This work provides a new technique for surface oriented volumetric image analysis. The method makes no assumptions about topology, instead constructing a local neighborhood from image information, such as a segmentation or edge map, to define a surface patch. Neighborhood constructions using extrinsic and intrinsic distances are given. This representation allows one to estimate differential properties directly from the image's Gauss map. We develop a novel technique for this purpose which estimates the shape operator and yields both principal directions and curvatures. Only first derivatives need be estimated, making the method numerically stable. We show the use of these measures for multi-scale classification of image structure by the mean and Gaussian curvatures. Finally, we propose to register image volumes by surface curvature. This is particularly useful when geometry is the only variable. To illustrate this, we register binary segmented data by surface curvature, both rigidly and non-rigidly. A novel variant of Demons registration, extensible for use with differentiable similarity metrics, is also applied for deformable curvature-driven registration of medical images.