Integrating existing cone-shaped and projection-based cardinal direction relations and a TCSP-like decidable generalisation

Abstract

We consider the integration of existing cone-shaped and projection-based calculi of cardinal direction relations, well-known in QSR. The more general, integrating language we consider is based on convex constraints of the qualitative form $r(x,y)$, $r$ being a cone-shaped or projection-based cardinal direction atomic relation, or of the quantitative form $(α,β)(x,y)$, with $α,β\in [0,2π)$ and $(β-α)\in [0,π]$: the meaning of the quantitative constraint, in particular, is that point $x$ belongs to the (convex) cone-shaped area rooted at $y$, and bounded by angles $α$ and $β$. The general form of a constraint is a disjunction of the form $[r_1\vee...\vee r_{n_1}\vee (α_1,β_1)\vee...\vee (α_{n_2},β_{n_2})](x,y)$, with $r_i(x,y)$, $i=1... n_1$, and $(α_i,β_i)(x,y)$, $i=1... n_2$, being convex constraints as described above: the meaning of such a general constraint is that, for some $i=1... n_1$, $r_i(x,y)$ holds, or, for some $i=1... n_2$, $(α_i,β_i)(x,y)$ holds. A conjunction of such general constraints is a $\tcsp$-like CSP, which we will refer to as an $\scsp$ (Spatial Constraint Satisfaction Problem). An effective solution search algorithm for an $\scsp$ will be described, which uses (1) constraint propagation, based on a composition operation to be defined, as the filtering method during the search, and (2) the Simplex algorithm, guaranteeing completeness, at the leaves of the search tree. The approach is particularly suited for large-scale high-level vision, such as, e.g., satellite-like surveillance of a geographic area.