This paper analyses the stochastic geometry of anisotropic random surfaces. Explicit expressions are derived for computing three-dimensional surface spectral moments from two-dimensional profile spectral moments. A new approach for obtaining good estimates of profile spectral moments from digitized measurements is presented. Application to actual data on five profiles of coated abrasive papers illustrates that the procedure is simple, efficient, and statistically optimal and is applicable to any type of homogeneous surface. Some typical characteristics of coated abrasive surface geometry such as the density of summits, the maximum slope, and the principal direction are calculated and their physical interpretations are explained.