In this paper we consider the completeness problem of reasoning about planar spatial relationships in pictorial retrieval systems. We define a large class of two-dimensional scenes, the extended pseudo-symbolic pictures. The existing rule system R is proved to be complete for (ex- tended) pseudo-symbolic pictures. We also introduce a new iconic indexing, the (extended) pseudo- 2D string representation, for them. The (extended) pseudo-2D string has the good properties of the 2D string. It is unambiguous, like the augmented 2D string, and has a compact form suitable for image retrieval. We then present efficient algorithms to determine whether a given planar picture is (extended) pseudo-symbolic or not, and if it is, these algorithms also return its (extended) pseudo-2D string representation. Picture retrieval by (extended) pseudo-2D strings is also discussed. 1. Introduction. Image database systems have been much studied over the past 20 years. One of the most important problems in the design of image database systems is how images are stored in the image databases (5, 6, 9, 11, 12, 24). While the use of indexing to allow database accessing has been well established in traditional database systems, content-based picture indexing techniques need to be developed for facilitating pictorial information retrieval from a pictorial database. Tanimoto (25) suggested the use of picture icons as picture indexes, thus intro- ducing the concept of iconic indexing. Subsequently, Chang et al. (11) developed the concept of iconic indexing by introducing the 2D string representation of the image. The 2D string approach is based on the idea that the spatial knowledge contained in a real picture can be suitably represented by a symbolic picture (i.e., a matrix of symbols) where every symbol corresponds to a significant element of the image. The position of a symbol in the grid corresponds to the position of the centroid of the rep- resented significant element. Depending on the application, the significant elements of the image can be pixels, lines, regions, and objects, etc. A 2D string representing a symbolic picture is derived from the picture by orthogonally projecting its symbols by columns and by rows. This approach gives an efficient and natural way to con- struct iconic indexes for two-dimensional pictures. With the 2D string approach, the problem of pictorial information retrieval for 2D pictures becomes a problem of 2D
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