A view of three decades of linear filtering theory
IEEE Transactions on Information Theory
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Let U(x) = s(x) + n(x), where s(x) and n(x) are random fields, s(x) is the useful signal and n(x) is noise. Assume that s(x) = n(x) = 0, where the bar stands for mean value. Suppose that the covariance functions R(x, y)@?U^*(x)U(y) and @?(x, y)@?U^*(x)s(y) are known, where the asterisk stands for complex conjugate. Assume that U(x) is observed in a finite region D @? R^r of the Euclidean space R' and @C is the smooth boundary of D. The estimation problem consists of finding an optimal (in the sense of minimum of variance) linear estimate of the signal As(x), where A is a known operator, given the observed values of U(x) in D and the covariance functions R(x, y) and @?(x, y). If A = I, the identity operator, then the estimation problem is the filtering problem. If A = I and the optimal estimate U@^@?@?"Dh(x, y)U(y) dy, i.e. |s(x) - Ucirc;(x)|^2 = min, where the minimum is taken over all linear estimates of s(x) given the observed values of U(x) in D, then h(x, y) is the solution of the equation (A)@?"DR(x,z)h(y,z) dz = @?(x,y), x,y @e D@?D@?@? A class R of random fields is introduced for which equation (A) is solved analytically. Qualitative properties of the solution are obtained, such as the order of singularity of h and the singular support of h. Robustness of the estimate (filter) is established. Numerical methods for computing the singular solution to equation (A) are given. Examples of the numerical results are presented. For the class R of random fields the estimation problem is equivalent to finding the solution to equation (A) which has minimal order of singularity. The class R consists of random fields whose covariance functions R(x, y) are positive rational functions of arbitrary selfadjoint elliptic operations in L^2(R'). The theory developed is a natural generalization of the Wiener filtering theory to the case of random fields. It gives stochastic modelling of a wide class of estimation problems. Applications to image processing in optics and geophysics are discussed.