Concise Papers __________________________________________________________________________________________ An Object-Oriented Fuzzy Data Model for Similarity Detection in Image Databases Arun K. Majumdar, Senior Member, IEEE, Indrajit Bhattacharya, and Amit K. Saha Abstract—In this paper, we introduce a fuzzy set theoretic approach for dealing with uncertainty in images in the context of spatial and topological relations existing among the objects in the image. We propose an object-oriented graph theoretic model for representing an image and this model allows us to assess the similarity between images using the concept of (fuzzy) graph matching. Sufficient flexibility has been provided in the similarity algorithm so that different features of an image may be independently focused upon. Index Terms—Image databases, fuzzy data model, spatial and topological relations, similarity detection. æ 1 INTRODUCTION WITH the advancement of multimedia applications, image data- bases have emerged as a major field of research [7], [9], [10], [11]. Unlike the size of a record in a conventional database, the size of an image is usually very large. This has necessitated the development of efficient storage techniques for image databases. Moreover, the queries in an image database often require retrieval based on the similarity with given images or based on certain attributes of the objects present in the images. Thus, it is necessary to support queries based on image semantics rather than based on mere pixel- to-pixel matching. The image database systems should therefore allow adequate abstraction mechanisms for capturing higher level semantics of the images in order to support content addressability as far as possible. Over the last decade, there has been considerable interest in representing spatial and topological relations among objects present in a scene. Chang and Jungert [6] have proposed 2D Strings and Interval Projection Strings to represent spatial relation- ships among objects in a scene. Similarly, Gudivada [1] has suggested a geometric approach, called R Strings, for capturing spatial relationships. These representation schemes using 2D or R Strings, however, are not efficient for capturing topological relations like meet, overlap, cover, contain, etc. Considerable work has been done in this particular direction by Egenhofer and Herring [2] with GIS design and developments. In real-life images, it is often not possible to precisely identify the boundaries of objects in a scene. Thus, object identification by image processing techniques is difficult and identification of binary relations, like left-of, contain, or overlap, inexact. In the image processing literature, statistical or fuzzy set theoretic approaches have been advocated to deal with noise or uncertainty in images. Several algorithms have been developed to filter out noise and identify boundaries of objects in such unfavorable environments [14]. In this paper, we will be using a fuzzy set theoretic approach to represent objects and their spatial and topological relationships. This will enable us to deal with the semantics of real-life images having objects with noncrisp boundaries. Moreover, object relationships, such as left-of or covered-by, with fuzzy qualifiers, like more-or-less, almost, etc., can also be handled. 2 SPATIAL AND TOPOLOGICAL RELATIONS Spatial relations like left, right, above, and below define the spatial orientations of the domain objects with respect to each other and help in ascribing meaning to a scene. On the other hand, topological relations deal with the nature of overlap between objects and are invariant under rotations of the scene. Egenhofer and Herring [2] have defined interior ðA o Þ and boundary ðAÞ of a region (n-cell) A. The topological relations between two regions (objects) A and B are defined in terms of the intersections (or nonintersections) of the interior and boundary of the regions under consideration. This model is called the 4-intersection model [2] and can be concisely represented by a 2  2 matrix: RðA;BÞ¼ A o \ B o A o \ B A \ B o A \ B   : Egenhofer and Herring used this approach to represent eight basic topological relations among the regions without holes, namely, disjoint, cover, meet, inside, contains, coveredby, overlap, and equal. Some pairs such as cover and coveredby are duals of each other (the same is true for inside and contains). Further, disjoint, meet, overlap, and equal are symmetric. The four empty/nonempty intersections describe a set of relations that provides complete coverage. These relations are mutually exclusive so that the union (OR) of all specifications is true and the intersection (AND) of any two specified relations is identically false. 3 AFUZZY OBJECT DATA MODEL FOR IMAGES We have already mentioned that, with imprecise or noisy data, it is often not possible to say definitely whether a pixel belongs to an object. Rather, an object may be looked upon as a fuzzy subset of the set of pixels. Accordingly, let O be the set of objects present in a two-dimensional scene. An object A 2 O is treated as a fuzzy subset of pixels of the scene with membership function 1 A . Thus, for a pixel located at a point z in the scene, the possibility of this pixel belonging to the object A is 1 A ðzÞ. In the following, we use the notations z X and z Y to denote the X and Y coordinates of a pixel z. Let z þ denote the set of 8-neighbors of a pixel at point z. Then, the boundary (A) of A may also be defined as a fuzzy subset with membership function 1 A ðzÞ¼ minf1 A ðzÞ;max u2zþ f1  1 A ðuÞgg::::::::ðaÞ: Following this approach, the definition of membership function for the interior A o of an object A is given by 1 A o ðzÞ¼ minf1 A ðzÞ; 1  1 A ðzÞg::::::::ðbÞ: In the proposed framework, the spatial and topological relations among the objects will also be imprecise and will be treated as fuzzy subsets. 3.1 Fuzzy Spatial Relations The spatial relations left  of , right  of , above, or below between a given pixel and an object would be imprecise and will be treated as fuzzy subsets. The corresponding relations between any two given 1186 IEEE TRANSACTIONS ON KNOWLEDGE AND DATA ENGINEERING, VOL. 14, NO. 5, SEPTEMBER/OCTOBER 2002 . A.K. Majumdar is with the Department of Computer Science and Engineering, Indian Institute of Technology, Kharagpur, West Bengal, 721 302, India. E-mail: akmj@cse.iitkgp.ernet.in. . I. Bhattacharya is with the Department of Computer Science, University of Maryland, College Park, MD 20742. E-mail: indrajit@cs.umd.edu. . A.K. Saha is with Rice University, 6100 Main, Houston, TX 77005. E-mail: amsaha@rice.edu. Manuscript received 1 Dec. 1999; revised 9 Jan. 2001; accepted 29 May 2001. For information on obtaining reprints of this article, please send e-mail to: tkde@computer.org, and reference IEEECS Log Number 111010. 1041-4347/02/$17.00 ß 2002 IEEE