In the modern life, the need for personal security and access control is becoming an important issue. Biometrics is the technology which is expected to replace traditional authentication methods that are easily stolen, forgotten and duplicated. Fingerprints, face, iris, and voiceprints are commonly used biometric features. Among these features, face provides a more direct, friendly and convenient identification method and is more acceptable compared with the individual identification methods of other biometrics features. Thus, face recognition is one of the most important parts in biometrics.
An embedded hidden Markov model (EHMM) is an extension of the previous 1-D HMM in order to deal with two-dimensional data such as images and videos. The EHMM was previously used for character recognition by Kuo and Agazzi (Kuo and Agazzi, 1994). In fact, a fully connected 2-D HMM would be desirable for modelling a 2-D face image. However, the computational complexity for a fully connected 2-D HMM increases explosively as the number of states grows. In this work, we adopt the Samaria's Pseudo 2-D HMM (P2D-HMM) obtained by linking 1-D left-right HMMs to form super states. The structure includes one internal 1-D HMM in each state in the outer 1-D HMM. While transitions in a vertical direction are allowed between two adjacent super states which include the super state itself, transitions in a horizontal direction are only allowed between two adjacent states in a super state including the state itself. From this structure, we know that the sequence of super states are used to model a horizontal slice of face image along the vertical direction and that the sequence of states in a super state are used to model a block image along the horizontal direction. We call this model an embedded HMM. It differs from a 2-D HMM since a transition between the states in different states are allowed. Figure 1 illustrates a 5-super states EHMM for face modelling, where each super state represents vertical facial features, such as forehead, eyes, nose, mouth, and chin in the face and each state in the super state represents horizontal local block features in the corresponding facial feature.
An illustration of 2-D EHMM with 5 super states for face modeling
The EHMM is defined by specifying the following parameters (Samaria and Young, 1994):
N0 is the number of super states in the vertical direction of EHMM.
is the initial super state distribution, where π0,i is the probability of observation sequence being in the ith super state.
is the super state transition probability matrix, where is the probability of transition from the ith to the jth super state.
is the set of left-right 1-D HMMs in each super state. Each is specified by the standard 1-D HMM parameters as follows.