After computing the centroids of the head in each

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Unformatted text preview: 1 . After computing the centroids of the head in each frame, the di erence in the absolute co-ordinates in successive frames was found. dxi ; dyi are the di erence in centroids of the head over successive frames. dxi = xi+1 , xi dyi = yi+1 , yi 1 X = dx1 ; dx2 ; : : : ; dxn Y = dy1 ; dy2 ; : : : ; dyn 3 2 The feature vectors in our case are the di erence in centroids of the head over successive frames. 4 where X and Y are the feature vectors for the di erence in x and y coordinates of the head respectively. Since there are n + 1 frames in each sequence, each feature vector is n elements long. Thus each feature vector is an n dimensional vector. Next, the mean and covariance matrix for the feature vector was found. This was repeated for all the monocular grayscale sequences. 2.4 Recognition of input sequence 2.2 Computing probability density functions We assume independence of the feature vectors X and Y and a multi-variate normal distribution for all sequences. From the independence assumption we have: We assume in the recognition of our input sequence that each sequence is uniquely described by the value of its a posteriori probability. For our problem, we assume all a priori probabilities the probability of any of the actions occurring to be equal and, thus, nd density functions for each of the classes where each class is an action. Thus, twenty such densities were found, corresponding to the ten di erent actions in the two orientations. Having obtained these twenty values for each of the classes, the most likely action is the class with the highest value. pX; Y  = pX pY  P = max P1 ; P2 ; P3 : : : Pm 5 where pX  = e 2n=2 jX j1=2 1 ,1 X ,X t X ,1 X ,X  2 6 pY  = e 2n=2 jY j1=2 1 ,1 Y ,Y t Y ,1 Y ,Y  7 2 where X is the n-component feature vector in the x direction, Y is the n-component feature vector in the y direction, X and Y are the mean vectors of the normal distribution and X and Y are the n , by , n covariance matrices. Unbiased estimates for X and Y are supplied by the sample covariance matrices 8 . 1 CX = n , 1 n X X ,  X Xi , X t 8 1 CY = n , 1 Yi , Y Yi , Y t i=1 9 i=1 n X i 2.3 Bayesian formulation of the approach Using the feature vector obtained from the test sequence, a posteriori probabilities are calculated using each of the training sequences. This is done using Bayes rule, which is a fundamental formula in decision theory. In the mathematical form it is given as 8 pX; 10 P !i =X; Y  = P !ipX; YY =!i  where X , Y m the extracted feature vectors, and, P are X; Y  = i=1 pX; Y =!i P !i . P !i =X; Y  is the a posteriori probability of observing the class !i given the feature vectors X and Y . P !i  is the a priori probability of observing the class !i , pXi ; Yi =! is the conditional density and m refers to the number of classes. 11 where P is the probability of the most likely class and P1 ; P2 ; P3 : : : Pm are the probabilities of m di erent actions. The frontal and lateral views...
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