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Assessing the Best Integration between Distance-Function and Image-Feature to Answer Similarity Queries∗ Pedro H. Bugatti, Agma J. M. Traina, Caetano Traina Jr. Department of Computer Science – ICMC, University of São Paulo at São Carlos – USP Av. do Trabalhador Sãocarlense, 400, 13560-970 – São Carlos, SP – Brazil

{pbugatti, agma, caetano}@icmc.usp.br

ABSTRACT The retrieval of multimedia data relies on a feature extractor to provide the intrinsic characteristics (features) from the data, and a measure to quantify the similarity between them. A challenge in multimedia database systems is how to best integrate these two key aspects in order to improve the quality of the retrieved selection when answering similarity queries. In this paper, we analyze and compare a set of distance functions and feature extractors with regard to the association and dependencies among them. The results show that the most widely used and well-known distance functions, such as the Euclidean distance, do not reach a desirable similarity assessment, and reveal that a careful choice of a distance function considerably improves the retrieval of multimedia data, which in our experiments reached up to 92%.

Categories and Subject Descriptors I.4.7 [Image Processing and Computer Vision]: Feature Measurement—feature representation; H.3.1 [Information Storage and Retrieval]: Content Analysis and Indexing—indexing methods

Keywords Content-Based Image Retrieval, Distance Function, Feature Extraction

1.

INTRODUCTION

The amount of multimedia and complex data (images, videos, sounds, time series, DNA sequences, among others) generated or managed in the nowadays computational systems grows in a very fast pace. Thus, the challenge on dealing with them concerns on two main aspects: • How to organize (index) the data for fast access; ∗This work has been supported by FAPESP and CNPq.

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• How to retrieve the data to comply with the users’ expectation. The first aspect is tackled by access methods for multimedia data, which are frequently multidimensional or even non-dimensional data, providing fast and efficient access to the data [5]. Multimedia data cannot be indexed and retrieved using the traditional databases approaches, because this data does not comply with the total order property. That is, it does not make much sense to compare if a video or image is exactly equal to another one. However, these data are typically compared by similarity. The second aspect relies on the initial mechanism accomplished in order to compare two complex data to extract the most meaningful characteristics (features) from them, regarding a specific criterion. Thus, the data is “seen” by these features, which are organized in feature vectors. Then, the complex data is indexed, retrieved and compared through their feature vectors. A distance function is employed to measure the similarity (or more properly, the dissimilarity) between the data. There is a close relationship between the features and the distance function used to compare the data, in order to return what the human beings would expect from such comparison [15]. However, the majority of the works concerning indexing and retrieval of multimedia data overlook this relationship and go for the most known and used distance functions, such as the Euclidean or other members of the Lp family. The most suited access methods to perform similarity queries are the Metric Access Methods (MAM), since they are built based on the distances between the data elements [5]. A distance function to be employed in a MAM must comply to the three axioms of a metric (non-negativity, symmetry and triangular inequality). The development of distance functions depends on which features extracted from the data will be used. This fact leads to the parallel elaboration and improvement among these two aspects. Currently, the Metric Access Methods basically use the Lp and LEdit distances to compare pairs of feature vectors, and for example until now mostly the Euclidean (L2 ) distance was tested using medical images [4] [10]. In this paper we will show that a careful choice of a distance function improves to a great extent the quality of retrieving complex data. To do so, we perform similarity queries (k -nearest neighbors) on a dataset composed of medical images obtained from magnetic resonance and angiogram exams using different distance functions, and comparing the differences on the retrieval ability. We employ

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images as an example of a complex data, since it is widely employed in multimedia systems. However, all the experiments and conclusions can be straightforwardly extended to other types of multimedia data. The remainder of this paper is structured as follows. Section 2 presents the main concepts need to follow the paper. Section 4 details experimental measurements obtained from the analysis of the integration between distance-function and image-feature intending to reach better answers to a given similarity query, and discusses the results obtained. Finally, Section 5 presents the conclusions of this work.

2.

BACKGROUND

A feature vector can be seen as an n−dimensional point in a vector space. Due to the closely related relationship between distances and features, we will present and briefly discuss them in this Section. Also, their main application in the literature will be highlighted. The majority of the works in the literature do not present a clear reason why a distance function was chosen to compare a specific kind of data [19]. In other words, one can only conclude that the distance functions are mostly empirically selected. In [20] an evaluation of a set of commonly used distance functions is performed, and two shape descriptors on an image dataset selected with the purpose of contour analysis (MPEG-7 contour shape database), but not covering others rq Oq descriptors, such as the traditional and widely used graylevel histogram, the texture-based descriptors (i.e. Haralick descriptors), as well as some not broadly used distances are analyzed. A drawback of high-dimensional spaces is that the con(d) cept of proximity lose its mean. In [1] distance functions are evaluated in terms of retrieval effectiveness when used in high dimensional spaces. Specifically, the Lp family behavior is analyzed and concluding that the dimensionality is sensitive to the variation of the p value. Likewise, considering high-dimensional spaces and knearest neighbor queries (k NN), in [12] the Euclidean distance is compared with the Cosine Angular distance that only deals with the cosine value between two feature vectors, and does not consider their magnitude. In [14] it was proposed an hybrid feature extraction algorithm that takes advantage of the compression power of Wavelet transform and the invariable characteristics of Discrete Fourier Transform, such as rotation, scaling, translation and reflection.

2.1

Distance Functions

The efficiency and efficacy of multimedia data retrieval will be significantly affected by the inherent ability of the distance function to separate data. Thus, we will briefly describe some of the distance functions that we evaluated in this paper. For the following definitions will be considered two feature vectors X = {x1 ,..., xn } and Y = {y1 ,..., yn }.

2.1.1

The Minkowski Family

The most widely used distance function are those of the Minkowski family (or Lp norm), which is employed over vector spaces. In a vector space, the objects are identified with n real-valued coordinates {x1 ,..., xn }. Thus, the Lp distances are defined as:

v u n uX p |xi − yi |p Lp ((x1 , ..., xn ), (y1 , ..., yn )) = t

(1)

i=1

The well-known Euclidean distance corresponds to L2 . According to the value assigned to p we obtain the Lp family variations. The Lp distances are additive, in the sense that each feature contribute independently to the measure of distance. The L1 distance also called City Block or Manhattan, corresponds to the sum of the differences along the coordinates. In this case, the set of equidistant points to a center point and a given range radius rq forms a diamond in a two dimensional (2D) space with diameters parallel to the coordinates axis (see Figure 1(a)). The L1 distance is often used for computing similarity between color images [10].

rq

rq

(b)

(a)

rq

(c)

Figure 1: Examples of spatial representations of distance functions in a 2D space: (a)L1 ; (b)L2 ; (c)L∞ . The L2 (Euclidean) distance is commonly used to calculate the distance between vectors, and corresponds to the human-being notion of spatial distance. This makes it the most employed in comparisons. The set of equidistant points using L2 distance forms in a 2D space a circumference (see Figure 1(b)). In an n-dimensional space it forms a hypersphere. The L∞ also known as Infinity or Chebychev corresponds to taking the limit of equation 1 when p tends to infinity. The set of equidistant points form in a 2D space a square with edges parallel to the coordinate axis (see Figure 1(c)). In an n-dimensional space it forms a hypercube. The result obtained by the computing of L∞ distance is the maximum difference of any of its coordinates.

2.1.2

Jeffrey Divergence

Empirically derived from the Kullback-Leibler divergence (DKL), the Jeffrey divergence is a modification from DKL. The Jeffrey divergence is symmetric and presents a better numerical behavior, it is stable and robust with regard to noise and the size of histogram bins [13]. The Jeffrey divergence is formally defined as:

dJ (X, Y ) = where mi =

2.1.3

 n  X xi yi + yi log xi log mi mi i=1

(2)

xi +yi . 2

Statistic Value χ2

This distance function is calculated as the difference between each observed and theoretic frequency for each possible result, rising it to square, and dividing each one by the theoretic frequency. Finally, the χ2 is defined as:

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n X (xi − mi )2 dχ2 (X, Y ) = mi i=1

(3)

i . This distance function emphasizes the elwhere mi = xi +y 2 evated discrepancies between two feature vectors compared, and measures how improbable the distribution is.

2.1.4

Quadratic Form

The Minkowski family, afore described, treats each attribute of the feature vector independently, do not taking into account the fact that certain pairs of attributes, correspond to features that are more important for the perceptual notion of similarity than others. The Quadratic Form distance was introduced as an attempt to solve this problem, and is formally defined as:

maximum, minimum and intermediate points called control points are identified based on gradient values. A region enveloped by a segment located among two control points is called a bucket. Thus, a bucket agglutinates a set of bins, performing a considerable dimensionality reduction, and at same time keeping the histogram shape. The number of buckets and their width are variable as a result of the piecewise linear approximation of the traditional gray-level histogram. Hence, it is not viable to predict the number of buckets for a specific Metric Histogram. Due to this particular characteristic of unpredictable number of buckets, the new MHD distance was proposed in [17]. This distance computes the overlapping area between two Metric Histograms acquired from two images (e.g. A and B). The MHD distance is formally defined as: Z

bm

|MH (A, x) − MH (B, x)| dx

M HD(A, B) = q dQ (X, Y ) = (x − y)T A (x − y)

where A = [aij ] is a N × N similarity matrix; aij denotes the similarity between the feature i and j. For our experid ments we used aij = 1 − maxi,jd , where di,j = |xi − yi | is [ i,j ] the distance between the feature i and feature j of the two vectors being compared.

2.1.5

Canberra

The Canberrra distance is comprehended as a comparative Manhattan distance, since the absolute difference in the feature values is divided by their absolute sum. This makes this distance the most restrictive one. It is formally defined as: dC (X, Y ) =

n X |xi − yi | |x i | + |yi | i=1

(5)

This distance is very sensitive to small changes, and it has been used in different areas such as DNA sequences in bioinformatics, and also in computer intrusion detection [6].

2.2

Feature Extractors

The feature vectors extracted from the multimedia data is the other key aspect for the similarity comparison between complex data.

2.2.1

Traditional Gray-Level Histogram

One of the most common technique used to represent an image regarding to its gray-level (color) content is the traditional histogram. It gives the frequency of occurrences of a specific gray-level obtained from the pixels of the image. Its omni-presence is mostly due to its nice properties of linear cost to be obtained, as well as its invariance on rotation, translation and scale, for normalized histograms. It can also be used as a first step on selecting the most relevant images for a query, thus reducing the candidate set, before applying a more costly feature extractor to compare the images [11].

2.2.2

Metric Histogram

The Metric Histogram is a method applied to the traditional gray-level histogram, which allows a faster comparison between images. The Metric Histogram is obtained through a piecewise linear approximation over the normalized histogram. To get the linear parts of the Metric Histogram, the

(6)

x=0

(4)

where MH (Image, x) is the metric histogram of the image that is represented by a continuous function; bm is the breadth of the larger histogram.

2.2.3

Texture Descriptors

Among the intrinsic features used to represent medical images, texture and shape stand out as effective descriptors to discriminate tissues. The Haralick descriptors [8], based on statistics moments and obtained from co-occurrence matrix have been largely used to a texture-based image representation [16]. The features obtained from the Haralick descriptors to our experiments are variance, entropy, energy, homogeneity, 3rd order moment, inverse variance and a new descriptor that measures the suavity in the transition among pixels alongside the image, which was called Gradient descriptor [7]. All these descriptors were combined in a single feature vector.

2.2.4

Shape Descriptors: Zernike Moments

A set of complex polynomials that delineate a complete orthogonal base defined over the interior of a unit circle (i.e. x2 + y 2 = 1) is defined as Zernike polynomials. Such polynomials can be denoted by polar coordinates. The Zernike moments are a combination among the Zernike polynomials formulation and the general moments theory, which generated a set of orthogonal moments that were named Zernike moments [9]. The great advantage of using a set o orthogonal moments consists on the fact that such moments practically have no redundancy. Thus, it can be said that each moment accumulates a distinct image characteristic, requiring a smaller number of moments to store the same information regarding to the non-orthogonal moments. The Zernike moments theory has the similar idea of the Fourier transformation of functions that aims at expanding the signal (or image) to orthogonal bases series. It is interesting to note that such moments give a broad representation of the objects shape present in the image.

2.2.5

Shape Descriptors: Improved EM/MPM

The improved EM/MPM algorithm proposed in [3] segmented the images using a technique that combines a Markov Random Field and a Gaussian Mixture Model to obtain a texture-based segmentation. The segmentation of

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images are accomplished according to a fixed number of different texture regions, where six features are extracted for each region: the mass m (or size); the centroid (xo and yo ); the average gray level (µ), the Fractal dimension (D); and the linear coefficient used to estimate D. In our experiments we segmented the images in five regions. Thus we obtained feature vectors composed of 30 positions.

3. METHODOLOGY To assess the distance function ability on properly separate the images, we have generated graphs based on the approach of precision and recall [2], obtained from the results of sets of similarity queries. Precision is the proportion of retrieved and relevant images to all the images retrieved. Recall is the proportion of relevant images that was retrieved by the query, out of all relevant images available. They are formally defined as: precision =

|RA | |R|

recall =

|RA | |A|

(a) angiogram

(b) Axial Pelvis MRI

(c) Axial Head MRI

(d) Axial Abdomen MRI

(e) Coronal Head MRI

(f) Coronal Abdomen MRI

(7)

where, RA is the number of retrieved and relevant images; R is the number of all images retrieved by the query; A is the number of relevant images in the dataset. Precision is 100% when every image returned to the user is relevant to the query, and recall reaches 100% when all images from the dataset have been returned. Both precision and recall (P&R) have a fixed range between 0.0 and 1.0 (100%). It is important to observe that when analyzing a precision and recall graph, the closer the curve to the top, the better is the retrieval technique. To build the P&R graphs, we applied sets of k -nearest neighbor (k NN) queries, using randomly selected query images from the dataset, and varying the values of k, for all feature sets generated by the feature extractors discussed. A k NN query consists in searching the dataset for the k closest images to the query image, comparing their feature vectors by using a defined distance function to compute how similar the images are. The distance functions used in the experiments were the ones described in Section 2.1. For each dataset analyzed, all images were employed as query centers. The feature vectors were acquired from the gray-level histogram, Zernike Moments, Haralick descriptors, metric histograms, and the improved EM/MPM algorithm. Each set of feature vectors obtained were indexed using the Metric Access Method (MAM) Slim-tree [18], to accelerate the similarity queries processing.

4. EXPERIMENTS In this section we present the experimental results on the comparisons performed with the distance functions and feature extractors described in Sections 2.1 and 2.2 respectively. These experiments aimed at gathering the best combination between the features and distances to separate the images.

4.1

that use a contrast, allowing the visualization of blood vessels in some region of the body. The dataset contains MR images divided in 8 classes according to the region of body examined and the type of specified section. Figure 2 illustrates an image example of each class: (a) Angiogram (36 images), (b) Axial Pelvis MRI (86 images), (c) Axial Head MRI (155 images), (d) Axial Abdomen MRI (51 images), (e) Coronal Head MRI (36 images), (f) Coronal Abdomen MRI (23 images), (g) Sagittal Head MRI (258 images) and (h) Sagittal Spine MRI (59 images). The images are represented by 8 bits, resulting in 256 gray-levels and comprising dimensions of 256 x 256 pixels.

The Dataset Description

To perform the experiments we have used a variety of medical image datasets from the real world, provided by the Hospital of our university. Due to space limitations, in the present paper there are only results obtained from one dataset. The dataset consists of 704 images of magnetic resonance (MR) and angiogram exams. The MR exams accentuate soft tissues, while angiograms are radiologic exams

(g) Sagittal Head MRI

(h) Sagittal Spine MRI

Figure 2: Examples of images from the dataset.

4.2

Results

In order to evaluate the best configuration between distance-function and image-feature to answer similarity queries, we performed sets of k -nearest neighbor (k NN) queries in the image dataset, and generated precision and recall graphs as described in Section 3. Figure 3 illustrates the precision and recall graphs considering the described dataset in five-level intervals of recall, comparing all the distance functions mentioned in Section 2.1 with the texture extractor (i.e. Haralick descriptors). Since the dataset is composed of 704 MR Images, we processed 704 queries using each image as a query center. The average values obtained from the precision/recall calculation was used to generate the graphs in Figure 3. The graphs in Figure 3 shows that Canberra distance presents a considerable gain in precision compared with the others, approximately 80% at a 40% of recall. The next one, χ2 distance, is followed by Jefrrey Divergence and L1 to a 40% recall level, respectively resulted in a precision up to 60% and 55%. Quadratic Form, L1 and Jeffrey Divergence give almost identical results. It is interesting to note that MHD distance for low recall values performs close to the others, but decreases the precision more slowly than the last ones. The commonly used Euclidean (L2 ) and Chebychev (L∞ ) presented the poorest results. The difference in precision reaches values of 92% when Canberra and Chebychev are compared. This value would make a huge difference in the response set returned to the users.

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Precision (%)

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L1 L2 Linf Quadratic X2 Jeffrey Canberra Metric Histogram MD 0.8 0.2

Feature Vector - Texture

Zernike features, since it also has low computational cost. Note that the gain for a careful choice of distance can reach 43% when comparing the MHD distance and χ2 distance at 45% of recall.

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Figure 3: Precision and Recall graphs, illustrating the retrieval ability for texture features considering0.4 several distance functions.

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L1 0.2 L2 Linf Quadratic X2 Jeffrey Canberra Metric Histogram MD

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Figure 4L1presents the precision and recall graphs for the L2 Linf histogram extractor considering the same distance functions. Quadratic X2 Analyzing the graphs of Figure 4 we observe that the clear 0 0 Jeffrey Canberra winner is MD the MHD distance, which is also the fastest disMetric Histogram tance to compute, being faster than 0.2 0.4 0.6 in average 0.8 four times 1 Recall (%) L1 . Also χ2 , Quadratic Form, Jeffrey Divergence and L1 distance functions presents almost the same behavior until 25% of recall. The difference in precision when MHD and L∞ are compared, reaches values of aproximately 75%. For these features the Canberra distance presents initially the lower levels of precision, improving its behavior for recall levels above 35%. We present the behavior of the MHD distance also in the other Figures 3, 5 and 6 as a baseline to provide comparison among the features. Feature Vector - Histogram 1

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Figure 5: Precision and Recall graphs, illustrating the retrieval ability when using the Zernike feature 0.2 0.4 0.6 0.8 1 vector for several distance functions. Recall (%) Figure 6 illustrates the precision and recall graphs, comparing the retrieval ability of the features generated by using the EM/MPM approach (see Section 2.2) considering the distance functions being evaluated. From the graphs of Figure 6 we can see that again the Canberra distance presented the higher values of precision, up to 95% to a recall level of 60%. Next, the Quadratic Form, L1 and Jeffrey Divergence one more time present almost identical results. The L2 and L∞ gave lower results compared with the others distances, and the lowest values of precision was given by the MHD distance. It is important to note that the gain in precision for a careful choice of distance can reach values, up to 90% when comparing the MHD distance and χ2 at a 55% of recall. The difference in precision when Canberra and Chebychev are compared, reaches values of aproximately 36% at the same recall level afore mentioned.

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L1 L2 Linf Quadratic X2 Jeffrey Canberra Metric Histogram MHD

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Figure 4: Precision and Recall graphs, giving the retrieval ability when using the normalized gray-level histogram feature vector for several distance functions. Figure 5 illustrates the precision and recall graphs obtained on the Zernike features. Analyzing the graphs of Figure 5, we observed that the all distances, but L∞ and MHD distance, tie. That is, the best distance when considering histograms became the worst for Zernike feature vectors. Canberra would be a nice choice for both texture and

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Figure 6: Precision and Recall graphs, illustrating the retrieval ability when using the improved EM/MPM algorithm with the several distance functions.

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It is important to emphasize that k NN queries usually ask for small values of k. That is, the user wants to know the five or ten most similar images to a reference one. Therefore, the low values of recall are the most relevant. Figure 7 summarizes the results obtained by the comparisons performed between the distance functions and feature vectors regarding to their precision evaluation. Distance Functions Higher Precision

Lower Precision

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+ L1 Quadratic Jeffrey

L2

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Canberra

L2

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Figure 7: Illustration of comparisons between the distance functions and feature extractors.

5.

CONCLUSIONS

In this paper we demonstrated the importance of choosing the best suited distance to a kind of feature. Due to space limitations we presented the comparison between only the main distances employed for multimedia data, and presented the graphs of precision and recall for illustrating the relationship between the distances and the features. From the experiments, it is straightforward to notice that such relationship is indeed strong. The L1 , Quadratic Form and Jeffrey Divergence distance functions presented a quite close behavior for all feature vectors. This fact can be explained for Quadratic Form and L1 distance functions, because Quadratic Form is a weighted distance function where the weight is defined by a similarity matrix A = [aij ] (see equation 4). We employed the L1 distance to compute this similarity matrix. The L1 and Jeffrey Divergence gives a similar spatial region where the elements at the same distance are. Thus, this explain their close behavior. For the Shape and Texture features used in the experiments, the Canberra distance presented the better precisions. Moreover, the texture features and the Canberra distance provided a considerable gain in comparison with the other distances. Just for the histogram feature vectors the Canberra distance presented a worse performance. It is important to highlight that the L2 and the L∞ distances, widely used in the literature presented very poor precision values for all features. The Canberra distance, a not broadly used distance and rarely employed in multimedia data retrieval, presented very promising results with regard to precision. Moreover, its computational cost is as inexpensive as the widely used distance function L1 .

6.

REFERENCES

[1] C. C. Aggarwal. On the effects of dimensionality reduction on high dimensional similarity search. In ACM PODS, Santa Barbara, CA, 2001.

[2] R. A. Baeza-Yates and B. Ribeiro-Neto. Modern Information Retrieval. Addison-Wesley Longman Publishing Co., Inc., Boston, MA, USA, 1999. [3] A. G. R. Balan., et al. Fractal analysis of image textures for indexing and retrieval by content. In IEEE CBMS, pages 56–63, Dublin, Ireland, 2005. ¨ [4] T. Bozkaya and Z. M. Ozsoyoglu. Indexing large metric spaces for similarity search queries. ACM TODS, 24(3):361–404, 1999. [5] E. Ch´ avez., et al. Searching in metric spaces. ACM Computing Surveys, 33(3):273–321, 2001. [6] S. M. Emran and N. Ye. Robustness of canberra metric in computer intrusion detection. In IEEE Workshop on Information Assurance and Security, West Point, New York, NY, USA, 2001. [7] J. C. Felipe., et al. Retrieval by content of medical images using texture for tissue identification. In IEEE CBMS, pages 175–180, New York, NY, USA, 2003. [8] R. M. Haralick, K. Shanmugam, and I. Dinstein. Textural features for image classification. IEEE TSMC, 3:610–621, 1973. [9] A. Khotanzad and Y. H. Hong. Invariant image recognition by zernike moments. IEEE TPAMI, 12(5):489–497, 1990. [10] F. Long, H. Zhang, and D. D. Feng. Fundamentals of Content-Based Image Retrieval. Multimedia Information Retrieval and Management. 2003. [11] H. M¨ uller., et al. A review of content-based image retrieval systems in medical applications-clinical benefits and future directions. International Journal of Medical Informatics, 73(1):1–23, 2004. [12] G. Qian, S. Sural, Y. Gu, and S. Pramanik. Similarity between euclidean and cosine angle distance for nearest neighbor queries. In ACM SAC, pages 1232–1237, Nicosia, Cyprus, 2004. [13] Y. Rubner and C. Tomasi. Perceptual Metrics for Image Database Navigation. The Kluwer International Series in Engineering and Computer Science. 2001. [14] C. L. Sabharwal and S. R. Subramanya. Indexing image databases using wavelet and discrete fourier transform. In ACM SAC, pages 434–439, New York, NY, USA, 2001. [15] S. Santini and R. Jain. Similarity measures. IEEE TPAMI, 21(9):871–883, 1999. [16] A. W. Smeulders., et al. Content-based image retrieval at the end of the early years. IEEE TPAMI, 22(12):1349–1380, 2000. [17] A. J. M. Traina., et al. The metric histogram: A new and efficient approach for content-based image retrieval. In Proc. of the IFIP Sixth Working Conf. on Visual Database Systems, pages 297–311, Deventer, The Netherlands, 2002. [18] Jr. Traina, Caetano., et al. Fast indexing and visualization of metric datasets using slim-trees. IEEE TKDE, 14(2):244–260, 2002. [19] N. Vasconcelos and A. Lippman. A unifying view of image similarity. In ICPR, page 1038, Washington, DC, USA, 2000. [20] D. S. Zhang and G. Lu. Evaluation of similarity measurement for image retrieval. In IEEE ICNNSP, pages 928–931, Nanjing, China, 2003.

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