，Hongzhang Jinand Liying Zheng

（1．College of Automation，Harbin Engineering University，Harbin 150001，China；2．College of Computer Science and Technology，Harbin Engineering University，Harbin 150001，China）

1-D Directional Filter Based Texture Descriptor in Fractional Fourier Domain

Kai Tian1?，Hongzhang Jin1and Liying Zheng2

（1．College of Automation，Harbin Engineering University，Harbin 150001，China；2．College of Computer Science and Technology，Harbin Engineering University，Harbin 150001，China）

Texture analysis is a fundamental field in computer vision.However，it is also a particularly difficult problem for no universal mathematical model of real world textures.By extending a new application of the fractional Fourier transform（FrFT）in the field of texture analysis，this paper proposes an FrFT-based method for describing textures.Firstly，based on the Radon-Wigner transform，1-D directional FrFT filters are designed to two types of texture features，i.e.，the coarseness and directionality.Then，the frequencies with maximum and median amplitudes of the FrFT of the input signal are regarded as the output of the 1-D directional FrFT filter.Finally，the mean and the standard deviation are used to compose of the feature vector.Compared to the WD-based method，three benefits can be achieved with the proposed FrFT-based method，i.e.，less memory size，lower computational load，and less disturbed by the cross-terms.The proposed method has been tested on 16 standard texture images.The experimental results show that the proposed method is superior to the popular Gabor filtering-based method.

fractional Fourier transform；texture analysis；Radon-Wigner transform；1-D directional window

1 Introduction

The fractional Fourier transform（FrFT）initially proposed by Candon in 1937 was reintroduced and reinterpreted by Namias in 1980［1］and by McBride and Kerr［2］，respectively.From then on，great attention has been paid to the FrFT，and many useful characteristics have been derived［3］.So far，as a tool for investigating complex signals and images，the FrFT has been applied to many respects of optical engineering and signal processing，such as filter design［4］，joint timefrequency offsets detection［5］，cross-terms suppression in the Wigner distribution［6］，and signal compressing［7］.However，there are few studies of applying the FrFT to texture analysis.

The texture analysis is particularly difficult for no universal mathematical model of real world textures［8］. Stimulated by the evidence from the psychophysical studies that visual perception is achieved by the accurate analysis of spatial frequency contents in local areas，many time-frequency tools are used for texture analysis.Such tools include Gabor transformation［9］，wavelet analysis［10］，and Wigner distribution（WD）［11-12］.Among them，the WD-based texture image analysis method，which possesses good resolution in both spatial and frequency，is more consistent with the human vision system.The studies of Zhu［11］and Reed［12］have shown that the 2-D WD presents the advantage of efficiency for extracting the pertinent features of a textured image.However，the WD-based texture analysis suffers from heavy load of computation and large memory size，as well as the interferences in the WD.To solve these problems，an FrFT-based method for analysis textured images is proposed in this paper.Compared to the WD-based method，the proposed method requires less memory size，lower computational load，and less disturbed by the crossterms.

2 Directional FrFT Filter

2.1 Basics of the FrFT

The FrFT is a linear transformation with the transform orderα∈（0，4］．Mathematically，it maps a signal，f（t），onto（Fαf）（u）by the following equation［3］：

Here the transform kernel

One of the most important relations of the FrFT to the time-frequency representations is that of the Radon-Wigner transform：

whereWfis the WD off（t）．Eq.（3）indicates that the projection of the WD of a signal onto an axis making angleφαwith theuaxis is equal to the squared magnitude of theαth order FrFT of the signal.Thus，it can be deduced that｜（Fαf）｜2possesses timefrequency distributing characteristics off（t）．

2.2 1-D Directional FrFT Filter

This paper extracts texture features from the squared magnitude of the FrFTs of a texture image based on the following two considerations：1）the squared magnitude of the FrFT of a signal is real，leading to an easy process to characterize textures；and 2）the squared magnitude of the FrFT of a signal is the Radon transform of the WD of the signal，from which some important texture characteristics can be revealed.

Because the memory size as well as the computational load will be increased greatly for the feature extraction based on 2-D window，a 1-D directional FrFT filtering that is inspired by Gabarda et al.［13］is proposed in this correspondence.Firstly，several 1-D windows of lengthNWalong different directions and centered at an image pixel are employed to produce directional 1-D series.Let｛I（px，py）｝be the 1-D series of pixel（m，n）along angleθ．Then，mathematically，the position（px，py）can be computed with Eqs.（4）and（5）.

Then each 1-D series of a pixel is transformed intoNFrfractional Fourier domains with Eqs.（1）and（2），gettingNFrFrFT series ofNWlength.Here，NFris the number of FrFT performed on each series.It is clear now that the proposed 1-D directional FrFT filtering consists of two sequential steps：1-D directional series calculation followed by several FrFTs.

3 Texture Feature Extraction

The success of classification largely depends on the ability of the method to describe textures.Because the coarseness and directionality are two essential perceptual cues used by the human visual system for discriminating different textures［14-15］，the orientation and the spatial-frequency are used as texture features.

The above mentioned 1-D directional FrFT filters is used to measure those two types of features.Firstly，the 1-D directional FrFT filtering appliesNDr1-D windows along different directions to the textured image，gettingNDr1-D directional series ofNWlength for each pixel（NDris the number of 1-D windows）.By doing so，the orientation characteristics of the textured image can be measured.Each of the 1-D series is then transformed intoNFrfractional Fourier domains，gettingNFr1-D FrFT series ofNWlength.From Eq.（3），it is known that the squared magnitude of the αth order FrFT of a signal intimately related to its Wigner distribution.Thus，the above obtainedNFr1-D FrFT series must contain spatialfrequency characteristics of the textured image. Furthermore，to reduce the size of features，the frequencies with maximum and medium amplitudes of each FrFT series are selected as the feature parameters.

Now the length of the feature vector for each pixel isNDr×NFr，whereNDrandNFrare related to the orientation and space／spatial-frequency distribution of the texture，respectively.Then，the mean and the standard deviation of the featured image are used to compose of the feature vector of the class represented by the input image，i.e.

wherei＝1，2，…，NDr×NFr，andμiandσiare the mean and standard deviation of the class；WandHthe widthand height of the image；I-Feathe featured image. The feature vectorVis constructed as

Fig.1 summarizes the proposed FrFT-based texture description method.Firstly，NDr1-D windows are applied to the input image to get the 1-D directional series of each pixel with Eqs.（4）and（5）.Then， each 1-D directional series is transformed intoNFrfractional Fourier domains，and the frequencies with maximum and medium amplitudes are chosen as the feature parameters.Next，the mean and the standard deviation of the featured image are calculated with Eqs.（6）and（7），based on which the feature vector of the input image can be constructed with Eq.（8）.

Fig.1 Main stages of the proposed method

There are two benefits which can be achieved with the above FrFT-based method.Firstly，｜（Fαf）｜2is real leading to an easy process to characterize textures. Secondly，the Wigner transform of a 1-D signal is a 2-D distribution，but theαth order FrFT of such a 1-D signal is still one dimensional，resulting in a less memory size required for the FrFT-based method.

4 Experimental Results and Analysis

To demonstrate the performance of the above texture descriptor，it is applied to texture classification. Here，the nearest neighbor classifier with the standardized Euclidean distance computed by Eq.（9）is employed to classify the input texture image.

whereVaandVbare two feature vectors；Pthe length of the feature vector andCithe standard deviation of theith class.

The tested 16 texture images，each of which is with the size 128×128，are shown in Fig.2.Each input texture image is divided into 49 overlapped sub-images with size of 32×32.21 of them are as the training samples，while the other 28 are test samples.

Firstly，to select suitable transform orders，we letNFr＝1，and the transform order，α，varies from 0.1 to 0.9 with step 0.1.FixNW＝8 andNDr＝4，（the orientations of the 1-D windows are 0，π／4，π／2 and 3π／4，respectively）.Table 1 lists the classification rates with different values ofα．

Then，setNW＝8，NDr＝4，andNFr＝2 to evaluate the effects ofNFr．The classification rates are listed in Table 2 in which the first three groups are selected from the best threeαin Table 1，while the fourth one corresponds to the best and the worstαin Table 1.The last group is the combination of the twoαwith the two worst classification rates.

Table 1 Classification rates withNDr＝4，NW＝8，NFr＝1

Table 2 Classification rates withNDr＝4，NW＝8，NFr＝2

The comparison result between Table 2 and Table 1 shows that the performance ofNFr＝2 is better thanNFr＝1.However，one should note that the memory size as well as the computational load required forNFr＝2 are greater thanNFr＝1.It is worth mentioning thatthe setting ofα＝｛0.2，0.9｝gives the algorithm the best classification rate forNFr＝2．From the view of the phase plane，the 0.2th order FrFT possesses more characteristics from the time domain，whereas the 0.9th order FrFT has more characteristics from the frequency domain.This provided us with this insight：when selecting the transformation domains，one may make some of them approach the time domain，while others approach the frequency domain.

Finally，we perform a comparison to the Gaborbased method that is a very popular method for texture analysis.The parameter settings areNDr＝4，NW＝8，andα＝｛0.2，0.9｝.The results in Ref.［16］gives 22 misclassified patches with the quadratic distance，i.e.（as shown in Fig.2），95.1%classification rate，while our proposed method gives 14 misclassified patches，i.e.96.88%classification rate.The results show that the FrFT-based method is better than the Gabor filters based method.

Fig.2 16 textures in Ref.［16］

5 Conclusions

Stimulated by the evidence from the psychophysical studies that visual perception is achieved by the accurate analysis of spatial frequency contents in local areas，many time-frequency tools are used for texture analysis.In this paper，considering the close relationship between the FrFT and the Wigner distribution，a new texture description method is proposed.It enjoys an easy process to characterize textures，less memory size，and less disturbed by the cross-terms.The method has been tested on 16 benchmark textures.From the simulation results，we can conclude that when selecting the transformation domains，one may make some of them approach the time domain，while others approach the frequency domain.Furthermore，the simulation results also show that the classification rate based on the quadratic distance can reach 96.88%which is better than that of the popular texture analysis method，i.e.，Gabor filtering based method.

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TP391.4

：

：1005-9113（2015）05-0125-04

10.11916／j.issn.1005-9113.2015.05.019

2014-09-12.

Sponsored by the National Natural Science Foundation of China（Grant No.61003128）.

?Corresponding author.E-mail：tiankai＠hrbeu.edu.cn.