B Tree: an Efficient Index Structure for Fast Search

International Review on Computers and Software (I.RE.CO.S.), Vol. 11, N. 10 ISSN 1828-6003 October 2016

BLBTree: an Efficient Index Structure for Fast Search Abdelkhalak Bahri1, Hamid Zouaki1, Rachid Oulad Haj Thami2 Abstract – Due of its interest in different areas, similarity search in high-dimensional spaces is one of the principal research axes today for CBIR Systems. The properties for high-dimensional data demand some adequate search methods to have a research in optimal time. Due to the curse of dimensionality, search time in the index structure exponentially increases, according the dimension of the descriptor. However, the performances of classical index structures become less than the sequential scan of data in search time when answering exact nearest neighbor queries. To overcome this problematic, two major approaches for high-dimensional data in CBIR systems have been proposed: The first approach permits to speed up the search by using multiple levels of lower bound distances, the second approach exploits the BTree index structure to speed up the search. We propose to combine both techniques to search for large nearest neighbors in a highdimensional space. We develop a new multidimensional index structure, called BLBTree. In BLBTree, instead of computing the distances in the high-dimensional original space to find the nearest neighbor, lot of candidates are to be rejected based on the lower bound distances: the research in BLBTree does not calculate the distances in the high-dimensional original space to find the nearest neighbor if the lower bounding distance exceeds the threshold, because the distances in the high-dimensional cannot be lower than the distances in the low-dimensional. We note that each object in BLBTree is described by a pyramid structure of histograms, with each histogram representing the lower resolution of the other histogram; due to this property, our index structure is well-suited for high-dimensional and large-scale problems. We have shown that our approach provides interesting and powerful experimental results. Copyright © 2016 Praise Worthy Prize S.r.l. - All rights reserved.

Keywords: Bag of Visual Word, High-Dimensional Space, B-Tree, Va-File, Idistance, Pcdistance, EBVW, Blbtree

The indexing technique permits to assign for each image a code to identify it in the database. However, the first approach has several limitations. It requires an effort in input and manual maintenance annotations that become expensive. In some cases, several solutions have been proposed to index data automatically. Thus, on a website, it is conceivable to describe an image from extracts keywords in the text boxes located around the image. In addition, there is not any guarantee on the line with the content of the image. Thus, with this approach an expert of database should know the keywords that can be used to index and retrieve effectively images. Despite the drawbacks of this approach, searching by keywords is much faster. On the other hand, image has a rich content, which often serves to illustrate a talk; more than this, a share of emotions is better than using many keywords. This is why the media and encyclopedias use images to describe objects, people, events, etc. The problems faced by text based retrieval for image search have been overcome by CBIR (Content Based Image Retrieval) and the solutions were given by many researchers in various ways. Boomilingam T. et al. [1] have presented review on CBIR Trends and Techniques to Upgrade Image

Nomenclature dbli(Q,P) dist(Qhi ,Phi) dist(Q , P)

hi Phi

ith Lower bound distance (Euclidean distance between Qhi and Phi) Euclidean distance between Qhi and Phi Euclidean distance between Qh4 and Ph4 (i.e. in this paper we have four resolutions, we notate also Ph4 = Phi) Histogram corresponds to ith resolution (see section 3) ith histogram of object P

I.

Introduction

Today with the development of multimedia systems, visual content is becoming the most support of communication used in different areas. Information research covers the digital document processing involving the structure, analysis, storage and data organization. Among the research techniques, there are two main approaches: the textual approach (keyword search) and the numerical approach (search by content). Both techniques require achieving a preprocessing consisting in index database. Copyright © 2016 Praise Worthy Prize S.r.l. - All rights reserved

DOI: 10.15866/irecos.v11i10.9711

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Retrieval. In this paper, we focus on indexing images by their visual content. The visual characteristics are various. Smeulders et al.[2] propose a state of the art of the main descriptors used for CBIR systems. These descriptors are grouped according to whether they are more interested in a color [3], [4], a texture [5] or a shape. The most famous shape descriptors is the SIFT [6]. Many articles have been proposed to improve the performances of SIFT by modifying some points of the algorithm which include the SIFT color detectors [7], [8], PCA-SIFT [9] and Gloh-SIFT[10]. Another shape descriptor based on the same approach as SIFT is the SURF [11] descriptor. After extracting the visual characteristics of the image, they have to be processed to define the signature or the index of the image. Then, each image can be represented by a vector, or by a set of vectors as in the case of SIFT and SURF descriptors. In this case, the search time is especially considered in the case of research in large image databases. To cure this problem, the use of suitable index structures is strongly desired. We briefly review the index structures that are widely used to process the nearest neighbor queries in multimedia databases in the next section. On the other hand, as we have noted above, searching by keywords is faster. With an analogy between the textual description and by visual content, the image is composed of visual words, and it can be represented as a bag of visual words. Thanks to its efficiency in the field of image classification and annotation, the model bag of visual words has become popular in recent years [12]-[16]. This visual content synthesis was presented firstly by Sivic and Zisserman [16] in the case of the videos search; and Csurka and their colleagues [17] in the image classification. Thus, for each image database, it is possible to calculate a histogram on this common reference. In this context, the image is represented by a single vector instead of multiple vectors (several SIFT descriptors). Based on this approach, we firstly presents a multi-resolution BOF (Bag of Feature) to describe an image database. Secondly, we present a new, efficient index structure for fast search in high dimensional space called BLBTree. The rest of this paper is organized as follows: Section 2 describes the related work. Section 3 presents image description using multi-resolution approach. In section 4, we present the details of our index structure BLBTree. Section 5 evaluates the performances of BLBTree experimentally. Finally, Section 6 concludes the paper.

II.

unordered set of vocabulary words. Computer vision researchers have developed a similar idea for image representation by visual content. That is to say, an image can be processed as a text document. The definition of vocabulary words is a little more complex than in the case of natural language, it involves different stages: detection of characteristics, description of these characteristics and creation of visual vocabularies. The visual dictionary (set of visual vocabularies) is usually determined by vector quantization of visual characteristics of the entire database [18]. The quantization step uses a clustering algorithm. The most famous quantization is k-means clustering method [19]. However, using k-means as an algorithm to create the visual dictionary, which takes a long time, especially in the case where the number of quantized vectors is large. Several works are proposed in the literature to overcome this problem as it is cited in [20]-[26], [42]-[45]. A recent work that we have proposed in [27], allow reducing the time of visual dictionary construction. In this paper, we focus our contributions on this method by describing an image through several histograms using multi-resolution approach and BTree structure to create an index structure called BLBTree to make the search much faster in high dimensional spaces. With an increase in the size of image databases, similarity search in a large image collection has become a crucial problem for similarity search systems. The use of index structures is strongly requested. A large number of index structures has been proposed in the literature to perform fast researches. The type of index structure considered depends on the type of visual features extracted from images [46]-[50]. Conventional methods for multidimensional indexing have been proposed to organize feature vectors of images to avoid sequential scan for a large database, allowing an access and a fast search when searching by visual content. In this context, the major problem of multidimensional indexing methods is how to index effectively a large multidimensional data collection to respond quickly and effectively to user queries. A first classification of multidimensional indexing methods is performed by Gaede [28]. Indeed, Gaede has classified these Methods according types of data that they support. In addition to that, numerous other indexing techniques proposed in the literature are considered less effective in information retrieval systems especially for multimedia data. The large dimensions of the space descended from many descriptors give back an actual problematic to index multimedia data. This problem is known as "the curse of dimensionality [29]." The indexing methods that have shown their limits facing this dimensionality problem must take into account a number of parameters, namely the volume of data, the nature of data, data distribution and ensure a balance between indexing quality and the response time for data access. Several studies, such as those of Weber et al. [30] and the Berrani thesis [31] studied the difficulties encountered when using multidimensional data.

Related Work

The visual vocabulary method in computer vision comes from a method called bag-of-words (BoW) developed automatically through natural language processing. In this context, the BoW model provides a representation of a text document. This method creates a vocabulary, in which each document is represented as an

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Indexing techniques are adapted to vectors of small dimensions, well, in real applications the data is usually represented by a number of features which make the use of the most difficult multi-dimensional index. To surpass the problems associated with a large, highdimensional data, approximate research methods (LSH)[32] and those of one-dimensional transformation are proposed in [33]-[35]. One recent index structure for the latter type is introduced by J. Cui et al. called pcDistance [35] which is a 1-dimensional mapping method. PcDistance [35] is a method that is similar to the iDistance [34] method. It maps d-dimensional points to a 1-dimensional value. In the iDistance [34], the dataset is partitioned and a reference point for each partition which is defined. The reference points are used to calculate the index key for the B+Tree. The index key for pcDistance [35] and iDistance[34] is calculated in the same way. In pcDistance [35], data partitions are found in the data set and Principal Component Analysis [36] is applied. In this paper, we considered the one-dimensional index structures type for their interests as in the case of indexing and searching in large database.

Fig. 1. Schema of EBVW: Points embedded in Euclidean space

III. Image Description Using MultiResolution Approach As we have previously reported, the description by bag visual words approach is more efficient in the levels of memory usage and search speed. In this paper, we will adapt the Multi-resolution approach for image description. In this context, we use a Pyramid Structure for image features (histograms) using a EBVW method that we have introduced in [27]. The schema of this method is presented in Fig. 1. The purpose of describing image by more than one signature (histogram) is to exploit these signatures in the research step to make it much faster. The principle of this description is based on the embedding method that is introduced in [16]. This principle is described as follows: Given a set of descriptor vectors, the first step that determines the diagonal for each dimension of all vectors in the database, and the number of resolutions (correspond to the number of dictionaries) that we want to create. The division is done according the median for each dimension. For the first resolution, each dimension is divided into two segments according to the median. In this resolution, we have 2d cells, where d represents the size of image feature (histogram). The average of elements in each cell represents a visual word. Then, for the first resolution we have a dictionary of 2d visual words. This process is repeated recursively until the end of each cell in the next resolutions. i.e for the ith resolution we have a dictionary of 2 i×d visual words. In the end, each image of the database is described according to the set of visual dictionaries that we have obtained. I.e each image is represented by a Pyramid structure features (histograms). An example of this technique is shown in Fig. 2.

Fig. 2. Image description using EBVW method and multi-resolution approach

This example shows a set of two-dimensional points: A, B,..., and Q. After embedding them in a Euclidean space, we have the following presentation: In this scheme, each dimension is divided into 22i segments, where i represents the ith resolution. The resulting dictionary consists of 12 words (empty cells are ignored) for the 2th resolution. The vocabulary word is built for each resolution. A word is represented by the average points for the corresponding cell. In the case where each dimension is divided into 21 (first resolution) segments (blue line), we have a dictionary of four words. Consider an object "O" described by the points (A, B, C, F, H, Q). The representation of this "O" in the dictionary of four words (first resolution) is: h1 = (3, 1, 0, 2), and this "O" is represented in the twelve (second resolution) words by h2 = (1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 2) (see Fig. 3). We notate that h2 is the lower resolution of h1. In this example, if we consider just two resolutions, the object "O" is described by a pyramid structure of two histograms: P = {h1, h2}. In this paper, we consider four resolutions for image description, i.e. Pyramid Structure for image features will be in the form of four histograms. The reason why we have considered this approach (Pyramid Structure for image features (or histograms)) in this paper is to exploit it for creating a new index structure called BLBTree in order to improve the search time and to make the research more faster. Details of this structure are presented in the next section.

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International Review on Computers and Software, Vol. 11, N. 10

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m

x1 i  

m

 y1 i  j 

and x2 i  

 y2 i  j 

j 1

j 1

n

dist  x1 ,x2  

  x1 i   x2 i 

2

i 1

    i 1  n

 y1 i  j   y2 i  j    j 1 j 1  m

m

 n



2



m

  y1 i  j   y2 i  j 

2

i 1 j 1

 dist  y1 , y2  So dist(x1, x2) represents the lower bound of dist (y1, y2). Based on the lemma 1, and the BTree structure [38], we have proposed a new multidimensional index structure called BLBTree for Binary Lower Bound Tree. The BTree structure adapted to store the data based on their keys. In our proposed structure, each image is represented by a few multidimensional vectors. The key of each image is calculated by a method that we have proposed in the definition 1. We note, the images that belong to the same category (i.e. that contain the same visual words) have the same key. Definition 1: Let P be an m-dimensional (size of visual vocabulary) vector descriptor for an image. The image key is defined by: Fig. 3. The ”O” description in the first and second dictionary using EBVW method and multi-resolution approach

IV.

m

key  P  

 P  i   2i i 1

The BLBTree Index Structure

where P[i]=1 if P[i] is not null, 0 otherwise. For example: let P = [0, 5, 0, 12] be a 4-dimensional vector, it key is:

This section presents the BLBTree index structure. This one uses the lower bound distances that we have described in lemma 1. The goal of BLBTree is to make the search in high dimensional space and in a large database quickly. The principal idea of The BLBTree is to skip the distance calculations between an image query and image databases that they have a small probability to be similar. This skipping is based on lower bound distances technique. Lemmas 2 and 3 describe this one.

key (P) = 0×20 + 1×21 + 0×22 + 1×23 = 10 The range key for m-dimensional vectors varies between 0 and 2m. So, the use of a hash table as an index structure is inefficient because it needs to reserve a big space memory for a small number of vectors. Further a big space memory reserved will be not used. In this paper, we propose to exploit the BTree [38] structure to create a new index structure called BLBTree to index a set of images. We have chosen to use this structure, because it's more adapted for this type of keys and their search complexity is O(log(n)), where n represents the database size.

Lemma 1: lower bound distance Given two objects (images) O1 and O2 represented by two Pyramid structures (pyramid of histograms as they are explained in section 3) PO1 and PO2 respectively, with PO1 = {x1, y1} and PO2 = {x2, y2} where y1 and y2 are the lower resolutions of x1 and x2 respectively. A lower bound on the squared Euclidean distance between y1 and y2 denoted by dist(y1,y2) can be represented as follows: dist(x1, x2) ≤ dist(y1, y2)

IV.1. Structure Node in BLBTREE Each image indexed in BLBTree is described by a pyramid of histograms (see section 3), such that this pyramid includes four histograms. The basic histogram in this pyramid is the most accurate signature for an image. The entries for each node in BLBTree contain the

Proof: Let y1 and y2 be two n-dimensional vectors that represent the lower resolutions of two P m-dimensional vectors x1 and x2 respectively, i.e.:

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following fields: - n: represents the number of stored keys, - Key []: array of n lists of image rows in ascending order of their keys. The images that have the same key are stored in the same list. - Leaf: Boolean value indicating whether the node is a leaf or not. - Child []: represents an array of n + 1 pointers to the node's children when it's not a leaf.

similarity queries. During query processing, the key of query object is firstly compared with the key of hierarchical nodes in the BLBTree. The search is then directed to the partition that has the same key to the query object. We compute the distances between the query object and the list of objects in the partition that have the same key of a query object. The algorithm starts by computing the sequences of lower bound distances between query object and all objects candidates before calculating exact distances (see lemma 1, lemma 2 and lemma 3). For clarity, algorithms 2 and 3 illustrate how to answer a range search query with exactly three levels of lower bound distances. It should be obvious how to extend them to work with any number of lower bound distances.

IV.2. BLBTREE Construction Given an object O to be inserted into BLBTree, first, we calculate its key using the formula 1. In the next step, we need to find the node in the tree where its key is equal to the key of O and insert it in the corresponding list in the node. If we don't find any node that has the same key of O in the tree, we insert it in the BLBTree with the same procedure as in the case of the BTree. The pseudo code of the algorithm is presented as follows:

Lemma 2: Basic filtering Let Q and P be two objects (images) represented by two pyramids (pyramid of histograms as it is explained in section 3), like P = {Ph1, Ph2, Ph3, Ph4} and Q = {Qh1, Qh2, Qh3, Qh4} where Phi+1 and Qhi+1 are the lower resolutions for Phi and Qhi respectively and R is a range search. If dist(Qhi, Phi)> R (with i < 4) then the distance dist(Qhi+1,Phi+1) will not be calculated and P is automatically not relevant to Q (see Figs. 4, 5, 6 and 7).

Algorithm 1 (Insert algorithm) InsertBLBTree (BLBTree:T, Object:O) { /*O is described by four histograms h1, h2, h3, h4 with hi+1 is a lower resolution of hi.*/ Key = key (O):/* the key is obtained by using the formulas 1 on the most lower histogram (h4)*/. r = T.root /* root Node of T*/ Node find= serach(r,key) if not find then if r.n = 2k-1 then s =allocateNode() T.root = s s.leaf= false s.n = 0 s.Child[1] = r shareChildBLBTree(s, 1, r) insertBLBTreeIncomplete(s, O) else insertBLBTreeIncomplete(r, O) end if else i = 0; while (i < find.n and key > find.key[i]) do i++; end while if (i < find.n and key = find.key[i]) then Append O in ListeObject[i] of find Node end if end if }

Proof: By means of Lemma 1 we have: dist(Qh1 ,Ph1 ) ≤ dist(Qh2,Ph2) ≤ dist(Qh3,Ph3) ≤ dist(Qh4 ,Ph4 ) So if dist(Qh1 ,Ph1 )> R we have dist(Qh4 ,Ph4 )> R. Lemma 3: inner filtering Let Q and P be two objects (images) represented by two pyramids (pyramid of histograms as it is explained in section 3), like P = {Ph1, Ph2, Ph3, Ph4} and Q = {Qh1, Qh2, Qh3, Qh4} where Phi and Qhi are the lower resolutions of Phi+1 and Qhi+1 respectively and R is a range search. If dist(Qhi ,Phi )≤ R (with i < 4) then dist(Qhi+1,Phi+1) will be calculated and it’s will be compared with R using the procedure of Lemma 2 (see Fig. 8).

The shareChildBLBTree and insertBLBTreeIncomplete functions are defined in the same way as in BTree. IV.3. Similarity Search in BLBTREE Fig. 4. Object P is not relevant to Q. The distances dbl2(Q, P), dbl3(Q, P) and dist(Q, P) will not be calculated since dbl1(Q,P) > R

BLBTree uses early termination strategies to answer Copyright © 2016 Praise Worthy Prize S.r.l. - All rights reserved

International Review on Computers and Software, Vol. 11, N. 10

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A. Bahri, H. Zouaki, R. Oulad Haj Thami

Algorithm 2 (range search algorithm ) rangeSearchBLBTree (Node:N, Query:Q, threshold){ /* Q described by four histograms Qh1 ,Qh2 ,Qh3 ,Qh4 */ key=keyCalcul(Qh4 ) /* by usnig the formula 1*/ i=1 while i ≤ N.n and key > N.key[i] do i= i+1 end while if i ≤N.n and key > N.key[i] then return lowerBoundSimilaritySearch(N, i,Q, threshold) end if if N is a leaf then return NIL else readDisque(N:Child[i]) return rangeSearchBLBTree(N:Child[i], Q, threshold) end if

Fig. 5. Object P considered not relevant to Q. The distances dbl3(Q, P) and dist(Q, P) will not be calculated because dbl2(Q,P) > R.

Algorithm 3 (Similarity search using lower bound distances ) lowerBoundSimilaritySearch(Node:N, i, Query:Q, threshold) { j=0 result=; ; List L= N.ListeObjects[i] /* list of objects that have the key Key[i]*/ ObjetImage obj=L.element(j) /* obj represent the jth object in L */ while obj not nil do dbl1=dist( objh1 ,Qh1 ) if (dbl1 ≤threshold) then dbl2=dist( objh2 ,Qh2 ) if (dbl2 ≤ threshold) then dbl3=dist( objh3 ,Qh3 ) if (dbl3 ≤ threshold) then d=dist( objh4 ,Qh4 ) if (d ≤ threshold) then append obj to result end if end if end if end if j=j+1 obj=L.element(j) end while return result }

Fig. 6. Object P is not relevant to Q. The distance dist(Q, P) will not be calculated since dbl3(Q, P) > R

Fig. 7. Object P is not relevant to Q because dist(Q, P) > R

IV.4. Complexity Analysis

Fig. 8. Object P is relevant to Q, sure enough dist(Q, P) ≤ R

The similarity search is done by computing a similarity score between the query image and all image databases using a similarity function. The complexity of this approach is O(n) where n represents the number of elementary calculations or the number of calculations of similarity scores. It is interesting that n is also the size of the database. So, we estimate the evolution of the search time based on the size of the database.

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International Review on Computers and Software, Vol. 11, N. 10

853

A. Bahri, H. Zouaki, R. Oulad Haj Thami

BLBTree using the lower bound distances technique as shown in search algorithms 2 and 3. Figure 9 presents range query processing on the Wang [39], COIL-100[40] and FLICKR [41] databases. The figure shows the number of distance computations needed to perform range query features. Firstly, we tune the parameters for BLBTree, the range search, and the pyramid of histograms for each object in the databases.

Research is called linear or exhaustive. The aim beyond designing BLBTree is to reduce the search time. Thus, when performing a search using a query image, the calculation of the similarity between the query image and all images in this structure is to be approximated by calculating a subset of images. The index structure selects a pool of candidates, and the system only calculates the similarity between a subset of this pool images and the query image using the lower bound distances. We know that the search complexity of BTree is O(log(n)). So that, we deduce that the search complexity in BLBTree which is in O(log(p)), where p is the number of calculated distances between the object and the selected partition, in which p is more less than n. It is interesting to note that, if the search threshold is small, the number of exact distance calculations is small. And hence, the number p tends toward a small constant. And therefore research in BLBTree becomes more faster.

V.

Experimental Results

For the evaluation of our contributions, we used three datasets: The Wang [39], the COIL-100[40], and the FLICKR database [41]. The Wang dataset [36] consists of 10 classes, and each of them contains 100 color images of 384× 256 pixels. The COIL-100[40] image database of Columbia University, which contains 100 categories, each category contains 72 images of 128×128 pixels. The last set is the FLICKR database [41] which consists of one million images that are extracted from the Flickr website (www.flickr.com). The conducted experiments are designed to compare our index structure BLBTree to other structures of the state of the art. The implementation is done in Java. Performance evaluation is made with a quad core of an Intel processor running at 1.6 GHz PC with 10 GB memory, Windows 7 operating system and 400 GB of local disk. Multidimensional index structures are evaluated on multidimensional vectors. Created data sets are stored in memory to evaluate the performance of index structures using an average of several queries. The set of queries is the same for each index structure. We measure in our experiments the number of calculated distances and total response time for a range search according to two criteria: the size of the database and the size of the multidimensional data. Addition to that, SURF [34] descriptor is firstly used as local features for image databases. A method proposed in section 3 is used to build a pyramid of bag of visual words for each image. The search times of algorithms are measured using the getInstance() and getTimeInMillis() methods of the Calendar class. V.1.

Distance Computations and Response Time in BLBTree

In this experiment, we evaluate the number of distance computations between query object and objects stored in

Fig. 9. Distance computations for 1024-dimensional data in various datasets

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International Review on Computers and Software, Vol. 11, N. 10

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A. Bahri, H. Zouaki, R. Oulad Haj Thami

We set size of data in 1024 (we choose the higher dimension) and we vary the number of lower bounds and range search from 0.1 to 0.9. The choice of high dimensional space in this experiment is designed to show the effectiveness of our method in high dimensional space, it is the reason why our method was developed. Besides, we show the effect of a number of lower bound distances that are used and the range search on the performance of BLBTree. Fig. 9 shows that there is an increasing in the number of distances computed when less of the number of lower bounds are used. This is the expected result because if more lower bound distances are used, it is more set of images will be skipped. Fig. 10 shows the average CPU times (in milliseconds) of our method during the range search in the above databases using several numbers lower bound distances. This figure shows that there is an increasing in CPU times when the less number of lower bounds are used. This result is explained as follows: which speeds up when we use more lower bound distances, we have a less of distance computations and then we have less CPU times. For example, in the range 0.4, by using the first lower bound, speed up the search in the COIL-100 database by 80%. Using both the first and second lower bounds increases the speed to 100%. For that way, we have opted to use three lower bounds in our method to have a compromise between the computation time and memory usage required to achieve a large-scale research. V.2.

evaluations, especially for high-dimensional spaces. From Fig. 12, we notice that the response time in VAFile and iDistance are linearly with descriptors size. Note that response time in iDistance is faster than the one in VAfile, and that of pcDistance is also faster than the one of the iDistance. For BLBTree, response time it more slowly compared to other methods. On the VA-file, it is evident that the iDistance response time and pcDistance are improved by the structuring of strategy adopted by each method.

Search Time in BLBTree, VA-File, iDistance and pcDistance

In this subsection, we compare the search time in our index structure compared to the one in the index structures VA-File [30], iDistance [34] and pcDistance [35] on a FLICKER database (big database). In our experiments, the approximate bit length for VA-File is 6 bits per dimension. For iDistance, we choose 64 reference points. In pcDistance we choose also 64 partitions and 50 max number of childs per node. The experimental results for the 64, 128, 256, and 512-dimensional data are shown in Figs. 11 and 12. They have clearly shown that the BLBTree outperforms the pcDistance, iDistance and VA-File in the whole databases, especially for the big databases. This is the result of the use of lower bound distances and the way of collection objects in our structure. This is owed, as we have already explained it above, in our research strategy which allows to avoid the counting of useless distances by using the counting of lower bound distances. Fact to have a big number of calculated distances, increase the research time in the index structure and then decrease their performances. However, some distance functions are so expensive to compute in terms of CPU time that the overall search time is dominated by the total number of distance

Fig. 10. Range search running time for 1024-dimensional data in various datasets

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Fig. 12. Search time comparison for several size of data.

Indeed, the VA-File approximates the vectors per cell which is an improvement of the sequential search. Search time is brows all files of approximations for associated cells containing the vectors. In the case of iDistance and pcDistance, search time is reduced to the course and exploration of research areas. In our method as we have already presented, using a number of lower bound distances can avoid several distance computations between query images and candidate images in the database, and hence we have a faster research. Therefore, our BLBTree is more suitable for high multi-dimensional data in comparison with the VA-File, iDistance and pcDistance indices.

VI.

Conclusion

As part of an application to set up image search tools in a database, we have proposed a new multidimensional index structure called BLBTree that enables fast high dimensional nearest neighbor search with excellent quality. BLBTree must take into account the fact that the images are described by a pyramid of histograms using the multi-resolutions approach. We take advantage of the benefits of BTree structure that provides excellent performance for the detection of mono-dimensional data. The basic idea of our structure (BLBTree) is to avoid additional computation distances between the query image and the image databases that have a small probability to be similar by using the lower bound distances, and then to reduce the search space concerned by a query image. We have shown by our experimental results that our approach gives good performance compared to other index structures in the state of the art. BLBTree aims to fill such a gap by creating an index structure that fits the data distribution. This strategy allows to minimize the number of distance calculations using lower bound distances to answer similarity queries. These benefits are summed up in reducing the query time. In the feature work, we plan to adapt BLBTree to support the multi-metrics for the nearest neighbor research. In the other future work, we will try to combine BLBTree and the LSH technique to speed up more the search time.

Fig. 11. Search time comparison for several size of data and databases

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Authors’ information 1

LIMA laboratory, Faculty of Sciences Choua¨ıb Doukkali University, El Jadida, Morocco. E-mails: [email protected] [email protected] 2

SI2M Laboratory, ENSIAS University Mohammed V Souissi,Rabat, Morocco. E-mail: [email protected] Abdelkhalak Bahri: I received my Ph.D. degree from the Department of Mathematics and Computer Science at the University Chouaib Doukkali, Faculty of Sciences, ElJadida, Morocco in 2014. My research interests focus on CBIR and multimedia database.

Hamid Zouaki received his PhD. In Mathematics from the institute of Applied Mathematics of Grenoble, France, 1991. He is currently Professor at the department of Mathematics of the University Chouaib Doukkali, El Jadida (Morocco). His research interests are representation tools for shape description, convex geometry, image analysis and optimization.

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