Fire Detection Algorithms Using Multimodal ... - Bilkent University

CHAPTER 6. WILDFIRE DETECTION 93where λ is a step size. In the well-known LMS algorithm, the ensemble averageE[e(x, n)D(x, n)] is estimated using the instantaneous value e(x, n)D(x, n) or itcan be estimated from previously processed pixels as follows:ê(x, n) ˆD(x, n)] = 1 ∑e(x, n)D(x, n) (6.14)Lwhere L is the number of previously processed pixels which is equal to the numberof terms inside the summation. The LMS algorithm is derived by noting thatthe quantity in Eq. 6.13 is not available but its instantaneous value is easilycomputable, and hence the expectation is simply replaced by its instantaneousvalue [73]:x,nw(n + 1) = w(n) + λe(x, n)D(x, n) (6.15)Eq. 6.15 is a computable weight-update equation. Whenever the oracle providesa decision, the error e(x, n) is computed and the weights are updated accordingto Eq. 6.15. Note that, the oracle does not assign her/his decision to each andevery pixel one by one. She/he actually selects a window on the image frame andassigns a “1” or “−1” to the selected window.Convergence of the LMS algorithm can be analyzed based on the MSE surface:where P yE[e 2 (x, n)] = P y (x, n) − 2w T p − w T Rw (6.16)= E[y 2 (x, n)], p = E[y(x, n)D(x, n)], and R = E[D(x, n)D T (x, n)],with the assumption that y(x, n) and D(x, n) are wide-sense-stationary randomprocesses. The MSE surface is a function of the weight vector w. Since E[e 2 (x, n)]is a quadratic function of w, it has a single global minimum and no local minima.Therefore, the steepest descent algorithm of Eqs. 6.13 and 6.15 is guaranteed toconverge to the Wiener solution, w ∗ [73] with the following condition on the stepsize λ [93]:0 < λ < 1α max(6.17)where α max is the largest eigenvalue of R.In Eq. 6.15, the step size λ can be replaced byµ||D(x, n)|| 2 (6.18)