1. 4.7 TIME ALIGNMENT AND NORMALIZATION
Linear time normalization:

2. 4.7 TIME ALIGNMENT AND NORMALIZATION

3. 4.7 TIME ALIGNMENT AND NORMALIZATION
m(k): a nonnegative (path) weighting factor
Mϕ: a (path) normalizing factor

4. 4.7 TIME ALIGNMENT AND NORMALIZATION

5. 4.7.1 Dynamic Programming-Basic Considerations
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
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.
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.

6. 4.7.1 Dynamic Programming-Basic Considerations
To determine the minimum cost path between points i and j,
the following dynamic program is used:

7. 4.7.1 Dynamic Programming-Basic Considerations
A recursion allowing the optimal path search to be conducted incrementally

8. 4.7.1 Dynamic Programming-Basic Considerations

9. 4.7.1 Dynamic Programming-Basic Considerations
1- Initialization
2- Recursion

10. 4.7.1 Dynamic Programming-Basic Considerations
3- Termination
4- Path backtracking

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

12. Endpoint Constraints

13. 4.7.2.2 Monotonicity Conditions

14. 4.7.2.3 Local Continuity Constraints
We define a path P as a sequence of moves, each specified by a
pair of coordinate increments,

15. 4.7.2.3 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:

16. 4.7.2.3 Local Continuity Constraints

17. 4.7.2.4 Global Path Constraints
For each type of local constraints, the allowable regions can be
Defined using the following two parameters:
Normally, Qmax=1/Qmin

18. Values of and for different types of paths2
ITAKURA 3
VII VI V IV
III II I
TYPE

19. 4.7.2.4 Global Path Constraints
We can define the global path constraints as follows:
The first equation specifies the range of the points in the (ix, iy) 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 (Tx , Ty)

20. Slope Weighting

21. 4.7.2.4 Global Path Constraints

22. 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

23. Slope Weighting

24. 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,
Typically, for types (a) and (b) slope weightings, we arbitrarily set:

25. 4.7.3 Dynamic Time-Warping Solution
Due to endpoint constraints, we can write:

26. 4.7.3 Dynamic Time-Warping Solution
Similarly, the minimum partial accumulated distortion along a path
Connecting (1,1) and (ix, iy) is:

27. 4.7.3 Dynamic Time-Warping Solution

28. 4.7.3 Dynamic Time-Warping Solution

29. 4.7.3 Dynamic Time-Warping Solution

30. 4.7.3 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: