Rectangle Reasoning: A Qualitative Spatial Reasoning with Superposition

Qualitative spatial reasoning (QSR) is a method for treating images or figures qualitatively by extracting only the information required by a user for a specified purpose (Aliello et al. 2007). It is widely considered a promising method for reducing memory and workspace requirements for computations that do not involve strict data. However, few studies have developed practical applications. We propose rectangle reasoning as a framework for the application of QSR through an autonomic window placement system. A window can be considered as spatial data of rectangular shape, with changeable size and ratio of edges. Moreover, multiple windows can be displayed in a superposed manner. If we place multiple windows, such that the relevant parts of each are visible and the unnecessary parts are not visible, it provides a display in the most useful form to the user, effectively using a monitor of limited size. The rectangle reasoning proposed here addresses the relative positional relationships of rectangles with superposition. Each rectangle is represented symbolically using two simple objects, regions and lines, and their relationships, as well as visibility. In this study, we formalize rectangle reasoning and propose a reasoning algorithm. This algorithm determines whether a figure exists that satisfies all the given visibility requirements in a two-dimensional (2D) plane with foreground/background, and, if such a figure is determined to exist, derives the superposed locations of the rectangles. This paper is organized as follows. In section 2 we formalize rectangle reasoning; in section 3 we describe the reasoning algorithm; and in section 4 we show our conclusions.