Hidden Markov Model

To cope with nonuniform changes in the speed of gesture, HLAC feature vectors extracted from the sequences of PARCOR images are fed into a recognizer based on Hidden Markov Models.

Let each gesture be represented by a sequence of HLAC feature vectors defined as $X = \{{\bf x}_1,\cdots,{\bf x}_T\}$. Each vector ${\bf x}_t$contains HLAC features extracted from PARCOR images at time t. Then the gesture recognition problem can be formulated as that of finding the class C_k which has the maximum posterior probability P(C_k|X). By using Bayes' rule, the posterior probability can be written as

$$P(C_k\vert X) = \frac{P(C_k)P(X\vert C_k)}{P(X)}.$$ (6)

Thus, for a given prior probabilities P(C_k), the most probable gesture can be found by estimating the likelihood P(X|C_k). In HMM based recognition, it is estimated by assuming a parametric model of gesture production as a Markov model with hidden states $\{S_j\}$. In the following experiments, we tried a simple left to right HMM with 7 states shown in Figure 3 for all gestures. To improve the recognition rate, we have to determine suitable model for each gesture. We also assumed the HLAC feature vectors are generated from Gaussian densities

$$p({\bf x}\vert S_j) = \frac{1}{\sqrt{(2\pi)^M \vert\Sigma_j\v... ...\bf x} - {\bf\mu}_j)^T \Sigma_j^{-1} ({\bf x} - {\bf\mu}_j)\}.$$ (7)

  

Figure 3: State transition model.

To learn the parameters of the HMM from the training examples, we used the well-known Baum-Welch algorithm.