have a legal path to the ending pint (T x, T y )

Slope Weighting

Global Path Constraints

Slope Weighting The weighting function can be designed to implement an optimal discriminant analysis for improved recognition accuracy. Set of four types of slope weighting proposed by Sakoe and Chiba

Slope Weighting

Slope Weighting Typically, for types (a) and (b) slope weightings, we arbitrarily set: The Accumulated distortion also requires an overall normalization. Customarily: For type (c) and (d) slope weighting,

Dynamic Time-Warping Solution Due to endpoint constraints, we can write:

Dynamic Time-Warping Solution Similarly, the minimum partial accumulated distortion along a path Connecting (1,1) and (i x, i y ) is:

Dynamic Time-Warping Solution

Dynamic Time-Warping Solution

Dynamic Time-Warping Solution

Dynamic Time-Warping Solution The ratio of grid allowable grid points to all grid points, when a k-to-1 time scale expansion and contraction is allowed: