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Extraction of Shift Invariant Wavelet Features for Classification of Images with Different Sizes Chi-Man Pun and Moon-Chuen Lee Abstract—An effective shift invariant wavelet feature extraction method for classification of images with different sizes is proposed. The feature extraction process involves a normalization followed by an adaptive shift invariant wavelet packet transform. An energy signature is computed for each subband of these invariant wavelet coefficients. A reduced subset of energy signatures is selected as the feature vector for classification of images with different sizes. Experimental results show that the proposed method can achieve high classification accuracy of 98.5 percent and outperforms the other two image classification methods.
æ 1
INTRODUCTION
THE issue of extracting invariant image features is always an important problem in content-based image analysis. In the last decades, much research has been focused on the issue of shift invariance and various methods have been proposed in the literature to cope with the problem. Some of the methods require either high oversampling rates [1], [2], [3] or intensive computational complexity, such as the matching pursuit algorithm [4]. In some other approaches, the resulting shift invariant feature representations are nonunique and involve approximate signal reconstructions, such as zero-crossing or local maxima methods [5], [6], [7]. Another approach has relaxed the requirement for shift-invariance by limiting the conditions on the scaling function [8], [9], [10]. Since the beginning of the 90s, different fast shift invariant wavelet decomposition algorithms [11], [12], [13], [14], [15] have been proposed to address the problem of shift invariance; some of them produce highly redundant wavelet coefficients. Concerning joint invariant features, many proposals can be found in the literature which specifically address the problem of rotation invariant and scale invariant image analysis [16], [17], [18], [19], [20], [21]. Xiong et al. [22] proposed a joint translation and scale invariant adaptive wavelet transform for texture identification and achieved good results. However, the texture images are assumed to be in a uniform background. To deal with the joint rotation, shift, and scale invariance problem, some methods [23], [24] employed the neural classifier to recognize simple images such as characters and numbers. Chen and Bui [25] proposed an invariant Fourier-wavelet descriptor for Chinese character recognition. But, their performance may drop significantly for databases of more general images. A nonlinear approach has been investigated in [26] for joint invariance of uniform scale, rotation, and shift nonorthonormal wavelet transform. However, its high redundant representation and computational complexity make it not suitable for robust image classification. In this paper, we propose an effective shift invariant wavelet feature extraction method for classification of images with
. C.-M. Pun is with the Department of Computer and Information Science, Faculty of Science and Technology, University of Macau, Macau, S.A.R. E-mail:
[email protected]. . M.-C. Lee is with the Department of Computer Science and Engineering, The Chinese University of Hong Kong, Hong Kong, S.A.R. E-mail:
[email protected]. Manuscript received 13 June 2003; revised 25 Feb. 2004; accepted 5 Mar. 2004. Recommended for acceptance by R. Basri. For information on obtaining reprints of this article, please send e-mail to:
[email protected], and reference IEEECS Log Number TPAMI-0131-0603. 0162-8828/04/$20.00 ß 2004 IEEE
Published by the IEEE Computer Society
NO. 9,
SEPTEMBER 2004
different sizes. The feature extraction process involves a normalization followed by an adaptive shift invariant wavelet packet transform. The normalization converts a given image into a size invariant image which is then passed to the adaptive shift invariant wavelet packet transform to generate adaptively some subbands of shift invariant wavelet coefficients with respect to an information cost function. An energy signature is computed for each subband of these wavelet coefficients. In order to reduce feature dimensionality, only the most dominant wavelet energy signatures are selected as feature vector for classification. The other wavelet approach proposed in [21] employed the log-polar wavelet energy signatures to tackle a different problem of joint rotation and scale invariant texture classification with a different wavelet transform.
2 Index Terms—Shift invariance, wavelet packet transform, normalization, image classification.
VOL. 26,
SHIFT INVARIANT WAVELET TEXTURE FEATURES
Discrete wavelet and wavelet packet transforms have been shown to be useful for image analysis in [27], [28], [29]. However, the problem of extracting shift invariant wavelet features for classification of images with different sizes was seldom addressed. In this section, we propose an algorithm to extract the shift invariant wavelet energy features from images of different sizes. The extraction of shift invariant wavelet features can be divided into three main steps. First, image normalization is applied on a given image to produce a standard size image. Second, an adaptive shift invariant wavelet packet transform is applied to the normalized image to produce the shift invariant wavelet coefficients. Third, energy signatures are computed for each subband of wavelet coefficients and a number of dominant energy signatures are selected to form the image feature vector.
2.1
Normalization
The proposed shift invariant image features are invariant to arbitrary shift and/or size changes to any rectangular image; this means that we have the same image features for different images obtained by applying only shift and/or size changes to any rectangular image. The normalization algorithm converts a given M N image into a class or standard size R S image so that two images (based on the same original image, as illustrated in Fig. 1) in different sizes can be matched accurately. Formally, the normalized form normði; jÞ of the M N rectangular image fðx; yÞ can be computed as follows: i ðM 1Þ j ðN 1Þ ; ; ð1Þ normði; jÞ ¼ f R1 S1 where R S is the class or a standard image size, i ¼ 0; ; R 1, and j ¼ 0; ; S 1. However, different images (especially texture images) in very different sizes can have similar visual content after normalization. As discussed in Section 3, the performance of classification of images with different sizes is only good for a limited range of size ratios.
2.2
Adaptive Shift Invariant Wavelet Packet Transform
After applying the normalization operation, a shifted image can be converted into a normalized image which is size invariant only. Next, we need to also resolve the shift problem by a shift invariant wavelet packet transform. The traditional 2D discrete wavelet packet transform is very efficient and can compute the wavelet packet coefficients with a complexity of OðnÞ, n being the number of pixels in the given image [32]. However, the wavelet packet coefficients thus obtained are not shift invariant. Recently, a number of shift-invariant wavelet decomposition algorithms [11], [12], [13], [14], [15] have been proposed and the image features are invariant to shift changes in both the rows and the columns of the image concerned. Nevertheless, such algorithms either involve high computational complexity or produce redundant wavelet coefficients. Therefore, we propose here an adaptive shift invariant
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p is two columns, the coefficients appear the same if the image Ck;ði;jÞ n n circularly shifted by 0; 2; 4; . . . ; 2 rows and 0; 2; 4; . . . ; 2 columns, respectively. The four periodic images of wavelet coefficients with one row shift:
n n
pþ1 S16kþ4;ði;jÞ pþ1 S16kþ7;ði;jÞ
oi¼I;j¼J n oi¼I;j¼J n oi¼I;j¼J pþ1 pþ1 ; S16kþ5;ði;jÞ ; S16kþ6;ði;jÞ ; i;j¼0 i;j¼0 i;j¼0 oi¼I;j¼J i;j¼0
can be computed as follows: XX pþ1 p ¼ hðmÞhðnÞCk;ðmþ2iþ1;nþ2jÞ S16kþ4;ði;jÞ
n n
pþ1 S16k;ði;jÞ
oi¼I;j¼J n oi¼I;j¼J n oi¼I;j¼J pþ1 pþ1 ; S16kþ1;ði;jÞ ; S16kþ2;ði;jÞ ; i;j¼0 i;j¼0 i;j¼0 oi¼I;j¼J
pþ1 S16kþ3;ði;jÞ
can be computed as follows: XX pþ1 p ¼ hðmÞhðnÞCk;ðmþ2i;nþ2jÞ S16k;ði;jÞ m pþ1 S16kþ1;ði;jÞ ¼
n
XX m
pþ1 S16kþ2;ði;jÞ
¼
pþ1 S16kþ3;ði;jÞ
¼
pþ1
p gðmÞhðnÞCk;ðmþ2i;nþ2jÞ
ð4Þ
n
XX m
ð3Þ
n
XX m
p hðmÞgðnÞCk;ðmþ2i;nþ2jÞ
p gðmÞgðnÞCk;ðmþ2i;nþ2jÞ ;
ð5Þ
n
pþ1
0 C0;ði;jÞ
¼ xði;jÞ is where I ¼ M=2 1, J ¼ N=2 1, and given by the intensity levels of the given image at row i and column j. Since we just keep one out of two rows and one out of
m
n
m
ð6Þ
p hðmÞgðnÞCk;ðmþ2iþ1;nþ2jÞ
ð7Þ
p gðmÞhðnÞCk;ðmþ2iþ1;nþ2jÞ
ð8Þ
p gðmÞgðnÞCk;ðmþ2iþ1;nþ2jÞ :
ð9Þ
n
p here is one row shifted and we just keep one Since the image Ck;ði;jÞ out of two rows and one out of two columns, the coefficients p is circularly shifted by would appear the same if Ck;ði;jÞ n 1; 3; 5; . . . ; 2 þ 1 rows and 0; 2; 4; . . . ; 2n columns, respectively. The four periodic images of wavelet coefficients with one column shift:
n n
pþ1 S16kþ8;ði;jÞ
oi¼I;j¼J n oi¼I;j¼J n oi¼I;j¼J pþ1 pþ1 ; S16kþ9;ði;jÞ ; S16kþ10;ði;jÞ ; i;j¼0 i;j¼0 i;j¼0 oi¼I;j¼J
pþ1 S16kþ11;ði;jÞ
i;j¼0
can be computed as follows: XX pþ1 p ¼ hðmÞhðnÞCk;ðmþ2i;nþ2jþ1Þ S16kþ8;ði;jÞ pþ1 S16kþ9;ði;jÞ ¼
m
n
m
n
XX
pþ1 S16kþ10;ði;jÞ ¼ pþ1 ¼ S16kþ11;ði;jÞ
p hðmÞgðnÞCk;ðmþ2i;nþ2jþ1Þ
XX m
ð10Þ ð11Þ
p gðmÞhðnÞCk;ðmþ2i;nþ2jþ1Þ
ð12Þ
p gðmÞgðnÞCk;ðmþ2i;nþ2jþ1Þ :
ð13Þ
n
XX m
n
p here is shifted by one column and we just Since the image Ck;ði;jÞ keep one out of two rows and one out of two columns, the p is circularly shifted by coefficients would appear the same if Ck;ði;jÞ n n 0; 2; 4; . . . ; 2 rows and 1; 3; 5; . . . ; 2 þ 1 columns, respectively. Finally, the four periodic images of wavelet coefficients with one row shift and one column shift:
n
ð2Þ
n
XX
pþ1 ¼ S16kþ7;ði;jÞ
n
i;j¼0
m
XX
pþ1 S16kþ6;ði;jÞ ¼
wavelet packet decomposition to generate shift invariant and nonredundant wavelet coefficients. A redundant set of wavelet packet coefficients for all possible circular shifts of a given image is first computed. Then, we eliminate the redundancy by summing and averaging the corresponding coefficients of all circular shifts to form a shift invariant wavelet packet representation which has exactly the same number of coefficients as the number of coefficients formed by the traditional wavelet packet decomposition. The details of the process are as presented below. In generating the shift invariant wavelet coefficients of a given M N image, a pair of quadrature mirror filters is employed to obtain an orthonormal representation and the periodic boundary handling method is applied during the wavelet packet decomposition process. Besides generating the zero shift coefficients like those generated by the standard decomposition, the proposed decomposition method generates three additional sets of redundant coefficients for three different circular shifts: one row shift, one column shift, and one row and one column shift. The three additional sets of wavelet coefficients are redundant because the wavelet coefficients of zero shift alone can fully represent the given image. They do not provide more information to represent the image and are used to obtain shift invariant wavelet coefficients. The four sets of coefficients are summed and averaged to produce one set of nonredundant and shift invariant wavelet coefficients for each decomposition level p. The four periodic images of wavelet coefficients with no shift:
n
XX
pþ1 ¼ S16kþ5;ði;jÞ
Fig. 1. (a) The original sample texture (D56) from the Brodatz album; (b) the image of (a) shifted 30 pixels to the left; (c) the image of (a) shifted 30 pixels to the left and upward, respectively; (d) the vertical dimension of image (a) resized with ratio 0.8; (e) the horizontal dimension of image (b) resized with ratio 0.8; and (f) both the vertical and the horizontal dimensions of image (a) resized with ratio 1.2.
m
pþ1 S16kþ12;ði;jÞ pþ1 S16kþ15;ði;jÞ
oi¼I;j¼J n oi¼I;j¼J n oi¼I;j¼J pþ1 pþ1 ; S16kþ13;ði;jÞ ; S16kþ14;ði;jÞ ; i;j¼0 i;j¼0 i;j¼0 oi¼I;j¼J i;j¼0
can be computed as follows: XX pþ1 p ¼ hðmÞhðnÞCk;ðmþ2iþ1;nþ2jþ1Þ S16kþ12;ði;jÞ pþ1 S16kþ13;ði;jÞ ¼ pþ1 ¼ S16kþ14;ði;jÞ pþ1 ¼ S16kþ15;ði;jÞ
m
n
m
n
m
n
XX XX XX m
ð14Þ
p hðmÞgðnÞCk;ðmþ2iþ1;nþ2jþ1Þ
ð15Þ
p gðmÞhðnÞCk;ðmþ2iþ1;nþ2jþ1Þ
ð16Þ
p gðmÞgðnÞCk;ðmþ2iþ1;nþ2jþ1Þ :
ð17Þ
n
p As the image Ck;ði;jÞ here is shifted by one row and one column and we just keep one out of two rows and one out of two columns, the
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Fig. 2. Full decomposition of a discrete image (C00 ) into shift invariant wavelet packet coefficients, a sample texture image D1 being included as an example. p coefficients would appear the same if Ck;ði;jÞ is circularly shifted by n n 1; 3; 5; . . . ; 2 þ 1 rows and 1; 3; 5; . . . ; 2 þ 1 columns, respectively. With the above 16 periodic images of wavelet coefficients together, a shift invariant representation can be obtained. However, there are, altogether, four sets of coefficients; any one set would provide sufficient information for the image; when compared with the traditional wavelet packet decomposition, the proposed decomposition method generates three additional sets of redundant coefficients which are required to generate the shift invariant coefficients. So, for each subband, the wavelet coefficients of the 16 periodic images subjected to different shifts are summed and averaged to produce four periodic images of shift invariant and nonredundant wavelet coefficients for next level decomposition. Formally, the four periodic images are computed as follows: pþ1 ¼ C4k;ði;jÞ pþ1 C4kþ1;ði;jÞ pþ1 C4kþ2;ði;jÞ pþ1 C4kþ3;ði;jÞ
1 pþ1 pþ1 pþ1 pþ1 S ð18Þ þ S16kþ4;ði;jÞ þ S16kþ8;ði;jÞ þ S16kþ12;ði;jÞ 4 16k;ði;jÞ 1 pþ1 pþ1 pþ1 pþ1 S ð19Þ ¼ þ S16kþ5;ði;jÞ þ S16kþ9;ði;jÞ þ S16kþ13;ði;jÞ 4 16kþ1;ði;jÞ 1 pþ1 pþ1 pþ1 pþ1 S ð20Þ ¼ þ S16kþ6;ði;jÞ þ S16kþ10;ði;jÞ þ S16kþ14;ði;jÞ 4 16kþ2;ði;jÞ 1 pþ1 pþ1 pþ1 pþ1 S :ð21Þ ¼ þ S16kþ7;ði;jÞ þ S16kþ11;ði;jÞ þ S16kþ15;ði;jÞ 4 16kþ3;ði;jÞ
When performing a full decomposition of discrete wavelet packet transform, the periodic subband images thus obtained can be organized in the form of a quad-tree. A sample full decomposition, up to level 3, of a discrete image into shift invariant wavelet packet representation for Brodatz texture D1 is shown in Fig. 2. In order to obtain a more effective and concise representation, we need to select the best basis representation for the image (D1). Similar to the approach proposed by Coifman and Wickerhauser [30], we can adaptively select some subband images to decompose further, instead of decomposing every subband image. The basic idea is to compute the information cost of each subband image and compare it with that of the sum of all next level subband images. If the information cost of the current subband image is less than that of the sum of all next level subband images, then the current subband image will not be decomposed; otherwise, we decompose the current subband image further and do comparison again until a maximum level or a specific level is reached. Hence, the best basis representation can be obtained by an efficient recursive selection process which determines the best decomposition of the given image based exclusively on the local minimization of the information cost function. Formally, let the
best basis representation for the subband image at level j be Ajk . Then, the best basis A00 for the given image x can be computed recursively by: 8 3 P > > < Ckp ; if MðCkp Þ MðApþ1 4kþi Þ p i¼0 Ak ¼ ð22Þ 3 > pþ1 > : A4kþi ; otherwise: i¼0
The recursive computation proceeds down to the maximum or specified level J, where AJk ¼ CkJ ; 0 k < 4J :
ð23Þ
A sample adaptive decomposition to obtain the best basis representation for Brodatz texture D1 is as depicted in Fig. 3. The best basis representation is optimal according to a given information P cost function M, where Mð0Þ ¼ 0 and Mðfxi gÞ ¼ Mðxi Þ. At each i level of the adaptive shift wavelet packet decomposition, we produce four 2D periodic images from one level to the next higher level, producing at most 4l 2D periodic images for a decomposition up to level l, which are the same number as those obtained from standard wavelet packet decomposition. Moreover, this decomposition can be performed efficiently. While performing the decomposition from one level to the next higher level, we quadruple the number of 2D periodic images and quarter the size of each of them. By repeating this procedure recursively to all levels, we can get the wavelet packet coefficients for all circular row and column shifts in p steps for p-level decomposition (p minðlog2 M; log2 NÞ). So, the computational complexity of our proposed adaptive shift invariant wavelet packet decomposition is quite efficient with only Oðn log nÞ complexity, n being the number of pixels in the image.
2.3
Extraction of Shift Invariant Wavelet Energy Signatures
The extraction of shift invariant wavelet features can be divided into three main steps. In Step 1, a normalized image is formed for a given M N image by applying the normalization algorithm as described in Section 2.1. In Step 2, an adaptive shift invariant wavelet packet transform, as described in Section 2.2, is applied to the normalized image to eliminate the shift effects, producing the shift invariant wavelet coefficients for the given image. In Step 3, in order to reduce the feature dimensionality of the wavelet coefficients, an energy signature is computed for each subband
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Fig. 3. Adaptive decomposition of a discrete image (C00 ) into shift invariant wavelet packet coefficients, a sample texture image D1 being included as an example.
of wavelet coefficients. Then, a fixed number of the most dominant energy signatures are selected to form the shift invariant wavelet feature vector. The details of the algorithm are as follows: Shift Invariant Wavelet Feature Extraction Algorithm Step 1. For a given M N image, apply the normalization algorithm to produce a class or standard size R S normalized image. Step 2. Apply the adaptive shift invariant wavelet packet transform to the normalized R S image, producing m subbands of wavelet coefficients Apk;ðr;sÞ , where p minðlog2 ðRÞ; log2 ðSÞÞ, k 2 f0; ; 4p 1g, and r ¼ 0; 1; . . . ; 2log Rp 1, s ¼ 0; 1; . . . ; 2log Sp 1. Step 3. Compute an average energy signature ( ) P p 1 Ei ¼ KL Ak;ðr;sÞ r;s
circular shifts (10 to 100 with 10 pixels interval in all four directions: left, right, top, and bottom). So, a set of 4,000 (25 4 40) texture images was created as data set I. For size invariance only, we extract from each region 27 subsamples with different size ratios (0.6 to 1.4 with 0.1 intervals in the vertical dimension, horizontal dimension, and both dimensions, respectively). Thus, a set of 2,700 (25 4 27) texture images was created as data set II. For joint shift and size invariance, we extract from each region 180 subsamples with different circular shifts (10 to 50 with 20 pixels interval in all four directions: left, right, top, and bottom) and different size ratios (0.8 to 1.2 with 0.1 interval in vertical dimension, horizontal dimension, and both dimensions, respectively). In this way, a set of 18,000 (25 4 180) texture images was created as data set III. In the first series of experiments, the performance of shift invariance and size invariance were investigated with 256 energy
i
for each subband i of wavelet coefficients Apk;ðr;sÞ , where i ¼ 1; ; m, K ¼ 2log Rp 1, and L ¼ 2log Sp 1. Step 4. Arrange all energy signatures in descending order and choose the first M most dominant energy signatures (with highest energy values) as feature vector, f ¼ ðE1 ; E2 ; ; EM Þ, where M m. Step 5. Output the feature vector f as image feature for the given image.
3
EXPERIMENTAL RESULTS
In order to demonstrate the effectiveness of our proposed shift invariant wavelet features in classification of images with different sizes, a number of experiments have been performed using a Euclidean classifier. All the experiments are based on a set of 25 classes of natural texture images (of size 128 128 ) obtained from the Brodatz texture album [33] shown in Fig. 4. Each texture from the album is first scanned with 150 dpi resolution to a digital image of size 256 256 pixels with 256 gray levels. Each of the output texture images is then divided into four 128 128 nonoverlapping regions. From these regions, we created three data sets for testing the shift invariance only, size invariance only, and joint shift and size invariance. For shift invariance only, we extract from each region 40 subsamples of size 128 128 with different
Fig. 4. Twenty-five classes of textures from the Brodatz album. Row 1: D1, D4, D6, D19, D20. Row 2: D21, D22, D24, D28, D34. Row 3: D52, D53, D56, D57, D66. Row 4: D74, D76, D78, D82, D84. Row 5: D102, D103, D105, D110, D111.
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TABLE 1 Classification Results of the Proposed Method for Different Size Ratios on Data Set II
TABLE 2 Classification Results of the Proposed Method for Different Number of Dominant Energy Signatures on Data Set III
features on data set I and II, respectively. Using data set I, we obtained a perfect classification result of 100 percent accuracy for the shift invariance testing, which shows that the proposed adaptive shift invariant wavelet packet transform can produce robust shift invariant image features. The classification results of size invariance testing on data set II are presented in Table 1, where the performance is very satisfactory for the size ratios from 0.8 to 1.2. However, the performance can drop significantly for size ratios greater than 1.2 or less than 0.8. This is possibly due to the fact that large size ratio differences can cause the different texture images to have visually similar texture content. In the second series of experiments, we tested reducing the dimension of the feature space by selecting the most dominant energy features from our proposed wavelet energy feature vector on data set III. Based on different number of dominant wavelet energy signatures, the overall classification results are as presented in Table 2. The best overall accuracy rate is 98.5 percent, which is better than that using a full energy feature set (256 features) for classification. It shows 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 significant discriminating information. The proposed image classification method was further compared using the same data set III and number of features (144) with two other classification methods: the traditional wavelet packet signatures (WPS) method [29] and the shift invariant wavelet packet signatures (SIWPS) method [15]. In the experiments, the three classification methods all used the same Euclidean classifier and the same data set. The classification results are as presented in Table 3, which demonstrates that the proposed image classification method outperforms significantly the other two methods.
4
ACKNOWLEDGMENTS This work was supported in part by the research grant RG067/0203S/PCM/FST funded by the Research Committee of the University of Macau, the Hong Kong Research Grants Council under Grant CUHK4377/02E, and the RGC Direct Grant Project ID 2050260.
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TABLE 3 Classification Results of the Proposed Method and Two Other Image Classification Methods: The Traditional Wavelet Packet Signatures (WPS) Method and the Shift Invariant Wavelet Packet Signatures (SIWPS) Method on Data Set III
CONCLUDING REMARKS
We address the problem of different kinds of invariance (rotation, shift, and scale) in image classification and propose a scheme to extract shift invariant wavelet features for classification of images with different sizes. The proposed wavelet energy features, which were obtained from the result of a normalization and an adpative shift-invariant wavelet packet transform, were very effective (Oðn log nÞ) with only 144 energy values. Experimental results show that the proposed classification scheme outperforms the other two image classification methods: the traditional wavelet packet signatures (WPS) method and the shift invariant wavelet packet signatures (SIWPS) method. The highest classification accuracy for 25 kinds of texture images was 98.5 percent. The accuracy could be even better if the neural classifier is used instead of the Euclidean classifier.
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