A two-dimensional signal processing algorithm is developed to obtain smoothed estimates of a vector random field when observed in the presence of additive two-dimensional white noise. The vector random field consists of a two-dimensional scalar random field and its two orthogonal derivatives, each of which can be scaled arbitrarily by a constant. The signal model consists of the Laplacian operator acting on the two-dimensional scalar random field excited by two-dimensional white noise. Such a signal model that permits a nonstationary and nonisotropic random field is "spatially dynamic" as opposed to a static model specified by autocorrelation functions or power spectral densities. The model is also "bilateral" such that any given spatial point’s dependency upon all its neighboring points is preserved and no unnecessary limitation of causality is enforced on the two-dimensional system. To preserve the noncausality of the problem, the estimation algorithm must not order the two-dimensional data since there is no sense in which any data point can be said to be "prior" to any other data point. To this end, a two-dimensional smoothing algorithm is developed first in the continuous domain by making use of a two-dimensional Karhunen-Loeve expansion that makes no specific reference to either the geometry or the ordering of the parameter space.