Although a manipulator must interact with objects in terms of their full complexity, it is the qualitative structure of the objects in an environment and the relationships between them which define the composition of that environment, and allow for the construction of efficient plans to enable the completion of various elaborate tasks. In this paper we present an algorithm which redescribes a scene in terms of a layered representation, from labeled point clouds of the objects in the scene. The representation includes a qualitative description of the structure of the objects, as well as the symbolic relationships between them. This is achieved by constructing contact point networks of the objects, which are topological representations of how each object is used in that particular scene, and are based on the regions of contact between objects. We demonstrate the performance of the algorithm, by presenting results from the algorithm tested on a database of stereo images. This shows a high percentage of correctly classified relationships, as well as the discovery of interesting topological features. This output provides a layered representation of a scene, giving symbolic meaning to the inter-object relationships useful for subsequent commonsense reasoning and decision making.