Some Integral Geometry Tools to Estimate the Complexity of 3D Scenes

Many problems in computer graphics deal with complex $3D$ scenes where visibility, proximity, collision detection queries have to be answered. Due to the complexity of these queries and the one of the models they are applied to, data structures most often based on hierarchical decompositions have been proposed to solve them. As a result of these involved algorithms/data structures, most of the analysis have been carried out in the worst case and fail to report good average case performances in a vast majority of cases. The goal of this work is therefore to investigate geometric probability tools to characterize average case properties of standard scenes such as architectural scenes, natural models, etc under some standard visibility and proximity requests. In the first part we recall some fundamentals of integral geometry and discuss the classical assumption of measures invariant under the group of motions in the context of non uniform models. In the second one we present simple generalizations of results describing the interactions between random lines and planes, and random bodies. And in the third one we report experimental results obtained for five fairly complex real-world models. These results exhibit a very good match between the theory and the practice, thus giving credit to the scene model used all along.