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Maintaining Spatial Relations in an Incremental Diagrammatic Reasoner Maintaining Spatial Relations in an Incremental Diagrammatic Reasoner Ronald W. Ferguson, Joseph L. Bokor, Rudolph L. Mappus IV and Adam Feldman College of Computing Georgia Institute of Technology 801 Atlantic Avenue Atlanta, GA 30332 {rwf, jlbokor, cmappus, storm} @cc.gatech.edu Abstract gram. This work extends the GeoRep diagrammatic rea- soner (Ferguson & Forbus, 2000), which is described in Because diagrams are often created incrementally, a qualitative section 3. After describing GeoRep, we discuss how Ge o- diagrammatic reasoning system must dynamically manage a po- Rep was modified to allow incremental processing, and tentially large set of spatial interpretations. This paper describes cover a number of implementation issues: how to handle an architecture for handling spatial relations in an incremental, nonmonotonic diagrammatic reasoning system. The architecture composite objects, the interface between low-level and represents jointly exhaustive and pairwise disjoint (JEPD) spatial high-level reasoning, and a modified default assumption relation sets as nodes in a dependency network. Examples of these mechanism. We also describe extensions to a user interface spatial relation sets are interval relations, relative orientation rela- allowing a user to create diagrams and update the infer- tions, and connectivity relations. The network caches dependen- ences of the reasoner. We then discuss future challenges cies between low-level spatial relations, allowing those relations for this architecture. to be easily assumed or retracted as visual elements are added or removed from a diagram. We also describe how the system sup- ports high-level reasoning, including support for creating default 2. Related Work assumptions. Finally, we show how this system was integrated In qualitative spatial reasoning, researchers have explored with an existing drawing program and discuss its possible use in how to process qualitative spatial vocabularies incremen- diagrammatic and geographic reasoning. tally. Notably, Hernández (1993) proposed mechanisms for maintaining consistent spatial knowledge that include a 1. Introduction propagation heuristic for inserting relations and reason Diagrams are useful across a wide variety of reasoning maintenance mechanisms for deleting relations. These tasks. Because a single diagram conveys many spatial rela- mechanisms use a dependency network, similar to that in a tions at a glance, it provides a rich medium for many do- truth-maintenance system. The system uses a combined set mains, including geography, architecture, and engineering. of orientation and topological spatial relations to represent A body of research exists showing how diagrams are used qualitative 2-D positions. in many cognitive tasks (Glasgow, Narayanan, & Other research has explored the use of conceptual Chandrasekaran, 1995). neighborhoods (Freksa, 1992) and topological distances These characteristics are more interesting when we con- (Egenhofer & Al-Taha, 1992) to understand gradual change sider that the spatial relations in a diagram may not be in the context of qualitative spatial vocabularies. Egenhofer static. Diagrams frequently change over time. The addition, and Al-Taha, for example, show how an analysis of topo- remo val, and modification of elements also changes the set logical distance between members of a relation set can be of spatial relations. Handling such incremental changes used to construct a graph that links the closest qualitative without significant reprocessing of previously established spatial relations. This graph can be used to show the set of spatial relations is key to making spatial and diagrammatic necessary intervening qualitative states that must occur be- reasoning efficient. tween two given states. An incremental processing approach is also be moti- Along with this research in qualitative spatial reasoning, vated other factors. For example, problem solving may there are a number of sketching systems that must maintain drive changes to a particular diagram as new ideas or sub- knowledge of the links between individual visual elements tasks emerge. Also, if the conceptual interpretation of the and the inferred relations, although few of them have ex- spatial relations is computationally expensive, we do not plicit frameworks for handling dependencies between spa- want to recompute the interpretation when only minor tial relations. A family of sketch interpretation systems by changes are made. Finally, a more practical benefit of in- Davis and colleagues (Davis, 2002) use blackboard systems cremental processing is that it distributes the processing to integrate a low-level reasoner with a high-level descrip- burden evenly over the extent of the task, which is useful tion language, as with GeoRep, but can handle sketched on low-end devices, such as personal digital assistants. shapes as well as vector graphics. In this paper, we describe an architecture for the main- Another approach is found in the Geometer’s Sketchpad, tenance of incremental, nonmonotonic changes to a dia- which uses a constraint system to dynamically enforce a set of constraints over abstract geometric LLRD Modules vector elements, including lines, rays, and cir- parallel graphics file basic prox. segments cles (Scher, 2000). These constraints Input shapes Proximity segments Segment Interval include geometric relations such as the Filter detector Orientation Relations attachment of a segment endpoint along prox. a circle perimeter and the bisection of prox. segments Segment corners Corner shapes an angle. The user can then move Connectivity Angularity Shape points in the figure as desired, subject Orientation corners only to the geometric constraints. This Concavity Index Line Detect polys allows visualization of simple geomet- (simple) polygons corner segments Segments segment concavities ric conjectures, such as the (true) con- index Detect polys Protrusion, jecture that the bisectors of a triangle’s (breakdown) Indentation angles intersect at a single central point. Detection However, unlike those in GeoRep, the indentation, spatial relations in Geometer’s Sketch- arc, circle, beside, above, parallel, polygon, concave, protrusion, ellipse, hort- perpendicular polyline, convex pad are not discovered by the system, prot-above line, point, equidistant, surrounding- but are entered explicitly by the user. glyph, vert-equidistant polygon mid- spline, text before, overlaps, acute, obtuse, connect, starts, ends, linear, perpen- Generated Low-Level Relations intersects, during, interval- dicular-to corner 3. The GeoRep Reasoner equal, meets This work extends an existing dia- grammatic reasoner, GeoRep, to make Figure 2: Data flow diagram of GeoRep’s Low-Level Relation Describer (LLRD). The processing incremental. GeoRep top portion shows the collections of visual operators and the visual elements they proc- (Ferguson & Forbus, 2000) has been ess. The lower portion shows the set of spatial relations produced by each set of visual used in a numb er of diagrammatic rea- operators. soning domains, including military dia- grams (Ferguson, Rasch, Turmel, & Forbus, 2000), simple physical diagrams (Ferguson & For- The first stage, the Low-Level Relational Describer bus, 1998), and logic circuits (Ferguson, 2001). In addition, (LLRD; Figure 2) represents a set of low-level visual rela- it has been used as the visual representation system for sev- tions. These low-level relations are generated starting with eral cognitive mo deling simulations (Ferguson, 2000; Fer- proximity calculations and ending with more complex rela- guson, Aminoff, & Gentner, 1996). tions, such as interval relations between parallel line seg- GeoRep’s input is a line-drawn diagram, given as a vec- ments. tor graphics file. The vector graphics file may contain a va- These particular spatial relations are also designed to riety of visual element types, including line segments, cir- model those qualitative spatial relationships that are de- cles, ellipses, arcs, spline curves and positioned text. tected in early vision. For example, it is well-known that GeoRep’s output is a predicate calculus representation. The humans are sensitive to relative angles (such as perpendicu- representation has three parts: the low-level spatial rela- lar lines), indentations in figure boundaries (Hoffman & tions, the high-level interpretation of the diagram, and in- Richards, 1984), and to vertical and horizontal orientations termediate spatial and conceptual relations that are pro- in the assumed frame of reference (Rock, 1973). Interest- duced in the process of interpretation. ingly, one relation set used that has not been tested for early To generate this representation, GeoRep uses a two- vision is interval relations (Allen, 1983) between parallel stage architecture (Figure 1). lines. In practice, these relations are extremely useful in modeling aspects of visual perception such as the detection of qualitative symmetry (Ferguson, 1994). A simple atten- High-level tion model uses a proximity detector to limit visual relation Line Low-level relations drawing relations (place tests to proximate visual elements. This acts as a limited vocabulary) focusing mechanism that keeps processing tractable. LLRD HLRD The system also models some aspects of attention and perceptual organization, though in a domain-dependent fashion. Grouping rules can be used to simulate simila rity- Visual Visual based grouping, and multiple LLRDs can be used to simu- operation domain library theory late visual separation
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