Fusion methods in multiple sensor systems using feedforward sigmoid neural networks

Consider a system of N sensors (S₁, S₂, …, S_N), where the sensor S_j outputs Y^(j) ϵ in response to input X ϵ, according to an unknown probability distribution P_y^(j)_IX. A training n-sample (X₁, Y₁), (X₂, Y₂),…, (X_n, Y_n) is given where Y_i, = (Y_i⁽¹⁾, Y_i⁽²⁾,…, Y_i^(N)) and Y_i^(j) is the output of S_j in response to input X_i ϵ. The problem is to choose a fusion function f: from a family, based on the sample, to minimize the expected square error where Y = (Y⁽¹⁾, Y⁽²⁾…, Y^(N)). We consider to be the set of feedforward neural networks of sigmoid units with a single hidden layer and bounded weights. The computation of f* ϵ that exactly minimizes I(f) is not possible in general since the underlying distributions are unknown. Under the boundedness of X and Y, we show that for a sufficiently large sample, a neural network estimate fˆ can be obtained such that P[I(fˆ) — I(f*)>ϵ] <δ for any ϵ>0, δ, 0<δ<1, and any distribution P_{Y, X}. Using various properties of the feedforward neural networks we obtain three different estimates for the required sample sizes.