Descriptor Combination using Firefly Algorithm - Iris.sel.eesc.sc.usp.br

computed as follows:{0 if s ∈ S,f max (〈s〉) =+∞ otherwisef max (π ·〈s,t〉) = max{f max (π),d(s,t)}, (8)in which d(s,t) means the distance between samples s andt, and a path π is defined as a sequence of adjacent samples.In such a way, we have that f max (π) computes the maximumdistance between adjacent samples in π, when π is not a trivialpath.The OPF algorithm works with training and testing phases.In the former step, the competition process begins with theprototypes computation. We are interested into finding theelements that fall on the boundary of the classes with differentlabels. For that purpose, we can compute a Minimum SpanningTree (MST) over the original graph and then mark as prototypesthe connected elements with different labels. Figure 3bdisplays the MST with the prototypes at the boundary. Afterthat, we can begin the competition process between prototypesin order to build the optimum-path forest, as displayed inFigure 3c. The classification phase is conducted by takinga sample from the test set (blue triangle in Figure 3d) andconnecting it to all training samples. The distance to alltraining nodes are computed and used to weight the edges.Finally, each training node offers to the test sample a costgiven by a path-cost function (maximum arc-weight along apath - Equation 8), and the training node that has offeredthe minimum path-cost will conquer the test sample. Thisprocedure is shown in Figure 3e.0.30.41.50.40.51.10.00.91.8(a)0.01.11.40.70.70.30.40.41.10.00.9(b)0.00.90.70.7V. DESCRIPTOR COMBINATION USING FIREFLYALGORITHMMansano et. al [14] have proposed an extension to Faria’set al. work [6] by introducing a set of parameters β =(β 1 ,β 2 ,...,β M ) in order to allow a greater variability ofarithmetic computations, which was limited by the linearformulation proposed by Faria et al. [6] (Eq. 9) . Equation10 presents the proposed formulation.δ ∗ D =δ ∗ D =N∑α i δ Di , (9)i=1N∑i=1α i δ βiD i, (10)in which −2 ≤ α i ,β i ≤ 2, β i ∈ R.The optimization algorithms are responsible for computingthe best values for α and β that maximize the accuracy rateover the validation set which is our objective function. As it isknown, each agent (particle, harmony or firefly) represents adifferent solution which changes at the end of each iteration.Each pair (α, β) provides a different training set which isassessed using the validation set. The process of computingnew solutions and validation each of them repeats until thenumber of iterations is reached and the best training set of allcomputed is saved in order to run the tests. The purpose ofvalidation the training set is to obtain the set that may providethe highest accuracy rates for a given problem.Thus, the methodology uses three sets: (i) training set whichwill become the final training set, (ii) validation set usedto evaluate the descriptor combination parameters at eachiteration of the optimization algorithms, and (iii) testing setwhich is used to assess the final training set. Then, the wholeprocess can be split in two phases: (i) the design phase wherethe final training set is modeled by means of optimizationalgorithms which compute the best parameters for the set ofdescriptors, and (ii) classification phase in which each sampleis classified using the training model computed in the previousphase. Figure 4 shows the methodology proposed by Mansanoet. al [14].0.40.40.70.50.6(c)0.0(d)0.40.70.00.40.6(e)Fig. 3. OPF pipeline: (a) complete graph, (b) MST and prototypes bounded,(c) optimum-path forest generated at the final of training step, (d) classificationprocess and (e) the triangle sample is associated to the white circle class. Thevalues above the nodes are their costs after training, and the values above theedges stand for the distance between their corresponding nodes.Fig. 4. Proposed methodology for descriptor combination. Extracted fromMansano et. al [14]