Embedded lattices tree: An efficient indexing scheme for content based retrieval on image databases

Abstract

One of the challenges in the development of a content-based multimedia indexing and retrieval application is to achieve an efficient indexing scheme. To retrieve a particular image from a large scale image database, users can be frustrated by the long query times. Conventional indexing structures cannot usually cope with the presence of a large amount of feature vectors in high-dimensional space. This paper addresses such problems and presents a novel indexing technique, the embedded lattices tree, which is designed to bring an effective solution especially for realizing the trade off between the retrieval speed up and precision.

The embedded lattices tree is based on a lattice vector quantization algorithm that divides the feature vectors progressively into smaller partitions using a finer scaling factor. The efficiency of the similarity queries is significantly improved by using the hierarchy and the good algebraic and geometric properties of the lattice. Furthermore, the dimensionality reduction that we perform on the feature vectors, translating from an upper level to a lower one of the embedded tree, reduces the complexity of measuring similarity between feature vectors. In addition, it enhances the performance on nearest neighbor queries especially for high dimensions. Our experimental results show that the retrieval speed is significantly improved and the indexing structure shows no sign of degradations when the database size is increased.

Introduction

The development of internet and multimedia devices have caused a rapid increase in the size of digital information that is used, and stored via several applications. In order to overcome such problems, efficient image retrieval tools are required in various domain including medicine, publishing, architecture, etc. For this purpose, many content based image retrieval systems (CBIR) [1], [2], [4], [3] have been developed. These systems have a common characteristics: the images are represented as vectors of d numeric features and similarity retrieval are performed by conducting nearest neighbor queries in the feature space. To apply the content based image retrieval to large size image databases, it is needed to develop multidimensional indexing structures efficiently supporting nearest neighbor retrieval. A straightforward way to perform similarity matching is the sequential search algorithm (SSA) where every feature vector in the database is scanned to find if it satisfies the query requirement or not. However, SSA can be very costly since the running time of SSA is proportional to the feature space dimension and the database size. There are many methods to organize the feature vectors of images in the database such that a ranked list of nearest neighbors can be retrieved without performing an exhaustive comparison with all the database image feature vectors. Actually, many researches are done on indexing high-dimensional vectors focused mainly on the vector quantization approach and the multidimensional indexing approach.

We can divide the multidimensional indexing techniques into two categories, space-partitioning methods and data partitioning methods. These techniques are formed mostly in a hierarchical tree structure used to divide the data space progressively into smaller partition. The main difference between the two categories is in the way the partitioning is performed.

In the first category, the kd-tree [5] divides the feature space into predefined hyper-planes regardless of the feature vectors distribution. Such regions are disjoint and their union covers the entire space. The major inconvenient of the kd-tree is that it identifies the position of the feature vector in the data space using, in each level of the tree, one coordinate at one time. Besides, the rigid partitioning of the space, can lead to the consultation of few populated or empty clusters especially in high dimensions.

In the second category, the most popular data partitioning techniques include the R-tree [7], SS-tree [8] and SR-tree [9]. Unlike the kd-tree structure, the feature space is divided according to the database items distribution. The R-tree applies hyper-rectangles, represented as nodes in the tree, to divide the space. Children of a node then divide the space inside the hyper-rectangle with smaller hyper-rectangles. It is known in previous works [5] that the R-trees outperform the kd-trees in high dimensional nearest neighbor queries. However, the most serious problem of the R-tree is that bounding rectangles can overlap in higher dimensions. In order to prevent this, White and Jain proposed the SS-tree [6], an alternative to R-tree structure, which uses minimum bounding spheres instead of rectangles. Even though SS-tree outperforms R-tree, the overlapping in the high dimensions still occurs. The SR-tree is an improvement of the R-tree and SS-tree. One of its most important features is that it employs the intersection of hyper-spheres and hyper-rectangles to determine the shape of a partition.

Generally, the main drawback of the multidimensional indexing techniques is that they do not scale up well to high-dimensional spaces due to the phenomenon called “the curse of dimensionality”. Furthermore, the number of partitions increases exponentially with the dimensionality. As it is reported in [10], these multidimensional indexing structures are mostly useful for medium dimensional feature spaces. To avoid the curse dimensionality problem, the principal component analysis (PCA) [11] and the latent semantic indexing (LSI) method [12], inspired from the text retrieval field and adapted to the image retrieval, are used to reduce the dimensionality of feature vectors. Lower dimensional transformed feature vectors are used to approximate the original feature vectors. However, the major drawback of these methods is the risk of the pertinent information loss in the feature space resulting from the reduction of the dimensionality. An other multidimensional indexing structures shortcoming is that the dissimilarity distance between two points has to be based on a distance computation which is costly and CPU intensive especially for high dimensional data spaces. Moreover, the performance of a nearest neighbor query algorithm degrades if a query point is located near a partition border because there is two decisions to take. The first is to ignore the neighboring partitions decreasing the retrieval precision. The second is to take into account the neighboring partitions resulting in increasing computational requirements. In addition to the multidimensional indexing technique inconveniences stated above, one of their major limit is the incremental construction of the indexing tree that could lead, depending on the order of the objects insertion, to significantly varying performances during the indexing phase.

As an alternative to the multidimensional techniques, many vector quantization schemes have been proposed in order to produce an image index. Existing solution which improve over standard k-means include the use of mean shift based clustering [13], hierarchical k-means [14], agglomerative clustering [15], randomized trees [16] and self-organizing maps (SOM) [17]. Indeed, these vector quantization algorithm provide a partitioning of the data into groups or clusters so that data items into a certain group are more similar to each other than to data items in other groups. Each group is then represented by its centroid or sometimes a single representative data item. Thus, instead of the original data items, the query point is compared to the centroids or the group representatives. The best group or groups, according to the used similarity measure, are then selected and the data items belonging to those groups are evaluated to return the k nearest neighbors. These based clustering algorithms suffer from the expensive distance computation especially if the dimensionality of the input vectors is high. Moreover, many of these clustering algorithms assumed that the number of clusters, the initial partitions and the learning weights were known prior to clustering, but this is rarely the case, especially in the indexing framework. Also, except the SOM algorithm, the majority of the data clustering algorithms does not preserve the topological ordering of the data space.

Taking into account the drawbacks cited above, and with an aim to accelerate nearest neighbors search, we propose an accurate and fast solution based on the lattice vector quantization and which differs significantly from existing solutions, as it does not rely on clustering techniques. Due to the good geometric and algebraic properties of the lattice ${\mathbb{Z}}^n$ [18], [19], the proposed indexing structure does not need learning parameters initialization and expensive on-line distance computation. Besides, it allows us to explore efficiently feature space and neighborhood relations between feature vectors preserving the topological ordering. By using the proposed indexing structure, the overlapping is avoided and the quality of clusters is preserved. Moreover, the hierarchy structure of the proposed indexing tree and the dimensionality reduction proposed method allow us to obtain an efficient partitioning of the feature space.

The paper is organized as follows. In Section 2, we present the adapted method to extract a fuzzy feature vector that describes the visual content of the database images. The extracted fuzzy feature vector is based on local descriptors of the prominent objects. Besides, the competitive agglomeration algorithm (CA) [20] is applied in order to group the obtained local descriptors into categories of visually similar regions. In Section 3, we outline the proposed algorithm used to index the obtained fuzzy feature vector. We present our proposed hierarchical indexing method based on the regular lattice in Section 4. In Section 5, we describe the proposed method to build the proposed hierarchical embedded lattices tree. Some experimental results are presented in Section 6 with the aim to illustrate the effectiveness of the proposed indexing method comparing it to SR-tree [9] and SOM [17].