stem/AI/Pattern Matching/Dynamic Time Warping.md

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Deterministic Pattern Recogniser Allows timescale variations in sequences for same class

D(T,N)=\min_{t,i}\sum_{\substack{t\in1..T \\ i\in1..N}}d(t,i)

• d(t,i) is distance between features from $t$-th frame of test to $i$-th frame of template

D(t,i)=\min[D(t,i-1),D(t-1, i-1),D(t-1,i)]+d(t,i)

• Allowing transition from current and previous frame only
• Recursive

Problems

• How much flexibility to allow?
• How to penalise warping?
• How to determine a fair distance metric?
• How many templates to register?
  • How to select best ones?

Basic Algorithm

1. Initialise the cumulative distances for t=1

D(1,i)=\begin{cases}d(1,i) & \text{for }i=1, \\ D(1, i-1)+d(1,i) & \text{for }i=2,...,N\end{cases}

2. Recur for t=2,...,T

D(t,i)=\begin{cases}D(t-1,i) + d(t,i) & \text{for }i=1, \\ \min[D(t, i-1), D(t-1, i-1),D(t-1,i)] + d(t,i) & \text{for }i=2,...,N\end{cases}

3. Finalise, the cumulative distance up to the final point gives the total cost of the match: D(T,N)

• Euclidean distances

Distortion Penalty

1. Initialise the cumulative distances for t=1

D(1,i)=\begin{cases}d(1,i) & \text{for }i=1, \\ d(1,i)+D(1, i-1)+d_V & \text{for }i=2,...,N\end{cases}

2. Recur for t=2,...,T

D(t,i)=\begin{cases}d(t,i)+D(t-1,i1)+d_H & \text{for }i=1, \\ \min[d(t,i)+D(t,i-1)+d_V,2d(t,i)+D(t-1,i-1),d(t,i)+D(t-1,i)+d_H] & \text{for }i=2,...,N\end{cases}

• Where d_V and d_H are costs associated with vertical and horizontal transitions respectively

3. Finalise, the cumulative distance up to the final point gives the total cost of the match: D(T,N)

• Allows weighting for dynamic penalties when moving horizontally or vertically
  • As opposed to diagonally

Store Best Path

1. Initialise distances and traceback indicator for t=1

D(1,i)=\begin{cases}d(1,i) & \text{for } i=1,\\ d(1,i)+D(1,i-1) & \text{for }i = 2,...,N\end{cases}

\phi(1,i)=\begin{cases}[0,0] & \text{for } i=1,\\ [1,i-1] & \text{for }i = 2,...,N\end{cases}

2. Recur for cumulative distances at t=2,...,T

D(1,i)=\begin{cases}d(t,i)+D(t-1,i) & \text{for } i=1,\\ d(t,i)+\min[D(t,i-1),D(t-1,i-1),D(t-1,i)] & \text{for }i = 2,...,N\end{cases}

\phi(1,i)=\begin{cases}[t-1,i] & \text{for } i=1,\\ \arg\min[D(t,i-1),D(t-1,i-1),D(t-1,i)] & \text{for }i = 2,...,N\end{cases}

3. Final point gives the total alignment cost D(T,N) and the end coordinates of the best path z_K=[T,N], where K is the number of nodes on the optimal path
4. Trace the path back for k=K-1,...,1,z_k=\phi(z_{k+1}), \text{ and }Z=\{z_1,...,z_K\}

• Stores best path

• Vary allowable movements through grid
• Second row for blocking multiple of the same movements in succession

Search Pruning

• Speed up algorithm for real-time
• Kill bad options

Gross Partitioning

• Too far from diagonal
• Probably wrong or bad

Score Pruning

• Examine existing branches
• See which scores are really bad