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Hidden Markov Model based 2D Shape Classification Ninad Thakoor 1 and Jean Gao 2 1 Electrical Engineering, University of Texas at Arlington, TX-76013, USA 2 Computer Science and Engineering, University of Texas at Arlington, TX-76013, USA

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Introduction Problem of object recognition Shape recognition Shape classification Shape classification techniques Dynamic programming based Hidden Markov Model (HMM) based Advantages of HMM Time warping capability Robustness Probabilistic framework

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Introduction (cont.) Limitations of HMM Unable to distinguish between similar shapes No mechanism to select important parts of shape Does not guarantee minimum classification error Proposed method deals with these limitations by designing a weighted likelihood discriminant function and formulates a minimum error training algorithm for it.

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Terminology S, set of HMM states. State of HMM at instance t is denoted by q t. A, state transition probability distribution. A = {a ij }, a ij denotes the probability of changing the state from S i to S j. B, observation symbol probability distribution. B={b j (o)}, b j (o) gives probability of observing the symbol o in state S j at instance t. , initial state distribution. = { i }, i gives probability of HMM being in state S i at instance t = 1. C j is j th shape class where j=1,2, …,M. HMM for C j can be denoted compactly as

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Shape description with HMM Shape is assumed to be formed by multiple constant curvature segments. These are hidden states of HMM. Each state is assumed to have Gaussian distribution. Mean of the distribution is the constant curvature of the segment. Noise and details of the shape are standard deviation of the state distribution.

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HMM construction Preprocessing Filter the shape Normalize the shape length to T Calculate discrete curvature (,i.e., turn angles) which will be treated as observations for the HMM Initialization Gaussian mixture model with N clusters built from unrolled example sequences

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HMM construction (cont.) Training Individual HMM are trained by Baum-Welch algorithm for varying number of states N Model selection (,i.e, optimum N) is carried out with Bayesian Information Criterion (BIC) N is selected to maximize BIC.

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Weighted likelihood (WtL) discriminant Motivation Similar objects can be discriminated by comparing only part of the shapes No point wise comparison is required for shape classification Maximum likelihood criterion gives equal importance to all shape points WtL function weights likelihoods of individual observations such that the ones important for classifications are weighted higher.

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WtL discriminant (Cont.) Log likelihood of the optimal path Q* followed by observation O is given by Where A simple weighted likelihood discriminant can be defined as

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WtL discriminant (Cont.) We use the following weighting function which is sum of S Gaussian windows Parameter p i,j governs the height, i,j controls the position, while s i,j determines spread of i th window of j th class.

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GPD algorithm Misclassification measure Cost function Re-estimation rule

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Experimental results Plane shapes: Classification accuracies (in %):

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Experimental results (cont.) Discriminant function comparison: HMM ML HMM WtL

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Questions? Please email your questions to ninad.thakoor@uta.edu OR ninad.thakoor@ieee.org ninad.thakoor@uta.edu ninad.thakoor@ieee.org Copy of the presentation is available at http://visionlab.uta.edu/~ninad/acivs2005/ THANK YOU!!!!!

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