Alpha-Divergence for Image Indexing and Retrieval

Motivated by Chernoff’s bound on asymptotic probability of error we propose the alpha-divergence measure and a surrogate, the alpha-Jensen difference, for indexing and retrieval in image and other databases. The alpha-divergence, also known as Renyi divergence, is a generalization of the Kullback-Liebler divergence and the Hellinger/Battacharya distance between the probability density characterizing image features of the query and the density characterizing features of candidates in the database. As in any divergence-based classification problem, the alpha-divergence must be estimated from the query or reference object and the objects in the database. The surrogate for the alpha-divergence, called the alpha-Jensen difference, can be simply estimated using non-parametric estimation of the joint alpha-entropy of the merged pairs of feature vectors. Two methods of alpha-entropy estimation are investigated: (1) indirect methods based on parametric or non-parametric density estimation over feature space; and (2) direct methods based on combinatorial optimization of minimal spanning trees or other continuous quasi-additive graphs over feature space. We analyze convergence rates and establish that the bias convergence rates of the MST entropy estimator can be better than that of an indirect estimator implemented with minimax adaptive kernel density estimation. We illustrate the MST estimator for geo-registration of images. 1 Indexing and Retrieval A database of images X = fXigi=1 is queried for content which is closely related to a reference image X0. The answer to the query is a partial re-indexing of the database in decreasing order of similarity to the reference image using an index function. This content-based retrieval problem arises in geographical information systems, digital libraries , medical information processing, video indexing, multi-sensor fusion, and multimedia information retrieval [1, 2, 3, 4]. Common methods for image indexing and retrieval are color histogram matching and texture matching using cross correlation. While these methods are computationally simple they often lack accuracy and discriminatory power. There are three key ingredients to image retrieval and indexing which impact the accuracy and computation efficiency: 1. selection of image features which discriminate between different image classes yet posess invariances to unimportant attributes of the images, e.g. rigid translation, rotation and scale; 0Alfred Hero heroeecs.umich.edu is with the Departments of Electrical Engineering and Computer Science (EECS), Biomedical Engineering, and Statistics at the University of Michigan Ann Arbor, MI 48109-2122. Bing Ma was with the Dept. of EECS at UM and is now with Intervideo, Inc., Mountain View, CA. Olivier Michel omichelunice.fr is with the Department of Astrophysics, University of Nice, France.