Indexing through laplacian spectra M. Fatih Demirci a,b, * , Reinier H. van Leuken a , Remco C. Veltkamp a a Institute for Information and Computing Sciences, Utrecht University, 3584CH Utrecht, The Netherlands b Department of Computer Engineering, TOBB University of Economics and Technology, Ankara, Turkey Received 31 October 2006; revised 3 May 2007; accepted 5 September 2007 Abstract With ever growing databases containing multimedia data, indexing has become a necessity to avoid a linear search. We propose a novel technique for indexing multimedia databases in which entries can be represented as graph structures. In our method, the topolog- ical structure of a graph as well as that of its subgraphs are represented as vectors whose components correspond to the sorted laplacian eigenvalues of the graph or subgraphs. Given the laplacian spectrum of graph G, we draw from recently developed techniques in the field of spectral integral variation to generate the laplacian spectrum of graph G þ e without computing its eigendecomposition, where G þ e is a graph obtained by adding edge e to graph G. This process improves the performance of the system for generating the subgraph sig- natures for 1.8% and 6.5% in datasets of size 420 and 1400, respectively. By doing a nearest neighbor search around the query spectra, similar but not necessarily isomorphic graphs are retrieved. Given a query graph, a voting schema ranks database graphs into an index- ing hypothesis to which a final matching process can be applied. The novelties of the proposed method come from the powerful repre- sentation of the graph topology and successfully adopting the concept of spectral integral variation in an indexing algorithm. To examine the fitness of the new indexing framework, we have performed a number of experiments using an extensive set of recognition trials in the domain of 2D and 3D object recognition. The experiments, including a comparison with a competing indexing method using two dif- ferent graph-based object representations, demonstrate both the robustness and efficacy of the overall approach. Ó 2007 Elsevier Inc. All rights reserved. Keywords: Spectral graph theory; Indexing; Laplacian spectrum; Spectral integral variation; Information retrieval 1. Introduction Shape matching is one of the fundamental problems in computer vision. In a typical matching problem the objec- tive is to compute an overall similarity value between an unknown shape (query) and a model, and to find the corre- spondences between their feature sets. The similarity value between two shapes can be used for shape recognition by using stored exemplars for different shape classes as mod- els. A linear search of a database, i.e., computing the sim- ilarity between the query and each database entry and selecting the closest one, is inefficient for large database systems. Therefore, an effective and efficient indexing mechanism is essential to select a small collection of candi- dates to which the actual matching process is applied. Criminology, medicine, trademark retrieval, and content- based image retrieval on the web are only a few examples of applications which are likely to contain large collections. For recognition purposes, it is very common to represent object views by graphs whose nodes correspond to image fea- tures and whose edges indicate relations between these fea- tures, e.g., [46,28,51,26,40]. Both nodes and edges may be labeled by attributes. These graph representations express many significant object properties such as geometric or hier- archical structures. Such representations, however, have drawbacks: matching two graphs is a difficult problem. Graph matching problems are often formulated as larg- est isomorphic subgraph problems, for which a rich body of research exists in the literature, such as pattern recogni- tion [30,29], chemical structures [37], or computer vision 1077-3142/$ - see front matter Ó 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.cviu.2007.09.012 * Corresponding author. Address: Department of Computer Engineer- ing, TOBB University of Economics and Technology, Ankara, Turkey. Fax: +90 312 292 4180. E-mail addresses: mfdemirci@etu.edu.tr (M.F. Demirci), renier@cs. uu.nl (R.H. van Leuken), remco.veltkamp@cs.uu.nl (R.C. Veltkamp). www.elsevier.com/locate/cviu Available online at www.sciencedirect.com Computer Vision and Image Understanding xxx (2008) xxx–xxx ARTICLE IN PRESS Please cite this article in press as: M.F. Demirci et al., Indexing through laplacian spectra, Comput. Vis. Image Understand. (2008), doi:10.1016/j.cviu.2007.09.012