In the first part of this article, we analyze the relation between local image structures (i.e., homogeneous, edge-like, corner-like or texture-like structures) and the underlying local 3D structure (represented in terms of continuous surfaces and different kinds of 3D discontinuities) using range data with real-world color images. We find that homogeneous image structures correspond to continuous surfaces, and discontinuities are mainly formed by edge-like or corner-like structures, which we discuss regarding potential computer vision applications and existing assumptions about the 3D world. In the second part, we utilize the measurements developed in the first part to investigate how the depth at homogeneous image structures is related to the depth of neighbor edges. For this, we first extract the local 3D structure of regularly sampled points, and then, analyze the coplanarity relation between these local 3D structures. We show that the likelihood to find a certain depth at a homogeneous image patch depends on the distance between the image patch and a neighbor edge. We find that this dependence is higher when there is a second neighbor edge which is coplanar with the first neighbor edge. These results allow deriving statistically based prediction models for depth interpolation on homogeneous image structures.