Relative phase in dual tree shearlets

Signal Processing 96 (2014) 241–252

Contents lists available at ScienceDirect

Signal Processing journal homepage: www.elsevier.com/locate/sigpro

Relative phase in dual tree shearlets Zhang Jiuwen, Zhang Runpu n, He Lulu, Dong Min School of Information Science and Engineering, Lanzhou University, 222 Tianshui Road, Lanzhou 730000, PR China

a r t i c l e i n f o

abstract

Article history: Received 7 March 2013 Received in revised form 22 July 2013 Accepted 11 September 2013 Available online 7 October 2013

The relative phase is an efficient approach to exploit the phase information of complex wavelet coefficients. However, the relative phase original generated from the pyramidal dual tree directional filter bank (PDTDFB) has three defaults. Firstly, its texture retrieval performance does not simultaneously improve in general as the scale increases. Secondly, it is not accurate enough when the directional subbands are not uniformly downsampled along row and column. Thirdly, its 2n number of directions are not optimal. In this paper, we propose a new multiscale and multidirection transform for relative phase, named as dual tree shearlets. The transform is based on the discrete shearlet transform, but a dual tree Laplacian pyramid is adopted to create a real-imaginary pair structure for deriving phase information under multiscale framework. The dual tree shearlets have the properties of uniform downsampled subbands; higher directional sensitivity and the 2-D Hilbert transform relationship between two channels like the dual tree complex wavelet transform (DTCWT). The numerical experiments presented in this paper demonstrate that the relative phase of our proposed method outperforms that of the PDTDFB in texture retrieval application both in terms of performance and computational efficiency. The results show that relative phase of the dual tree shearlets amends the above mentioned defaults. & 2013 Elsevier B.V. All rights reserved.

Keywords: Shearlets Dual tree Uniform downsampling Complex wavelet Relative phase Texture retrieval

1. Introduction For image compression or content-based image retrieval, a rich, reliable and precise representation of the location of features is essential. Wavelet transforms have emerged as a popular basis for image content analysis due to their ability of isolating image energy concisely into scalar, directional and spatial. Compared with conventional real-valued wavelets, complex wavelets which are able to provide both magnitude and phase information have shown a consistent representation to the structures in images. The magnitudes of complex wavelet coefficients indicate the amplitude of features, and the phases indicate the locations of these features.

n

Corresponding author. Tel.: þ86 13619310825. E-mail address: [email protected] (Z. Runpu).

0165-1684/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.sigpro.2013.09.020

In the earlier research, through the example of reconstructing a signal with phase of the Fourier coefficients alone, Oppenheim and Lim present the importance of phase information in [1]. Then Morrone and R.A. Owens [2] further demonstrate that phase information is a crucial component in representing the structures of image. Afterwards, Gabor phases have been used in iris and palm-print identification [3,4], face recognition [5] and texture discrimination [6]. Some other applications suggest the relationships of interscale phase in complex wavelet domain, such as texture synthesis [7], blurred detection [8] and face recognition [9]. More recent studies concentrate on exploiting the phase information of complex wavelet coefficients. Suggesting the use of a modified product of coefficients at the same location and adjacent scales in dual tree complex wavelet transform (DTCWT) [10,11]. M. Miller and N. Kingsbury [12] provide an approach to capture local phase relationship across scale and spaces. Consequently,

242

Z. Jiuwen et al. / Signal Processing 96 (2014) 241–252

an efficiently denoising algorithm using the modified coefficients according to [12] has been presented in [13]. Another method investigated in [14] attempts to utilize the difference of phase between adjacent complex wavelet coefficients namely relative phase for texture image retrieval. Then in [15], the feature orientation of all subbands is linearly proportional to the relative phase in the pyramidal dual tree directional filter bank (PDTDFB) [16], DTCWT [10,11] and uniform discrete curvelet transform (UDCT) [17]. The probability density function of relative phase is proposed in [18]. As a complementary to the magnitude, the relative phase information of complex wavelet coefficients has been studied to obtain new features in texture image retrieval when images are decomposed by different complex wavelet transforms including PDTDFB and DTCWT. However, through further research we found that the relative phase original generated from PDTDFB has the following defaults. Firstly, the retrieval accuracy would not be simultaneously higher in general as the scale and direction increases as shown in Tables 1 and 2 if the features are derived only from the relative phase. Thus it cannot fully utilize the multiscale information though the relative phase is originally defined in the multiscale framework. Secondly, the relative phase is not accurate enough when the directional subbands are obtained through nonuniform downsampling along row and column respectively. Moreover, PDTDFB and DTCWT can capture the geometry of images only in 2n directions [16] and in six fixed directions [11] respectively, and these numbers of decomposing directions are both not optimal for phase information. In this paper, we preliminarily focus on the structure of complex wavelet transforms in the implementation of relative phase. We develop a new complex wavelet transform in which subbands in the relative phase model are accurate enough in multiscale and multidirection. Derived from the discrete shearlet transform [19,20] which has been used in some applications such as magnetic resonance imaging (MRI) reconstruction

[21], we develop a new dual-channel multiscale decomposition named dual tree shearlets. It has the properties of uniform downsampled subbands, the arbitrary number of directions for decomposing and 2-D Hilbert transform relationship between its two channels like DTCWT. We compare the texture retrieval performances of relative phase of DTCWT, PDTDFB and of our dual tree shearlets. The outline of the rest of this paper is as follows. The relative phase, the related multiresolution and multidirection complex-valued transforms are reviewed in Section 2. In Section 3, firstly we discuss in detail the requirements of relative phase and the properties of directional subbands in DTCWT and PDTDFB, and then we propose a new kind of complex shearlet transform named as dual tree shearlets, with the properties of dual-channel, multiscale, uniform downsampling along row and column in subbands and the arbitrary number of directions for decomposing. A performances comparison of the relative phase using DTCWT, PDTDFB and our dual tree shearlets in texture image retrieval is presented in Section 4. Finally, we conclude this paper in Section 5.

2. Background 2.1. Relative phase In [7], the authors have pointed out that the local phase varies linearly with distance from features, and recently the proof for this relationship was given in [15]. For an ideal complex filter with one-side frequency support, it has been proven that the phase in the vicinity of the features such as a step or a ramp has a linear relationship with the distance in the condition of jwt j{1, in which w is the frequency in the support of complex filter and t is the distance to feature. We refer [14] for the deep understanding of the relative phase. The relationship between the angle of an edge and the adjacent coefficients located

Table 1 The comparison of PDTDFB and the dual tree shearlets with features extracted from different number of scale in the VisTex database. Vector

Level

Method

RP-mean (%)

RP-var (%)

RP (%)

RP-MAG (%)

MAG (%)

[8, 8, 8]

S¼ 1

PDTDFB D-Shearlets PDTDFB D-Shearlets PDTDFB D-Shearlets

56.17 45.62 58.87 62.69 56.54 64.78

43.41 60.13 45.11 72.09 44.34 73.44

63.27 62.76 63.52 73.82 61.13 75.48

76.84 75.56 77.87 79.79 78.81 80.70

65.35 63.88 72.00 69.85 74.60 73.43

PDTDFB D-Shearlets PDTDFB D-Shearlets PDTDFB D-Shearlets

35.08 50.50 37.09 65.31 36.32 65.72

31.98 62.16 33.97 70.51 32.79 70.60

43.58 63.85 44.76 73.24 42.97 73.71

70.33 75.73 72.88 79.57 74.21 80.20

66.09 64.40 72.20 70.07 74.22 73.26

PDTDFB D-Shearlets PDTDFB D-Shearlets PDTDFB D-Shearlets

35.08 50.50 45.98 62.91 51.03 67.87

31.98 62.16 38.97 71.46 43.95 73.02

43.58 63.85 53.24 73.17 58.53 75.53

70.33 75.73 75.21 79.32 78.35 81.37

66.09 64.40 72.59 70.07 75.10 73.35

S¼ 2 S¼ 3 [16, 16, 16]

S¼ 1 S¼ 2 S¼ 3

[4, 8, 16]

S¼ 1 S¼ 2 S¼ 3

Z. Jiuwen et al. / Signal Processing 96 (2014) 241–252

243

Table 2 The comparison of PDTDFB and the dual tree shearlets with features extracted from different number of scale in the Brodatz database. Vectors

Levels

Methods

RP-mean (%)

RP-var (%)

RP (%)

RP-MAG (%)

MAG (%)

[8, 8, 8]

S¼1

PDTDFB D-Shearlets PDTDFB D-Shearlets PDTDFB D-Shearlets

76.04 66.81 81.32 81.02 86.61 82.80

75.72 82.72 81.83 87.37 81.10 88.15

83.00 83.30 84.79 86.82 81.61 87.54

84.92 84.21 86.37 86.61 84.47 86.70

76.56 75.79 83.34 81.04 84.63 82.88

PDTDFB D-Shearlets PDTDFB D-Shearlets PDTDFB D-Shearlets

60.60 71.68 66.95 82.94 67.40 83.42

71.94 82.34 76.40 87.51 74.08 87.93

72.69 82.20 75.64 86.92 75.94 86.99

82.55 84.16 84.49 85.98 84.93 86.38

78.86 75.45 82.69 80.69 83.93 82.50

PDTDFB D-Shearlets PDTDFB D-Shearlets PDTDFB D-Shearlets

60.60 71.68 71.00 81.73 74.36 83.57

71.94 82.34 79.08 86.89 81.25 88.24

72.69 82.20 79.59 86.37 81.69 87.38

82.55 84.16 85.09 85.91 86.03 86.66

78.86 75.45 82.43 80.44 84.10 81.88

S¼2 S¼3 S¼1

[16, 16, 16]

S¼2 S¼3 S¼1

[4, 8, 16]

S¼2 S¼3

tion rienta o d n a Subb dge pe B Ste

A

Ds ,k

k

If K Z6, jθjr ðπ=12Þ, we can make an approximation: θ  tan θ. Thus the feature orientation αk can be approximated by: αk  γ k  tan γ k þ

B

H

M k

k

A

k

Fig. 1. Relationship between the angle αk of an edge and the distances from two horizontally adjacent coefficients located at A and B to the edge in the direction normal to the subband orientation γk (0 r γk r(π/4)) at some arbitrary scale.

tan θ ¼ tan ðαk  γ k Þ ¼ ¼

MA′ AA′  HM  AH ¼ MB′ MB′

AA′ BB′  Ds;k sin γ k AA′ BB′ ¼  tan γ k Ds;k cos γ k Ds;k cos γ k

ð1Þ

Similarly, the feature orientation αk of other subbands in complex wavelet domain can be approximated as given by [1]: 8  BB′ K K γ k  tan γ k  DAA′ if > 4 o k r 2; > s;k cos γ k > <  BB′ γ k þ cot γ k þ DAA′ if K2 o k r 3K ð2Þ αk  4; s;k sin γ k > > > γ þ tan γ  AA′  BB′ if 3K ok r K: : k

in its neighborhood in directional subband is given in Fig. 1. The relative phase model is originally generated from the pyramidal dual tree directional filter bank (PDTDFB), but it is still valid in other complex wavelet transforms. Let an image be decomposed by complex wavelet with S scales and K orientations in each scale, and the angle difference between two consecutive directional subbands is π/K. For an edge at angle αk in the directional subband k with the center angle γk, the subband k contains directional information at angels αk ¼γk þθ, where  π/2Koθoπ/2K. In particular, let us consider an edge at angle αk in the supported region of subband k with 1 r k r (K/4), in which case the center angle of subband γk is an acute angle. We take two horizontally adjacent coefficients A and B located in the neighborhood of the edge as shown in Fig. 1. The distance between A and B at scale s in direction k is Ds,k, then their distances to the edge in the direction normal to subband orientation are represented by AA′ and BB′. The angle of the edge αk can be determined by determining θ in terms of AA′ and BB′:

AA′  BB′ Ds;k cos γ k

k

Ds;k sin γ k

4

From the above relationships, it can be stated that the feature orientation αk is linearly proportional to AA′ BB′. The term AA′  BB′ has a linear relationship with the difference of the phases at A and B i.e. AA′  BB′ p ∠A ∠B [15]. Furthermore, for a coefficient in the subbands of ‘mostly vertical’, the next coefficient in the vertical direction is taken to compute the difference of the phases and other subbands are done in the horizontal direction. Thus, for the coefficient c at position (i,j) in scale s and orientation k, let ∠c(i,j) represent the phase of it, the relative phase is defined as in [14]: ( ∠csk ði; jÞ  ∠csk ði; j þ 1Þ if 1 r k r K=2; RP sk ði; jÞ ¼ ð3Þ ∠csk ði; jÞ  ∠csk ði þ 1; jÞ if K=2 o k r K:

2.2. A review of complex wavelet transforms 2.2.1. The dual tree complex wavelet transforms The dual tree complex wavelet transform (DTCWT) [10,11] is an enhancement to conventional real-valued wavelets, with important additional properties of approximate shift invariant, directional selectivity and a limited amount of redundancy. A dual tree of real wavelet filters has been used in DTCWT to generate the real and

244

Z. Jiuwen et al. / Signal Processing 96 (2014) 241–252

imaginary parts of complex wavelet coefficients. The filters employed in the two trees are designed in such a way that the aliasing in one branch in the first tree is approximately canceled by the corresponding branch in the second tree. The dual tree filter banks (FB) is designed such that the dual FB is approximately a half-sample shift of the primal FB and thus the two channel FB act as a Hilbert transformer [22]. In two dimensions, the DTCWT has six directional subbands at each scale, which is better than the real wavelet with three subbands in directional selectivity. A comprehensive explanation of the transform and details of the dual tree FB design of DTCWT is given in [11]. 2.2.2. Pyramidal dual tree directional filter banks The pyramidal dual tree directional filter banks (PDTDFB) has been proposed in [16]. The transform provides a shiftable and scalable multiresolution directional description to image by combing the Laplacian pyramid and the complex directional filter bank (DFB). The complex DFB is constructed with the dual tree structure of real fan filters. Each primal directional filter and its dual filter are designed to have the 2-D Hilbert transform relation as show in [23]. As a result, for each subband, the equivalent directional complex filter has a one-sided frequency support. The PDTDFB can have 2n complex-valued subbands and each subband is represented by the separate real and imaginary part, where n should be positive integer. The most desirable property of the PDTDFB is that there is no significant aliasing in all decimated complex directional subbands. Therefore, the complex directional subbands provide a shiftable representation of the image in scales and directions. In particular, for the image, the magnitude and phase information of a coefficient in a complex subband provide local spatial information on the edge at a specific scale and direction. In addition, it cannot be ignored that the PDTDFB has lower computation efficiency for its implementation. More details of the transform construct and complex DFB design are referred to [16]. 2.3. Discrete shearlet transform The discrete shearlet transform has been introduced in [19,20] As a much more flexible multiscale directional representation, this transform is especially designed to address anisotropic and directional information at various scales. Unlike traditional wavelet transform, the analyzing functions associated with the discrete shearlet transform are defined at various scales and highly anisotropic. Thus through discrete shearlet transform, an image can be decomposed into various frequency regions whose supports are contained in a pair of trapezoidal regions symmetric with respect to the origin and are directionally oriented. Compared with other ‘directional wavelets’ including the DTCWT [10,11], contourlets [24] and the PDTDFB [16], an important advantage of the discrete shearlet transform is that there are no restrictions on the number of directions for the shearing. Moreover, it should be noted that in discrete shearlet transform the directional subbands are undecimated at a specific scale, in contrast, the DFB used in the contourlets and PDTDFB decimates 2-D data along vertical and horizon. The discrete shearlet

transform is still computationally efficient despite being highly redundant. Refer to [19,20] for additional details about the mathematical framework and the implementation of the discrete shearlet transform. 3. Dual tree shearlets 3.1. The discussion of relative phase model In [15], the author points out that the relative phase models are not accurate enough when the size of the subband is too small, for example smaller than 32  32. As a result, the relative phase feature can only be used in few conditions in the original PDTDFB domain. We demonstrate this default in texture retrieval as shown in Tables 1 and 3. In experiment, we select 40 texture images as [15] from the VisTex database [25,26]. Each of these 512  512 grayscale images is treated as a single class and then divided into sixteen 128  128 non-overlapping subimages. Thus the database contains 640 texture samples from 40 classes, and each class contains 16 samples. All samples are normalized to zero mean and unit variance to reduce the intensity correlation. The normalized Euclidean distances between the query image and each database image are measured according to the extracted feature vectors and the database images that give the smallest distance are retrieved. The query pattern should be any one of the texture patterns from the database. For each image, we select 16 nearest neighbors, and the number of these textures belonging to the same class as the query texture, containing itself, is counted. This number is divided by 16 to calculate the retrieval rate. The average of all classes is the overall performance of the transform in special scales and directions. To represent the number of decomposed orientations in each scale, we provide the decomposition vector, for example [8, 8, 8]. The scales in a decomposition vector are arranged from coarse to fine, which means the last number of the decomposition vector represents the number of orientations in the finest scale. Afterward, each sub-image is decomposed with different complex wavelet transforms as the decomposition vector and the relative phase matrix are computed. The circular mean and standard deviation of the relative phase matrix are named as RP-mean and RP-var, and the combination of them is treated as the relative phase feature (RP). Parallel, the mean and standard deviation of the absolute values of the complex coefficients are calculated as the magnitude feature (MAG). The RP features and the combination of RP features and MAG features are represented by RP and RP-MAG separately. Results imply that there are three aspects of this problem have to be noticed. The first question relates to the property of downsampling in multiscale decomposition. For the Laplacian pyramid (LP) used in PDTDFB, a coarse approximation of the original image is derived by filtering and sampling along the two dimensions separately and the detail is obtained as the same size of original image, then the process is iterated on the coarse version. Thus the size of the image in each level is twice as long as the next level image in two dimensions. Then the complex directional

Z. Jiuwen et al. / Signal Processing 96 (2014) 241–252

filter bank (DFB) is applied on each scale to obtain the 2  2n directional subbands where n should be a positive integer. In addition, each channel of the complex DFB is critically sampled, which means that one dimension of directional subbands will be halved as n increases. Because of the restriction that the sizes of direction subbands must be more than 32 in both row and column, the accuracy of the relative phase model cannot be ensured in the subbands of second or higher scale. Table 1 presents the texture retrieval rates of relative phase features in PDTDFB when two or more scales are considered, which are equal or lower than the ones using only the finest scale. Therefore in the PDTDFB domain, the relative phase model can only utilize the phase information for subbands at the finest scale though the relative phase is originally defined in multiscale framework. The second aspect involves the nonuniform downsampling along row and column in the directional decomposition. Reviewing the relative phase model proposed in [14], the distance between the two adjacent coefficients is calculated as 2s at s scale which is uncorrelated with the number of decomposed orientations. In the derivation of relative phase, its effectiveness is based on the connotive assumption that the edges in the original image can be represented exactly in directional subbands. As shown in Fig. 1, the edge at angle αk is captured in the complex wavelet directional sub and k as the same structure and the distance between two adjacent coefficients is the gap between them in the original image. While in PDTDFB domain, the sizes of directional subbands are related to the number of decomposed orientations 2n in each scale. Because of the nonuniform downsampling in DFB, one dimension of directional subbands will be halved as n increases. Thus there exists structure distortion for the edge represented in directional subbands. Parallelly, according to [15], linear relationship between local phase and distance must satisfy the restriction jwt j{1, in which w is the frequency in the support of complex filter and t is the distance to feature. While the operator of nonuniform downsampling in complex DFB nonuniform enlarges the distance between the two adjacent coefficients, the coefficients near the feature do not restrict satisfy the restriction of jwt j{1. Table 3 presents the retrieval rates of the relative phase feature in the finest scale with the decomposition vector of [4, 4, 4], which is the only condition in which all the directional subbands are uniformly downsampled along vertical and horizon in PDTDFB. Compared with Table 1, the results suggest that although fewer directional subbands are obtained, the retrieval rate is higher than other conditions when directional subbands are uniformly downsampled along vertical and horizon. The third problem is that the number of decomposing directions in conventional complex wavelet transform is not optimal for phase information. The DTCWT has fixed six directions and PDTDFB has 2n directions, but we expect the more flexible multiscale directional representation. In the application of Gabor phases, the best performance happens when each scale is decomposed into six directions, but this number of directions cannot be achieved in the PDTDFB domain. For the reason that every subband

245

Table 3 Average retrieval accuracy using various complex wavelet transform with relative phase and magnitude features when the decomposition vector is [4, 4, 4] in the VisTex database. Levels Methods

RP-mean (%)

RP-var (%)

RP (%)

RP-MAG (%)

MAG (%)

S ¼1

PDTDFB DTCWT D-Shearlets

51.55 39.33 35.43

44.48 47.06 57.90

64.22 76.12 59.12 73.01 60.22 74.88

60.75 60.79 56.72

S ¼2

PDTDFB DTCWT D-Shearlets

61.11 34.03 59.74

54.13 63.57 65.90

69.30 78.10 58.63 74.30 72.23 77.80

68.44 68.48 66.33

S ¼3

PDTDFB DTCWT D-Shearlets

60.12 31.57 65.33

54.89 63.64 65.73

69.34 79.81 59.48 77.36 74.00 79.65

73.68 71.61 70.74

in DTCWT is square, we use the relative phase model in it as shown in Table 3. The performance of DTCWT is acceptable but it is limited because of its poor directional selection. From the above discussion, we can conclude that the relative phase originally generated from PDTDFB has three defaults. Firstly, the relative phase cannot fully capture the multiscale information though it is originally defined in multiscale framework. Secondly, the relative phase is not accurate enough when the directional subbands are obtained through nonuniform downsampling along row and column respectively. At last, PDTDFB can capture the geometry of images only in 2n directions, and this number of decomposing directions is not optimal for phase information. The DTCWT is also restricted by its poor directional selection. Consequently, we need a new transform which should have the following properties:

 Structure of multiscale with dual-channel which  

can represent image in multiscale and provide phase information. Uniform downsampled directional subbands which can ensure the unbiased reference direction of each subband. More flexible directional representation which can provide the optimal decomposing directions for getting phase features.

3.2. The structure of dual tree shearlets As introduced in Section 2, the discrete shearlet transform is much more flexible in directional decomposition and its directional subbands are undecimated at a specific scale. However, the existing discrete shearlet transform is real-valued which means it does not have explicit phase information. To obtain the phase information based on the shearlets, we propose a new structure called the dual tree shearlets. Fig. 2 shows its structure. For the discrete shearlet transform at fixed resolution levels, the procedure can be concisely regarded as follows: first, apply the Laplacian pyramid (LP) scheme to decompose an image into a low-pass subband and a band-pass subband, and then decompose the high pass image with the shearing filters [19] to obtain the directional subbands.

246

Z. Jiuwen et al. / Signal Processing 96 (2014) 241–252

Real tree part

Shearing filters

Directional subbands

Direction 1

Direction 1

Direction K

Direction K

Direction 1

Direction 1

Direction K

Direction K

Bandpass subband Lowpass subband Image

Dual tree LP Lowpass subband Bandpass subband Imaginary tree part

Shearing filters

Directional subbands

Real tree part

Directional subbands

Shearing filters

Subband 1

Filter1

Lowpass subband Inverse Real tree LP sum

Subband K

Im -R

Bandpass subband

Filter K

Image

Subband 1

Im-R Im-I 2

Filter1

sum Subband K

Bandpass subband

Filter K

Lowpass subband Directional subbands

Inverse Imaginary tree LP

Im -I

Shearing filters Imaginary tree part

Fig. 2. The structure of dual tree shearlets (a) decomposition and (b) reconstruction.

The key in constructing complex shearlets is the designing of complex filters: each pair of them should satisfy the Hilbert transform relationship. In this paper, we focus on the portion of LP to implement the complex-valued spatial decomposition. Consequently three aspects of this problem have to be addressed. The first question involves constructing a special structure subsampled LP with two channels named dual tree LP transform. The two channels of this LP are designed to satisfy the relationship of dual tree shearlets. As a result, the two obtained pyramids can be regarded as the real and imaginary parts of LP decomposition. We know that the Hilbert transform relationship within the complex filters in DTCWT can be obtained by filter phase delays between the two trees. It turns out that the low-pass filters should satisfy a very simple property except in the first stage: one of them should be approximately a half-sampled delay of the other [27]. For the first stage we can use the same set of filters in each of the two trees, it is necessary only to translate one set of filters by one sample with respect to the other set [11]. Therefore, we design the dual tree LP with a structure similar to the DTCWT. Fig. 3(a) and (b) show a single level of it. Filters h0(n) and h1(n) are the low-pass filters of the dual tree LP and they satisfy a

certain phase constraint. The multiscale decomposition is obtained by applying the same processes on coarse scale signal c1 and c2 iteratively. Filters g0(n) and g1(n) are also low-pass filters. Subbands c1 and c2 are up-sampled and filtered by filters g0(n) and g1(n), then they are subtracted by x to get the difference d1 and d2, which are the bandpass subbands of the real tree and the imaginary tree respectively. The reconstruction of the dual tree LP is just the inverse of the analysis process, besides we average the final output results of the two channels as the reconstructed image. The filters of the dual tree LP are designed as below: g 0 ðnÞ ¼ h0 ðN  nÞ

ð4Þ

g 1 ðnÞ ¼ h1 ðN  nÞ

ð5Þ

At level 1 filters h0(n) and h1(n) satisfy the following relationship: h1 ðnÞ ¼ h0 ðn 1Þ

ð6Þ

At level 2 and the following levels, filters h0(n) and h1(n) must satisfy the following relationship: h1 ðnÞ ¼ h0 ðn 1=2Þ

ð7Þ

Z. Jiuwen et al. / Signal Processing 96 (2014) 241–252

c1

h0 n

2

2

g0 n

h1 ( n )

2

2

g1 (n )

d1

x

d2 c2

c1 d1

h0 n

2

2

g0 n

x1 x

h1 ( n )

d2

2

2

g1 ( n )

x1 x2 2

x2

c2 Fig. 3. The structure of dual tree LP (a) analysis and (b) synthesis.

We use “9-7” and “Q-shift” filters [28] in the implementation of dual tree LP. To get the limited spectrum which is approximately half of the Fourier space, we apply different filters to the two dimensions of the 2-D data respectively. For the row, we use the low-pass filter of “9-7”. Parallel for the column, at level 1, we use the low-pass filter of “9-7” ash0(n), filter h1(n) is just a sample delay of filter h0(n); at level 2 and the following level, filters h0(n) and h1(n) satisfy the following relationship: h0(n)¼h(n  1/4) and h1(n)¼h(n  3/4), where h(n  1/4) and h(n  3/4) are filters pair designed by “Q-shift” algorithm. In this way, filters h0(n) and h1(n) satisfy the phase constrained relationship of 1/2 sampling delay. The above mentioned structure is directly derived from DTCWT, so under the above designing, d1 and d2 form an approximately analytic signal as DTCWT, which means they satisfy the 901 phase orthogonal constraint. Therefore, the dual tree LP is nearly shift-invariance. The second problem relates to the directional decomposition in each scale of the complex-valued pyramid. Through the dual tree LP transform constructed above, a real-valued two-dimensional image is decomposed to multiscale analytic signal. For an analytic signal, it is well known that applying band-pass filters on its real part and imaginary part respectively will not cause aliasing. Therefore we filter the two channel outputs of dual tree LP separately using shearing filters [11] at each scale. As a result, the obtained directional subbands in each scale are also analytic. Consequently, the amplitude and phase information of complex coefficients in each subband provide local information on the directional feature of the image at a fixed scale and direction. Unlike the DFB used in PDTDFB which is the combination of critically sampled fan filter banks and pre/post re-sampling operations, the directional filters used in our dual tree shearlets are obtained through a different approach. Similar to the discrete shearlet transform, a local variant such as the window function is implemented in the

247

directional decomposition. In practice, we use a Meyer window to construct the shearing filters. As shown in [19], we can perform the shearing filters “directly” in the time-domain using the convolution-operator. With the small sized shearing filters, we use the overlap-add method to compute the convolutions which are the obtained directional subbands in each scale. For the reason that the convolution is restricted to be of the same size as the band-pass image, the directional decomposition is overcompleted and each directional subband has the same size of pyramid at a special scale. That is, there is no downsampling operation in directional decomposition. Thus the decimating operator is only occurred in the stage of dual tree LP decomposition, which can be treated as that the two channels are independent under the relationship of phase constrain. For the LP used in each channel, a coarse approximation of the original image is derived by filtering and sampling along the two dimensions separately and the detail is obtained as the same size of the original image, then the process is iterated on the coarse version. Consequently the size of the image in each level is twice as long as the next level image in the two dimensions which means that all the decimating operators are uniformly downsampled in the dual tree LP. Furthermore, each directional subband has the same size as the given band-pass image. Therefore, all the directional subbands are uniformly downsampled at different scales in different directions as we wish. It should be noted that we can express the formulation of the windowing with a non-dyadic spacing. That is, unlike PDTDFB, there are no restrictions on the number of direction for the shearing which is impossible when the DFB is used. In practice, the number of orientations at each scale is constructed as even concerning the directional sensitivity. In addition, the only one restriction on the shearing filters is that the size of the filter should be more than the maximum number of directional subbands. While the DFB used in PDTDFB does not have such flexibility as shown in [24]. Furthermore, although the dual tree shearlets is a cumbersome approach, the small sized filters in the time-domain allow us to use a fast overlap-add method to compute the convolutions [29]. Compared with PDTDFB, we can see that our dual tree shearlets is most efficient computationally in Table 4. The third aspect deals with the reconstruction of our dual tree shearlets as shown in Fig. 2 (b). The processing of reconstruction has two steps: inverting the directional decomposition and dual tree LP. Firstly, the inversion of directional filtering only requires a summation of the shearing filters rather than inverting a DFB as PDTDFB. Consequently, for each channel of the dual tree LP, reconstruction is just the inverse of the analysis process. Besides we average the final output results of the two channels as the reconstructed image. 4. Experiments In this section, we extract the statistical features of relative phase in our dual tree shearlets for texture image retrieval. For comparison, the results of texture retrieval in other complex wavelet transform use the same features

248

Z. Jiuwen et al. / Signal Processing 96 (2014) 241–252

Table 4 Average retrieval accuracy using various complex wavelet transform with relative phase and magnitude features when the decomposition vector is [4, 4, 4,] in the Brodatz database. Levels Methods

RP-mean (%)

RP-var (%)

RP (%)

RP-MAG (%)

S¼1

PDTDFB DTCWT D-Shearlets

57.99 67.41 57.16

68.38 63.09 76.42

75.95 83.34 76.41 81.53 79.77 82.96

74.96 68.58 72.33

S¼2

PDTDFB DTCWT D-Shearlets

50.93 78.68 76.79

79.14 76.26 84.30

69.82 84.16 82.76 86.06 85.04 85.97

82.25 80.04 80.34

S¼3

PDTDFB DTCWT D-Shearlets

46.71 78.96 79.89

81.86 79.23 86.65

75.43 85.50 83.80 86.90 86.36 86.72

84.61 82.81 82.63

1 sc ¼ 1  ∑ð1  cos ðθi;j  θÞÞ N i;j

MAG (%)

 1  m ¼ ∑xði; jÞ N i;j

4.1. Database In our experiment, we select 40 texture images from the VisTex database [25,26] as in [15]. Each of these 512  512 grayscale images is treated as a single class and then divided into sixteen 128  128 non-overlapping sub-images. Thus we create a database consisting of 640 texture samples from 40 classes, and each of them contains 16 samples. All samples are normalized to zero mean and unit variance to reduce the intensity correlation. Afterwards, we decompose each image with the complex wavelet transform and compute the corresponding feature vectors. Parallel, to check the algorithm in other types of textures, we select 40 texture images from the Brodatz database [30] which are divided and normalized as above. 4.2. Measure of descriptive statistics Since the relative phase is circular data, we will present the definitions of circular mean and circular variance to make comparisons between circular distributions. The two features provide statistical information of the relative phase in each subband at special scale. Let θi,j be the circular observation at position (i, j) given in terms of angle. The circular mean direction of this set, denoted by θ is defined as follows: ! i;j

ð8Þ

i;j

Thus, the circular mean direction is given by: ∑i;j sin θi;j mc ¼ arctan ∑i;j cos θi;j

ð10Þ

Parallelly, the magnitude features are extracted based on the dual tree shearlets. Firstly, we combine the coefficients of the two corresponding subbands to create the complex coefficients in each complex subband. Then, for each directional subband, the mean and standard deviation of the absolute values of the complex coefficients are calculated as follow:

extracting method as above. A complex wavelet transform is used to decompose a texture image into multiscale and multidirectional complex subbands. Then we extract the statistical features of relative phase for classification in these subbands as defined in [14]. The comparison transforms we used are DTCWT [10,11] and PDTDFB [16], both of them have directionality and approximate one-side supports in the frequency domain, which are similar with our dual tree shearlets.

θ ¼ arg ∑ cos θi;j þ j∑ sin θi;j

The circular variance represents how concentrated circular data are towards this mean direction. If N is the number of elements, the circular variance is defined as:

ð9Þ

s¼

1 ∑ðjxði; jÞj  mÞ2 N i;j

ð11Þ !ð1=2Þ ð12Þ

where x(i, j) is the coefficient at position (i, j), N is the number of elements. 4.3. Features Each image in the database is applied to the following three decompositions: the DTCWT, the PDTDFB and our dual tree shearlets. The DTCWT is applied with three scales and fixed six orientations per scale, while others have three scales of more flexible orientations. To represent the orientations of each scale, we provide the decomposition vector, for example [8, 8, 8]. The scales in a decomposition vector are arranged from coarse to fine which means the last number of the decomposition vector represents the number of orientations in the finest scale. For the dual tree shearlets, we adjust the size of the shearing filters, which are represented by the filters size vector of the same order as the decomposition vector, for instance [8, 8, 16]. We combine the coefficients of the two corresponding subbands to create the complex coefficients in each complex subband. The relative phase matrix of each complex subband in the complex wavelet domain is computed as in (1) and (2), and circular mean and standard deviation of this relative phase matrix will be estimated by (9) and (10) to form the relative phase feature (RP). After that, the mean and standard deviation of the absolute values of the complex coefficients are calculated by (11) and (12) as the magnitude feature (MAG). Furthermore, we combine the RP feature and the MAG feature to create the RP-MAG feature. In the first experiment, with results shown in Tables 1 and 2, we compare the PDTDFB and our dual tree shearlets with the decomposition vector of [8, 8, 8], [16, 16, 16], [4, 8, 16] and the filters size vector of [8, 8, 16], [16, 16, 16], [8, 8, 16]. In the second experiment with results shown in Tables 3 and 4, we use different number of scales in DTCWT, PDTDFB and dual tree shearlets (D-shearlets). The DTCWT has six subbands in each scale and the decomposition vector for PDTDFB and dual tree shearlets is [4, 4, 4], thus all the subbands are square. The filters size

Z. Jiuwen et al. / Signal Processing 96 (2014) 241–252

249

vector used in dual tree shearlets is [8, 8, 16]. In Tables 5 and 6, we present some decomposition vectors that the PDTDFB cannot achieve such as [6, 6, 6], [10, 10, 10], [6, 8, 10], [12, 12, 12], [14, 14, 14] with the filters size vector of [6, 6, 12], [10, 10, 10], [6, 8, 10], [12, 12, 12], [14, 14, 14].

between the two images computed by the RP-MAG feature vector. Thus, we can compute the distance as [31]:     mc ðxÞ mc ðyÞ sc ðxÞ  sc ðyÞ  i;j   i;j  i;j i;j c dxy ¼ ∑ ð13Þ  þ ∑   i;j   αðmci;j Þ αðsci;j Þ i;j 

4.4. Distance

    mi;j ðxÞ mi;j ðyÞ    þ ∑si;j ðxÞ  si;j ðyÞ dxy ¼ ∑    Þ Þ αðm αðs i;j i;j i;j i;j

The normalized Euclidean distances between the query image and each database image are measured and the database images that give the smallest distance are c retrieved. Let dxy and dxy respectively be the distance between two RP feature and MAG feature vectors of image x and y obtained from complex subbands. The sum of the above two distances is Dxy which represents the distance

Table 5 Average retrieval rates for dual tree shearlets with the decomposition vectors that the PDTDFB cannot achieve in the VisTex database. Vectors

Levels RP-mean (%)

RP-var (%)

RP (%)

RP-MAG (%)

MAG (%)

[6, 6, 6]

S¼ 1 S¼ 2 S¼ 3

47.51 64.24 66.42

60.44 64.24 74.11

63.87 75.63 74.53 78.98 76.05 80.83

62.37 67.57 72.10

[10, 10, 10] S¼ 1 S¼ 2 S¼ 3

51.85 69.01 69.46

61.65 71.03 70.83

65.69 75.94 75.37 79.90 75.76 81.12

62.11 68.92 73.06

[6, 8, 10]

S¼ 1 S¼ 2 S¼ 3

51.85 66.31 69.41

61.65 71.91 74.74

65.69 75.94 74.72 79.93 77.26 81.78

62.11 69.49 73.23

[12, 12, 12] S¼ 1 S¼ 2 S¼ 3

49.12 64.35 65.90

61.79 71.22 72.14

63.06 75.60 73.56 79.78 74.87 80.51

64.01 70.25 73.49

[14, 14, 14] S¼ 1 S¼ 2 S¼ 3

51.40 67.30 68.02

62.71 71.04 70.57

65.09 76.29 74.84 79.86 75.26 80.67

62.90 69.17 73.14

Table 6 Average retrieval rates for dual tree shearlets with the decomposition vectors that the PDTDFB cannot achieve in the Brodatz database. Vectors

Levels RP-mean (%)

RP-var (%)

RP (%)

RP-MAG (%)

MAG (%)

[6, 6, 6]

S¼ 1 S¼ 2 S¼ 3

69.21 82.34 83.57

81.42 85.88 87.61

83.37 84.63 86.30 87.02 87.56 87.29

74.93 81.21 83.46

[10, 10, 10] S¼ 1 S¼ 2 S¼ 3

72.81 83.66 84.53

83.13 87.43 88.34

84.07 84.58 87.38 87.07 87.97 87.34

75.27 81.25 83.44

[6, 8, 10]

S¼ 1 S¼ 2 S¼ 3

72.81 82.44 84.45

83.13 87.42 88.58

84.07 84.58 87.29 86.75 88.03 87.36

75.27 81.10 83.05

[12, 12, 12] S¼ 1 S¼ 2 S¼ 3

71.13 82.17 83.16

83.22 88.15 88.65

83.02 83.79 87.29 86.17 87.64 86.61

74.75 80.55 82.83

[14, 14, 14] S¼ 1 S¼ 2 S¼ 3

72.65 83.96 84.06

83.66 87.88 88.21

83.58 84.64 87.54 86.65 87.68 86.87

76.17 81.08 82.79

c

Dxy ¼ dxy þ dxy

ð14Þ

ð15Þ

where αðmci;j Þ,αðsci;j Þ,α(mi,j) and α(si,j) are the standard deviations of mci;j ðdÞ,sci;j ðdÞ, mi;j ðdÞ and si;j ðdÞ of the entire database respectively. The query pattern should be any one of the texture patterns from the database. The distances between the query and each database image are measured according to the extracted feature vectors. Then we select 16 nearest neighbors, and the number of these textures belonging to the same class as the query texture, containing itself, is counted. This number is divided by 16 to calculate the retrieval rate. The performance of the entire class is obtained by averaging this rate over the 16 members who belong to the same class of texture. The average of all classes is the overall performance of the algorithm.

5. Results In the experiments, we use all the subbands contained in different scales. From fine to coarse, the number of scale we use is represented by s. The decomposition vectors are shown in the column of the vector. Each subband is extracted by two RP feature parameters and two MAG feature parameters. The circular mean and standard deviation of the relative phase matrix are applied to experiments, which are represented as RP-mean and RP-var. The RP feature, MAG feature and the combination of RP feature and MAG feature are represented by RP, MAG and RP-MAG separately. In the first experiment, the average retrieval rates are based on both phase and magnitude. The performance of our dual tree shearlets is compared with the performance of PDTDFB in the condition of more decomposition vector in two textures database separately. Table 1 shows results in the VisTex database and Table 2 is based on the Brodatz database. As shown in Tables 1 and 2, in the condition of more than one scale and finer directional selection, the RP and RP-MAG features of dual tree shearlets have consistently improved compared with PDTDFB. It should be noted that based on the dual tree shearlets, the retrieval rates increase with the increase in the number of scales as we expected. That means we can efficiently utilize the multiscale phase information with the dual tree shearlets. While in the PDTDFB domain, the relative phase model is only accurate enough for limited directional subbands at the finest scale because of the operator of nonuniform downsampling. A demonstration is that a higher retrieval rate can be obtained for RP features of PDTDFB when the decomposition vector is [4, 4, 4] as shown in Tables 3 and 4, in which condition, although fewer directions

250

Z. Jiuwen et al. / Signal Processing 96 (2014) 241–252

are obtained, all the directional subbands are uniformly downsampled. In the second experiment with the same method and the same two databases, compared with other decomposition vector as presented in Tables 1 and 2, the most significant difference is that all PDTDFB subbands are uniformly downsampled along the vertical and horizon with the decomposition vector of [4]. The results in the VisTex and Brodatz database are shown in Tables 3 and 4 separately. Let θ represent the difference between the arbitrary edge whose angle is αk in the directional subband k and the center angleγk. For the reason that the number of decomposed orientations is only 4 per scale, it can be obtained that |θ| r(π/8) in which case the approximation θ  tan θ that was used in the relative phase model is not accurate enough. Although the accuracy of the model is not much fine and fewer PDTDFB directions are obtained with the decomposition vector of [4, 4, 4], the retrieval rate of RP feature is higher than other conditions shown in Tables 1 and 2. While our dual tree shearlets has the inherent property that all its directional subbands are uniformly downsampled, thus the retrieval rate is highly affected by the unsophisticated approximation and the fewer directions. But it also performs better in the condition of two or more scales based on RP features. Because of the poor directional selecting, the DTCWT cannot improve the performance in spite of its subbands being uniformly downsampled. In the third experiment as shown in Tables 5 and 6 which is based on the VisTex and Brodatz database respectively, some dual tree shearlets decomposition vectors that the PDTDFB cannot achieve are presented. Only 2n number of directions can be achieved in the

PDTDFB domain, but we expect the more flexible multiscale directional representation such as the six directions which shows best performance in the application of Gabor phases. The number of orientations at each scale of our dual tree shearlets is constructed as even concerning the directional sensitivity. Compared with the results as shown in Tables 1 and 2, most results of these decomposition vectors in this experiment are significant improvements. For example, higher retrieval rate can be obtained if the images are decomposed with six, ten and fourteen orientations rather than the eight orientations as shown in Tables 1 and 2 in per scale. Results indicate that the best performance does not occur in the traditional 2n decomposed directions. Thus the flexibility in directions of our dual tree shearlets is an important property. We should point out that the retrieval rate does not strictly improve with the increase of the directions. Because of the restriction that the size of shearing filter must be longer than or equal to the number of directions, the designing of the shearing filters is not flexible enough in the condition of too many directions. Thus for the dual tree shearlets, the retrieval rate increases with the directions only in limited numbers. We note that the best results are obtained when decomposition vector is [6, 8, 10]. Therefore, it is possible that each scale has its own best number of decomposed directions. The results suggest that the performance of the RP feature is consistently significant compared with the MAG feature. The circular standard deviation of the RP feature is most significant. We noticed that in the Brodatz database the retrieval rate of RP feature in the condition of two or more scales is even higher than that of the RP-MAG feature which means the phase information is dominant

Fig. 4. Three images in the VisTex database whose retrieval rates are the lowest when the decomposition vector is [6, 8, 10] in dual tree shearlets domain.

Fig. 5. Three images in the VisTex database whose retrieval rates are 100% when the decomposition vector is [6, 8, 10] in dual tree shearlets domain.

Z. Jiuwen et al. / Signal Processing 96 (2014) 241–252

compared with the magnitude information. And our algorithm perform the same trend in the two databases which means that our dual tree shearlets can be applied to diverse types of textures database. We have to point out that there are limitations of our algorithm in some types of textures. Fig. 4 shows the three images in the VisTex database whose retrieval rates are the lowest when the decomposition vector is [6, 8, 10] in dual tree shearlets domain, and Fig. 5 shows three images whose retrieval rates are 100% in the same condition. Results imply that our algorithm do not perform so well for the images which have random textures. Notwithstanding its limitation, this study does confirm our previous suspicions on the formal relative phase. Comparing with the relative phase defined in PDFTFB and DTCWT, our work exploits the multiscale characteristic, with uniform downsampled subbands and higher directional sensitivity. Thus our dual tree shearlets does have advantages and provides a more appropriate representation for the implementation of relative phase. We compare the retrieval rate of RP feature in our dual tree shearlets and other relevant complex-valued wavelet transforms in Fig. 6. Differing from the above experiments, the group of textures consists of N images from the 40 texture images in the VisTex database used above, and they are divided and normalized. Thus, we create a database consisting of 16  N texture samples from N classes, and each class contains 16 samples. For each sample, the DTCWT is applied with 3 scales, while the dual tree shearlets and the PDTDFB are applied with 3

Fig. 6. Average retrieval rate of RP feature according to the number of the classes in database.

251

scales and 8 orientations per scale. Their RP feature vectors are computed as in the above experiments. Fig. 6 shows the overall performances for the case of N from 1 to 40. The retrieval rate reduces as the increasing of class contained. It is clear that the performance of the RP feature in our dual tree shearlets is consistently better than that in the DTCWT and the PDTDFB. 5.1. Time Computational efficiency is an important property in many image processing. Whereas, the PDTDFB has higher amount of computation for its implementing scheme which many times contains Fourier transforms and inverse Fourier transforms. The DTCWT has high computational efficiency, but the poor directional selecting limits its application. Thus, we compare the computational efficiency of PDTDFB and our dual tree shearlets in Table 7. All the experiments are implemented in MATLAB R2009b with an Intel core 2 CPU 2.20 GHz machine. The relative phase and magnitude features extracting time in the PDTDFB and the dual tree shearlets in different number of scales are shown in Table 7. It is clear that the time of RP-MAG feature vector extracted in dual tree shearlets is much less than that in the PDTDFB domain. Despite the fact that the decomposition of dual tree shearlets is more redundant than the PDTDFB, but its computational efficiency is significant. 6. Conclusion Aim at the problem that the relative phase model is not accurate enough in PDTDFB domain when multiscale and multidirection are considered. A new complex wavelet transform, which we called dual tree shearlets, is proposed for phase information extraction. The transform is constructed based on the discrete shearlet transform, which has several attractive properties including multiscale, dual tree structure, flexible directional decomposition and uniform downsampled subbands. The complex-value decomposition is implemented with the dual tree LP inspired by the DTCWT. By applying the relative phase model to capture the phase information from the dual tree shearlets, the relative phase feature is used in texture image retrieval. The simulation results show that the feature from the dual tree shearlets perform much better than the exiting complex wavelet transform such as PDTDFB and DTCWT when more than one scale is considered. In addition, the number of direction in each scale is

Table 7 Evaluation of relative phase and magnitude features vector extracting time in the PDTDFB and the dual tree shearlets. Vectors

[4, 4, 4] [8, 8, 8] [16, 16, 16] [4, 8, 16]

S¼1

S¼2

S¼ 3

PDTDFB (s)

D-Shearlets (s)

PDTDFB (s)

D-Shearlets (s)

PDTDFB (s)

D-Shearlets (s)

154.13 249.16 496.90 496.90

57.41 100.61 201.87 201.87

270.96 403.11 698.54 616.57

77.48 144.07 285.66 242.60

334.86 508.02 803.22 728.23

92.15 173.25 346.35 251.08

252

Z. Jiuwen et al. / Signal Processing 96 (2014) 241–252

more flexible. Thus our transform can efficiently utilize the multiscale and multidirection phase information. Moreover, we demonstrate that the dual tree shearlets is computational efficient, which enable the relative phase feature to be more efficient in searching and browsing texture images. References [1] A.V. Oppenheim, J.S. Lim, The importance of phase in signals, in: Proceedings of the IEEE 69 (1981) 529–541. [2] M.C. Morrone, R.A. Owens, Feature detection from local energy, Pattern Recognition Letters 6 (5) (1987) 303–313. [3] J.G. Daugman, High confidence visual recognition of persons by a test of statistical independence, IEEE Transactions on Pattern Analysis and Machine Intelligence 15 (11) (1993) 1148–1161. [4] D. Zhang, W.K. Kong, J. You, M. Wong, Online palmprint identification, IEEE Transactions on Pattern Analysis and Machine Intelligence 25 (9) (2003) 1041–1050. [5] B. Zhang, S. Shan, X. Chen, W. Gao, Histogram of Gabor phase patterns (HGPP): a novel object representation approach for face recognition, IEEE Transactions on Image Processing 16 (1) (2007) 57–68. [6] J.M.H. du Buf, P. Heitkämper, Texture features based on Gabor phase, Signal Processing 23 (3) (1991) 227–244. [7] J. Portilla, E.P. Simoncelli, A parametric texture model based on joint statistics of complex wavelet coefficients, International Journal of Computer Vision 40 (1) (2000) 49–70. [8] Z. Wang, E.P. Simoncelli, Local phase coherence and the perception of blur, Advances in Neural Information Processing Systems 16 (2004) 786–792. [9] X. Zhang, Y. Jia, Face recognition with local steerable phase feature, Pattern Recognition Letters 27 (16) (2006) 1927–1933. [10] N.G. Kingsbury, Complex wavelets for shift invariant analysis and filtering of signals, Applied and Computational Harmonic Analysis 10 (3) (2001) 234–253. [11] I. Selesnick, R. Baraniuk, N. Kingsbury, The dual-tree complex wavelet transform, IEEE Signal Process Magazine 22 (6) (2005) 123–151. [12] M.A. Miller, N.G. Kingsbury, Image modeling using interscale phase properties of complex wavelet coefficients, IEEE Transactions on Image Processing 17 (9) (2008) 1491–1499. [13] M.A. Miller, N.G. Kingsbury, Image denoising using derotated complex wavelet coefficients, IEEE Transactions on Image Processing 17 (9) (2008) 1500–1511. [14] A. Vo, S. Oraintara, T.T. Nguyen, Using phase and magnitude information of the complex directional filter bank for texture image retrieval, in: Proceedings of IEEE International Conference on Image Processing, vol. 4, 2007, pp. IV-61–IV-64.

[15] A. Vo, S. Oraintara, A study of relative phase in complex wavelet domain: property, statistics and applications in texture image retrieval and segmentation, Signal Processing Image Communication 25 (1) (2010) 28–46. [16] T.T. Nguyen, S. Oraintara, The shiftable complex directional pyramid Part 1: theoretical aspects, IEEE Transactions on Signal Processing 56 (10) (2008) 4651–4660. [17] T.T. Nguyen, H. Chauris, Uniform discrete curvelet transform for seismic processing, in: EAGE Conference and Technical Exhibition, Rome, 2008, pp. 5495–5498. [18] A. Vo, S. Oraintara, N. Nguyen, Vonn distribution of relative phase for statistical image modeling in complex wavelet domain, Signal Processing 91 (1) (2011) 114–125. [19] G. Easley, D. Labate, W.Q. Lim, Sparse directional image representations using the discrete shearlet transform, Applied and Computational Harmonic Analysis 25 (1) (2008) 25–46. [20] D. Labate, W.Q. Lim, G. Kutyniok, G. Weiss, Sparse multidimensional representation using shearlets, in: Proceedings of SPIE Wavelets XI, San Diego, California, vol. 5914, 2005, pp. 254–262. [21] J. Aelterman, H.Q. Luong, B. Goossens, A. Pižurica, Augmented Lagrangian based reconstruction of non-uniformly sub-Nyquist sampled MRI data, Signal Processing 91 (12) (2011) 2731–2742. [22] A. Potamianos, P. Maragos, A comparison of the energy operator and the Hilbert transform approach to signal and speech demodulation, Signal Processing 37 (1) (1994) 95–120. [23] T.T. Nguyen and S. Oraintara, A shift-invariant multiscale multidirection image decomposition, in: Proceedings of IEEE International Conference on Acoustics, Speech and Signal Processing, France, vol. 2, 2006, pp. 153–156. [24] M.N. Do, M. Vetterli, The contourlet transform: an efficient directional multiresolution image representation, IEEE Transactions on Image Processing 14 (12) (2005) 2091–2106. [25] M. Kokare, P.K. Biswas, B.N. Chatterji, Texture image retrieval using new rotated complex wavelet filters, IEEE Transactions on Systems, Man, and Cybernetics 35 (6) (2005) 1168–1178. [26] M.N. Do, M. Vetterli, Wavelet-based texture retrieval using generalized Gaussian density and Kullback–Leibler distance, IEEE Transactions on Image Processing 11 (2) (2002) 146–158. [27] I.W. Selesnick, Hilbert transform pairs of wavelet bases, IEEE Signal Processing Letters 8 (6) (2001) 170–173. [28] N. Kingsbury, Design of Q-Shift complex wavelets for image processing using frequency domain energy minimization, in: Proceedings of IEEE International Conference on Image Processing, 1 (2003) 1013–1016. [29] S. Mallat, A Wavelet Tour of Signal Processing, Academic Press, San Diego, 1998. [30] P. Brodatz, Textures: A Photographic Album for Artists & Designers, Dover, 1966. [31] B. Manjunath, W. Ma, Texture features for browsing and retrieval of image data, IEEE Transactions on Pattern Analysis and Machine Intelligence 18 (8) (1996) 837–842.