Fire Detection Algorithms Using Multimodal ... - Bilkent University

CHAPTER 6. WILDFIRE DETECTION 97Figure 6.4: Geometric interpretation: Weight vectors corresponding to decisionfunctions at each frame are updated as to satisfy the hyperplane equations definedby the oracle’s decision y(x, n) and the decision vector D(x, n). Lines in the figurerepresent hyperplanes in R M .This hyperplane will probably not be the same as y(x, n) = D T (x, n)w(n) hyperplaneas shown in Fig. 6.4. The next set of weights, w(n + 2), are determinedby projecting w(n + 1) onto the hyperplane in Eq. 6.31. Iterated weights convergeto the intersection of hyperplanes [10], [20] for 0 < µ < 2 according to theprojections onto convex sets (POCS) theory [13], [95], [56].If the intersection of hyperplanes is an empty set, then the updated weightvector simply satisfies the last hyperplane equation. In other words, it tracks decisionsof the oracle by assigning proper weights to the individual subalgorithms.Another weight update algorithm can be developed by defining hyperslabs inR M as follows:y(x, n) − ɛ ≤ D T (x, n)w ≤ y(x, n) + ɛ (6.32)where ɛ > 0 is an artificially introduced positive number such that hyperslabs definedat different time instants produce a nonempty intersection set. In Eq. 6.32,if y(x, n) = 1 then y(x, n) − ɛ ≥ 0 and if y(x, n) = −1 then y(x, n) + ɛ < 0.Furthermore, when y(x, n) = 1, the upper bound can be simply removed andprojections can be performed onto the half-spaces. A half-space can be defined