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2016 23rd International Conference on Pattern Recognition (ICPR)
Cancún Center, Cancún, México, December 4-8, 2016
Texture Classification with Discrete Fractional
Fourier Transform
Liying Zheng* and Yan Chu*
School of Computer Science and Technology
Harbin Engineering University
Harbin, China
zhengliying@hrbeu.edu.cn
chuyan@hrbeu.edu.cn
Abstract—Based on the fact that the discrete fractional Fourier proposed by Candon in 1937. The idea was then reintroduced
transform (DFrFT) of a signal consists of not only spatial but and re-interpreted by Namias [12] and McBride and Kerr [13],
frequency characteristics, a novel texture classification algorithm respectively. So far, great attention has been paid to the FrFT,
is presented in this paper. A 1-D local DFrFT is proposed and the resulting in many applications of the FrFT on the areas such as
fractional frequency histogram is established to represent a radar signal processing[14-16], medical ultrasound imaging
texture. Moreover, the conceptions such as the between-class [17, 18], and image encryption[19-21]. Recently, some efforts
scatter and the within-class scatter, the coarse-to-fine searching, have been done on applying the FrFT to the field of pattern
2-statistic distance are also adopted to
and the minimum χ recognition. Kong et al. employed the amplitudes of the 2-D
complete the classification. The simulation results on four groups FrFT to represent the emotional state of a person [19]. Jing et
of benchmark texture images validate the performance of the al. proposed a face recognition approach by combining the
proposed texture classification method. FrFT with the discrimination analysis technique [20]. Based on
Keywords- discrete fractional Fourier transform; texture the 2-D discrete FrFT (DFrFT) and the principal component
classification; fractional frequency histogram analysis (PCA), Liu and Wang also proposed a method to
address the problem of face recognition [21]. Wang and Wang
I. INTRODUCTION applied the personal features derived from the FrFT, i.e. mean
energy within critical bands, to identify speakers [22]. Singh et
Image texture which contains important information for al. completed high-resolution SAR images categorization by
object recognition and scene interpretation is referred to as the using the log cumulants of the FrFT coefficients[23].
arrangement of certain basic elements. The ability of Studies have shown that the FrFT is intimately related to
discriminating almost all types of textures of human visual the time frequency representation of the Cohen class.
system attracts the researchers from various fields, especially Specifically, the FrFT of a signal produces a rotation of its WD
from computer vision society, to pay much attention to the task in phase space [24].It can be deduced that many important
of texture classification. local characteristics of a signal can be revealed from its FrFT.
One problem in texture analysis is the description of texture In this sense, the FrFT can potentially be used for analyzing
characteristics. Some popular texture descriptors are based on textures that have obviously spatial/spatial frequency
statistical methods, such as gray level co-occurrence characteristics. In this paper, an efficient texture classification
matrices[1], while others are based on modeling techniques, method based on the DFrFT is presented. Several 1-D local
such as Markov random field [2, 3] and fractal based modeling directional DFrFTs are firstly performed on a locally centered
[4]. Studies on human visual system show that the process of texture image. The DFrFT frequencies are then used to
texture segmentation requires high resolution in both spatial generate a histogram. Next, the scatter matrices of between-
and frequency domains. Thus, to describe a texture, many class and within-class, as well as a coarse-to-fine searching
time-frequency analysis tools have been employed. For technique are adopted to estimate the optimal DFrFT order.
example, Reed and Wechsler proposed a texture analysis 2
Finally, the texture is classified by using the minimum χ -
method by using the 2-D pseudo-Wigner distribution (PWD). statistic distance classifier.
They concluded that the WD-based method enjoys superior The remainder of this paper is organized as follows.
joint resolution [5]. Zhu et al described a framework for the Section 2 briefly introduces the DFrFT. Section 3 describes the
WD-based texture analysis [6]. furthermore, other popular 1-D local directional DFrFT and the fractional frequency
techniques such as Gabor transform[7, 8], wavelet analysis[9, histogram of a texture image in detail. Section 4 presents the
10], and windowed Fourier transform[11] have also been proposed DFrFT-based texture classification method. Section 5
employed to analyze textures. gives the experimental results followed by some conclusions in
As a time-frequency analysis technique, the fractional Section 6.
Fourier transform (FrFT) is a generalization of the
conventional Fourier transform, and its concept was initially
* Corresponding author
978-1-5090-4846-5/16/$31.00 ©2016 IEEE 2032
α
II. DISCRETE FRFT (DFRFT) Since I (x, y, m) is complex, one can use amplitude, phase
d
It is well known that a digital image is in discrete form. angle, or both, as the features to describe a texture. In this
Thus, in this paper, the discrete form of the FrFT has been paper, the amplitudes are employed for simplicity. Specifically,
employed. Let vector x = [ x(0), x(1), …., x(N-1) ] represent an the fractional Fourier frequencies corresponding to the sorted
α α th amplitudes are used. Firstly, |I(x,y,m)| are sorted in descending
N-length discrete time signal, and x and F be the α order d
DFrFT of x and the DFrFT matrix, respectively. Then, we have order. Then fractional Fourier frequencies corresponding to the
sorted |I(x,y,m)| are recoded. In accordance to the results shown
x FxT (1) d
in fig.1, theoretically, the difference of the sorted fractional
By now, several definitions of the DFrFT have been Fourier frequencies of two signals will be more obvious than
proposed [25-28]. In this paper, the DFrFT proposed by the traditional Fourier frequencies.
Ozaktas et al. [28] is employed for its excellent approximation
to the continuous FrFT. The DFrFT matrix of Ozaktas’ B. Fractional Frequency Histogram and Its Similarity
definition is given by To represent a texture, the fractional frequency histogram
F DΛHlpΛJ (2) of the above 1-D local directional DFrFT is constructed. Let
I(x,y) be a texture image with size of W by H. The fractional
where D and J are matrices representing the decimation and frequency histogram of I(x,y) is given by
interpolation operations, respectively. Λ is a diagonal matrix H(I,l,z) {H(I,l,z),H(I,l,z)...H(I,l,z)} (5)
that corresponds to the chirp multiplication, and H 1 2 D
corresponds to the convolution operation [28]. lp
with H (I,l,z) (zu(x,y,l)) (6)
Unlike the discrete Fourier transform (DFT) which d d
represents a signal in pure frequency domain, the DFrFT of a xy
signal with 0 1is an intermediate status between the spatial where D is the number of directions. δ(.) is the Dirac delta
function. z 0,1,2,...,| N | with |N | being the length of
and the frequency domains. d d
neighborhood sets at direction d and in this paper we
III. LOCAL DIRECTIONAL DFRFT AND THE FRACTIONAL let| N0 || N1 || ND |. u(x, y,l) is the fractional Fourier
FREQUENCY HISTOGRAM d
frequency corresponding to the sorted| I(x, y,m)|.
d
A. 1-D Local Directional DFrFT 2
The similarity of two textures is measured by the χ -statistic
Let I (x, y, n) be the nth directional neighbor of an image ~
d distance [29]. Let I and I be two texture images with the same
I(x,y) with direction d. To eliminate the effects due to ~
illumination, I (x, y, n) is firstly locally centered by using size, and H(I, l, z) and H(I ,l,z)be the fractional frequency
d ~
ˆ (3) histograms of I and I , respectively. The similarity of I and
Id (x, y,n) Id (x, y,n) I(x, y) ~
Define the 1-D local directional DFrFT as I is given by
D |I| |Nd| ~ 2
ˆ (4) 2 ~ (Hd(I,l,z)Hd(I,l,z))
Id (x, y,m) F (m,n)Id(x,y,n) (I,I ) ~ (7)
nNd d1l 0 z0 Hd(I,l,z)Hd(I,l,z)
th α
where Id (x,y,m) is the α order DFrFT, F is the DFrFT matrix where | I |W H .
given in (2), and N is the neighborhood indices in direction d.
d
As shown in fig.1, for each pixel P, there are four directional IV. TEXTURE CLASSIFICATION WITH THE DFRFT
0 0 0 0
neighborhood sets which are along 0 , 45 , 90 and 135 ,
α
respectively. According to the property of the DFrFT, I d (x, y, A. Summary of the Proposed Method
m) can reveal both spatial and frequency characteristics of the Fig.2 summarizes the major stages of the proposed texture
neighborhoods of I(x, y). classification method. The neighbors along direction d of each
pixel of a texture image I are firstly obtained with the similar
template shown in Fig.1, resulting in a 1-D series {I(x,y,
n )|d=0, 1, 2, 3 and n N }. Then, the locally centered
d d d
ˆ
series I(x, y,nd ) are obtained by using (3). Next, the optimal
th
α order 1-D local directional DFrFT is performed according
to (1) and (2), getting a 1-D series I(x, y,m) in the fractional
d
Fourier domain. Next, the fractional Fourier amplitudes
| I(x, y,m) | are sorted in descending order followed by
d
recording the corresponding fractional frequencies
Figure1. Illustration of directional neighborhoods. corresponding to the sorted amplitudes. Next, the fractional
frequency histogram is calculated with (5) and (6). Finally, a
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type of classifier can be adopted for completing the where S and S are the scatter matrices of between-class and
b w
classification. Here, since one of our targets is that of within-class, respectively. tr(.) means the trace of a matrix.
evaluating the performance of the proposed DFrFT texture Here, S and S are computed according to the fractional
b w
descriptor, the minimum distance classifier is adopted. frequency histogram obtained by (6) and (8). Please refer to
[20] For details.
B. Representation of a Texture Here, to get the optimal transform order, a coarse-to-fine
Assuming that there are C classes of textures in the training searching procedure which is similar to the algorithm in [30] is
set, the average fractional frequency histogram is employed to adopted. Let Δα0, λ, and ε be three positive numbers less than
th
represent the c class, i.e. 1, and αopt denote the optimal transform order .The coarse-to-
Nc fine searching procedure consists of the following five steps.
1 (c,i) Step1, α = 0, α = 0.995, Δα = Δα ;
H(c,l,z) c H(I ,l, z) (8) begin end 0
N i1 Step 2, Let α = {α , Δα+α , 2Δα+α , …, α };
begin begin begin end
c th
where c = 1,2,…C. N is the training samples in the c class. Step 3, Compute J with (9) for each α;
(c,i) th th
I denotes the i sample in the c class. To classify a texture,
2 Step 4, Select the α which maximizes the J as α ;
we compute the χ -statistic distance between its spectral opt
histogram and the average spectral histogram of each class by Setp5, α = max{0, α – 0.5Δα}, α = min{1, α +
using (8). Then, the image is classified into the class with the begin opt end opt
2 0.5Δα}, Δα= Δα×λ, go to Step 2 until Δα≤ε.
minimum χ -distance. Comparing to [30], the range of α is [0,0.995] rather than
C. Optimal Transform Order Estimation [0,1.0], since we find that when α =1.0, J may achieve the
The optimal DFrFT order α is a critical parameter for the extreme, but the performance of the proposed classification
proposed texture classification algorithm. To estimate this algorithm is probably poor.
parameter, the between-class and the within-class scatter V. E
matrices have been adopted [20]. An α which can maximize (9) XPERIMENTAL RESULTS AND ANALYSIS
is selected as the optimal transform order. A. Experimental Data
J tr(Sb)/tr(Sw) (9) The texture classification method discussed in Section 3
and Section 4 has been implemented in Matlab R2007b. The
experimental dataset are consists of four groups of benchmark
texture images which have been shown in Fig.3, being named
as Group1, Group2, Group3, and Group4, respectively.
Group1 whose images are the same with those in fig.1 of [11]
consists of 16 textures with the size of 128×128. The first four
textures in Group1 were generated by Gaussian random fields,
and the other 12 textures were chosen from Brodatz album. 10
Brodatz textures in Group2 are D4, D9, D19, D21, D24, D28,
D29, D36, D37 and D38 with the size of 320×320. These
textures are the same with fig.11(h) of [31]. Group3 is from
MIT Vision Texture database consisting of Fabric.0009,
Fabric.0016, Fabric.0019, Flowers.0005, Food.0005,
Leaves.0003, Misc.0000, Misc.0002, Sand.0000, and
Stone.0004 with the size of 256×256. The textures in Group3
are the same with fig.11(i) of [31]. Group4 is from the fifth test
suite of Outex texture database, consisting of
Outex_TC_00005, whose 24 textures are canvas001, 002, 003,
005, 006, 009, 011, 021, 022, 023, 025, 026, 031, 032, 033,
035, 038, 039, carpet002, carpet004, carpet005, carpet009,
tile005 and tile006. The size of each texture in Group4 is
320×320.
To generate training and test data, each texture in
Group1~Group4 was divided into a number of patches. A total
of 21 patches of each texture in Group1 were randomly
selected as training data, while the other 28 were used for test
data. The randomly selected half patches of each texture in
Group2, Group3 and Group4 were used for training while the
Figure2. Flowchart of the DFrFT-based texture classification. other half part for testing. All experiments were repeated 10
times and the average performance of the proposed method has
been evaluated.
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TABLE I. CLASSIFICATION RATES WITH DIFFERENT METHODS (%)
Texture DFrFT Liu Azencott DFT Singh
Groups
Group1 98.21 99.78 95.1 87.1 71.6
Group2 100 83.1 100 90 100
Group3 85.62 79.1 85.94 81.8 87.2
Group4 88.93 -- 55.86 83.2 51.3
The worst result of the proposed DFrFT method is on
Group3. These textures which is the same with fig.11(i) in[31]
is rather difficult to discriminate. In fact, all the heuristic
methods selected in [31] yield the classification rate of about
10% which is much lower than our method.
Moreover, table I illustrates that from the view of the
uniformity of the results, the proposed DFrFT method is better
than other methods. The mean and the standard deviation of the
DFrFT are 0.932 and 0.07, while they are 0.843 and 0.198 for
Figure3. Textures. (a) Group1, (b) Group2, (c) Group3, and (d) group4. Azencott’s method, 0.855 and 0.0373 for the DFT, and 0.775
and 0.3 for Singh’s method.
B. Optimal Transform Order for Each Group Finally, we compared the computational cost of the
The coarse-to-fine searching shown in section 4.3 is proposed method to three of the above mentioned methods.
adopted to estimate the optimal DFrFT order for each group of The average computational cost for classifying a texture with
textures. For Group1 and Group3, Δα = 0.1, λ = 0.1, and ε the proposed DFrFT, Azencott’s method, the DFT, and Singh’s
0 method are 155.2 milliseconds (ms), 6.2ms, 21.3ms, and
=0.01. And for Group2 and Group4, α = 0.1, λ = 0.1, and ε
0 61.5ms, respectively. Clearly, the proposed method is much
| N || N ||N |8
=0.1. Here, we let 0 1 3 .The optimal slower than other methods, since both calculating the 1-D local
DFrFT orders are 0.84, 0.3, 0.91 and 0.4 for Group1, Group2, DFrFT and establishing the fractional frequency histogram are
Group3, and Group4, respectively. time consuming. At this stage, therefore, our method is not fit
C. Comparison to Other Methods for real time applications where the speed of an algorithm is
crucial. However, its time cost for processing a texture with
We compared the proposed DFrFT-based method with size of 320 by 320 is about 155 ms, meaning 39 textures per
Liu’s spectral histogram method [29], Azencott's windowed second, which may still satisfy most applications that don’t
Fourier transform [11], traditional DFT, and Singh’s FrFT[23]. have a strict requirement on the speed.
Here, | N0 || N1 || N3 |8 , the DFrFT orders are 0.84,
VI. C
0.3, 0.91, and 0.4 for Group1, Group2, Group3, and Group4, ONCLUSIONS
respectively. Here, we let the DFrFT order be equal to 1 to get Based on the fact that the DFrFT of a signal reveals both
the results of the traditional DFT. TableI lists the classification spatial and frequency characteristics, a novel texture
rates. Note that the result of Liu’s method for Group4 is missed classification method has been proposed in this paper. A local
since all results of this method are from [29]. Furthermore, the directional DFrFT is proposed and the fractional frequency
result of Azencott for Group1 is from [11]. histogram is generated from the discrete fractional Fourier
For the textures in Group1, the methods in [11] and [29] frequencies corresponding to the sorted amplitudes of the local
give 22 and 2 misclassified patches, i.e., the classification rates directional DFrFTs of a texture. Furthermore, by using the
of 95.1% and 99.78%. The proposed method gives 8 matrices of the between-class scatter and the within-class
misclassified patches with classification rate of 98.21%. The scatter, as well as the coarse-to-fine searching strategy, the
classification rates of other two methods on Group1 are lower optimal DFrFT order has been estimated. The comparison to
than 90%, especially for Singh’s method whose rate is only the four existing methods validates the proposed DFrFT-based
71.6%. The main reason for the poor performance of Singh’s texture classification method. One problem with the proposed
method is that “it is dependent upon the application”[23]. For method is that of its time cost. For the future work, we need to
the textures in Group2 and in Group3, the proposed method study how to optimize the method to improve its efficiency.
outperforms the DFT and Liu’s method, and is competitive to
R
the other two methods, i.e. Azencott’s method and Singh’s EFERENCES
method. For Group4, the performance of the proposed method [1] R. M. Haralick, "Statistical and structural approaches to texture," in
is much better than the DFT, Singh’s or Azencott’s methods. It Proceedings of the IEEE, 1979, pp. 786-804.
is worth to mention that the best result of Group4 submitting to [2] H. Derin and W. S. Cole, "Segmentation of textured images using Gibbs
the Outex website is 86% [32], which is still lower than the random fields," Computer Vision, Graphics, and Image Processing vol.
result of the proposed DFrFT method. 35, pp. 72-98, 1985.
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