Fire Detection Algorithms Using Multimodal ... - Bilkent University

CHAPTER 5. FLAME DETECTION USING PIR SENSORS 76monitoring fire compared to human motion.5.2.1 Threshold Estimation for State TransitionsThe thresholds T 1 and T 2 in the wavelet domain determine the state transitionprobabilities for a given sensor signal. In the training step, the task is to findoptimal values for T 1 and T 2. Given (T 1, T 2) and ground-truth fire and non-firewavelet training sequences, it is possible to calculate the transition probabilitiesfor each class. Let a ij denote the transition probabilities for the ‘fire’ class andb ij denote the transition probabilities for the ‘non-fire’ class.The decision about the class affiliation of a state transition sequence C of sizeL is done by calculating the two joint probabilities P a (C) and P b (C) correspondingto fire and non-fire classes, respectively:P a (C) = ∏ ip a (C i+1 |C i ) = ∏ ia Ci ,C i+1(5.4)andP b (C) = ∏ ip b (C i+1 |C i ) = ∏ ib Ci ,C i+1(5.5)where p a (C i+1 |C i ) = a Ci ,C i+1, and p b (C i+1 |C i ) = ∏ i b C i ,C i+1, and i = 1, ..., L .In case of P a (C) > ξP b (C), for ξ > 0, the class affiliation of state transitionsequence C will be declared as ‘fire’, otherwise it is declared as ‘non-fire’. In ourimplementation, we take ξ = 1 without loss of generality.Given N a training sequences A 1 , ..., A Na from ‘fire’ class and N b training sequencesB 1 , ..., B Nb from ‘non-fire’ class, the task of the training step is to findthe tuple (T1, T2) which maximizes the dissimilarity D = (S a − S b ) 2 , whereS a = ∑ i P a(B i ) and S b = ∑ i P b(A i ).This means that, for each given tuple (T 1, T 2), there is a specific value of thedissimilarity D, so that D is a function of (T 1, T 2)D = D(T 1, T 2) (5.6)