IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART B: CYBERNETICS, VOL. 35, NO. 4, AUGUST 2005 801 Euler Vector for Search and Retrieval of Gray-Tone Images Arijit Bishnu, Bhargab B. Bhattacharya, Malay K. Kundu, C. A. Murthy, and Tinku Acharya, Senior Member, IEEE Abstract—A new combinatorial characterization of a gray-tone image called Euler Vector is proposed. The Euler number of a binary image is a well-known topological feature, which remains invariant under translation, rotation, scaling, and rubber-sheet transformation of the image. The Euler vector comprises a 4-tuple, where each element is an integer representing the Euler number of the partial binary image formed by the gray-code representation of the four most significant bit planes of the gray-tone image. Computation of Euler vector requires only integer and Boolean operations. The Euler vector is experimentally observed to be ro- bust against noise and compression. For efficient image indexing, storage and retrieval from an image database using this vector, a bucket searching technique based on a simple modification of Kd-tree, is employed successfully. The Euler vector can also be used to perform an efficient four-dimensional range query. The set of retrieved images are finally ranked on the basis of Mahalanobis distance measure. Experiments are performed on the COIL database and results are reported. The retrieval success can be improved significantly by augmentiong the Euler vector by a few additional simple shape features. Since Euler vector can be computed very fast, the proposed technique is likely to find many applications to content-based image retrieval. Index Terms—Content-based image retrieval (CBIR), Euler number, feature extraction, Mahalanobis distance, range query. I. INTRODUCTION A COMPACT and easily computable image feature is highly desirable for efficient management of image database, search and retrieval. The characteristic parameters of the image should also preferably remain invariant to various transfor- mations, such as translation, rotation, scaling, rubber-sheet shearing, degradation by noise, compression, etc. Existing methods of feature extraction usually employ either geometric features of an object or luminance signature of an object [35], [41]. A. Geometric Features Various elementary geometric shapes are used to provide characteristic features such as edge, corner, line, curve, hole, and boundary curvature to define individual features of an Manuscript received June 8, 2004; revised November 3, 2004. This work was supported by a grant from Intel Corporation. This paper was presented in part at the International Conference on Information Technology: Coding and Com- puting, Las Vegas, NV, April 2002. This paper was recommended by Associate Editor S. Sarkar. A. Bishnu is with the Japan Advanced Institute of Science and Technology, Ishikawa, Nomi-gun 9231292, Japan. B. B. Bhattacharya, M. K. Kundu, and C. A. Murthy are with the Indian Sta- tistical Institute, Kolkata 700 108, India (e-mail: bhargab@isical.ac.in). T. Acharya is with Avisere, Inc., Chandler, AZ 85226 USA. Digital Object Identifier 10.1109/TSMCB.2005.846642 image [5], [9]. Several transformations e.g., medial axis trans- formation, morphological transforms, may be applied for analyzing the structure of the shape patterns. These geometric features can be roughly categorized into three types [9]. 1) Global parameters are extracted from the geometric and topological parameters of the entire image, like perimeter, area, centroid, curvature, Euler number, contour points, convex hull, etc. Geometric features calculated from moments, like center of mass, orientation, bounding rectangle, etc., are also used. A combination of global features using Euler number, convex hull and its deficien- cies has been in use lately [38]. 2) Structural parameters are the features which are local in nature, each describing a portion of the object. Features like line segment, arc segment with constant curvature, corner specifications, etc., that define pieces of an object’s boundary are widely in use. 3) Relational parameters represent geometrical relations among local features using graph representations. Dis- tance and relative orientation of substructures and regions of an object are interrelated using mainly graph-based methods. B. Luminance Features In this category, features are based on the luminance informa- tion represented by the intensity values [16]. 1) Spatial features are characterized by the gray-levels or colors and their distributions like amplitude and his- tograms [28], [39]. 2) Transform features provide the frequency domain infor- mation of the image, obtained by zonal filtering in the se- lected transform space e.g., Fourier descriptor, DCT, with applications to shape analysis [20], [29], [44]. 3) Edges and boundaries characterizing object boundaries and shape may be extracted using gradient operators. Boundaries are extracted by edge linking techniques like contour following, edge linking, and heuristic graph searching [16]. 4) Invariant moments [16] are used for shape and scene matching applications [15], [31], [40]. Zernike moments provide features for rotation-invariant image recognition [21]. 5) Texture features are mostly based on structural, statistical, or spectral properties. There are several methods for tex- ture extraction using gray-level co-occurrence statistics [13], Gabor filters [17], windowed Fourier filters [2], as- sociation rules [34]. 1083-4419/$20.00 © 2005 IEEE