Fire Detection Algorithms Using Multimodal ... - Bilkent University

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CHAPTER 2. FLAME DETECTION IN VISIBLE RANGE VIDEO 15(a)(b)Figure 2.2: (a) A sample flame pixel process in RGB space, and (b) the spherescentered at the means of the Gaussian distributions with radius twice the standarddeviation.
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
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Chapter 5Flame Detection Using PIRS
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Chapter 7Conclusion and Future Work
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Bibliography[1] B. Albers and A. Ag
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BIBLIOGRAPHY 112[18] D. Chetverikov
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BIBLIOGRAPHY 114[38] G. Healey, D.
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BIBLIOGRAPHY 116[56] U. Niesen, D.
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