Fourth International Conference on Networked Computing and Advanced Information Management
Design and Application of All Phase Pyramidal Directional Filter Bank Li Li1),2), Zhengxin Hou1), Chengyou Wang1), Fan He1) 1) School of Electronic Information Engineering, Tianjin University, Tianjin 300072, China 2) Department of Electronic Engineering, Tianjin University of Technology and Education, Tianjin 300222, China
[email protected] subband decomposition filters and directional filters are designed based on all phase discret cosine transform (APDCT). It is shown in this paper that the pure 2-D APDCT filters have better frequency and direction selectivity and can be faster implemented. Moreover, unlike the wedge-shaped frequency partition that are tilting in contourlet, the proposed DFB can obtain right horizontal and vertical frequency partition, which in space domain are more proper for human vision system and can represent more nature scene characteristics. The paper is structured as follows. In Section 2, we describe APPDFB and its building block, especially the construction of APDFB. In Section 3, we study the issues associated with the all phase DCT filter design and give some examples. Application of the APPDFB in image denoising is discussed in Section 4. Concluding remarks are drawn in Section 5.
Abstract Based on the essence of contourlet and all phase DCT sequency filtering theory, a novel all phase pyramidal directional filter bank is developed and applied to image denoising. In all phase datum space, pure 2-D directional filters are designed directly from its spectrum characteristics, so the directional filter banks can be constructed flexibly without rotation and resampling of image so as to preserve more image details and decrease the computational complexity. In addition, with the all phase directional filter bank proposed, horizontal and vertical directional subbands are divided, which can not be obtained by contourlet and other directional filter banks. It is shown in our experimental results that the proposed approach is superior to both wavelet and contourlet for most of the images and has especially better visual quality for images with many horizontal and vertical edges.
2. Structure of APPDFB
1. Introduction
2.1 General structure of APPDFB
As a typical example of 2-D signal, image differs from 1-D signals with its directional characteristic. And it is well known that there is limitation of wavelet transform in capture the geometry of image edges, which is just the separable extension of 1-D transforms. Being a multiscale and multidirectional transform constructed by combining the Laplacian pyramid (LP) [1] with directional filter bank (DFB) [2], contourlet transform (CT) [3] has shown its advantages in image denoising [4], enhancement [5], texture retrieval [6] and many other fields over wavelet, because it can capture significant information about an object, especially because of its directionality and anisotropy. Based on the essence of contourlet transform, a novel all phase pyramidal directional filter bank (APPDFB) are developed and applied to image denoising. In the proposed structure, both the LP
The basic structure of APPDFB is very similarly with that of contourlet. It consists of two stages, namely LP decomposition and directional decomposition illustrated in Figure 1. With this structure, the 2-D frequency plane is divided into one lowpass subband and multiple bandpass directional subbands. In designing the structure, we focus on the following two aspects to decrease distortion and computation. Firstly, the structure proposed is actually a redundant representation without downsampling and upsampling operators. By allowing redundancy, it is possible to enrich the set of basis functions so that the representation is more efficient in capturing some of the signal behavior [7]. In addition, redundancy representations are generally more flexible and easier to design. On the other hand, redundancy keeps it away
This work was support by the Natural Science Fund of Tianjin Municipal, China (No. 07JCYBJC13800) and the Research Fund for the Doctoral Program of Higher Education of China (No. 20060056051).
978-0-7695-3322-3/08 $25.00 © 2008 IEEE DOI 10.1109/NCM.2008.85
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from the application that requires lower data capacity like image compression. Secondly, all the filters in the structure are designed directly from their spectrum shapes and are purely 2-D filters. The subband decomposition in the proposed structure is obtained by LP without downsampling and upsampling. And with APDCT, the LP decomposition filters at each stage can be very easily designed and expanded. Figure 1 illustrates the decompositions with J = 2 ( J denotes the number of LP decomposition stages). The image is divided into one lowpass subband and J highpass subbands. The redundancy of this structure is much higher ( J ) than that of contourlet (less than 1/3), which limit its application to applications that has not limitation on redundancy such as denoising, enhancement, and controur detection.
2.2 Construction of APDFB
Lowpass subband
Highpass
Bandpass directional
subband1
Image
subbands Highpass
Bandpass directional
subband 2
subbands
Stage1 Pyramid
Stage2 decomposition
Directional
decomposition
Figure 1. Structure of APPDFB
ω2 7 6 5 4
0 1 2 3
(π, π) 4 5 6 7
ω1
ω2 2 3 4 5 1 0 7 6 5 4 3 ( − π, − π) (b)
6 7 0 1 2
(π, π)
ω1
3 2 1 0 ( − π, − π) (a) Figure 2. Idealized frequency partitioning obtained by CT and the proposed structure
After LP decompositon, the highpass subbans should be further decomposed into different directional subbands. There may be different decomposition approaches when different DFB structure is used. The proposed all phase DFB (APDFB) is a tree structured filter bank without downsamplers and upsamplers in each channel. As an example, the first highpass subband is divided into two directional subbands and the second highpass subbuand is divided into four directional subbands in Figure1. The number of directions can be flexibly selected for each subband according to the characteristics of image.
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DFB plays a very important role in the construction of contourlet and APDFB differs from other kinds of DFB mainly by its construction, the frequency partitions and the design of its basic filters (which will be talked about in detail in part 3). So the construction of APDFB is highlighted in this part. Bamberger and Smith have constructed a 2-D DFB that can be maximally decimated while achieving perfect reconstruction in 1992. The original construction of the DFB in [2] involves modulating the input image and resampling operations, which may cause geometry distortion and frequency shift. And to obtain the desired frequency partition, a complicated tree expansion rule has to be followed for finer directional subbands [8]. In contourlet, a different construction for DFB is used, which avoids modulation the input images and has a simpler rule for expansion the decomposition tree. And the key in the DFB is to use an appropriate combination of shearing operators together with twodirection partition of qunincunx filter bank (QFB) at each node in a binary tree-structured filter bank, to obtain the desired 2-D spectrum division as shown in Figure2 (a) [9]. And the rotation of input images with shearing and down-sampling still cause geometry distortion and lose of details. In an adapted version of contourlet: nonsubsampled contourlet transform (NSCT), the nonsubsampled DFB is achieved by switching off the downsamplers and upsamplers in each two-channel filter bank in the DFB tree structure and upsampling the filter accordingly [7]. This may also decrease the accuracy of filters and has some effect on the directional selectivity of NSCT. For all phase DFB, the biggest advantage is that the directional filters can be designed directly in 2-D according to the shape of its spectrum. So we don’t have to rotate images or upsample the filters. The frequency partition can be easily obtained by just a combination of 2-D filters with different frequency selectivity. By this way, the construction of DFB is more flexible and the expansion of it can be easier. In Figure 2 (b), the idealized 8 frequency partitions obtained by APDFB is given and compared with that obtained by contourlet given by Figure 2 (a). It is shown that the directions obtained by the proposed DFB are based on vertical and horizontal instead of the wedge-shaped tilting directions resulted from contourlet. Figure 3 shows the structure of APDFB and the 8 directional subbands of the image “Zoneplate” (a famous synthetic round picture). As demonstrated, the APDFB is a completed reconstructed DFB and the
reconstruction is implemented by just simply adding up all the directional subbands.
So APDCT filters can be very easily designed by constructing the sequence response matrix according to the frequency response required.
3.2. Design of All decomposition filter Q 3a
phase
LP
subband
As demonstrated in Figure.1, the ideal passband support of the low pass filter at the jth stage is the region [−(π / 2 j ,(π / 2 j )]2 . So we can design the sequence response matrix of the first stage accordingly, when N = 4 :
Q4a Q 3b
⎡1 ⎢1 F1 = ⎢ ⎢0 ⎢ ⎣0
Q2a Q4b Q1a Q2b
1 0 0⎤ 1 0 0 ⎥⎥ 0 0 0⎥ ⎥ 0 0 0⎦
Then we can get the low pass filter for the first subband decomposition from (2) as:
Q1b
Figure3. Structure of APDFB and the examples
of directional subbands
⎡ 0.00195 −0.0016 −0.0114 −0.0220 −0.0114 −0.0016 ⎢−0.00160 0.0013 0.0094 0.00183 0.0094 0.0013 ⎢ ⎢−0.01140 0.0094 −0.0660 0.1288 0.0660 0.0094 ⎢ Q1 = ⎢ −0.0220 0.0180 0.1288 0.2500 0.1288 0.0180 ⎢−0.01140 0.0094 0.0660 0.1288 0.0660 0.0094 ⎢ ⎢ −0.0060 0.0013 0.0094 0.0183 0.0094 0.0013 ⎢ ⎣ 0.00195 −0.0016 −0.0114 −0.0220 −0.0114 −0.0016
3. Design of APDCT filters 3.1. APDCT Based on the concept of all phase date space [10], sequences such as DFT, DCT/IDCT and Walsh can all be applied to design digital filters [10]-[12]. APDCT filter has shown its effectiveness in subband decomposition in [11]. The 2-D APDCT filter Q of the size (2 N − 1) × (2 N − 1) is linear-phased, so Q (m, n) = Q (−m, n) = Q(m, −n) = Q(− m, −n) 0 ≤ m ≤ N − 1,
0 ≤ n ≤ N −1
The sequence response matrix of for the second stage is just the expansion of F1 : ⎡1 ⎢1 ⎢ ⎢0 ⎢ 0 F2 = ⎢ ⎢0 ⎢ ⎢0 ⎢0 ⎢ ⎣⎢0
(1)
Then we can define the matrix:
Q1/4
Q (0,1) ⎡ Q (0, 0) ⎢ Q (1, 0) Q (1,1) ⎢ ⎢ ⋅ ⋅ =⎢ ⋅ ⋅ ⎢ ⎢ ⋅ ⋅ ⎢ ⎣⎢Q ( N -1, 0) Q ( N -1,1)
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
Q (0, N -1) ⎤ Q (1, N -1) ⎥⎥ ⎥ ⋅ ⎥ ⋅ ⎥ ⎥ ⋅ ⎥ Q ( N -1, N -1) ⎦⎥
(2)
F is the sequence response matrix, for APDCT the elements in G are calculated from (3): ⎧ N −i 0 ≤ i ≤ N − 1, j = 0 ⎪⎪ N 2 G (i, j ) = ⎨ ⎪ 1 ⎡( N − i) cos ijπ − csc jπ sin ijπ ⎤ 0 ≤ i ≤ N − 1 ⎪⎩ N 2 ⎢⎣ N N N ⎥⎦ 0 < j ≤ N − 1
1 0 0 0 0 0 0⎤ 1 0 0 0 0 0 0 ⎥⎥ 0 0 0 0 0 0 0⎥ ⎥ 0 0 0 0 0 0 0⎥ 0 0 0 0 0 0 0⎥ ⎥ 0 0 0 0 0 0 0⎥ 0 0 0 0 0 0 0⎥ ⎥ 0 0 0 0 0 0 0 ⎦⎥
By this way, the sequence response matrix for the subsequent stages is obtained by expanding the sequence response matrix for the first stage. Figure 4 (a) and (b) show the frequency responses of the designed all phase DCT low pass filters Q1 and Q2 :
And according to [11] Q 1/4 =GFG T
0.00195 ⎤ −0.00160 ⎥⎥ −0.00140 ⎥ ⎥ −0.02200 ⎥ −0.01140 ⎥ ⎥ −0.00160 ⎥ ⎥ 0.00195 ⎦
(3)
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1
0.8
0.6
0.6
Magnitude
Magnitude
1
0.8
0.4
0.2
sequency response, which are shown in Figure 5 (b) and (d) respectively. When N = 4 : 1 1 ⎤ ⎡ 0.5 0.75 0 0⎤ ⎡0.5 0 ⎢0.75 ⎥ ⎢ 1 0.5 0 ⎥ 0 0.25 0.75 0 ⎥ ⎥ F3a = ⎢ F2a = ⎢ ⎢ 1 0.25 0 0 ⎥ ⎢1 1 0.5 0 ⎥ ⎢ ⎥ ⎢ ⎥ 0.75 0 0 ⎦ 1 1 0.5⎦ ⎣ 1 ⎣1
0.4
0.2
0 1
0 1 1
0.5 0.5
0 0 -0.5 -1
Fy
1
0.5 0.5
0 0 -0.5
-0.5 -1
F
F
(a)
-0.5 -1
y
-1
(b)
F
Then we can get the fan filter from (2) as
Figure4. Frequency response of Q1 and Q 2
−0.0055 ⎡ 0 ⎢0.0055 0 ⎢ ⎢0.0055 0.0276 ⎢ Q2a = ⎢0.0092 0.0183 ⎢0.0055 0.0276 ⎢ 0 ⎢0.0055 ⎢ 0 −0.0055 ⎣
3.3. All phase directional filter It is shown that all phase filters can be easily and flexibly designed by constructing sequency response F . In a similar way, all phase directional filter can also be easily designed by designing sequency response F according to the spectrum requirement. ω1 ω1 (π,π)
As for Q2b , which has spectrum opposite to Q2a , its sequency response F2b can be designed similarly or just minus F2a from an all-one matrix with the same size. 1 1 ⎤ ⎡ 0.5 0.75 1 1 ⎤ ⎡0.5 1 ⎢ 0.75 ⎥ ⎢ 0 0.5 1 ⎥ 0 0.25 0.75 1 ⎥ F =⎢ ⎥ ⎢ 3a F2b = ⎢ ⎥ 1 0.25 0 0 ⎢ 0 ⎥ 0 0.5 1 ⎢ ⎥ ⎢ ⎥ 1 0.75 0 0 ⎣ ⎦ 0 0 0.5⎦ ⎣ 0
(π,π)
ω0
ω0
And for Q1a and Q1b , since they are not symmetric elements in the four quadrants, it is impossible to design them directly in all phase datum space. Fortunately however, they can be obtained by just rotate the fan filter Q2a to ±45D respectively. For the most complext filter Q4a and Q4b in Figure 3, N = 4 is not enough to describe the frequency spectrum of the filter. So we can enlarge the size of sequency response, and when N = 8 , F4a =
( − π, − π)
( − π, − π)
Q2a
F2a
Q3a
(a)
(b)
(c) ω1
ω1
(π,π)
F3a
(d ) (π,π)
ω0
ω0 ( − π, − π)
( − π, − π)
Q2b
F2b
Q3b
F3b
(e )
(f)
(g)
(h)
ω1
(π,π)
ω1
0.578125 0.625 1 1 1 1 1 ⎤ ⎡ 0.5 ⎢0.578123 0.3125 0.59375 0.25 0.125 0.375 0.625 0.875⎥⎥ ⎢ ⎢ 0.625 0.59375 0.09375 0.625 0.75 0.25 0 0 ⎥ ⎢ ⎥ 1 0.25 0.625 0 0.375 0.953125 0.75 0.25 ⎥ ⎢ ⎢ 1 0.125 0.75 0.375 0 0.15625 0.8375 1 ⎥ ⎢ ⎥ 1 0.375 0.25 0.953125 0.15625 0 0.046875 0.625⎥ ⎢ ⎢ 1 0.625 0 0.75 0.84375 0.046875 0 0 ⎥⎥ ⎢ ⎢⎣ 1 0.875 0 0.25 1 0.625 0 0 ⎥⎦
(π,π)
ω0
ω0
The frequency responses of the designed directional filters are shown in Figure6.
( − π, − π)
( − π, − π)
Q1a
Q1b
(e )
(g)
−0.0055 −0.0092 −0.0055 −0.0055 0 ⎤ −0.0276 −0.0183 −0.0276 0 0.0055 ⎥⎥ −0.1600 0 0 0.0276 0.0055 ⎥ ⎥ 0.1600 0.5000 0.1600 0.0183 0.0092 ⎥ 0 0 0.0276 0.0055 ⎥ −0.1600 ⎥ −0.0276 −0.0183 −0.0276 0 0.0055 ⎥ 0 ⎥⎦ −0.0055 −0.0092 −0.0055 0.0055
4. Experiment
Figure 5. Spectrum of directional filters and
In order to illustrate the potential of APPDFB, we study addictive white Gaussian noise removal from images by means of threshold estimators. Although not being the best denosing method available, this simple thresholding scheme [13, 14] can often be a good indication of the potential of different transform. For a fair comparison, we use the same method for wavelet and contourlet transform.
corresponding sequency response For the example of fan filter Q2a and more complex filter Q3a in Figure 3, the spectrums of them are shown in Figure 5 (a) and (c) respectively. Since all phase digital filter has symmetric elements in the four quadrants, only 1/4 of the spectrums are used to design
354
Threshold of the i-th directional subband of the j-th scale Ti , j is chosen as:
and APPDFB. The results show that the proposed APPDFB is superior to the others for most of the images. For the image “Building” with a lot of horizontal and vertical edges, APPDFB yields improvement in excess 1.97dB over the contourlet as well as better visual quality. Figure 7 displays the reconstructed images using APPDFB, Contoulet, and NSCT, improvement can be seen in APPDFB reconstructed image, particularly near the horizontal and vertical edges and on the roof.
Ti , j = Kσ i , j
(4) We set K = 4 for the finest scale and K = 3 for the remaining ones. σ i , j is the noise variance of the i-th directional subband of the j-th scale, which are the averaged variances of the respective subband of ten normalized noise images. -1
1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
5. Conclusion and discussion
1 Fy
5
0.8 0
Magnitude
g
Fx
5
0.6
In this paper, we proposed a novel pyramidal directional filter bank based on APDCT for image denosing. The structure of APPDFB is very similar with contourlet except for the absence of subsampling and upsampling operators. So APPDFB can preserve more image details and can be very easily reconstructed. In addition, the 2-D APDCT filters can be flexibly designed and have better frequency and direction selectivity. All these characteristics attributes to the advantages of APPDFB in denoising over contourlet and NSCT. The application of the proposed scheme should be further studied in fields like image enhancement and directional analysis.
0.4
0.2
0 1 0.5 0.5
0 0 -0.5
(a)
-0.5 -1
Fy
Q2 a
-1
(b )
F
Q3a
0.9
1.5 0.8 0.7
1 Magnitude
Magnitude
0.6 0.5 0.4
0.5
0.3
0
0.2
1
0.1 0 1
0.5 1
0.5
0 0.5
0.5
0 0 -0.5 F
0
-0.5
-0.5 -1
(c )
-0.5 Fy
-1
Q4 a
-1
-1
(d )
Fx
Q1a
Figure 6. Frequency response of directional
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Table 1. Denoising PSNR Comparison (dB) Image
Lena
Barbara
Building
σ
Wavelet
CT
NSCT
20
26.58
27.26
29.07
APPD FB 29.26
30
22.14
24.17
26.07
26.70
40
19.06
21.81
23.74
24.63
50
16.64
19.95
21.93
22.90
20
24.41
25.41
25.92
24.43
30
20.90
22.72
23.67
23.38
40
18.33
20.74
22.04
22.25
50
16.11
19.15
20.97
21.16
20
26.69
27.14
29.06
29.11
30
22.45
24.07
26.05
26.52
40
19.31
21.71
23 .78
24.51
50
17.10
19.91
21.95
22.81
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Table1 shows the PSNR results for various transforms and noise intension. For wavelet, contourlet, NSCT and APPDFB, LP decomposition level is set to be 1. And the directional decomposition for the only highpass subband is set to be 8 for contourlet, NSCT
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Wavelet denoisie (PSNR=26.69) APPDFB denoise (PSNR=29.11dB) denoise (PSNR = 2
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Figure7. Denoising experiments
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