Image and Vision Computing 28 (2010) 1524–1529
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Image and Vision Computing journal homepage: www.elsevier.com/locate/imavis
Pulse-coupled neural networks and one-class support vector machines for geometry invariant texture retrieval Yide Ma a,*, Li Liu a, Kun Zhan a, Yongqing Wu b a b
School of Information Science and Engineering, Lanzhou University, Lanzhou, Gansu Province 730000, People’s Republic of China School of Mathematics and Statics, Lanzhou University, Lanzhou, Gansu Province 730000, People’s Republic of China
a r t i c l e
i n f o
Article history: Received 2 November 2009 Received in revised form 4 March 2010 Accepted 5 March 2010
Keywords: Pulse-coupled neural network (PCNN) Intersecting cortical model (ICM) Texture retrieval Support vector machine (SVM) Feature extraction
a b s t r a c t The pulse-coupled neural network (PCNN) has been widely used in image processing. The outputs of PCNN represent unique features of original stimulus and are invariant to translation, rotation, scaling and distortion, which is particularly suitable for feature extraction. In this paper, PCNN and intersecting cortical model (ICM), which is a simplified version of PCNN model, are applied to extract geometrical changes of rotation and scale invariant texture features, then an one-class support vector machine based classification method is employed to train and predict the features. The experimental results show that the pulse features outperform of the classic Gabor features in aspects of both feature extraction time and retrieval accuracy, and the proposed one-class support vector machine based retrieval system is more accurate and robust to geometrical changes than the traditional Euclidean distance based system. Ó 2010 Elsevier B.V. All rights reserved.
1. Introduction Texture analysis plays a very important role in computer vision and pattern recognition. Texture is a salient and indispensable feature in indexing the content-based images, and it presents in almost all real world images with intrinsic properties of periodicity, coarseness, inherent direction and pattern complexity. How to extract significant, compact and effective geometry invariant image features is a hot issue on texture retrieval. Considerable researches have been done on invariant texture feature extraction with numerous algorithms being developed based on different models. Early approaches for texture feature retrieval are based on second-order statistic, such as gray level co-occurrence matrix (GLCM) [1–3]. GLCM constructs matrices depending on the orientation and distance between pixel pairs by space relevant of textures in gray level, but the retrieval accuracy may not be very high. More prevalent texture analysis method is Gabor filter [4,5], which skillfully performs a multichannel representation in line with the multichannel filtering mechanism of the human visual system. But since the conventional Gabor representation is variant to the different orientations and scales, its extracted features often produces a fairly unacceptable performance in retrieving the rotated and scaled versions of the query texture image. A rotation-invariant and scale-invariant Gabor representations * Corresponding author. Tel.: +86 0931 8912786/013993112999; fax: +86 0931 8912779. E-mail address:
[email protected] (Y. Ma). 0262-8856/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.imavis.2010.03.006
are proposed in [6], and each representation only involves a simple modification of the conventional Gabor filter. During the last decade, the pulse-coupled neural network (PCNN), which is derived from the phenomena of synchronous pulse bursts in the animal visual cortex, is developing rapidly in image processing. Using the characters of neuron capture and synchronous pulse activity, PCNN can resolve the feature extraction problems commendably. When being applied for feature extraction, PCNN can get features of translation, rotation, scaling and distortion invariance, and the accuracy is improved greatly [7–9]. The standard PCNN owns too many parameters and each parameter has its own specific physical significance, which make PCNN become very complex. A simplified version of the PCNN model, the intersecting cortical model (ICM), is investigated by Kinser [10] for the first time. It is based on neural network techniques and is especially designed for image processing. Since ICM inherits characters of PCNN and its computation is faster than the full PCNN model, it is becoming more and more popular in image processing [11–13]. Support vector machine (SVM) has achieved nearly perfect results in classification problem, but it involves both positive and negative information and which two are required to be balance in order to arrive at optimal classification status. In fact, the standard SVM is not very effective on one-class problem which uses only positive information. One-class classification tries to classify only one class of objective samples and distinguish it from all other possible samples. Müller et al. [14] and Schölkopf et al. [15] suggested an improved scheme of adapting SVM to solving one-class classification problem, and proposed one-class SVM (OCSVM)
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model. The strategy is to map the data into the feature space corresponding to a kernel, and then to separate them from the origin with maximum margin. We just need to estimate whether a test sample from a probability distribution lies in or out the subset of the feature space, which is bounded by some a priori specified value, so the complexity is reduced. For the speciality of solving the one-class problem, OCSVM is widely applied for image retrieval [16–19]. In this paper, OCSVM is utilized to train and predict the features, which are output of PCNN or ICM. Computational experiments show that the local-connected neural networks, such as PCNN and ICM, perform better than the previous feature extraction methods, such as the Gabor filter [6], and the proposed texture retrieval system based on one-class SVM is superior to ED based system. Here is the structure of this paper. Section 2 gives a brief introduction of the PCNN and ICM models for feature extraction. And then Section 3 states the process of OCSVM for one-class classification problem in detail. Section 4 describes the proposed invariant texture retrieval algorithm. In Section 5, the experimental results and comparisons are demonstrated and summarized. Finally, Section 6 consists of some concluding remarks. 2. PCNN and ICM for feature extraction
Fig. 1. The structural model of the pulse-coupled neuron.
which have the similar characteristics to fire synchronously. That is the capture character and synchronous pulse activity character. During the iterations, each neuron’s fire period is different and dynamic threshold is attenuating respectively. The neurons with the same weights fire at different time, that is the dynamic pulse release character. For each iteration, the total number of firings over the whole PCNN is computed and stored in an array G: P GðnÞ ¼ Nn¼1 YðnÞ, where n is the iteration ðn ¼ 1; . . . ; NÞ. That is the time series stated by Johnson [7], which display the features such as segments, edges and textures of original stimulus and have invariance of translation, rotation, scaling and distortion.
2.1. The PCNN model 2.2. The ICM model PCNN is a simplified model for the cortical neurons in the biological visual area of the cat’s brain [20]. Each neuron in PCNN is coupled to its adjacent neurons through synaptic weights, and a fired neuron is able to capture its adjacent neurons to fire synchronously. The structural model of the pulse-coupled neuron is shown in Fig. 1. There are three parts in the model: the input part, the linking part and the pulse generator part. The input part consists of feeding input F and linking input L channels. The neuron receives the input signals from F and L inputs. F is the primary input S from the neuron’s receptive field. L is the secondary input of lateral connections with the neighboring neurons. The linking part is a modulation process. L is added a constant positive unitary bias, and then it is multiplied by F, where b is the linking strength. In this way, the internal activity U is generated by the modulation of F and L. The pulse generator part includes a threshold adjuster, a comparison organ and a pulse generator. When state U is greater than the dynamic threshold H, PCNN produces the output pulse image Y. Johnson developed the full mathematical description of PCNN for image processing [7,21] as follows.
F ij ðnÞ ¼ eaF F ij ðn 1Þ þ V F Lij ðnÞ ¼ eaL Lij ðn 1Þ þ V L
X kl X
Mijkl Y kl ðn 1Þ þ Sij
W ijkl Y kl ðn 1Þ
kl
U ij ðnÞ ¼ F ij ðnÞð1 þ bLij ðnÞÞ
Hij ðnÞ ¼ eaH Hij ðn 1Þ þ V H Y ij ðn 1Þ 1 if U ij ðnÞ > Hij ðnÞ Y ij ðnÞ ¼
0
otherwise
where the ði; jÞ pair stands for the position of the neuron in the map and ðk; lÞ is that of its neighboring neurons. S is the stimulus (here is the input image). n ¼ 1; . . . ; N is the iteration number; aF ; aL and aH are the attenuation time constants of F, L, H , respectively; V F ; V L and V H denote the inherent voltage potential of F, L, H, respectively. M and W (normally M ¼ W) represent the constant synaptic weights which are computed by inverse square rule; b is the linking strength parameter. Y ij is the firing state of the ijth neuron and it gets either the binary value 0 or 1. Each neuron in PCNN is a dynamic neuron and may generate a pulse when stimulated. The pulse captures its neighboring neurons
In order to obtain minimum universal visual cortical neuron network models and reduce computational complexity, ICM was advanced by Kinser for the first time [10,11]. Compared with PCNN, there are no linking input L and internal activity U in the ICM, and thus the minimal system now consists of two coupled oscillators, a small number of connections and a nonlinear function. The state oscillators of all the neurons are represented by F while the threshold oscillators of all the neurons by H. Thus, the ijth neuron has state F ij and dynamic threshold state Hij , which decide whether the ICM neuron produces pulse sequence Y ij . The mathematical model of the ICM is described like this:
F ij ðnÞ ¼ fF ij ðn 1Þ þ
X
Mijkl Y kl ðn 1Þ þ Sij
kl
Hij ðnÞ ¼ g Hij ðn 1Þ þ hY ij ðnÞ 1 if F ij ðnÞ > Hij ðnÞ Y ij ðnÞ ¼
0
otherwise
where f ; g and h are scalars (examples of values are 0.9, 0.8, and 20.0, respectively); M is the connection function through which the neurons communicate and it also follows the inverse square rule. The scalars f and g are decay constants and thus less than 1. In order to ensure that the threshold Hij eventually falls below the state F ij and the neuron pulses, the relation between g and f must be g < f . The scalar h dramatically increases the threshold of firing neuron to make sure each neuron fires only one time and thus generally its value is large. The firings Y ij ðnÞ are the output pulse images of the ICM, which also represent unique features that are inherent in the input images and have invariance of translation, rotation, scaling and distortion. ICM is derived from several visual cortex models and is basically the intersection of these models. Therefore, it is more suitable for image processing and quite similar to the PCNN, even outperforms of the PCNN. Compared with the full PCNN model, the ICM has higher speed because of reducing the number of equations over the PCNN. Each neuron in ICM has two oscillators and a nonlinear operation. Thus, when stimulated, each neuron is capable of producing a spike sequence, and groups of locally connected neurons have the ability to synchronize pulsing activity. When stimulated
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by an image is that these collectives can represent inherent features of the stimulating image. To summary here, the PCNN or ICM can extract the image fundamentals inherently and have geometry invariance. Further more, the PCNN or ICM does not need training or adjustments to extract these fundamentals from a wide range of images. Therefore, it is an excellent choice of PCNN and ICM for texture feature extraction. 3. The OCSVM for one-class problem Schölkopf et al. [15] suggested the OCSVM algorithm to solve the one-class classification problem. The OCSVM algorithm maps training data into a high-dimensional feature space corresponding to a kernel and finds the optimal hyperplane to separates the training data from the origin with maximum margin. The OCSVM may be viewed as a regular two-class SVM when considering the origin as the only member of the second class. In this paper, let fx1 ; x2 ; . . . ; xl g be the training data belonging to the one class X; l is the number of training data and X is a compact subset of RN . Let / be a feature map: X ! F; F is a high-dimensional feature space. To separate the obtained data set f/ðx1 Þ; /ðx2 Þ; . . . ; /ðxl Þg in F from the origin with maximum margin, the following quadratic program is solved in [15]:
8 l P > < min 1 T x x q þ m1l ni 2 x2F;n2Rl i¼1 > : subject to xT /ðxi Þ > q ni ; ni P 0;
ð1Þ i ¼ 1; . . . ; l
where m 2 ð0; 1Þ, which is prior specified, has mainly two properties: (1) there are at least ml points are on or beyond the hyperplane and at most ð1 mÞl points are on the right side of the hyperplane; (2) m is related to the bound error of learning process and the number of support vectors, that means m bounds the fractions of outliers and support vectors from above and below, respectively. ni are nonzero slack variables which are penalized in the objective function. Using the Lagrange theory and kernel method [14], the dual problem of Eq. (1) is obtained as below:
8 > < min a
1 2
P
ai aj kðxi ; xj Þ
ij
> : subject to 0 < ai < m1l ;
P
ai ¼ 1
ð2Þ
i
where ai are Lagrange coefficient. And By solving Eq. (2) the kernel decision function is:
f ðxÞ ¼ sgn
l X
ai kðxi ; xÞ q
ð3Þ
i¼1
where ai is the optimal solution; q is the classification threshold; xi is the support vectors corresponding to ai > 0 in training data and x is the unknown sample (in this paper, it is the feature vector of retrieval images). f ðxÞ takes the value +1 in the database and takes the value 1 out the database. For an unknown sample x, we can decide whether it belongs to the database by calculating the value of decision function f ðxÞ in Eq. (3), which will positive for most testing samples with appropriate parameters. 4. Geometry invariant texture retrieval algorithm The outputs of PCNN and ICM are a series of binary images, which can represent unique features of original stimulus, such as texture, edge and segment. But they cannot be used directly as the feature of original image for later classification because of too large amount of data they contained. Therefore, in order to reduce the calculation complexity, some transforms for the binary images are desirable and finally the transformed data can still represent the original image. Time series is a good way to get unique
features, which counts the 1’s within the binary pulse images, that is the number of firing pixels. Besides the time series, there are other more effective methods for characterizing images. Entropy represents the energy information of images and reflects the quantity of information contained in the image. It is testified that a unique feature can be obtained by calculating entropy to each output pulse image of PCNN or ICM [8,9]. Considering the 1’s within the binary pulse images is G and the total number of pixels within the binary pulse images is Total, the probability of 1’s and 0’s in binary pulse images are calculated as P 1 ¼ G=Total and P0 ¼ 1 P1 , respectively. Then entropy can be obtained easily shown in Eq. (4), in which not only the 1’s but also the 0’s within binary pulse images are considered, so it can be used to evaluate pulse images.
Entropy ¼ P1 log2 P1 P2 log2 P2
ð4Þ
For an original image S, the N pulse images of PCNN or ICM can be compressed by Entropy into a 1 N feature vector, which is geometry invariant. In order to detect and classify images perfectly and accurately, after extracting effective features of image patterns, using an suitable classification method is very necessary and critical for texture retrieval. The distance similarity comparison is usually adopted for classification, such as Euclidean distance (ED), which calculates similarity between the query images and images in database and the top matched images can be retrieved. This method is very simple, but a predefined threshold is required and it is very difficult to find a suitable threshold value, which impacts on the results greatly. In this paper, the OCSVM is employed to train and predict the feature vector. The proposed geometry invariant texture retrieval system is shown in Fig. 2. To retrieve image is to compare the similarity between the query image and each image in the database. In the first step, crop all 112 Brodatz [22] images to a standard dimensions of 128 128 pixels as standard database and calculate feature vector for each images to achieve a signature database, which is performed off line. In the second step, it is the time for user to query, which is performed online. In order to investigate the rotation and scale invariant texture retrieval performance, the query images are geometry changed with joint rotation and scaling. After several experiments, the query images are rotation changed by bicubic interpolation and scaling changed by nearest neighbor interpolation. Of course the query images should also be resized or tailor part of the images to the standard dimensions of 128 128 pixels, though the standard image may be very different from the original image itself and the retrieval accuracy could be affected. Here in Fig. 2, OCSVM trains the features in signature database to achieve a training model. Then OCSVM uses the trained model to predict whether the features of the query image belong to the database, which returns the decision function in Eq. (3) that takes the value +1 in the database and 1 elsewhere. 5. Experiments 5.1. The setting of experimental parameters The parameters of PCNN are shown in Table 1, and the parameters of ICM are f ¼ 0:9; g ¼ 0:8; h ¼ 20; N ¼ 37 and M is the
Fig. 2. Diagram for invariant texture retrieval.
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Y. Ma et al. / Image and Vision Computing 28 (2010) 1524–1529 Table 1 The parameters set for PCNN.
Table 2 Retrieval rates of different methods based on ED (%).
Parameters
aF
aL
aH
VF
VL
VH
b
N
r, s
PCNN + ED
ICM + ED
Gabor + ED
Value
0.1
1.0
1.0
0.5
0.2
20
0.1
37
0, 0.8 0, 1.0 0, 1.2 30, 0.8 30, 1.0 30, 1.2 60, 0.8 60, 1.0 60, 1.2 90, 0.8 90, 1.0 90, 1.2
84.821 100 90.179 78.571 92.857 78.571 80.357 91.071 78.571 84.821 100 90.179
94.643 100 97.321 83.929 97.321 93.750 88.393 97.321 91.964 94.643 100 97.321
79.464 100 72.321 67.857 79.464 67.857 67.857 74.107 66.964 79.464 100 72.321
same as that of PCNN. M and W is given by [0.5, 1, 0.5; 1, 0,1; 0.5, 1, 0.5]. All of the model’s parameters are setup empirically, according to the following experimental database. The LIBSVM [23] is used in the part of OCSVM experiment. It includes five SVM models: C-support vector classification (C-SVC), m-support vector classification (m-SVC), distribution estimation (one-class SVM), -support vector regression (-SVR), and m-support vector regression (mSVR). In LIBSVM the following four basic kernels are provided:
Note: (1) r represents rotation angles, s represents scales; (2) PCNN + ED, ICM + ED and Gabor + ED are retrieval algorithms from [8,9,6], respectively.
linear : Kðxi ; xj Þ ¼ xTi xj
d polynomial : Kðxi ; xj Þ ¼ cxTi xj þ r ;
c>0
radial basis function ðRBFÞ : Kðxi ; xj Þ ¼ expðckxi xj kÞ; sigmoid : Kðxi ; xj Þ ¼ tanh cxTi xj þ r
c>0
Here, c, r and d are kernel parameters. In this paper, the one-class SVM (here is OCSVM) model and RBF kernel function are utilized. The involved parameters are m of OCSVM and c of RBF kernel. In order to produce robust results, it is quite important to set these parameters. For different samples m and c have different optimal value, and there are no uniform and auto methods to calculate them at present. In general, the cross-validation is conducted to find the best values of m and c, but when for large data with high dimension the cross-validation is not very effective and will run slowly. Here m is calculated by numerous experiments and the one with best accuracy is selected. Generally, c is the reciprocal of sample dimension [24], and here it is 1=N in Table 1. 5.2. Experimental results and comparisons The rotation and scale invariant texture retrieval performance is investigated based on the feature extraction algorithms described in previous section. The experiments are carried out to calculate retrieval accuracy using different models and different measures. The retrieval accuracy is defined as: A ¼ R=T, where R is the number of correct retrieval images and T is the total number of images in database. In the experiments, three methods are presented for invariant feature extraction: PCNN, ICM and Gabor filter. And the OCSVM and ED are employed as the classifier. Here the Gabor filter and ED are as the comparative algorithms. First of all, for a data set of the texture images with joint rotation and scale changes, we process each image in Brodatz database to standard dimensions of 128 128 pixels with different orientations (0°, 30°, 60° and 90°) and different scales (0.8, 1 and 1.2). Thus a data set of 1344 (112 4 3) texture images are created for the experiments. Then we predict their features by OCSVM compared with ED. The retrieval results of invariance performance for different models are summarized in Tables 2 and 3. The comparison of feature extraction time for different methods is exhibited in Table 4. Secondly, a further convincing modification is tried to demonstrate the performance of the proposed retrieval algorithm more intuitive, which crop images to standard size of 128 128 with fixed scaling of 1.2 and different orientations (0°–90° with 10° intervals), and predict their features by different retrieval methods aforementioned. Similarly crop images to standard size of 128 128 with fixed rotation of 60° and different scales (0.6–1.5
Table 3 Retrieval rates of different methods based on OCSVM (%). r, s
PCNN + OCSVM
ICM + OCSVM
Gabor + OCSVM
0, 0.8 0, 1.0 0, 1.2 30, 0.8 30, 1.0 30, 1.2 60, 0.8 60, 1.0 60, 1.2 90, 0.8 90, 1.0 90, 1.2
100 99.107 100 99.107 98.214 98.214 97.321 97.321 97.321 100 99.107 100
96.429 98.214 98.214 96.429 97.321 98.214 96.429 97.321 98.214 96.429 98.214 98.214
98.214 99.107 98.214 99.107 98.214 97.321 99.107 98.214 96.429 98.214 99.107 98.214
Note: (1) r represents rotation angles, s represents scales; (2) PCNN + OCSVM, ICM + OCSVM and Gabor + OCSVM are the proposed OCSVM based retrieval algorithms with parameter m ¼ 0:002; 0:01 and 0.012, respectively.
Table 4 The comparison of feature extraction time (s). r, s
PCNN
ICM
Gabor
0, 0.8 0, 1.0 0, 1.2 30, 0.8 30, 1.0 30, 1.2 60, 0.8 60, 1.0 60, 1.2 90, 0.8 90, 1.0 90, 1.2
35.151637 31.176427 33.133058 115.327992 158.700713 212.261748 113.799764 155.410759 207.942629 31.856408 31.302105 34.215897
24.096888 24.643524 23.112506 98.540965 136.981903 189.237544 97.586395 141.568160 192.420792 23.890143 24.091708 25.610726
84.230306 84.973476 85.173566 159.962541 200.312413 257.726232 161.727390 202.985878 252.740151 86.401981 86.147354 85.789743
with 0.1 intervals), and predict their features by different retrieval methods again. The results are showed in Figs. 3 and 4, which indicate the performance of proposed retrieval algorithm clearly. The results of invariance performance for different methods are shown in Tables 2–4, Figs. 3 and 4. It is clear to conclude from the results that: (1) Tables 2 and 3 show a comparison of the OCSVM based retrieval system with the ED based retrieval system according to the feature extraction methods of PCNN, ICM and Gabor. It is clear from these tables that the results of 0° and 90° rotations are higher than 30° and 60° rotations, and the results of 0° and 90° rotations are all about equal. When images are rotated with 90°, it can be matched more
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100
90
Accuracy
80
70
PCNN+ED PCNN+OCSVM ICM ICM+OCSVM Gabor Gabor+OCSVM
60
50
40
0
10
20
30
40
50
60
70
80
90
Angels Fig. 3. Performance for rotation invariance of different retrieval methods. The scale of images here is fixed to 1.2.
100
6. Conclusions
90
Accuracy
80
70
PCNN+ED PCNN+OCSVM ICM ICM+OCSVM Gabor Gabor+OCSVM
60
50
40 0.6
and more important, OCSVM has global optimal solution and solves the nonlinear classification problem in highdimension by kernels. All in all, the OCSVM shows better performance than ED. (3) In Table 4, under the same circumstances, PCNN and ICM cost less time than Gabor for feature extraction and ICM costs the least time in the three. It is because that PCNN and ICM are illumined from mammalian vision cortex neuron and closer to genuine biological neuron than Gabor filter, especially the ICM has lower complexity. (4) Figs. 3 and 4 exhibit the performance of different retrieval methods respectively from both the aspects of rotation invariance and scale invariance. It is obvious from these figures that the OCSVM based retrieval algorithms have stronger rotation and scale invariant features than ED based algorithm. In Fig. 3, PCNN + OCSVM and ICM + OCSVM have better performance of rotation invariance than Gabor + OCSVM. In Fig. 4, the accuracy followed by the order from high to low as ICM + OCSVM, PCNN + OCSVM and Gabor + OCSVM when the scales are greater than 1, and the accuracy followed by the order from high to low as Gabor + OCSVM, PCNN + OCSVM and ICM + OCSVM when the scales are lower than 1. Thus we can discover that PCNN + OCSVM has more steady scale invariance.
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
Scales Fig. 4. Performance for scale invariance of different retrieval methods. The rotation of images here is fixed to 60°.
accurate than 30° and 60° with standard database which is derived from 0°, because the coupling matrix is not able to be made completely circularly symmetric. (2) With comprehensive consideration, for ED based retrieval in Table 2, generally the retrieval rates decrease in turn for the following three methods: ICM + ED, PCNN + ED and Gabor + ED. In Table 3, for OCSVM based retrieval, PCNN + OCSVM performs better than ICM + OCSVM and Gabor + OCSVM for 0° and 90° rotations, but PCNN + OCSVM is more smooth and flat than the other two for 30° and 60° rotations with different scaling changes. Overall the accuracy of using OCSVM for classification are much better than ED. We believe there are two major reasons for this result: (1) the ED based similarity comparison need to set a threshold by hand and it is sensitive for the results, in addition, the whole similarity comparison between the query image and each image in database are carried out one by one, so it is time consuming especially for a large database; (2) however, in the OCSVM method the classification result only related to the support vectors, so it is fast for training and testing,
In this paper, for studying the problem of rotation and scale geometry invariance in texture retrieval, the algorithm of training pulse features based on OCSVM is illustrated. The series of pulse images represent information of its original image and are invariant to geometry changes. Extensive texture image retrieval experiments were conducted over the Brodatz texture database using the conventional Gabor features and pulse features based on OCSVM and ED classification methods respectively. The experimental results indicate that the proposed OCSVM based retrieval algorithm using pulse images is quite robust to geometry changes of texture patterns, and the retrieval performance is much superior to that of the ED based scheme. It arrives at a conclusion that the pulse features and statistical learning theory have a broad application prospects of image processing. Acknowledgment This paper is jointly supported by National Natural Science Foundation of China (Nos. 60572011 and 60872109), Program for New Century Excellent Talents in University (No. NCET-06-0900), and Project supported by the Natural Science Foundation of Gansu Province, China (No. 0710RJZA015). References [1] R.M. Haralick, K. Shanmugam, I. Dinstein, Textural features of image classification, IEEE Transaction on Systems, Man and Cybernetics 3 (1973) 610–621. [2] L. Soh, C. Tsatsoulis, Texture analysis of SAR sea ice imagery using gray level co-occurrence matrices, IEEE Transaction on Geoscience and Remote Sensing 37 (1999) 780–795. [3] D.A. Clausi, An analysis of co-occurrence texture statistics as a function of grey level quantization, Canadian Journal of Remote Sensing 28 (2002) 45–62. [4] B.S. Manjunath, W.Y. Ma, Texture features for browsing and retrieval of image data, IEEE Transaction on Pattern Analysis and Machine Intelligence 18 (1996) 837–842. [5] D.S. Zhang, A. Wong, M. Indrawan, G.J. Lu, Content-based image retrieval using Gabor texture features, IEEE Pacific-Rim Conference on Multimedia (PCM’00) (2000) 392–395. [6] J. Han, K.K. Ma, Rotation-invariant and scale-invariant Gabor features for texture image retrieval, Image and Vision Computing 25 (2007) 1474–1481.
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