Efficient algorithms for spatial configuration information retrieval
Introduction
Spatial configuration retrieval is an important research topic of content-based image retrieval in geographic information system (GIS), computer vision, and VLSI design, etc. A user of a GIS system usually searches for configurations of spatial objects on a map that match some ideal configuration or are bound by a number of constraints. For example, a user may be looking for a place to build a house. He wishes to have a house A north of the town that he works, in a distance no greater than 10 km from his child’s school B and next to a park C. Moreover, he would like to have a supermarket D on his way to work. Under some circumstances, the query conditions cannot be fully satisfied at all. The users may need only several optional answers according to the degree of configuration similarity. Of the configuration similarity query problem, the representation strategies and search algorithms have been studied in several papers [1], [2], [3], [7], [8], [16], [17], [21], [25], [26].
A configuration similarity query can be formally described as a standard binary constraint satisfaction problem which consists of: (1) a set of n variables, that appear in the query, (2) for each variable , a finite domain of m values, (3) for each pair of variables , a constraint which can be a simple spatial relation, a spatio-temporal relation or a disjunction of relations. In addition, unary constraints such as physical and semantical features can be added to the variables. The goal of query processing is to find instantiations of variables to image objects so that the input constraints are satisfied to a maximum degree. The dissimilarity degree of a binary instantiation is defined as the dissimilarity between the relation (between objects and in the image to be searched) and the constraint (between and in the query). The inconsistency degree can be calculated according to the principles such as conceptual neighborhood [25] or binary string encoding [26]. Given the inconsistency degrees of binary constraints, the inconsistency degree of a complete solution can be defined as:
Given the defined dissimilarity degree , the similarity degree , which is not affected by the problem scale and is within the range [0, 1], can be defined as:where is the dissimilarity degree of the solution S for a query, n is the number of variables in a query, n (n − 1) is the set of constraints between distinct variable pairs (including inverse and unspecified constraints), and D is the maximum dissimilarity degree between two constraint relations. Setting an appropriate minimum value MIN for can help to obtain the balance between the approximation degree of the solutions to query conditions and processing cost. The smaller the MIN, the more the solutions obtained, while the processing cost increases too.
In the real world, spatial data often have complex geometry shapes. It will be very costly if we directly to calculate the spatial relationships between them, while much invalid time may be spent. If N is the number of spatial objects, and n the number of query variables, the total number of possible solutions is equal to the number of n-permutations of the N objects: . Using minimum bounding rectangles (MBRs) to approximate the geometry shapes of spatial objects and calculating the relations between rectangles will reduce the calculation greatly. So we can divide the spatial configuration retrieval into two steps: firstly the rectangle combinations for which it is impossible to satisfy the query conditions will be eliminated, and then the real spatial objects corresponding to the remaining rectangle combinations will be calculated using computational geometry techniques. To improve the retrieval efficiency, the index data structure which is called R-tree[4]or the variants R+-tree [5] and R∗-tree [6] can be adopted.
The next section takes topological and directional relations as examples to study the mapping relationships between the spatial relationships for MBRs and the corresponding relationships for real spatial objects; Section 3 studies three spatial configuration retrieval algorithms; Section 4 presents the experimental system for comparing the three algorithms, designs the experiments, analyzes the experimental results and make a conclusion; the last section concludes this paper.
Section snippets
Spatial mapping relationships
This paper mainly concerns the topological and directional relations for MBRs and the corresponding spatial relationships for real spatial objects. The ideas in this paper can be applied to other relationships such as distance and spatio-temporal relations, etc.
Algorithms for spatial configuration information retrieval
As mentioned above, topological relations and directional relations can all be transformed into spatial relations for MBRs. Given a spatial configuration query, we can first transform the spatial relations constraints among spatial objects into spatial relations constraints among their MBRs, then some efficient retrieval algorithm and the R-tree that organizes the spatial data can be applied to pick out the MBR combinations that will later be checked using computational geometry technique (some
Experiments
The algorithms SFC-DVOSolver, RSFC-DVOSolver and HRSFC-DVOSolver are implemented in this paper and their performance is analyzed. The programmes are developed in Java language with the free Eclipse development environment [27], which have been run on a PC(1.7G) with 256 MB of RAM. The traditional CSP algorithms are implemented using the JCL (Java Constraints Library) algorithm library [19] developed by Bhattacharjee etc. The R-tree algorithms by Guttman [4] are implemented using
Conclusions
This paper has studied the spatial configuration information retrieval problem which includes (1) the mapping relationship among the spatial relations (topological and directional relations) for real spatial objects, the corresponding spatial relations for the corresponding MBRs and the corresponding spatial relations between intermediate nodes and the MBRs in R-tree, and (2) three systematic search algorithms. Through experiments two of the three algorithms are shown to be feasible, i.e.
Acknowledgements
We would like to thank the editors and anonymous reviewers for their valuable comments and helpful suggestions for improvements. This work is supported by research project of “SUST Spring Bud”.
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