Fire Detection Algorithms Using Multimodal ... - Bilkent University

CHAPTER 2. FLAME DETECTION IN VISIBLE RANGE VIDEO 16A Gaussian mixture model with D Gaussian distributions is used to modelthe past observations {Q 1 , ..., Q n }P (Q n ) =D∑η(Q n |µ d,n , Σ d,n ) (2.6)d=1where D is the number of distributions, µ d,n is the mean value of the d-th Gaussianin the mixture at time step n, Σ d,n is the covariance matrix of the d-th Gaussianin the mixture at time step n, and η is a Gaussian probability density functionη(Q|µ, Σ) =1(2π) n 2 |Σ| 1 2e − 1 2 (Q−µ)T Σ −1 (Q−µ)(2.7)In our implementation, we model the flame color distribution with D = 10 Gaussians.In order to lower computational cost, red, blue and green channel valuesof pixels are assumed to be independent and have the same variance [77]. Thisassumption results in a covariance matrix of the form:where I is the 3-by-3 identity matrix.Σ d,n = σ 2 dI (2.8)In the training phase, each observation vector, Q n , is checked with the existingD distributions for a possible match. In the preferred embodiment, a match isdefined as an RGB vector within 2 standard deviations of a distribution.none of the D distributions match the current observation vector, Q n , the leastprobable distribution is replaced with a distribution with the current observationvector as its mean value and a high initial variance.The mean and the standard deviation values of the un-matched distributionsare kept the same. However, both the mean and the variance of the matchingdistribution with the current observation vector, Q n , are updated. Let the matchingdistribution with the current observation vector, Q n , be the d-th Gaussianwith mean µ d,n and standard deviation σ d,n . The mean, µ d,n , of the matchingdistribution is updated as:and the variance, σd,n 2 , is updated as:µ d,n = (1 − c)µ d,n−1 + cQ n (2.9)σ 2 d,n = (1 − c)σ 2 d,n−1 + c(Q n − µ d,n ) T (Q n − µ d,n ) (2.10)If