This paper presents representation and logic for labeling contrast edges and ridges in visual scenes in terms of both surface occlusion (border ownership) and thinline objects. In natural scenes, thinline objects in- clude sticks and wires, while in human graphical communication thin- lines include connectors, dividers, and other abstract devices. Our analy- sis is directed at both natural and graphical domains. The basic problem is to formulate the logic of the interactions among local image events, speciﬁcally contrast edges, ridges, junctions, and alignment relations, such as to encode the natural constraints among these events in visual scenes. In a sparse heterogeneous Markov Random Field framework, we deﬁne a set of interpretation nodes and energy/potential functions among them. The minimum energy conﬁguration found by Loopy Belief Prop- agation is shown to correspond to preferred human interpretation across a wide range of prototypical examples including important illusory con- tour ﬁgures such as the Kanizsa Triangle, as well as more difﬁcult ex- amples. In practical terms, the approach delivers correct interpretations of inherently ambiguous hand-drawn box-and-connector diagrams at low computational cost.