Over recent years, the use of homogeneous Gibbs prior models in image processing has become widely accepted. There has been, however, much discussion over precisely which models are most appropriate. For most applications, the simplest Gaussian model tends to oversmooth reconstructions, so it has been rejected in favor of various edge-preserving alternatives. We claim that the problem is not with the Gaussian family, but rather with the assumption of homogeneity. In this article we propose an inhomogeneous Gaussian random field as a general prior model for many image-processing applications. The simplicity of the Gaussian model allows rapid calculation, and the flexibility of the spatially varying prior parameter allows varying degrees of spatial smoothing. This approach is in the spirit of adaptive kernel density methods where only the choice of the variable window width is important. The analysis of real single-photon emission computed tomography data is used to illustrate the methods, and simulated data are used to demonstrate that the proposed procedures lead to more accurate reconstruction than edge-preserving homogeneous alternatives. The inhomogeneous model allows greater flexibility; small features are not masked by the smoothing, and constant regions obtain sufficient smoothing to remove the effects of noise.