mlib_SignalDTWKVector_S16

- perform dynamic time warping for K-best paths on vector data

Synopsis

cc [ flag... ] file... -lmlib [ library... ]
#include <mlib.h>

mlib_status mlib_SignalDTWKVector_S16(mlib_d64 *dist, 
     const mlib_s16 **dobs, mlib_s32 lobs, mlib_s32 sobs, 
     void *state);

Description

The mlib_SignalDTWKVector_S16() function performs dynamic time warping for K-best paths on vector data.

Assume the reference data are

    r(y), y=1,2,...,N

and the observed data are

    o(x), x=1,2,...,M

the dynamic time warping is to find a mapping function (a path)

    p(i) = {px(i),py(i)}, i=1,2,...,Q

with the minimum distance.

In K-best paths case, K paths with the K minimum distances are searched.

The distance of a path is defined as

            Q
    dist = SUM d(r(py(i)),o(px(i))) * m(px(i),py(i))
           i=1

where d(r,o) is the dissimilarity between data point/vector r and data point/vector o; m(x,y) is the path weighting coefficient associated with path point (x,y); N is the length of the reference data; M is the length of the observed data; Q is the length of the path.

Using L1 norm (sum of absolute differences)

             L-1
    d(r,o) = SUM |r(i) - o(i)|
             i=0

Using L2 norm (Euclidean distance)

                    L-1 
    d(r,o) = SQRT { SUM (r(i) - o(i))**2 }
                    i=0

where L is the length of each data vector.

To scalar data where L=1, the two norms are the same.

    d(r,o) = |r - o| = SQRT {(r - o)**2 }

The constraints of dynamic time warping are:

1. Endpoint constraints

       px(1) = 1
       1 ≤ py(1) ≤ 1 + delta

   and

       px(Q) = M
       N-delta ≤ py(Q) ≤ N

2. Monotonicity Conditions

       px(i) ≤ px(i+1)
       py(i) ≤ py(i+1)

3. Local Continuity Constraints

   See Table 4.5 on page 211 in Rabiner and Juang's book.

   Itakura Type:

       py
       |
       *----*----*
       |p4  |p1  |p0
       |    |    |
       *----*----*
       |    |p2  |
       |    |    |
       *----*----*-- px
             p3

   Allowable paths are

       p1->p0    (1,0)
       p2->p0    (1,1)
       p3->p0    (1,2)

   Consecutive (1,0)(1,0) is disallowed. So path p4->p1->p0 is disallowed.

4. Global Path Constraints

   Due to local continuity constraints, certain portions of the (px,py) plane are excluded from the region the optimal warping path can traverse. This forms global path constraints.

5. Slope Weighting

   See Equation 4.150-3 on page 216 in Rabiner and Juang's book.

A path in (px,py) plane can be represented in chain code. The value of the chain code is defined as following.

    ============================
    shift ( x , y ) | chain code
    ----------------------------
        ( 1 , 0 )   |     0
        ( 0 , 1 )   |     1
        ( 1 , 1 )   |     2
        ( 2 , 1 )   |     3
        ( 1 , 2 )   |     4
        ( 3 , 1 )   |     5
        ( 3 , 2 )   |     6
        ( 1 , 3 )   |     7
        ( 2 , 3 )   |     8
    ============================

        py
        |
        *  8  7  *
        |
        *  4  *  6
        |
        1  2  3  5
        |
        x--0--*--*-- px

where x marks the start point of a path segment, the numbers are the values of the chain code for the segment that ends at the point.

In following example, the observed data with 11 data points are mapped into the reference data with 9 data points

        py
        |
     9  | * * * * * * * * * *-*
        |                  /
        | * * * * * * * *-* * *
        |              /
        | * * * * * * * * * * *
        |            /
        | * * * * *-* * * * * *
        |        /
        | * * * * * * * * * * *
        |       |
        | * * * * * * * * * * *
        |      /
        | * * * * * * * * * * *
        |    /
        | * * * * * * * * * * *
        |  /
     1  | * * * * * * * * * * *
        |
        +------------------------ px
          1                   11

The chain code that represents the path is

    (2 2 2 1 2 0 2 2 0 2 0)

See Fundamentals of Speech Recognition by Lawrence Rabiner and Biing-Hwang Juang, Prentice Hall, 1993.

Parameters

The function takes the following arguments:

dist

The distances of the K-best paths.

dobs

The observed data array.

lobs

The length of the observed data array.

sobs

The scaling factor of the observed data array, where actual_data = input_data * 2**(-scaling_factor).

state

Pointer to the internal state structure.

Return Values

The function returns MLIB_SUCCESS if successful. Otherwise it returns MLIB_FAILURE.

Attributes

See attributes(5) for descriptions of the following attributes:

See Also

mlib_SignalDTWKVectorInit_S16(3MLIB), mlib_SignalDTWKVector_S16(3MLIB), mlib_SignalDTWKVectorPath_S16(3MLIB), mlib_SignalDTWKVectorFree_S16(3MLIB), attributes(5)