Interactive image retrieval using fuzzy sets

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1 Pattern Recognition Letters ) 1021± Interactive image retrieval using fuzzy sets Hichem Frigui * Department of Electrical and Computer Engineering, The University of Memphis, 206 Engineering Science Bldg, Memphis, TN , USA Received 11 August 2000; received in revised form 29 January 2001 Abstract We present an image retrieval system which permits the user to submit a coarse initial query and continuously re ne it. The user's relevance feedbacks is modeled by fuzzy sets, and is used to discover and use the more discriminatory features for the given query. The proposed system uses a dissimilarity measure based on the fuzzy integral. Ó 2001 Elsevier Science B.V. All rights reserved. Keywords: Content-based image retrieval; Relevance feedback; Similarity measures; Fuzzy integral; Choquet integral; Fuzzy measures 1. Introduction The recent increase in the size of multimedia information repositories has made Content-based image retrieval CBIR) one of the most active research areas in the past few years. Unlike traditional database techniques which retrieve images based on exact matching of keywords, CBIR systems represent the information content of an image by visual features such as color, texture, and shape, and retrieve images based on similarity of features. Therefore, a good similarity or equivalently a dissimilarity) measure is essential for the e ective retrieval in such systems. Unfortunately, the meaning of similarity is rather vague and dif- cult to de ne. In general, di erent similarity measures capture di erent aspects of perceptual similarity between images. What makes the problem even harder is the fact that di erent features * Tel.: ; fax: address: hfrigui@memphis.edu H. Frigui). do not contribute equally, and therefore, cannot be considered equally important for computing the similarity or dissimilarity) between images. For example, when a user perceives two images as being similar, what he or she really means is that the images are similar either in an individual feature, or in some combination of features. In this paper, we present a novel method that models the user's positive and negative feedbacks by fuzzy sets. These fuzzy sets are then used to learn the feature relevance weights. We show that the learned weights can be used as fuzzy densities in a fuzzy integral based retrieval system. We also show that the learned feature relevance values can be used as weights in a weighted Euclidean distance-based retrieval system. The rest of this paper is organized as follows: Section 2 summarizes related work on similarity measures and feature relevance. Section 3 gives an overview of fuzzy integrals and fuzzy measures. Section 4 discusses how to use the Choquet integral as a dissimilarity measure. Section 5 describes /01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S )

2 1022 H. Frigui / Pattern Recognition Letters ) 1021±1031 how to model the user's feedback using fuzzy sets and how to update the feature relevance values. Experimental results over more than 3000 images for testing the learning behavior of the system are given in Section 6. Concluding remarks are given in Section 7. a priori does not utilize the full potential of the fuzzy integral, and does not re ect the user's perception of similarity. Instead of xing the fuzzy measures, our system learns them based on the user's positive and negative feedbacks. 2. Related research 2.1. Similarity measures The retrieval process of images involves evaluating the degree of similarity between the query image feature vector and the feature vectors of the images stored in the database. Several similarity measures have been used in the past few years. The Euclidean or weighted Euclidean) distance has been used in several CBIR systems Flickner et al., 1995; Bach et al., 1998). A hybrid neural network algorithm is used in Ma and Manjunath, 1996) to learn the visual similarity by clustering patterns in the feature space. Other similarity measures that have been suggested include cosine similarity Celentano and Sciascio, 1998), ``Crosstalk'' distance Agnew et al., 1997), ``relational distance measure'' and ``sum of minimum distance'' Tao et al., 1997). Unfortunately, Most of these measures are not always consistent with human perception of visual content, and their performance degrades as the dimensionality of the feature space increases. In Santini and Jain, 1999), the authors study several models that have been proposed in the psychological literature. Recently, the fuzzy integral has been used as a dissimilarity measure in CBIR Santini and Jain, 1999; Popescu and Gader, 1998). The fuzzy integral has also been shown to be a successful tool for evidence fusion, image segmentation, data classi cation, and feature extraction Grabisch et al., 1995; Tahani and Keller, 1990; Grabisch, 2000). The performance of the fuzzy integral is highly dependent on the way the fuzzy measures are speci ed. In most pattern recognition applications, a training data set is usually used to learn these measures Keller et al., 2000). So far, in CBIR, a training algorithm is not available, and the fuzzy measures have been xed heuristically. Unfortunately, xing these measures 2.2. Feature relevance In CBIR, each image is represented by a multidimensional feature vector, and during the retrieval process, the user selects the visual feature s) that he or she is interested in. Moreover, since the importance of each feature depends on the location of the query feature vector in the feature space and the user's preference, the user needs to also specify the weights associated with each feature. Unfortunately, the speci cation of weights requires the user to have a comprehensive knowledge of the feature representation, which is normally not the case. To alleviate this di culty, several interactive mechanisms that involve the user as part of the retrieval process have emerged Bhanu et al., 1998; Rui et al., 1998; Huang and Rui, 1997; Buckley and Salton, 1995; Bouet and Djeraba, 1998; Celentano and Sciascio, 1998). In the above systems, the burden of specifying the weights is removed, as the user needs only to categorize the retrieved images. In Bhanu et al., 1998), the user labels the retrieved images as ``desired'' or ``undesired''. In Rui et al., 1998) the user assigns a score varying from )3 highly non-relevant) to 3 highly relevant) to each retrieved image. The user's feedback is then used to re ne the feature weights for the next iteration of retrieval. Unfortunately, binary labeling, and even multilevel labeling does not re ect the nature of human concepts and thoughts, which tend to be abstract, uncertain, and imprecise. In this paper, we present an approach that uses fuzzy sets to model the user's feedback, and make the transition from ``desired'' to ``undesired'' gradual. This smooth transition will be modeled by continuous fuzzy membership functions. These functions are known to provide exibility in modeling linguistic expressions such as ``this image is very relevant'' or ``this image is more or less relevant''.

3 H. Frigui / Pattern Recognition Letters ) 1021± Fuzzy measures and fuzzy integrals 3.1. Fuzzy measures Let X ˆfx 1 ;...; x n g be an arbitrary set. A set function g : 2 X! 0; 1Š is a fuzzy measure if it satis es the following three axioms: 1. Boundary conditions: g ; ˆ 0; g X ˆ 1, 2. Monotonicity: if A; B X, and A B, then g A 6 g B, 3. Continuity: iffa i g is an increasing subsequence of subsets of X, then lim g A i ˆg [1 A i!: i!1 iˆ1 A fuzzy measure is a Sugeno measure or a k-fuzzy measure) if it satis es the following additional condition for some k > A; B X with A \ B ˆ; g A [ B ˆg A g B kg A g B : 1 For a nite set X, the fuzzy measure of a set A can be expressed by Sugeno, 1977) " # Y g A ˆ1 1 kg i 1 ; k 6ˆ 0; 2 k x i2a where g i ˆ g fx i g. The value g i is called the density of the measure, and is interpreted as the importance of the single information source x i. Using 2) and the fact that g X ˆ1, the value of k can be uniquely determined by solving the equation k 1 ˆYn iˆ Fuzzy integrals 1 kg i : 3 A fuzzy integral is an integral of a real function with respect to a fuzzy measure. The Sugeno integral Sugeno, 1977) and the Choquet integral Murofushi and Sugeno, 1991) are the most common ones. Let X be a set, g a fuzzy measure, and h : X! 0; 1Š be a function where h x denotes the con dence value of x, and let A a ˆfxjh x P ag Sugeno integral The Sugeno integral of h with respect to the fuzzy measure g can be de ned as Z S g h ˆ x h x g ˆ sup min a; g A \ A a Š: a2 0;1Š If X is a discrete set, the Sugeno integral can be computed as follows: n S g h ˆmax min h x i ; g A i Š; iˆ1 4 where h x 1 P h x 2 P P h x n, and A i ˆ fx 1 ;...; x i g. The Sugeno integral is not a proper extension of the Lebesgue integral. The Choquet integral was developed to overcome this drawback Choquet integral The Choquet integral of h with respect to the fuzzy measure g can be de ned as Z C g h ˆ x h x g ˆ Z 1 0 g A a da: If X is a discrete set, the Choquet integral can be computed as follows: C g h ˆXn iˆ1 h x i h x i 1 Šg A i ; 5 where h x 1 6 h x h x n, h x 0 ˆ0, and A i ˆfx i ;...; x n g. The performance of the fuzzy integral is heavily dependent on the fuzzy measures. Both the Sugeno and the Choquet integrals have been used in several applications with comparable performance. In fact, it has been shown that both integrals share many important properties Murofushi and Sugeno, 1991; Grabisch and Nicolas, 1994). For a more extensive theoretical background on fuzzy measures and fuzzy integrals, the reader can refer to Grabisch et al., 1995; Wang and Klir, 1992; Murofushi and Sugeno, 1991; Grabisch et al., 1992).

4 1024 H. Frigui / Pattern Recognition Letters ) 1021± The Choquet integral as a dissimilarity measure The fuzzy integral can be interpreted as a fuzzy expectation Sugeno, 1977), the maximal grade of agreement between two opposite tendencies Wierzchon, 1989), or the maximal grade of agreement between the objective evidence and the expectation Tahani and Keller, 1990). In this work, we develop a retrieval system that uses the Choquet integral as a dissimilarity measure. The system can be easily modi ed to use the Sugeno integral. Let X ˆff 1 ;...; f n g be a set of the n features used in the CBIR system, where f i can represent a color, texture, or shape feature. Let x k and x l be two n-dimensional feature vectors that represent the two images to be compared. To use the Choquet integral as a dissimilarity measure, we treat the di erence in each dimension between x k and x l as an information source the ``h'' function). The feature relevance values See Section 5) will be treated as the fuzzy densities. C g x k ; x l ˆXn iˆ1 h jx k i x l i j jxk i 1 xl i 1 j i g ff i ;...; f n g ; 6 where the notation i means that the feature indices have been permuted so that jx k 1 xl 1 j 6 jxk 2 xl 2 j 6 6 jxk n xl n j: 4.1. Feature normalization The Choquet integral in 5) requires that h x 2 0; 1Š. Therefore jx k i xl i j in 6) must also fall in this range. To guarantee this condition, we normalize all the feature vectors in the database as follows. Suppose that there are M images in the database, and let the feature vector representing the mth image be x m ˆ x m 1 ;...; xm n Š: If we combine all the feature vectors in matrix form, we have an M n matrix 2 3 x x 1 n 6 X ˆ : x M 1... x M n Let x i ˆ x 1 i ;...; xm i Š T denote the ith column of the matrix X. IfM is large enough, then we can assume the sequence x i to be a Gaussian sequence with mean l i and standard deviation r i. W e then normalize the original feature sequence as follows: x k i ˆ xk i l i 0:5 6r i for i ˆ 1;...; n and k ˆ 1;...; M: 7 According to the 3r rule, the probability of a normalized feature value being in the range of [0,1] is 99.7%. Out-of-range values will be mapped to either 0 or 1. The normalization in 7) ensures that each individual feature receives equal emphasis when calculating the dissimilarity, and that jx k i xl i j 6 1:0. l i and r i are stored in the database, and will be used to normalize the presented query feature vector. 5. Feature relevance Let g i ˆ g ff i g represent the relevance of feature i. The relevance values will also serve as the fuzzy densities to be used in the Choquet integral. In this work, we restrict g i to be a k-measure. This restriction will simplify the system considerably since only the fuzzy densities need to be learned in order to compute the fuzzy integral. Other measures, such as possibility measure, which provide a way to compute the measure of a union set from a pair of component measures, can also be used in the proposed system. The g i depend on the user's perception of similarity, and on the location of the query feature in the feature space. Let R be a linguistic variable that describes the degree of relevancy, and let R be another variable that describes the degree of irrelevancy. The degree of relevancy is used to integrate the user's positive feedback, while the degree of irrelevancy is used to integrate the

5 H. Frigui / Pattern Recognition Letters ) 1021± user's negative feedback. R and R can assume p di erent linguistic values, i.e., R ˆfv 1 ;...; v p g and R ˆfv 1 ;...; v p g: Each linguistic value is characterized by a membership function l v or l i v i. Figs. 1 and 2 display a typical example of such functions where R and R have three linguistic values. The relevance of feature i depends on the linguistic value assigned by the user, and the dissimilarity between feature i of the query image x Q i ) and feature i of the image being considered x m i ), i.e., jx Q i x m i j. Let SF be a fuzzy set, ``Similar Features'', that describes the similarity between x Q i and x m i. Fig. 3 shows a typical membership function, l SF, of the fuzzy set SF. We now discuss how to update the g i values according to user's relevance feedback Feature relevance updating All the fuzzy densities g i are considered equally important for the rst time retrieval, and are initialized to the same value for all features using g i ˆ 1=n; i ˆ 1;...; n: 8 These weights are used to compute the fuzzy measures of the sets gff i ;...; f n g, for i ˆ 1;...; n 1, using the following steps: 1. Use the fuzzy densities, g i, to compute k by using 3). 2. Compute the fuzzy measures recursively as gff i gˆg i gff i ;...; f n gˆgff n g gff i ;...; f n 1 g kgff n ggff i ;...; f n 1 g 9 Fig. 1. Typical membership functions of linguistic values v 1 ˆ ``very relevant'', v 2 ˆ ``moderately relevant'', and ˆ ``slightly relevant''. v 3 These fuzzy measures are then used to compute the Choquet integral in 6). Based on these dissimilarity values, the system retrieves the K most similar images. The user then classi es each retrieved image as relevant or irrelevant. We denote the set Fig. 2. Typical membership functions of linguistic values v 1 ˆ ``very irrelevant'', v 2 ˆ ``moderately irrelevant'', and Fig. ˆ ``slightly irrelevant''. v 3 3. Typical membership functions of the fuzzy set ``similar features''.

6 1026 H. Frigui / Pattern Recognition Letters ) 1021±1031 of relevant images by X, and the set of irrelevant images by X. The user also assigns a linguistic label to each image in X and X. Relevant images will be assigned a label from the set R,and irrelevant images will be assigned a label from the set R. These labeled images constitute the training data, and are used to update the feature relevance weights using g i ˆ g i gdg i gdg i ; 10 For each feature System estimates the membership function l SF x k System updates the fuzzy densities using 10), 11), and 12) End For End IF Until the user is satis ed where Dg i ˆ [ x k 2X l SF jx Q i x k i j \ l v xk q Experimental results 6.1. Data and features and Dg i ˆ [ x k 2X l SF jx Q i x k i j \ l v xk : 12 q In 11) and 12), ``S'' denotes any valid fuzzy union operator, ``\'' denotes any valid fuzzy intersection operator, and it is assumed that the feature vector of the kth retrieved image x k ) has been labeled v q if it was relevant, and v q otherwise. Notice that there is one membership function l v q or l v q for each retrieved image, and one membership function l SF for each feature of each retrieved image. These membership functions can be learned o ine by analyzing the images in the database, or speci ed by an expert. After updating the feature relevance weights, the system restarts a new iteration of retrieval based on the new weight vector of feature relevance. The above procedure is summarized below. Normalize the query feature vector x Q using 7) Initialize the fuzzy densities using 8) Repeat Compute k by solving 3) Compute gfx i ;...; x n g for i ˆ n 1; n 2;...; 1 using 9) For each image x m in the database Compute C g x Q, x m using 6) Retrieve the K most similar images IF user is not satis ed For each retrieved image x k User classi es x k as relevant or irrelevant User assigns a linguistic value v q or v q ) to x k The image database for our experiments consists of 3584 images of size obtained by dividing texture images into 64 subimages. The 56 original images 1 are texture images from the Brodatz album Brodatz, 1966), and are shown in Fig. 4. We use 24 Gabor lters 4 scales and 6 orientations) for feature extraction. The normalized mean and standard deviation of each ltered image are used as features. This provides us with a 48-D feature vector for each image. A more detailed description of these features along with a comparison retrieval performance and CPU time) can be found in Manjunath and Ma, 1996). In order to perform an objective evaluation of our system, we generate the ground truth by partitioning the image database into classes. Each original image constitutes a separate class. Thus, our database consists of 56 classes, and each class contains 64 subimages. The middle subimage of each image will be used to test the system as query image). For each query image, the system retrieves 50 images that are most similar to the query image. To simulate the user's feedback, the system uses the ground truth to partition the retrieved images into a set of relevant images X ) and a set of irrelevant images X ). X contains the set of images that have the same label as the query image, and X contains the other images. The system then assigns membership 1 These images are available at the MIT media at `` vismod.

7 H. Frigui / Pattern Recognition Letters ) 1021± Fig. 4. Texture images used in the experiment. functions as described in the next subsection), and updates the feature relevance weights Membership assignment We illustrate the performance of the system for the simple case where each linguistic variable has only one value: v 1 and v 1. Using the ground truth and the retrieved images, we de ne the membership functions of v 1 and v 1 as: l v x m 1 ˆ 1 1 max 0; C g x Q ; x m a ; l v x m 1 ˆ 1 1 max 0; b C g x Q ; x m ; where a and b are constants that control the core of the fuzzy sets v 1 and v 1. In this paper, we use a ˆ C g x Q ; x 1, the dissimilarity between the query image and the top retrieved image, and b ˆ C g x Q ; x 50, the dissimilarity between the query image and the last retrieved image.

8 1028 H. Frigui / Pattern Recognition Letters ) 1021±1031 Fig. 5. Performance curve. The membership function of the fuzzy set SF is de ned as: l SF x m i ˆ max 0; jx Q i x m i j d i : d i should be chosen to guarantee that there are su cient data with high membership values. We choose d i as the smallest value that allows 10% of the retrieved images to satisfy jx Q i x m i j 6 d i. To compute Dg i and Dg i, we use the sum and product for the fuzzy union and intersection respectively, and Eqs. 11) and 12) become Dg i ˆ X l SF x k i l v x k x k 2X 1 and Dg i ˆ X l SF x k i l v x k x k 2X 1 After updating g i using 10), values that are not in the range [0,1] are clipped to either 0 or 1. In this paper, we report the results of two experiments. In the rst one, we use weighted Euclidean distance as a dissimilarity measure, where the weights are the learned feature relevance values. In the second experiment, we use the Choquet integral as a dissimilarity measure, where the fuzzy densities are the learned feature relevance values Results and discussion Fig. 5 shows the performance curve of the Choquet integral based system and the weighted Euclidean distance based system. The performance is evaluated using the number of retrieved relevant images in each iteration. As can be seen, the learned feature relevance weights increase the number of relevant images after each additional iteration. Moreover, the Choquet integral based system performs better than the weighted Euclidean distance based system. From Fig. 5, we observe that in both cases, the performance increases the most in the rst iteration. This is a desirable property, since it means that the user is likely to get reasonable results after only one iteration. Fig. 6. Query image: class 55, subimage no. 33.

9 H. Frigui / Pattern Recognition Letters ) 1021± Fig. 7. The 40 most similar images before the relevance feedback. Each image is labeled by its class no. and subimage no.). Fig. 8. The 40 most similar images after three feedback iterations. Each image is labeled by its class no. and subimage no.). A typical retrieval results using the Choquet integral-based system, in response to the query image in Fig. 6, displayed in Fig. 7 without feedback), and in Fig. 8 with three feedback iterations). Table 1 shows sample retrieval results where learning feature relevance improves

10 1030 H. Frigui / Pattern Recognition Letters ) 1021±1031 Table 1 Typical results demonstrating the learning behavior of the system. Table entries show the no. of retrieved relevant images Class Euclidean distance Choquet integral No. of iterations No. of iterations performance over time. For example, for a query from class 7, and using the Choquet integral, the number of retrieved relevant images changes from 7 with no feedback) to 32 after three feedbacks). 7. Conclusion In this paper, we have proposed an image retrieval system where human and computer can interact with each other to improve the retrieval performance of a CBIR system. This approach permits the user to submit a coarse initial query and continuously re ne it using positive and negative relevance feedbacks. For both the Choquet integral-based dissimilarity and the Euclidean distance, our system requires the user to provide only a vague and more natural description of the retrieved images. These descriptions are modeled by fuzzy sets, and used to discover and use the more discriminatory features for the given query. The proposed retrieval system uses a novel dissimilarity measure based on the Choquet integral. Unlike previous attempts to use the Choquet integral as a dissimilarity measure where the fuzzy densities were xed a priori, our system uses the learned feature relevance values of each iteration as fuzzy densities. The Choquet integral dissimilarity measure is computationally more involved than traditional distance measures. However, it retrieves a comparable number of relevant images in a fewer number of iterations. If the response time for a given application is critical, the learned feature relevance values can be used as weights in the Euclidean or the ``city-block'' distance. The experimental results reported in this paper consider only texture features, and only one value for each linguistic variable. Future work will include using more features such as color and shape), and using the image database to estimate the optimal number of linguistic values, and learn their membership functions. Acknowledgements The authors would like to thank the anonymous reviewers for their valuable comments. Partial support of this research was provided by a grant from the University of Memphis New Faculty Research Initiation Awards. This support does not necessarily imply endorsement by the University of research conclusions. References Agnew, B., Faloutsos, C., Wang, Z., Welch, D., Xue, X., Multimedia indexing over the web. In: SPIE Conference, Vol Bach, J.R., Fuller, C., Gupta, A., Hampapur, A., Horowitz, B., Jain, R., Shu, C., The virage image search engine: an open framework for image management. In: SPIE Conf. on Storage and Retrieval for Image and Video Databases IV, San Jose, CA. Bhanu, B., Peng, J., Qing, S., Learning feature relevance and similarity metrics in image databases. In: IEEE Workshop on Content-Based Access of Image and Video Libraries, Santa Barbara, CA.

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