This paper proposes a new operator called the surfaceshape operator for describing image structure. We consider an image function as a bivariate surface, then established the surfaceshape operator for describing the shape of each pixel compared with its neighborhood. From a geometry viewpoint, the types of surface shapes are elliptic, hyperbolic, cylindrical, etc. To give clearer meanings, in general, here we label them based on the topographic structure such as hill, dale, ridge, valley, etc. We also extend the surface-shape operator to have the capability of describing multiscale topographic structures by incorporating in scale-space representation. The surface-shape operator has invariant properties under linear and monotonic gray-tone transformations, and is insensitive to additive noises modeled by linear functions, so it has robustness with brightness variations and shading effects. Finally, we illustrate applicability of the surface-shape operator with applications to image classifications. The proposed method is compared with the spatial gray-level-dependence method and the multiresolution simultaneous autoregressive method. The results show that the proposed method works better, and has much more robustness with brightness variations and shading effects than those methods.