Application of a hybrid orthogonal function system on trademark image retrieval

Transcription

1 Bulletin of the JSME Journal of Advanced Mechanical Design, Systems, and Manufacturing Vol.8, No.6, 4 Alication of a hybrid orthogonal function system on trademark image retrieval Xiaochun WANG *, Yena WANG**, Honglei SUN** and Ruixia SONG** * College of Sciences, Beijing Forestry University, No.35 Tsinghua East Road, Haidian District, Beijing 83, China wamgxiao@bjfu.edu.cn ** College of Sciences, North China University of Technology, No.5 Jinyuanzhuang Road, Shijingshan District, Beijing 4, China Received November 3 Abstract This aer resents a new class of discrete orthogonal moments, W-moments, based on a hybrid orthogonal function system, W-system of degree one. The W-moments have two main advantages, one is comutationally simle due to the iecewise linear structure of the basis functions, and the other is the reroducibility, namely shaes and even shae grous can be recisely reconstructed by finite number of the W-moments, due to the orthogonality of the basis functions on a discrete data set. These roerties make the W-moments referable to the conventional moments when used for describing comlex shaes consisting of multile disjointed curves. To evaluate the roosed moments, image retrieval exeriments are conducted on MPEG-7 CE and a self-constructed trademark data set. In the exeriments, the W-moments are first derived from the trademark images, similarity between query image and each trademark in the database is then evaluated using the Euclidean distance between their W-moments feature vectors, and the trademark retrieval is finally accomlished by comaring their similarities. The exerimental results show that the moments roosed in this aer give referable retrieval results in comarison to other five conventional moments, and the W-moments should be feasible and efficient in ractical alication. Key words : Hybrid function system, Trademark retrieval, W-system, Feature vector, W-moments, Orthogonal moment. Introduction A Trademark is a unique mark used to identify a unique roduct or service, which is a comany s valuable intellectual roerty asset. To avoid a comany s financial or creative losses caused by infringement, trademark rotection is necessary. One legal way to rotect a trademark is to register the trademark in a national trademark office. However, before trademark registration, a comrehensive check is required by searching the whole trademark database to make sure the trademark to be registered is not similar to any trademark already registered (Mohd Amuar, 3). The comlexity of searching for the desired trademark image in trademark databases increases as number of trademarks grows. Therefore, it has become extremely imortant to establish a highly accurate and effective automatic trademark retrieval system (TIR). Although TIR belongs to the technology category of content-based image retrieval (CBIR), it is one of the most challenging areas in CBIR (Eakins, et al., ). There is no universally acceted trademark retrieval scheme so far (Iwanaga, et al., ). CBIR technique has been roosed in the early 99s, in which the low-level visual features including colors, shaes, textures, and satial relations between objects are major features used. Among these visual features, shae is one of the most imortant and rominent low-level image features, and in many cases, shae in CBIR is more owerful for identification than color and texture (Chahooki and Charkari, ). For examle, in trademark image retrieval system, color and texture does not lay a useful role, since the design trademarks are often registered as binary images (Jain and Vailaya, 998). Moreover, the majority of users are interested in retrieval by shae (Alajlan, et al., 7). Therefore, many shae-based image retrieval techniques or aroaches have been develoed to imrove the erformance of image retrieval. Paer No.3-96

2 Journal of Advanced Mechanical Design, Systems, and Manufacturing, Vol.8, No.6 (4) The central issue of shae-based image retrieval is shae reresentation, or shae descrition. According to the definition given by Kendall, Shae is all the geometrical information that remains when location, scale and rotational effects are filtered out from an object. Shaes which are ercetually similar refer to the shaes obtained by rotating, translating, scaling and affinely transforming a given shae. Therefore, aart from being comact, identifiable and reliable, a good shae descritor should be invariant to rotation, scale, and translation, so that shaes which are found ercetually similar by human have the nearly same descritors. Besides, a good shae descritor should be robust to noise, distortion and occlusion. Roughly, Shae reresentation methods can be classified into two categories: contour-based and region-based. Contour-based methods use only the contour or border of the shae, while the region-based methods take into account the internal details besides the contour (Zhang and Lu, 4). Many shae reresentations are resented in the literatures, such as invariant moments (Belkasim, et al., 989), geometric moments, Zernike moments, seudo-zenike moments, Legendre moments (Teague,98), Fourier descritors (Bartolini, et al., 5), Wavelet descritors, histogram-based descritor (Iwanaga, et al., ). Region-based descritor does not need to know boundary information, which makes it suitable for more comlex shae reresentation. Previous studies have shown that region-based methods are more accurate than contour-based ones. Zernike moments, which have been adoted by MPEG-7 as region-based shae descritors, have shown great erformance in shae retrieval. Several aers have revealed that using the feature set of Zernike moment magnitudes or moment invariants can achieve better results than others. In trademark image retrieval system, color and texture does not lay a useful role, while the shae feature vectors are usually used to reresent the contents, so TIR belongs to the technology category of CBIR. However, most of the trademark images are artificially-roduced images consisting of several distinct geometric shaes, which makes shae matching more difficult. As we know, classical contour-based shae descritors can only efficiently describe simle shae with a single connected region, but not a comlex shae consisting of several disjoint regions since shae descritors can be changed drastically if there is a small crack on the contour (Mei and Androutsos, 9). Therefore, contour-based shae descritors may not be suitable for trademark images reresentation (Kim and Kim, ). Region-based orthogonal moments, such as Zernike moments and Legendre moments, allow for accurate reconstruction of the described shae by exanding an image into a series of orthogonal moment bases and have advantage in trademark image retrieval (Wei, et al., 9). Note that when using Legendre moments and Zernike moments to reresent a shae, the shae can be accurately achieved only when the whole series is considered. However, in alications we always use the artial sum of the series as an aroximation, and recision of shae reresentation deends on the number of moments used. If the number of moments decreases, the aroximation accuracy degrades. So accuracy and comutational comlexity are two goals needed to balance. To imrove aroximation accuracy, high order of moments are needed, which would result in high comutational comlexity; while to reduce the comutational comlexity, only low order moments can be used, which would increase the aroximation error. Tchebichef moment introduced by Mukundan (Mukundan, et al., ) is a class of orthogonal moment features. It can be used to exactly reconstruct the discrete-sace images since the basis set is directly orthogonal on the discrete domain of the image coordinate sace. It is suerior to Legendre moments and Zernike moments in terms of reserving the analytical roerties needed to ensure information redundancy in a moment set. However comutation of Tchebichef moments is time consuming, esecially for higher order moments. In this aer, we resent a new class of moment functions, called W-moments, and aly it on trademark image retrieval. The W-moments are based on an orthogonal hybrid function system, W-system, roosed by the authors (Wang and Song, 8). The great advantage of the W-moments is its characteristics of being able to accurately reconstruct information, no matter it is continuous or discontinuous. Therefore, the W-moments not only ensure the minimal information redundancy, but also eliminate the need for numerical aroximation when they are alied to describe the discrete sace images, esecially to describe shaes with several disjoint arts. Furthermore, the W-moments have low comutational comlexity due to the iecewise linear structure of the basis functions.. Calculation of the Shae Princial Orientation and Object Region The trademarks in the exerimental database used in this aer are all binary images without noise. This means that we don t need to erform image rerocessing of color to gray conversion, image binaryzation. However, binary

3 Journal of Advanced Mechanical Design, Systems, and Manufacturing, Vol.8, No.6 (4) images still come in different sizes, locations and orientations. To make the retrieval techniques have advantage of being invariant to the scaling, rotation and translation of trademarks, it is necessary to erform normalization to each trademark image. The normalization in this aer includes rincial direction normalization and scale normalization. () Princial direction normalization The orientation of a D shae is comuted based on the direction of the first rincial axis, which is the eigenvector of the matrix corresonding to the largest eigenvalue. Assume ( x, yis ) the center of mass of an image, According to K-L transform, the direction of the eigenvector corresonding to the largest eigenvalue is given as: m m ( m m) 4m arctan( ) m () q where mq is the center moment of order +q of the image, and it is defined as mq ( x x) ( y y) f ( x, y). The shae xy, rincial orientation is calculated as: when m 3, and when m 3. By erforming the shae transformation oeration x cos sin x y cos sin y () the shae is rotated horizontally. After translating its center of mass ( x, yto ) the origin, the sindle of the image coincides with the x coordinate axis. () Scale normalization The object region must be secified before the moments are calculated. First, the external rectangle of the target image is obtained by rojecting the image to the two coordinate axes. Next, enlarge the external rectangle to a square with side length being equal to the external rectangle s longer ones, kee the centroid of the target image in the origin, and set the square as the object region. Finally, standardize the image to a uniform size 8 8. This ste makes the resulting moments invariant to translation and scale. 3. Introduction to W-system Generally, the available sets of classical orthogonal functions can be divided into two classes: the class of continuous basis functions and the discontinuous class of iecewise constant basis functions. Piecewise constant basis functions are efficient in reresenting discontinuous functions, but when the functions to be reresented are continuous, it will always give a stair-case fit. To retain sufficient accuracy, the reresentations require a large number of exansion terms. Comared with iecewise constant basis functions, continuous basis functions are efficient in reresenting continuous functions, but when the functions to be reresented have discontinuities or jums, Gibb henomenon will occur. Therefore, these two classes of functions would not be well suited to reresent functions that ossess both continuity and discontinuity. However, there are situations in which the functions ossess both continuity and discontinuity, such as the function that describes a trademark shae defined by multile disjoint curves. To reresent functions with mixed features of continuities and jums, develoing hybrid orthogonal functions systems is imerative. Some efforts and roosals in constructing hybrid function systems have been made recently. The U-system (Feng and Qi, 984), V-system (Song, et al., 7) and W-system (Wang and Song, 8) are three such systems. Each of them consists of not only smooth functions but also discontinuous functions at multi-levels. The U-system is regarded as a generalization of the Walsh system and the Slant function set, and the V-system and W-system are generations of the Haar function set. They have been widely used in image and information rocessing. In this section, we give a brief introduction to the W-system. The W-system is an orthogonal function system on [,] with a set of mutually orthogonal iecewise olynomials as its basis functions, which is defined using a set of orthogonal Haar functions and Legendre olynomials. Let P ( x) be Legendre olynomial of degree defined on the interval [, ], which satisfies the following equation: 3

4 Journal of Advanced Mechanical Design, Systems, and Manufacturing, Vol.8, No.6 (4),, P ( t) P ( t) dx (3) Using Legendre olynomial of degree, we define a iecewise olynomial of degree on the interval [,], denoted by : ( x) n j j n ( x) P ( x j), x (, ), j,,, (4) n n Let N h ( x), h ( x),, h ( x) be the first N Haar functions and S ( x) j be the multilication of hj( x) and ( x), i.e. S ( x) ( x) h ( x) (5) j j Define k W N the union of function set L N, that is k k n W L, N, n,,3,, (6) N N where L N is defined as LN { S ( x), S ( x),, SN ( x)}. The W-system of degree k is denoted by W k, which is defined as k k W lim W (7) N N Obviously, the W-system of degree zero is just the Haar function system. Any function which is square integrable in interval [,] can be exanded in W-series, and certain truncated form of the series can accurately reconstruct the 3 function. The basis functions of W8 are deicted in Fig.. The reader is referred to [8] for more details about the W-system and W-matrix. Fig. Basis functions of 3 W 8 4

5 Journal of Advanced Mechanical Design, Systems, and Manufacturing, Vol.8, No.6 (4) 4. The W-moments Moments are quantities that characterize the shae of an object. One imortant roerty of the moments is their invariance under affine transformation. There are orthogonal moments and non-orthogonal moments. Orthogonal moments, such as Legendre moments, Zernike moments and seudo-zernike moments, have the minimum information redundancy to reresent the image, but the comutational comlexity is high. While Non-orthogonal moments, such as Hu moments, geometric moments, comlex moments, and rotational moments, resent a relatively low comutational cost, but are sensitive to noise and suffer from higher degree of information redundancy. Moreover, the reconstruction is not straightforward for non-orthogonal moments. Among the many moment shae descritors, Zernike moments and Legendre moments are two most imortant moments alied in the field of image shae reresentation. The (+q)th order geometric moments of function f ( x, y) on G are given by q mq x y f ( x, y) dxdy,, q,,, (8) G Geometric moments are not orthogonal, so it is comutational comlicated to reconstruct an image from the q geometric moments. Legendre moments and Zernike moments are given by relacing the monomial roduct x y with Legendre olynomial and Zernike olynomial, resectively. We take the Legendre moments as an examle to illustrate their definition, discrete aroximation and the inverse moment transform. The Legendre moments of order m+n for an image with intensity function are defined as f ( x, y) (m)(n) mn P ( ) ( ) (, ),,,,, 4 m x Pn y f x y dxdy m n (9) where Pm ( x) is Legendre olynomial of degree m. For the discrete sace version of the N N aroximated by image, Eq. (9) is usually NN (m )(n ) i N j N mn m n N i j N N P ( ) P ( ) f ( i, j), m, n,,, () The image function f ( i, j) can be written as an infinite series exansion in terms of Legendre olynomials over the square [,] [,] as i N j N f ( i, j) mnpm ( ) Pn ( ), i, j,,,, N N N () mn Motivated by Legendre moments and Zernike moments, we define the W-moments by relacing the monomial q roduct x y in Eq. (8) with the basis functions of the W-system. Note that the W-system of degree one W lim W is defined by orthogonal iecewise linear functions, where N N W L L N N N n { S ( x), S ( x), S ( x), S ( x), S ( x), S ( x)} ( N, n,,3, ) N N Linear functions are comutationally simle comared with olynomials of higher degrees. For this reason, we emloy the W-system of degree one to define the W-moments. For convenience, we rearrange WN as WN LN LN { W ( x), W ( x), W ( x)}, where W ( x) is defined by N k Sk ( x), k N Wk ( x) SkN( x), N k N () 5

6 Journal of Advanced Mechanical Design, Systems, and Manufacturing, Vol.8, No.6 (4) The basis functions of W 8 are deicted in Fig Given an image intensity function order +q by Fig. f ( x, y) Basis functions of W 8. Using the basis functions ( ) k W x in W N, we define the W-moment of q W ( x ) W ( y q ) f ( x, y ) dxdy,, q,,, (3) G where G is region [,] [,]. In fact, the W-moments are the rojection of the image function on the orthogonal basis functions Wk ( x). A comlete moments of order n consists of all moments q such that, q n and contains ( n ) moments. The W-moments are comutationally simle since functions W ( x) W ( y) are always iecewise olynomial of degree two in two variables for any nonnegative integer and q, namely, the calculating comlexity doesn t increase with the increase of number and q. It should be noted that we always set the maximum order of W-moments in secific alication. The reason for doing so is that the W-system is constructed by classes and we always use class as a unit, i.e. when we use the W-system, we always take the whole class into consideration. For the image with brightness function f ( i, j) ( i, j,,, ), Eq. (3) has the following discrete aroximation n q ( ) W ( ) f ( i, j) W i j (4) mn m n i j According to the definition and roerties of the basis function Wk ( x), we have i i, m n Wm( ) Wn( ) i, m n (5) For any bounded function f ( i, j ) ( i, j,,, ), the orthogonality roerty leads to the following inverse moment transform i j f ( i, j) mnwm ( ) Wn ( ) (6) m n 6

7 It should be noted that the Eq. (6) is different from the Eq. (); the former is sum of finite terms, while the latter is sum of a series. So we have the conclusion: an image can be exactly reconstructed from finite number of the W-moments using inverse moment transform (6). The W-moments comletely eliminate the need for any aroximation of continuous integrals. In addition to the advantage of accurate reconstruction, low comutational comlexity is the other redominate advantage of the W-moments, because basis functions Wk ( x) ( k,,, ) in Eq. (3) are iecewise linear function, this means the comutational comlexity of W-moments doesn t increase as k increases. Set Wm ( Wm (), Wm ( ),, Wm ( )) (7) The Eq. (4) can be rewritten as W FW (8) T mn m n where F is a we have matrix formed by the image brightness function, i.e F f i j (, ). Using matrix multilication, T T WFW, F W W (9) where W N W () W ( ) W ( ) N N N N W () W ( ) W ( ) N N,, N N NN N WN () WN ( ) WN ( ) N N N N () For a M M image matrix F f ( i, j) M M, if only the moments of order u to ( M ) ( M M) are comuted, we have the following aroximation M M i j f ( i, j) mnw m( ) Wn ( ) () M M mn Figure 3 shows the reconstruction results of a trademark using the W-moments, Zernike moments and Legendre moments. Obviously, the lower moments deict the image s outline, and with the order increasing, the reconstructed image becomes more and more accurate. The reconstruction results show the suerior feature reresentation caability of the W-moments. It is noted that the maximum order of moments and the number of moments used for the W-moments, Zernike moments, and Legendre Moments in Fig. 3 are not exactly the same. The reason why they are different is that a comlete moment of order n contains different number of moments for different methods. So the number of moments used in Fig. 3 is just very close to each other, not exactly the same. (a) Original image with 64x64=496 ixels

8 Journal of Advanced Mechanical Design, Systems, and Manufacturing, Vol.8, No.6 (4) (b) W-6 (6) (c) W-4(64) (d) W-3(56) (e) W-6(56) (f) Z-7 () (g) Z-5 (7) (h) Z-3 (7) (i) Z-63 (56) (j) L-7 (36) (k) L-5 (36) (l) L- (76) (m) L-45 (35) Fig.3 Reconstruction effects of a trademark using W-moments, Zernike moments and Legendre moments with different orders (The numbers after caital letters are the maximum orders of moments and the number in the brackets is the total number of moments corresonding) 5. Image feature descrition using W-moments For a given image, its W-moments are arranged in a zigzag scan way to form a vector{,,,, mn}. We call it W-moments feature vector. We ll use this feature vector to reconstruct the image. The similarity between two images is measured by the Euclidean distance between the acquired W-moments feature vectors. The smaller distance between the W-moments feature vectors is, the more similar these two images are. So image classification and retrieval can be erformed based on the distance. Even though the rincial axis of an image is determined, and the shae is rotated horizontally, one shae still has four ossible ositions as shown in Fig. 4. Accordingly, one shae may corresond to four feature vectors; the first vector corresonds to the normalized shae, the other vectors corresond to the generated shaes formed by erforming horizontal, vertical, both horizontal and vertical flis to the normalized one. Therefore, when comaring two shaes, A and B, we should comute the distance between one of the feature vectors of A and each of the four feature vectors of B. The smallest distance is defined as the distance between the two shaes. An examle (original image, normalized image, and its flied images) is resented in Fig. 4. (A) (A) (A) (A3) (A4) Fig. 4 The generated shaes formed by erforming horizontal and vertical flis to the shae with normalized rincial axis, (A) Original shae (A) Shae with normalized rincial axis (A) Image from u-down fli (A3) Image from right-left fli (A4) Image from u-down and right-left fli Generally, the W-moments feature vectors for the trademark images in the database are re-calculated off-line and stored. It should be noted that each trademark image in the database corresonds four moment feature vectors as mentioned above. At query time, the user enters a query trademark A into the system and the moment feature vector is extracted from the query object. The feature vector of query trademark is comared with feature vectors stored in the database, and then the system retrieves the relevant shaes from the database on the basis of the closeness of the extracted features, the best match trademarks are finally resented to the user. The secific retrieval rocess includes 8

9 Journal of Advanced Mechanical Design, Systems, and Manufacturing, Vol.8, No.6 (4) following stes: () Perform image rerocessing to the query trademark image A, including image binaryzation, rincial direction normalization and scale normalization, and obtain a horizontal image HA with centroid in the original. () Comute the W-moments feature vector of normalized image HA, and calculate the distances between query trademark image and images in the database. (3) Array all the acquired distances in increasing order, and outut the first m corresonding trademarks, i.e. the most similar m images to the query image. 6. Trademark retrieval exeriments To test the retrieval erformance of the W-moments, we conduct a comrehensive erformance comarison of the roosed algorithm with 5 classical moments, which are Zernike moments (ZM), Hu invariant moments (HM), Fourier-Mellin moments (OFMM), Legender moments (LM), and geometric centroid moments (CM). The retrieval efficiency varies with the number of moments used. However, increasing the number of moments doesn t mean imroved erformance. Table resents the retrieval efficiency of ZM, LM and W-moments (WM) with different maximum order of moments, and it shows that number of order u to 4 gives the highest retrieval efficiency for WM, 5 for ZM, and for LM. From our exerimental result, we also know that number of order u to 8 gives the highest retrieval efficiency for OFMM, and 7 for CM. So the Moments used in the exeriments resented in this section are from order zero u to order 5 for ZM, 7 for CM, 8 for OFMM, and for LM. The retrieval exeriments are conducted on two databases: one is benchmark database MPEG-7 CE (CE), the other is self-constructed trademark data set consisting of 75 trademarks. Database CE is comosed of images with comlex shaes and textures, and multi closed boundaries, which is emloyed to test the recognition ability of region-based algorithm to comlex images. The self-constructed data set trademark database consists of two class of trademark, one class is downloaded from the internet, and the other is created by making local changes to the downloaded ones. In the self-constructed trademark data set, we have designed 45 grous of images, for each. Images in the same grou are obtained by local changing a given trademark, they are visually similar. The self-constructed data set is created to test the recognition ability of the algorithm to highly homogeneous images. Figure 5 resents some images in the self-constructed trademark data set. The rerocessing methods for Zernike moments and orthogonal Fourier-Mellin moments include: () move the centroid of the image to the origin; () draw a circumscribed circle around the target; (3) draw the outer tangential square and cut out the image in the domain of the square; (4) normalize it to size 8 8. The rerocessing method for the W-moments, the geometric center moments and Legender moments is the same as the one described in section. For Hu moments the only rerocessing erformed is moving the centroid of the image to the origin. For erformance measure, we use the recision and recall of the retrieval for evaluation of the image retrieval efficiency. Recall R is the roortion of the relevant shae actually retrieved, and recision P is the roortion of retrieved shae that is relevant, i.e. number of retrieved relevant shes r Precision P total number of retrieved shaes n number of retrieved relevant shes r Recall R total number of relevant shaes in the whole database m Precision P measures the accuracy of the retrieval, while recall R measures the robustness of the retrieval erformance. As an evaluation method, PR curve is widely used in attern recognition. Table Performance comarison of different number of moments on the database CE-A (%) method WM ZM LM order Bull s eye

10 Journal of Advanced Mechanical Design, Systems, and Manufacturing, Vol.8, No.6 (4) (a) Trademarks downloaded from the internet (b) Self-constructed trademarks Fig. 5 Examles of trademark in the self-constructed trademark database 6. Retrieval exeriments on benchmark database CE Database MPEG-7 CE is comosed of five classes, A, A, A3, A4 and B, which is designed to test region-based shae descritor s erformance under different shae variations. There are 88, 9, 3, 3 and 8 shaes in A, A, A3, A4 and B, resectively. The use of the database is briefly described here. shaes in class A are designated as queries, which is classified into grous with 5 similar shaes in each grou to test scale invariance; 4 shaes in class A are designated as queries, which is classified into grous with 7 similar shaes in each grou to test rotation invariance;33 shaes in class A3 are designated as queries, which is classified into 3 grous with similar shaes in each grou to test scale and rotation invariance ; 33 shaes in class A4 are designated as queries, which is classified into 3 grous with similar shaes in each grou to test robustness to ersective transformation. In class B, 68 shaes have been manually classified into classes by MPEG-7. The number of similar shaes in each grou is 68, 48,, 8, 7,, 45, 45, 45, and 4, resectively. These 68 shaes are used as queries for subjective test. The exerimental results in terms of average recision-recall are given in Fig. 7(a-e). The exerimental results show that the roosed method on class A, A3 and B has the best erformance. It can be seen from Fig. 7 that there is only slight difference of retrieval erformance between WM, ZM and LM on class A. It means that the roosed algorithm is referable to recognize similarity transformed and visually similar shaes, the similarity transforms here include scaling, rotation and TRS hybrid transform. It can be also seen from Fig. 7 that erformance of the roosed method on database A4 is not satisfying, while orthogonal Fourier-Mellin moments offered the best erformance. It means that the new method is not suitable for indentifying the similarity after ersective transform. However, ersective similarity is not our focus comared with scaling similarity and rotation similarity. Figure 6 resents some images in CE-A4. Fig. 6 Examles from database CE-A4 which is done the ersective transform 6. Retrieval exeriments on self-constructed trademark data set The self-constructed trademark data set comrises 75 different trademarks, in which 3 are downloaded from the internet, and 45 are designed by local changing the existed trademarks. It is designed for testing the algorithms robustness against structural changes of trademark images. The designed trademarks are classified into 45 grous with visually similar shaes in each. These 45 trademarks are used as queries. The exerimental result in terms of average recision-recall is deicted in Fig.7 (f), which shows that the W-moments method has acquired the

11 Journal of Advanced Mechanical Design, Systems, and Manufacturing, Vol.8, No.6 (4) highest retrieval accuracy. (a) CE-A (b) CE-A (c) CE-A3 (d) CE-A4 (e) CE-B (f) self-constructed trademark data set Fig. 7 PR curves of different retrieval methods on the benchmark databases MPEG-7 CE and self-constructed trademark data set 6.3 Retrieval exeriments on the databases CE-A with eer noise added Artificial trademarks are easily corruted. Therefore, robustness against noise is imortant for trademark retrieval methods. Exeriments are conducted in this subsection to evaluate the robustness of the roosed algorithms against noise. In the exeriments, eer noise of density.3 (the density is adjustable) is firstly added to each trademark in database MPEG-7 CE-A, trademark retrieval on the olluted database is then erformed. The exerimental results in terms of average recision-recall are deicted in Fig.8. As it is ointed in exerimental results, the roosed algorithm is more robust against noise comaring to other 5 classical moments (ZM, HM, OFMM, LM, and CM). From Fig. 8, we can also see that Hu moments, geometric centroid moments are very sensitive to noise.

12 Journal of Advanced Mechanical Design, Systems, and Manufacturing, Vol.8, No.6 (4) Fig. 8 PR curves of different retrieval methods on the databases MPEG-7 CE-Awith eer noise added 6.4 Performance comarison in comutation time Average retrieval time for above exeriments is resented in Table, from which we can see that the erformance time for WM is less than that for ZM, CM, and much less than that for OFMM and LM in all exeriments. Although HM resents a relatively low comutational cost, it is very sensitive to noise and suffer from higher degree of information redundancy. Table Average retrieval time for trademark retrieval exeriments by different methods WM ZM CM HM OFMM LM CE-A CE-A CE-A CE-A CE-B Self-constructed database CE-Awith eer noise added Conclusion In this aer, we have roosed a class of new moment functions, W-moments, based on a hybrid orthogonal function system, W-system of degree one. The W-system of degree one is comosed of iecewise olynomials of degree one with multi-level jum discontinuities. Owing to orthogonality of the basis function set in the discrete domain of image coordinate sace, finite W-moments can be used to accurately reconstruct the corresonding image. The W-moments not only rovide a comact and nonredundant set of descritors, but also are robust against noise and comutationally simle, because each basis function is suorted only on a right-oen interval and exressed as an easy comuting linear function. The roosed algorithm has been tested on MPEG-7 CE and a self-constructed trademark data set. Comarison exerimental results show that our algorithm can kee excellent invariance in image translation, rotation and scaling and the retrieved results match human visual ercetion very well. The simlicity and reroducibility of our method makes it feasible and efficient in ractical alication. Acknowledgement This research is suorted by National Key Basic Research Project of China under Grant No. CB34, National Natural Science Foundation of China under Grant No. 676, No. 6379, No , Beijing Natural Science Foundation Program and Scientific Research Key Program of Beijing Municial Commission of Education under Grant No. KZ9.

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