HIDDEN MARKOV MODEL BASED WEIGHTED LIKELIHOOD DISCRIMINANT FOR MINIMUM ERROR SHAPE CLASSIFICATION Ninad Thakoor Electrical Engineering The Univerity of Texas at Arlington Arlington, TX 76019 Jean Gao Computer Science and Engineering The Univerity of Texas at Arlington Arlington, TX 76019 ABSTRACT The goal of this communication is to present a weighted likeli- hood discriminant for minimum error shape classification. Differ- ent from traditional Maximum Likelihood (ML) methods in which classification is carried out based on probabilities from indepen- dent individual class models as is the case for general hidden Markov model (HMM) methods, our proposed method utilizes informa- tion from all classes to minimize classification error. Proposed ap- proach uses a Hidden Markov Model as a curvature feature based 2D shape descriptor. In this contribution we present a General- ized Probabilistic Descent (GPD) method to weight the curvature likelihoods to achieve a discriminant function with minimum clas- sification error. In contrast with other approaches, a weighted like- lihood discriminant function is introduced. We believe that our sound theory based implementation reduces classification error by combining hidden Markov model with generalized probabilistic descent theory. We show comparative results obtained with our approach and classic maximum-likelihood calculation for fighter planes in terms of classification accuracies. 1. INTRODUCTION Object recognition is a classic problem in image processing, com- puter vision, and database retrieval. Among others, object recogni- tion based on shape is widely used. First step towards the design of a shape classifier is feature extraction. Shapes can be represented by their contours or regions [1]. Curvature, chain codes, Fourier descriptors, etc. are contour based descriptors while medial axis transform, Zernike moments, etc. are region based features. Con- tour based descriptors are widely used as they preserve the local in- formation which is important in classification of complex shapes. Feature extraction is followed by shape matching. In recent years, dynamic programming (DP) based shape matching is be- ing increasingly applied [2], [3], [4], [5]. DP approaches are able to match the shapes part by part rather than point by point, and are robust to deformation and occlusion. Hidden Markov Models (HMMs) are also being explored as one of the possible shape mod- eling and classification frameworks [6], [7], [8], [9], [10]. Apart from having all the properties of DP based matching, HMM pro- vides a probabilistic framework for training and classification. The current HMM approaches apply maximum likelihood (ML) as their classification criterion. Due to good generalization prop- erty of HMM, applying ML criterion to similar shapes does not provide good classification. Also, ML criterion is evaluated us- ing information from only one class and does not take advantage of information from the other classes. Generally shapes can be discriminated using only parts of the boundaries rather than com- paring whole boundary. ML does not provide such mechanism. To overcome these shortcomings, we propose a weighted like- lihood discriminant for shape classification. The weighting scheme emulates comparison of parts of shape rather than the whole shape. The weights are estimated by applying Generalized Probabilistic Descent (GPD) method. Unlike ML criterion, GPD uses informa- tion from all the classes to estimate the weights. As GPD method is designed to minimize the classification error, the proposed clas- sifier gives good classification performance with similar shapes. This paper is organized as follows: The description phase of the proposed method is discussed in Section 2, while Section 3 formu- lates discriminative training with GPD. Experimental results are presented in Section 4 and the paper ends with the conclusions and suggestions for further research in Section 5. 2. SHAPE DESCRIPTION WITH HMM Before proceeding to the detailed topology of HMM, we introduce the terminology used in the rest of the paper. 1. S, set of states. S = {S1,S2,...,SN }, where N is num- ber of states. State of HMM at instance t is denoted by q t . 2. A, state transition probability distribution. A = {a ij }, a ij denotes the probability of changing the state from Si to Sj . a ij = P [q t+1 = S j |q t = S i ], 1 ≤ i, j ≤ N. (1) 3. B, observation symbol probability distribution. B = {bj (o)}, b j (o) gives probability of observing the symbol o in state S j at instance t. bj (o)= P [o at t|qt = Sj ], 1 ≤ j ≤ N. (2) 4. π, initial state distribution. π = {π i }, π i gives probability of HMM being in state Si at instance t =1. π i = P [q 1 = S i ], 1 ≤ i ≤ N. (3) If Cj is j th shape class where j =1, 2,...,M and M is total number of classes then for convenience, HMM for C j can be com- pactly denoted as, λ j =(A,B,π). (4) An in depth description about HMM can be found in [11]. For the approach proposed in this paper, the description phase employs HMM topology proposed by Bicego and Murino [7]. The