Statistical Modeling of the Gabor Filter Magnitude Using Gamma Distribution for Effectively Vehicle

Verification

Jing-Ming Guo Department of Electrical Engineering

National Taiwan University of Science and Technology Taipei, Taiwan

E-mail: jmguo@seed.net.tw

Heri Prasetyo Department of Electrical Engineering

National Taiwan University of Science and Technology Taipei, Taiwan

E-mail: heri_inf_its_02@yahoo.co.id

Abstract— Vehicle verification based on still image feature can be considered as supervised classification problem. An image descriptor is directly derived from the Gabor filtered output statistics of a given image. In general, the magnitude of the Gabor filtered output is modeled as the Gaussian distribution. So that the image descriptor is composed from mean, standard deviation, and skewness value of the Gabor filter magnitude [5, 6, 8]. However, Arrospide et. al. [9] argued that the skewness parameter is not meaningful for the class separation. Then, the feature descriptor is well defined only using mean and standard deviation of Gabor output distribution which leads to lower feature dimensionality. Based on our observation, the magnitude of the Gabor filter has strong tendency to follow the Gamma distribution. We propose a new texture descriptor derived from the maximum likelihood estimation of the Gamma distribution for effectively vehicle verification task. Experimental result shows that the proposed method is superior to the former approach under several classifier techniques.

Keywords- Gabor filter; Gamma distribution; maximum likelihood estimation; supervised classification; vehicle verification.

I. INTRODUCTION The Gabor filter is a powerful tool to extract the main

feature of an image. The Gabor kernel exhibits the desirable characteristics of spatial locality, spatial frequency and orientation selectivity [1]. Gabor filter and its wavelet have been broadly applied in several image processing and computer vision applications, such as face recognition [3], traffic sign detection [4], and vehicle verification [5-9]. The Gabor filter with different scales and orientations can be used to build the different Gabor wavelet to enrich the image feature in the feature extraction stage. The Gabor filter can be applied for whole image or in smaller sub-window in an image.

The recent application of the Gabor filter usage is in the vehicle verification on road and highly traffic activity. The vehicle detection tries to recognize the occurrence of the moving object such as car moving, human moving, etc. The result of the vehicle detection is fed into the vehicle verification stage. The vehicle verification performs the class labeling by assigning the vehicle hypothesis obtained from the

vehicle detection stage as positive (vehicle) or negative class (non-vehicle). The classifier is involved in the verification stage to capture the data generalization and to make decision for the vehicle class assignment.

The method in [5-9] employed the Gabor filter to extract the feature descriptor of an image. An image is simply represented by their corresponding feature descriptor constructed from the Gabor filter output result. The vehicle verification verifies the correctness of the vehicle candidate using the extracted feature descriptor. In the former scheme, the feature descriptor is composed from the Gaussian statistical value [5-9]. They assumed that the Gabor filter magnitude have Gaussian distribution. The vehicle verification based on Gabor filter had a good result in order to validate the correctness of the vehicle occurrence.

The rest of this paper is organized as follows: The related work for vehicle verification is briefly introduced in Section II to facilitate the understanding of the prospective readers. Section III elaborates the modeling of the Gabor filter magnitude using Gamma distribution. The proposed method for vehicle verification is delivered in Section IV. Extensive experimental results are reported in Section V. Finally, the conclusion is drawn in Section VI.

II. RELATED WORK Gabor wavelets are widely and successfully applied in

computer vision application, such as face recognition [3], sign detection [4], and vehicle verification [5-9], due to their biological relevance and computational properties. The Gabor kernels have similarity with the receptive field profiles of the mammalian cortical simple cells [1]. The Gabor wavelet coefficient can be computed by performing the dilation and translation of the Gabor mother wavelet. In the spatial domain, a Gabor wavelet is a complex exponential filter modulated with a Gaussian filter function. The Gabor wavelets in the spatial domain �� � ���� �� can be defined as follows:

������ � ���� �

�� ��� �� ���� ��������� � �������������� � ��� !� ��

� "#, (1)

where $ and % define the orientation and scale of the Gabor kernel, $ � &'�(� ) � * � (+ and % � &'�(� ) � , � (+ . The

978-1-4799-0434-1/13/$31.00 ©2013 IEEE ICICS 2013

symbol �-� denotes norm operation and � � .�(. The wave vector �����can be calculated as:

���� � ������&�/�+, (2)

where �� � �01234� and /� � 5$367 �012.

The Gabor filter parameter can be set as * � 8, , � 6, 9 � :5, �012 � 53:, and 4 � .:, to achieve optimum result as suggested in [1]. Fig. 1 shows the real part of the Gabor filter. The Gabor filter representation of an on-road image (vehicle and non-vehicle) can be obtained by convolving the image with Gabor filters. Let ;��� be the gray level distribution of an image of size < = >. The convolution of an image with a Gabor kernel ������ is defined as follows:

?������ � ;��� @ ������, (3)

where @ denotes the convolution operator, and ?������ is the convolution result corresponding to the Gabor kernel at scale % and orientation $ . The set of Gabor wavelet representation of image ;��� is denoted as:

A � &?������B$ � '�) � * � (C �% � '�) � , � (+. (4)

The Gabor filter output magnitude can be computed as:

D������ � EF���?������ G ;H��?������ . (5)

Fig. 2 shows an example of the Gabor filter convolution result. Fig. 2 (a) and (b) denote the vehicle image and the real part of its corresponding Gabor convolution result.

The feature descriptor can be constructed from the Gabor filtering result. Several strategies utilize the raw Gabor responses, the thresholded Gabor features, Gabor energy features and grating cell operator features to construct an image descriptor. The main drawback of these strategies is in the high dimensionality of the feature descriptor (as large as the size of the image) causing heavy training and classification task. To avoid this problem, the statistical parameter of the Gabor filtered output is preferred to represent the entire distribution of the Gabor filters output. The simple idea is to choose an underlying assumption to specific distribution, and then the distribution parameters are computed to construct the feature descriptor.

In the former approach, the Gaussian distribution is chosen [5-9] to describe the statistical nature of the Gabor filter magnitude in the vehicle verification task. In [5-8], an image descriptor is constructed using the mean, standard deviation and skewness of the Gabor filter magnitude. Further investigation found that the skewness is less meaningful for class separation in the vehicle verification problem [9]. So that the feature descriptor is proposed based on Gaussian assumption by removing the skewness term as follow [9]:

4I�� � �JK�K� 9K�K� JK��� 9K��� ) � JLM��NM�� 9LM��NM� , (6)

where J��� � �OP Q Q D������P��R�O�SR� and 9��� �

E �OP Q Q TD������ � J���U�P��R�O�SR� .

III. MODELING GABOR FILTER MAGNITUDE USING GAMMA DISTRIBUTION

The probability density function (PDF) of Gamma distribution can be expressed as:

���C V� W � XYZ�[ �[M� ���&�V�+, for � \ ', (7)

where ]�^ is Gamma fuction which is defined as ]�^ �_ ����&��+�1M�`�aK for ^ b '.

Given the independent sample data as c � &��� ��� ) � �d+, where �e � D������ and f � <>. The likelihood function of this sample can be defined as:

g�cBV� W � h ���eBV� WdeR� �i XYZ�[j

d ����� )�d[M�����&�V���G�� G kG �d+, (8)

where V and W are parameters to be estimated. Then the log likelihood of the function g�cBV� W is given by:

l�cBV� W � mno h ���eBV� WdeR� , (9)

With simple algebraic manipulation, the equation (9) can be rewritten as:

l�cBV� W � f&W mno V � mno ]�W+ G �W � (Q mno ��edeR� � VQ �edeR� ,(10)

The log likelihood equations have a unique root. The Maximum-Likelihood Estimation (MLE) can find the probability parameter estimator of Vp and Wq from the sample data c.

By differentiating the log-likelihood function with respect to V, �W and setting them to zero, we will obtain: rrX �l�cBV� W � f [s

Xt � Q �edeR� � ', (11)

rr[ �l�cBV� W � f !mno Vp � Zu�[s

Z�[s " G Q mno ��edeR� � ', (12)

The equation (11) yields [sXt � �d Q �edeR� . By substituting this

value into equation (12), we can get:

f !mno Wq � mno �d Q �edeR� � Zu�[s

Z�[s " G Q mno ��edeR� � ', (13)

where Zu�[s

Z�[s is digamma function defined as Zu�[s

Z�[s v wxyzWq{ ���[s � �

��[s� G ���K[s| � k.

Defining the geometric mean as ��� !�d Q mno �edeR� " �

&h �edeR� +S}, then rewriting equation (13) as follow:

mno Wq � Zu�[sZ�[s � mno �

d Q �edeR� � mno ��� !�d Q mno �edeR� ",

mno Wq � Zu�[sZ�[s � mno �

d Q �edeR� � mno&h �edeR� +S},

mno Wq � Zu�[sZ�[s � mno

S}Q 2~}~�S�h 2~}~�S

S}, (14)

Let � � mnoS}Q 2~}~�S�h 2~}~�S

S}, then the equation (14) can be solved

numerically with the Newton-Raphson method using the following iteration scheme as:

Wq � Wq � ��� [sM�u�Ys��Ys M�

SYsM �

�Y�u�Ys��Ys

, (15)

The function ���

��������� is known as trigamma function.

To achieve faster convergence, the initial solution for the Newton-Raphson is given by:

�� v �M�����M��������� , (16)

To measure the fitting accuracy of each distribution model, we use the goodness-of-fit statistic value �� and Kullback-Leibler Divergence (KLD) as:

�� � Q T���~M���~CXt�[sU����~CXt�[s

�eR� , (17)

�g� � _ ���eC Vp� Wqe mno ���~CXt�[s���~ � Q ���eC Vp� Wqmno ���~CXt�[s

���~�eR� (18)

where ���e and ���eC Vp� Wq represent the empirical and expected frequency for the Gabor filter output magnitude D������, respectively. The lower value of the �� and KLD yield better fitting result. Fig. 3 shows the data fitting from the Gabor filter output magnitude using the Gaussian and Gamma distribution assumption. From this figure, we can make conclusion that the Gamma distribution is well matched for Gabor filter magnitude. In our experiment, there are more than 98% of the images in dataset have better fitting result with the Gamma distribution.

IV. PROPOSED VEHICLE VERIFICATION SYSTEM The general framework for vehicle detection is formerly

proposed in [5] consisting the vehicle hypothesis generation and hypothesis verification. The occurrence of moving object is detected in the vehicle hypothesis generation. Later on, the vehicle verification examines this hypothesis by transforming this problem into the two-class pattern classification problem (i.e., vehicle versus non-vehicle). In the pattern classification system, the feature is firstly extracted from an image, and then classifier makes decision from the extracted feature. The classifier learns the data behavior from the training set to make generative hypothesis for testing set. The vehicle verification problem can be regarded as supervised learning where a class label is given for each training instances.

Fig. 4 illustrates the schematic diagram of the proposed vehicle verification system. There are two major step involved in the proposed method, i.e. Gabor filtering and classification. Let ;��� be an input grayscale image. The color image is firstly converted into grayscale image. The Gabor feature extraction is performed by convolving image ;��� with Gabor filter bank ������ for all scales and orientations. Then the maximum likelihood estimation under Gamma distribution assumption is performed on the Gabor filter output magnitude D������ to obtain the distribution estimator Vp and Wq . Thus, the

feature descriptor for the vehicle verification is composed from Vp and Wq as follow:

4X�[ � �VK�K� WK�K� VK��� WK��� ) � VLM��NM�� WLM��NM� . (19)

In the classification procedure, the classifier learns the data generalization from a given training set. The classifier assigns a class label based on the extracted feature descriptor. The proposed method employs several classifiers such as K-Nearest Neighbor (KNN), Bayes decision classifier, and minimum distance classifier. The majority voting and different distance, i.e. Euclidean and Manhattan distance, are used in the KNN classifier for finding the K-closest neighbor. At the end of the proposed method, the class label (vehicle or non-vehicle) will be returned for an input image.

V. EXPERIMENTAL RESULT In this section, several experimental results were reported

to demonstrate the effectiveness of the proposed method. The experiments were conducted with the open access GTI vehicle image database [10], consisting 8000 color images of size �� = ��. All images in database are grouped into two classes, i.e. positive (vehicle rear) and negative class (non-vehicle and other elements in traffic sequences), and each class consist of 4000 images divided into four groups (far, left, middle close, and right). The Gabor filtering is applied to construct the feature descriptor by converting the color images in dataset into monochrome images. The proposed descriptor captures the texture information rather than color. Fig. 5 shows an example of vehicle and non-vehicle sample of GTI database. Each row presents sample in the far, left, middle close, and right pose, respectively from top to bottom.

Five-fold cross validation method is used to measure the performance accuracy by computing the proportion of correct prediction in the testing set given the training set. Several classifiers are chosen to predict the class label of the testing set.

Table I shows the vehicle verification accuracy under different classifier and vehicle pose. The verification stage for vehicle images is performed between the Gaussian [9] and Gamma texture descriptor under the same experimental setting. We use the same training and testing set for each experiment/cross validation fold. Our proposed feature descriptor based on Gamma maximum likelihood estimation significantly outperforms than the Gaussian assumption feature descriptor under all different classifier, i.e. minimum distance classifier, Bayes decision classifier, and k-NN. It is noteworthy that the proposed feature descriptor can offer a better result compared with [9] with the same dimensionality.

Fig. 6 shows a simple application of vehicle detection and verification as formerly suggested in [5]. First, the vehicle hypothesis is generated from a single image as illustrated in Fig. 6 (a). Subsequently, the vehicle verification inspects and concludes this hypothesis falls into the vehicle or non-vehicle object. Fig. 6 (b) exhibits the correct vehicle detection when the Gamma feature descriptor is utilized in the vehicle verification stage. The proposed feature descriptor can perform well for vehicle verification task.

VI. CONCLUSION In this paper, a new feature descriptor based on the

maximum likelihood estimation under the Gamma distribution has been proposed. The proposed feature descriptor is derived based on assumption that the magnitude of the Gabor filter output is well modeled using the Gamma distribution. The Gamma distribution gives the better statistic fitting compared with Gaussian distribution. The proposed feature descriptor outperforms than the Gaussian feature descriptor as reported in the experimental result section. In the future work, another separable filter can be employed to generate an image feature descriptor. The heavy-tailed distribution can be investigated to give the better statistics fitting which causes the better accuracy for the vehicle verification problem.

Fig. 1. Real part of the Gabor filter

(a)

(b)

Fig. 2. Gabor filter convolution result, (a) vehicle image, and (b) real part of the convolution result of (a) with Gabor filter.

Fig. 3. Gaussian and Gamma fitting for Gabor filter output magnitude.

Fig. 4. Schematic diagram of the proposed vehicle verification system.

(a)

(b)

Fig. 5. Example of vehicle (a) and non-vehicle image (b) from GTI database.

(a) (b)

Fig. 6. Example of vehicle verification, (a) hypothesis generation, and (b) hypothesis verification.

Table I. Average Accuracy Comparison

Vehicle Pose

Bayes Decision KNN-L2 KNN-L1 Minimum Distance

Gaussian Gamma Gaussian Gamma Gaussian Gamma Gaussian Gamma

Far 0.852�0.142

0.949�0.060

0.865�0.198

0.952�0.069

0.858�0.189

0.956�0.071

0.640�0.315

0.997�0.002

Left 0.942�0.057

0.973�0.035

0.969�0.027

0.989�0.015

0.960�0.036

0.992�0.012

0.696�0.263

1.000�0.000

Middle Close

0.960�0.058

0.977�0.026

0.913�0.120

0.997�0.004

0.916�0.118

0.992�0.012

0.784�0.249

1.000�0.000

Right 0.911�0.114

0.964�0.065

0.956�0.059

0.983�0.024

0.953�0.051

0.986�0.026

0.739�0.185

0.998�0.004

Average 0.916 0.966 0.926 0.980 0.922 0.982 0.715 0.999

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[4] J. G. Park, and K. J. Kim, “Design of visual perception model with edge-adaptive Gabor filter and support vector machine for traffic sign detection,” Expert Systems with Applications, vol. 40., no. 9, pp. 3679-3687, July 2013.

[5] Z. Sun, R. Miller, G. Bebis, and D. Dimeo, “A real-time precash vehicle detection system,” Proc. 6th IEEE Workshop on Applications of Computer Vision, pp. 171-176, 2002.

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[7] Y. Zhang, S. J. Kiselewich, and W. A. Bauson, “Legendre and Gabor moments for vehicle recognition in forward collision warning,” IEEE Intell. Transp. Syst. Conf., Canada, pp. 1185-1189, Sept. 2006.

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[10] GTI Vehicle Image Database, (2011), [Online] Available: http://www.gti.ssr.upm.es/data/

0 0.5 1 1.5 2 2.50

0.5

1

1.5

2

2.5

3

3.5

x

p(x)

Gamma:χ2=257.08,KLD=30.90, Gaussian:χ2

=244783.38,KLD=36.89

Original DataGamma DistributionGaussian Distribution

0 0.5 1 1.5 2 2.5 3 3.5 4 4.50

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

x

p(x

)

Gamma:χ2=111.70,KLD=10.77, Gaussian:χ2=56090.84,KLD=9.90

Original DataGamma DistributionGaussian Distribution

0 0.5 1 1.5 2 2.50

0.5

1

1.5

2

2.5

3

3.5

4

x

p(x)

Gamma:χ2=228.18,KLD=44.19, Gaussian:χ2

=16948.97,KLD=53.68

Original DataGamma DistributionGaussian Distribution

0 0.5 1 1.5 2 2.5 3 3.50

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

x

p(x)

Gamma:χ2=110.83,KLD=22.18, Gaussian:χ2=612.71,KLD=27.67

Original DataGamma DistributionGaussian Distribution