[9] P. J. Burt, âFast Filter transforms for image processingâ, Comput. Graphics and ... [16] R. Hummel and R. Moniot, âReconstructions from zero crossings in scale ...
Texture Classification Using Shift-Invariant Wavelet Packet Decomposition Pun Chi-Man Faculty of Science and Technology University of Macau, Macau Lee Moon-Chuen Department of Computer Science and Engineering The Chinese University of Hong Kong, Hong Kong Abstract: - This paper proposes a high performance texture classification method using dominant energy features based on shift-invariant wavelet packet coefficients obtained by 2D shift-invariant wavelet packet decomposition. Experiments employing a reduced feature set show that the proposed method involves a relatively small classification time while still achieving a high accuracy rate (95.6%) for classifying twenty classes of natural texture images. Key-Words: - Wavelet Packets, Shift-Invariance, and Texture Classification
1 Introduction Texture analysis plays a very important role in computer vision and pattern recognition, since most of real world objects consist of different kinds of texture surfaces. Texture is a low-level image feature and is around all of us. And there are many different applications involving texture analysis, including medical imaging, remote sensing, industrial inspection, document segmentation and texture-based image retrieval, etc. [1]. Texture analysis has been studied widely for over three decades. In early approaches for texture classification, attention was focused on analysis of the first-order or second-order statistics of textures, and stochastic models such as Gaussian Markov random fields (GRMFs) and autoregression [2]. Recent developments in the spatial / frequency analysis such as Gabor filters [3][4], and Wavelet transform [5][6] provide good multiresolution analytical tools for texture analysis and classification. Many experiments showed that these approaches can achieve a high accuracy rate. However, most of these methods do not perform well when the textures are shifted. The main reason is that there exists a major drawback in wavelet transform, namely their sensitivity to translations due to the dyadic structure of the wavelet expansions. Shift invariant representation is very useful and desirable for pattern recognition and classification. The issue of shift invariance of wavelet transforms has been addressed using different approaches. Some of the approaches either require high oversampling rates [8][9][10][11] or intensive computational complexity,
such as the matching pursuit algorithm [12]. In some other approaches, their resulting representations are non-unique and involve approximate signal reconstructions, such as zero-crossing or local maxima methods [14][15][16]. Another approach has relaxed the requirement for shift-invariance by limiting the conditions on the scaling function [17][18][19]. In this paper, we propose a 2D shift-invariant wavelet packet decomposition and a high performance texture classification method using dominant energy features from shift-invariant wavelet packet coefficients. In next section, we briefly review the standard wavelet packets decomposition technique. In Section 3, we introduce the proposed shift-invariant wavelet packets decomposition techniques for images. Then an application to texture classification together with some experimental results are presented in Section 4. Finally, conclusions are drawn in Section 5.
2 Wavelet Packets Wavelets have been shown to be useful for texture classification in literature, possibly due to their finite duration which provides both the frequency and spatial locality. The hierarchical wavelet transform uses a family of wavelet functions and its associated scaling functions to decompose the original signal / image into different subbands. The decomposition process is recursively applied to the sub-bands to generate the next level of the hierarchy. If an orthonormal wavelet basis has been chosen, the coefficients computed are independent and
possess a distinct feature of the original signal. Wavelet packets can be described by the collection of basis functions as follows[6][7]:
hh( k , l ) = h(k ) h(l ) hg (k , l ) = h( k ) g (l ) gh(k , l ) = g (k ) h(l )
W2 n (2 p −1 x − l ) = 21− p ∑ h(m − 2l ) 2 p Wn (2 p x − m)
gg ( k , l ) = g ( k ) g (l )
(1)
m
W2 n +1 ( 2
p −1
1− p
x − l) = 2
∑ g (m − 2l ) 2 Wn ( 2 x − m) p
p
m
(2)
And the wavelet packet coefficients can be decomposed by: 2 0,( r , s )
where p is a scale index, l is a translation index, h is a lower-pass filter , g is a high-pass filter with g (k ) = (−1) k h(1 − k ) . The function W0 ( x) can be identified with the scaling function φ and W1 ( x) with the mother wavelet ψ . The inverse relationship between wavelet packets of different scales can be specified as follows: 2 p W n (2 p x − k ) = ∑ h(k − 2l ) 2 p −1 W 2 n (2 p −1 x − l ) + l
∑ g (k − 2l )
2
l
p −1
W 2 n +1 (2
p −1
{C } {C } {C } {C } {C } {C }
{C
1 0,( r , s )
}
{C } 1 1,( r , s )
{C
0 0,( r , s )
} {C
}
{C
}
1 2,( r , s )
(3)
x − l)
1 3,( r , s )
Due to the orthonormal property, the wavelet packet coefficients at different scale and position of a signal f(x) can be easily computed via
C
p n,k
= 2
∞
∫ f ( x) ⋅ W
p
n
(2 p x − k )dx
(4)
−∞
{
0 where C0,( r ,s)
l
l
C 2pn−,1l = ∑ h(m − 2l ) ⋅ C np,m C 2pn−+11,l = ∑ g (m − 2l ) ⋅ C np,m
(6)
C
=
2 4,( r , s ) 2 5,( r , s )
...... ...... ......
......
} for r, s = 1, 2, ... , N are given by the
(9)
C 4jl +1,( r , s ) = ∑∑ h(n) g ( m)C l j,(−m1 + 2 r , n + 2 s )
(10)
(7)
is given by
∫ f ( x) ⋅ φ ( x − k )dx
(8)
−∞
The 2-D wavelet packet basis functions can be expressed by the product of two 1-D wavelet packet basis functions along the horizontal and vertical directions respectively. The corresponding 2-D filter coefficients have four groups:
n
C 4jl + 2,( r , s ) = ∑∑ g ( n)h(m)C l j,(−m1 + 2 r ,n + 2 s ) m
0 0, k
......
n
m
m
C
}
C 4jl ,( r , s ) = ∑∑ h(n)h(m)Cl j,(−m1 + 2 r ,n + 2 s ) m
m
∞
j 0,( r , s )
Similar to (6) and (7) for the 1-D case, the wavelet coefficients for the 2-D case can be defined by the following recursive relations:
Using (1) and (2), we have,
0 0,k
{C
N × N image with N = 2n.
For discrete signal , wavelet packet coefficients may be computed efficiently as follows: From (3), we have, (5) Cnp,k = ∑ h(k − 2l ) ⋅ C2pn−,1k + ∑ g (k − 2l ) ⋅ C2pn−+11,k
Note that
......
2 1,( r , s ) 1 2,( r , s ) 2 3,( r , s )
n
C 4jl +3,( r , s ) = ∑∑ g (n) g (m)Cl j,(−m1 + 2 r ,n + 2 s ) m
(11) (12)
n
j
and Cl ,( r ,s ) are circular periodic 2D images with period 2log N-j, j = 0, 1, ..., log N. and l = 0, 1 ,..., 4j-1. Most of the basis functions have the properties of smoothness, number of vanishing moments, symmetry, good time and frequency localization. In addition, they satisfy the admissibility condition and are absolutely square integrable functions. The discrete wavelets can be classified as non-orthogonal, biorthogonal or orthogonal wavelets. Non-orthogonal wavelets are linearly dependent and redundant frames. Orthogonal wavelets
are linearly independent, complete and orthogonal. In our experiment, we employ one of the most widely used wavelets constructed by Daubechies (db20). These wavelets are othonormal, compactly supported. They have maximum number of vanishing moments for the support, and are reasonably smooth. The low-pass and band (high)–pass filter coefficients satisfy the conditions of Orthogonality, Normality and Regularity.
2 Shift-Invariant Decomposition
Wavelet
Packet
The standard discrete wavelet transform (DWT) and wavelet packet decomposition are widely used in texture analysis recently, due to their high accuracy [5][6] and good performance, which compute the wavelet packet coefficients for a given N × N image in Ο( N 2 ) operations [7]. However, the coefficients are not shift invariant. In order to achieve the shift-invariant wavelet packet decomposition, we can build a redundant set of wavelet packet coefficients for all possible circular shifts of a polar transformed image, which is similar to Beylkin’s approach [8] except that we make the extension to 2-D wavelet packet decomposition. This shift-invariant wavelet packet decomposition algorithm is quite efficient with only Ο( N 2 log N ) complexity for a given N × N image. The details of the shift-invariant 2-D wavelet packet decomposition works as follows. For each scale j, (1 ≤ j ≤ log N), we compute 4j 2-D period sequences. However, in order to achieve the shift invariance, we compute four shifts: (0,0), (0,1), (1,0), (1,1) for each of the sequences for every scale j. The coefficients of each 2-D period sequences would be recursively computed as follows similar to (9)-(12):
C
j ,(0,0) 4 l ,( r , s )
= ∑∑ h(n) h(m)C m
j −1 l ,( m + 2 r , n + 2 s )
n
j −1 C4jl,(0,1) ,( r , s ) = ∑∑ h( n) h( m)Cl ,( m + 2 r , n + 2 s +1) m
(14)
n
j −1 C4jl,(1,0) ,( r , s ) = ∑∑ h( n) h( m)Cl ,( m + 2 r +1, n + 2 s ) m
(13)
(15)
n
j −1 C4jl,(1,1) ,( r , s ) = ∑∑ h( n ) h( m)Cl ,( m + 2 r +1, n + 2 s +1) (16) m
Similarly,
n
the
2-D
period
j ,(0,0) sequences C4l +i ,( r , s ) ,
j ,(1,0) j ,(1,1) C4jl,(0,1) +i ,( r , s ) , C4 l +i ,( r , s ) , C4 l +i ,( r , s ) for i=1,2,3 can be
computed in almost the same way, except the filters h(n)g(m), g(n)h(m) and g(n)g(m) are used instead for i = 1,2,3 respectively.
Since we quarter the size when we decompose the 2-D period sequence from one scale to the next higher scale, j ,( 0, 0) the 2-D period sequence C 4l ,( r , s ) contains all the j −1
coefficients that appear if C l ,( r , s ) is circulantly shifted by any one of (0,0), (0,2) (2,0), (2,2), ...., (2n,2n), and the j , ( 0 ,1) sequence C 4l ,( r , s ) contains all the coefficients for a circular shift by any one of (0,1), (2,1), (0,3), (2,3), ...., j , (1, 0 ) (2n,2n+1), and the sequence C 4l ,( r , s ) contains all the coefficients for a circular shift by any one of (1,0), (1,2), j , (1,1) (3,0), (3,2), ...., (2n+1,2n), and the sequence C 4l ,( r , s ) contains all the coefficients for a circular shift by any one of (1,1), (1,3), (3,1), (3,3), ...., (2n+1,2n+1). Therefore, j ,(0,0) with these four 2-D period sequences C4l ,( r , s ) , j ,(1,0) j ,(1,1) C4jl,(0,1) ,( r , s ) , C4l ,( r , s ) and C4l ,( r , s ) , we can achieve the
shift-invariant decomposition for the 2-D period j −1 sequence C l ,( r , s ) . The similar statements are true for the other
C
2-D
j ,(0,1) 4 l + i ,( r , s )
,C
period
j ,(1,0) 4 l + i ,( r , s )
,C
j ,(1,1) 4 l + i ,( r , s )
sequences
C4jl,(0,0) + i ,( r , s )
,
for i=1,2,3. Therefore, a set
of shift-invariant wavelet coefficients can be obtained for the given image. In addition, this decomposition can be still performed efficiently. Stepping from one scale to the next higher scale we quadruple the number of 2-D periodic sequences and quarter the size of each of them. By repeating this procedure recursively to all scales, we can get the wavelet packet coefficients for all circular shifts in log N steps with only Ο( N 2 log N ) complexity.
3 Texture Classification Application For texture classification, we perform a shift-invariant wavelet packet decomposition for each class texture up to level 2 and compute the energy measure as signatures Sn =
1 N2
N
∑C
p 2 n,k
for the coefficients. However, the full
k =1
wavelet packet decomposition will produce many coefficients and so a large feature set will be obtained. As proposed by Chang and Kou [5], the most dominant frequency channels obtained from tree-structured wavelet transform are very good for discriminating textures. Hence, we can reduce the feature set by choosing the signatures with highest energy values as features. And the classification process is performed by the Mahalanobis classifier [21]:
d ( x, vi ) = ln ∑ i + ( x − vi ) t ∑ i−1 ( x − vi )
(17)
where x is the feature vector of an unknown texture
image, vi and Σi are the mean vector and covariance matrix of the data in class i . For testing the proposed texture image classification scheme, we used twenty natural textures, as shown in Figure 1, from the Brodatz texture album [20] as our class images. Each texture is scanned with 150 dpi resolution, and each image, having the size 640×640 pixels and 256 gray levels, is divided into twenty-five 128×128 non-overlapping regions. So, a database of 500 (20 × 25) images was created for our testing. Table 1 shows the classification performance of our method for different number of dominant energy features. When 144 dominant energy features were used for classification, the best performance was obtained with 95.6% accuracy, which is better than using a full energy feature set (256 features) for classification. Also, the experimental results show that even if we used a small number of dominant energy features, a high accuracy could still be achieved. This implies that the low energy features do not convey any significant discriminating information.
Figure 1. Twenty class textures from Brodatz album. Row 1: D1, D4, D6, D20, D21. Row 2: D22, D28, D34, D52, D53. Row 3: D57, D74, D76, D78, D82. Row 4: D84, D102, D103, D105, D110 Number of dominant energy features 16
32
64
96
144
192
256
Corr. Err.
438 62
450 50
459 41
468 32
478 22
472 28
472 28
Acc (%)
87.6
90
91.8
93.6
95.6
94.4
94.4
Table1. Classification Results with different number of dominant energy features.
4 Conclusion
This paper presents an innovative shift-invariant wavelet packet decomposition method together with a dominant wavelet packet energy feature set for texture image classification. The proposed method has been shown to be invariant to shifts in texture images and reduces the feature set, while still archiving high accuracy rate. However, since our method assumes that the texture images have the same orientations and scales, it may not perform well when this assumption is not valid. Hence, we will focus on how to incorporate the proposed method with other rotation / scale-invariant texture classification schemes for further enhancement. References: [1] M. Tuceryan, and A.K. Jain. “Texture analysis. Handbook of Pattern Recognition and Computer Vision”, chapter 2.1, pages 235--276. World Scientific, 1993. [2] R. W. Conners and C. A. Harlow, “A theoretical comparison of texture algorithms,” IEEE Trans. PAMI, vol.2 pp.204-222, May 1980. [3] A. C. Bovik, M. Clark, and W. S. Geisler, “Multichannel texture analysis using localized spatial filters,” IEEE Trans. PAMI, vol.12, Jan. 1990. [4] A. Teuner , O. Pichler and B. J. Hosticka, “Unsupervised texture segmentation of images using tuned matched Gabor filters”, IEEE Trans. on Image Processing, Vol. 6, No. 4, 1995, pp. 863-870. [5] T. Chang, and C.C.J. Kuo. “Texture analysis and classification with tree-structured wavelet transform”, IEEE Trans. Image Processing, vol. 2, p.429-441, April 1993 [6] A. Laine, and J. Fan. “Texture classification by wavelet packet signatures”, IEEE Trans PAMI, vol. 15 p.1186-1191, Nov. 1993. [7] I. Daubechies, Ten Lectures on Wavelets, CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM Press, Philadelphia, Pennsylvania, 1992 [8] G. Beylkin, “On the representation of operators in bases of compactly supported wavelets”, SIAM J. Numer. Anal., Vol. 6, No. 6, Dec. 1992, pp. 1716-1740. [9] P. J. Burt, “Fast Filter transforms for image processing”, Comput. Graphics and Image Proc., Vol. 16, 1981, pp. 20-51. [10] O. Rioul and P. Duhamel, “Fast algorithms for discrete and continuous wavelet transforms”, IEEE Trans. Inf. Theory, Vol. 38, No. 2, Mar. 1992, pp. 569-586. [11] N. Saito and G. Beylkin, “Multiresolution representations using the auto-correlation functions
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