Fusion methods for multiple sensor systems with unknown error densities

Consider that the sensor S_i, i = 1,2,...,N, outputs y⁽ⁱ⁾ε{lunate}R;^d, according to an unknown probability density p_i(y⁽ⁱ⁾|x), corresponding to an object with parameter xε{lunate}R^d. For the system of N sensors, S₁, S₂,...,S_N, a training l-sample (x₁,y₁), (x₂,y₂),..., (x₁,y₁) is given where y_i = (y⁽¹⁾_i,y⁽²⁾_i,...,y^(N)_i) and y^(j)_i is the output of S_j in response to input x_i. The problem is to estimate a fusion rule f{hook}:R^Nd{mapping}R^d, based on the sample, such that the expected square errorI(f{hook}) = ∫[x - f{hook}(y⁽¹⁾,y⁽²⁾,...,y^(N))]²p(y⁽¹⁾,y⁽²⁾,...,y^(N)|x)p(x)dy⁽¹⁾dy⁽²⁾...dy^(N)dx is to be minimized over a family of fusion rules F based on the given l-sample. Let f{hook}^*ε{lunate}F minimize I(f{hook}). In general, f{hook}^* cannot be computed since the underlying probability densities are unknown. Using Vapnik's empirical risk minimization method, we show that if F has finite capacity, then under bounded error condition, for sufficiently large sample, f̂ can be obtained such thatP[I( f ̂)-I(f^*)>epsiv;] < δ for arbitrarily specified ε > 0 and δ, 0 < δ < 1. We obtain similar conditions for the case when F is a set of Lipschitz continuous functions with a fixed constant; these conditions are applicable to feedforward neural networks with a particular type of sigmoidal units. Then we identify sufficiency conditions for the composite system (of fuser and sensors) to be better than the best of the individual sensors. We then discuss linearly separable systems to identify objects from a finite class where f̂ can be computed in polynomial time using quadratic programming methods.