Several mathematical approaches for quantifying the three-dimensional topographical structure from scanning probe microscopy images are evaluated. Variational, i.e., scale-dependent, roughness based on root-mean-square roughness, Fourier deconvolution, and the two-dimensional autocovariance function are compared for surfaces with widely varying character in order to develop criteria for accurate quantification. Thermally evaporated gold, a calibration grid, polycrystalline Si3N4, and silicon fracture surfaces serve as models for these techniques. The role of image artifacts on each approach is detailed.