Design of Nonsubsampled Directional Filter Banks With Arbitrary Number of Channels

IN recent years, two-dimensional (2D) directional filter banks (DFBs) have been received much attention for their ability to capture the desired directional information of images. The original 2D DFB was proposed by Bamberger and Smith in 1992 [1]. It can decompose images into several wedge-shaped frequency subbands. However, since it is based on multi-level binary tree-structure, it can only realize the 2ⁿ subband decompositions. Later, some researchers improved the original DFBs on solving frequency scrambling [2] and lowering design complexity [3], but there is no discussion on the number of directional subbands. In [4], Do and Vetterli proposed the contourlet transform by combining the DFBs with the Laplacian pyramid [5], to achieve the multiresolution property. The number of directional subbands at each scale is still 2ⁿ. For more flexibility on direction partition, multiresolution DFB was proposed in [6]. It can provide 3 … 2ⁿ directional subbands. However, the expected 2D filters are very difficult to design. Due to the rich textures in images, DFBs with arbitrary number of channels are highly expected.

In this paper, we propose an efficient method for the design of M-channel DFBs, where the number of channels M can be arbitrary. It is based on the pseudo-polar Fourier transform (PPFT) which computes the Fourier transform at pseudo-polar grid. Such grid consists of equispaced points along rays, where different rays are equispaced in slope [7]. After taking a coordinate transformation, we modify the PPFT and convert the pseudo-polar grid to the Cartesian 2D discrete Fourier transform (DFT) grid where the vertical and the horizontal axes represent the radial and slope directions respectively. Then we employ one-dimensional (1D) FBs to the modified PPFT along the horizontal axis. With these operations, the 2D DFBs are obtained. It possesses two key features. One is that since the design of 2D DFBs is reduced to that of 1D FBs and a modified PPFT, many available design methods of 1D FBs can be applied. The other is that since 1D FBs can have arbitrary number of subbands, the DFBs by using the proposed method can have arbitrary number of subbands. In order for the proposed method to have the property of multiresolution, we combine the proposed DFBs with Laplacian pyramid [5]. Consequently, the proposed system can realize the multiresolution and multidirection image decompositions.

This paper is organized as follows. Section II reviews the PPFT, which is the basis for our method. In Section III, we present the theory and design of the proposed M-channel nonsubsampled DFBs. Experimental result is given in Section IV. This paper is concluded in Section V.

The PPFT proposed in [7] evaluates the Fourier transform on the pseudo-polar grid. Such grid is composed of equispaced points along rays, where different rays are equispaced in slope rather than angle. The pseudo-polar grid is separated into two groups: the basically vertical (BV) and the basically horizontal (BH) subsets, defined by TeX Source $$\eqalignno{BV &=\cases{\omega_y = {\pi l\over N}, &$-N\leq l < N,$\cr\omega_x = \omega_y{2m\over N}, &$-{N\over 2}\leq m <{N\over 2},$}&\hbox{(1)}\cr BH &=\cases{\omega_x = {\pi l\over N}, &$-N\leq l < N,$\cr\omega_y = \omega_x{2m\over N}, &$-{N\over 2}\leq m < {N\over 2},$}&\hbox{(2)}}$$Fig. 1(a) and (b) illustrates the BV and BH subsets, respectively, and Fig. 1(c) shows the complete pseudo-polar grid consisting of BV and BH subsets.

Fig. 1. Pseudo-polar grid. (a) Pseudo-polar grid on BV subset, (b) grid on BH subset, (c) complete grid consisting of BV and BH subsets.

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The PPFT of an N × N image x(n₁, n₂) in BV subset is expressed as TeX Source $$\eqalignno{X_{BV}(\omega_x,\omega_y) &=X_{BV}(m,l)=\sum^{N-1}_{n_1=0}\sum^{N-1}_{n_2=0}x(n_1,n_2)\exp(-i(n_1\omega_x+n_2\omega_y))\cr&=\sum^{N-1}_{n_1=0}\sum^{N-1}_{n_2=0}x(n_1,n_2)\exp\left(-i\left(n_1{2\pi m l\over N^2}+n_2{2\pi l \over N}\right)\right)&\hbox{(3)}}$$and in BH subset is TeX Source $$\eqalignno{X_{BH}(\omega_x,\omega_y) &=X_{BH}(l,m)=\sum^{N-1}_{n_1=0}\sum^{N-1}_{n_2=0}x(n_1,n_2)\exp(-i(n_1\omega_x+n_2\omega_y))\cr&=\sum^{N-1}_{n_1=0}\sum^{N-1}_{n_2=0}x(n_1,n_2)\exp\left(-i\left(n_1{\pi l\over N}+n_2{2\pi m l \over N^2}\right)\right)&\hbox{(4)}}$$In [8], Averbuch et al. developed a fast algorithm for the computation of PPFT, with the same complexity order as that of the Cartesian fast Fourier transform. It has been proved that the fast algorithm is stable, invertible and requires only 1D operations to realize the PPFT.

Based on the introduction to the PPFT, we notice that rays can be presented by one axis in PPFT domain. Therefore, a 1D filter with respect to the slope axis in PPFT domain represents a 2D filter with wedge-shaped support in Cartesian frequency domain. If we apply 1D perfect reconstruction (PR) FBs to the PPFT of images along slope axis, we can decompose images into several directional subbands, and therefore the DFBs with arbitrary number of channels can be obtained. Inspirited by this idea, we offer an efficient method for creating 2D M -channel nonsubsampled DFBs. It mainly relies on two steps. First, the PPFT in BV and BH subsets are modified, respectively, which is a precondition for employing 1D FBs. Second, a conversion from 1D FBs to 2D DFBs is performed.

B. DFBs With Arbitrary Number of Channels

Let us review the principle of 1D PR FBs first.

1D M-channel FBs have been widely studied in recent years [9], [10], [11], [12]. A general M-channel FB is shown in Fig. 2, where and are analysis and synthesis filters, respectively. N is the decimation factor which satisfies 1 ≤ N ≤ M. If the FB shown in Fig. 2 is PR, the overall transfer function T(z) and the aliasing gains A_l (z) l = 1,···,N−1, should satisfy the two conditions TeX Source $$\eqalignno{T(z) &= \sum^{M-1}_{k=0}\hat{h}_k(z)\hat{f}_k(z) = cz^{-n_0}&\hbox{(9)}\cr A_l(z) &= \sum^{M-1}_{k=0}\hat{h}_k \left(z W^l_N\right)\hat{f}_k(z) = 0,&\hbox{(10)}}$$where c and n₀ are constant, and . In this work, we treat the nonsubsampled system of which the decimation factor N is equal to 1. Thus, the aliasing gain A_l(z) does not exist. It leads to the more flexibility for designing 1D PR FBs.

Fig. 2. A general M-channel filter bank.

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Assume the filters in Fig. 2 have real coefficients. The frequency responses of analysis filters are sketched in Fig. 3, which are symmetric with respect to the vertical axis.

Fig. 3. Frequency responses of real-valued analysis filters.

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Back to (7), we apply the M-channel real-valued FB shown in Fig. 2 to X′_BV(n₁, n₂) along n₁ axis, resulting in the subbands X′_{BV_k}(n₁, n₂), TeX Source $$x'_{BV\_k}(n_1,n_2) = x'_{BV}(n_1,n_2) *h_k(n_1)\eqno{\hbox{(11)}}$$, where ‘*’ denotes the convolution operation and h_k(n₁) are analysis filters, k = 0, 1, ···, M − 1. In frequency domain, they are TeX Source $$X'_{BV\_k}(k_1,k_2) = X'_{BV}(n_1,k_2)\cdot \hat{h}_k(k_1).\eqno{\hbox{(12)}}$$ are the DFT of h_k(n₁).

Subsequently, we convert the from the PPFT domain to the spatial domain. We refer to the converted subbands in spatial domain are x_k(n₁, n₂), k = 0, 1, ···, M−1. According to the description on the PPFT in Section II, each subband x_k(n₁, n₂) has a wedge-shaped frequency support. The similar procedure can be done to X′_BH(k₁, k₂), producing the same number of directional subbands. Therefore, splitting [−π, π)² into 2M wedge-shaped regions is implemented. By taking the reverse operations of the decomposition procedure, the input image can be reconstructed. Take M = 5 as an example. The frequency supports of the resulting 10 directional subbands are shown in Fig. 4. The 5 subbands numbered as 0, 1, 2, 3, 4, are obtained by filtering X′_BV(k₁, k₂) with a 5-channel FB, and the remaining 5 subbands are achieved from X′_BH(k₁, k₂).

Fig. 4. Frequency supports of 10 directional subbands.

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It should be pointed that the PPFT can be inversed exactly with an iterative or direct algorithm [8]. Thus, if the involved FBs are PR, the original image can be reconstructed exactly.

From the above analysis, it can be seen that since the number of channels of 1D FBs can be arbitrary, the proposed method can decompose images into arbitrary number of directional subbands.

The proposed method can decompose images into arbitrary directional subbands. In order for the proposed method to have the ability of multiresolution analysis, we introduce the nonsubsampled Laplacian pyramid algorithm [5], leading to the lowpass and hihgpass subbands, and then we apply the proposed DFBs to the highpass subband. Multiresolution can be achieved by reiterating with the same or different FBs in lowpass subbands.

This section considers the two-level directional decompositions of the test image Zoneplate. The 256 × 256 original image x(n₁, n₂) is shown in Fig. 5(a). A ‘9/7’ biorthogonal filter [13] is chosen for the Laplacian pyramid stage. In the first level decomposition, we convert the highpass band x^l_d from spatial domain to the PPFT domain, resulting in the PPFT X′_BV(m,l) and X_BH(l, m). Then we separately modify them as X′_BV(k₁, k₂) and X′_BH(k₁, k₂) (see (5) and (6)). Finally, we apply a 5-channel nonsubsampled real-valued FB to the X′_BV(k₁, k₂) and X′_BH(k₁, k₂) along the slope axes, respectively. Such a FB is designed by cosine modulation method [11], of which all the filters have the same length of 40. The magnitude responses of the analysis filters are displayed in Fig. 5(b). Following the analysis in Section III-B, the DFBs designed by the 5-channel FB performs a frequency partition as shown in Fig. 5(c). Fig. 5(d) presents the 10 directional subbands in the spatial domain. This cannot be performed by the existing method [1], [2], [3], [4], [6]. For the highpass band of the second level, the 2-channel ‘9/7’ biorthogonal FB is used, producing 4 directional subbands which are displayed in Fig. 5(e). The lowpass band of the second level is given in Fig. 5(f). Taking the reverse operations of the forward part, the input image is reconstructed which are shown in Fig. 9(g). The peak-signal-to-noise-ratio (PSNR) between the input image and the reconstructed image is 237.74.

Fig. 5. Two level image decompositions. (a) Original image, (b) magnitude responses of analysis filters used in the first level decomposition, (c) frequency partitioning schemes of the 5-channel real-valued FB, (d) the 10 directional subbands of the first level decomposition, (e) the 4 directional subbands of the second level decomposition, (f) the lowpass subband, (g) reconstructed image.

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Since the Laplacian pyramid algorithm and the involved FBs are PR, the original image can be reconstructed exactly.

This paper addressed an efficient method for the design of 2D DFBs. This method is based on the PPFT and 1D PR FBs. It can decompose images into arbitrary number of directional subbands without designing 2D directional filters. The design complexity is reduced efficiently and the design flexibility is improved. Further, a Laplacian pyramid is combined with the proposed approach, leading to a multiresolution and multidirection system. In later work, we will study the applications of this proposed system, such as in image denoising, edge detection and image restoration and so on.