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.

### A. Modifying the PPFT in BV and BH Subsets

From (1), the definition of the BV subset, it is obvious that *m* axis represents the rays with the slope of *ω*_{x}/*ω*_{y}, where −π ≤ *ω*_{y} < π and −1 ≤ *ω*_{x}/*ω*_{y} < 1. Similarly, in (2), *m* axis represents the rays with the slope *ω*_{y}/*ω*_{x}, − π ≤ *ω*_{x} ≤ π, − 1 < *ω*_{y}/*ω*_{x} ≤ 1. In order to partition the whole frequency plane [− π, π)^{2} into several wedge-shaped regions with respect to the partition scheme of the employed 1D FBs, we attempt to modify the PPFT in BV and BH subsets, respectively.

It is known that 1D digital filters have the period of 2π, while the rays in the BV and BH subsets have the slope with the region of [−1, 1) and (−1, 1], respectively. 1D FBs cannot be used directly. Therefore, we firstly separate modify *X*_{BV}(*m*, *l*) in (3) and *X*_{BH}(*m*, *l*) in (4) as *X*′_{BV}(*k*_{1}, *k*_{2}) and *X*′_{BH}(*k*_{1}, *k*_{2}),
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$$\eqalignno{X'_{BV}(k_1,k_2) = \cases{X_{BV}\left(k_1+{N\over 2},k_2\right), &$-N\leq k_1 < 0, -N\leq k_2 < N,$\cr0, &$0\leq k_1 < N, -N\leq k_2 < N$},&\hbox{(5)}\cr X'_{BH}(k_1,k_2) = \cases{0, &$-N\leq k_1 < 0, -N\leq k_2 < N,$\cr X_{BH}\left(k_2,{N\over 2}-k_1\right), &$0\leq k_1 < N, -N\leq k_2 < N$},&\hbox{(6)}}$$Then, we separately regarded the *X*′_{BV}(*k*_{1}, *k*_{2}) and *X*′_{BH}(*k*_{1}, *k*_{2}) as the 2D DFT of the signals *x*′_{BV}(*n*_{1}, *n*_{2}) and *x*′_{BH}(*n*_{1}, *n*_{2}) where *n*_{1}, *n*_{2} = 0, 1, …, 2*N*−1,
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$$\eqalignno{X'_{BV}(\Omega_x,\omega_y) &= X'_{BV}(k_1,k_2)= \sum^{2N-1}_{n_1=0}\sum^{2N-1}_{n_2=0}x'_{BV}(n_1,n_2)\exp(-i(n_1\omega_x+n_2\omega_y))\cr&=\sum^{2N-1}_{n_1=0}\sum^{2N-1}_{n_2=0}x'_{BV}(n_1,n_2)\exp\left(-i\left(n_1{2\pi k_1 \over 2N}+n_2{2\pi k_2 \over 2N}\right)\right)&\hbox{(7)}\cr X'_{BH}(\Omega_x,\omega_y) &= X'_{BH}(k_1,k_2)= \sum^{2N-1}_{n_1=0}\sum^{2N-1}_{n_2=0}x'_{BH}(n_1,n_2)\exp(-i(n_1\omega_x+n_2\omega_y))\cr&=\sum^{2N-1}_{n_1=0}\sum^{2N-1}_{n_2=0}x'_{BH}(n_1,n_2)\exp\left(-i\left(n_1{2\pi k_1 \over 2N}+n_2{2\pi k_2\over 2N}\right)\right)&\hbox{(8)}}$$where and *k*_{1}, *k*_{2} = − *N*, −*N* + 1, …, *N*−1. Both the *X*′_{BV}(*k*_{1}, *k*_{2}) and *X*′_{BH}(*k*_{1}, *k*_{2}) have the period of 2π. As a result, we can multiply them with a 1D FB along the horizontal axes, respectively, which is equivalent to the convolution operations in spatial domain. This can decompose images into several directionally-oriented subbands. The next subsection will analyze the detailed operations.

### 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
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$$\eqalignno{T(z) &= \sum^{M-1}_{k=0}\hat{h}_k(z)\hat{f}_k(z) = cz^{-n_0}&\hbox{(9)}\cr A_l(z) &= \sum^{M-1}_{k=0}\hat{h}_k \left(z W^l_N\right)\hat{f}_k(z) = 0,&\hbox{(10)}}$$where c and *n*_{0} 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.

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.

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

Subsequently, we convert the from the PPFT domain to the spatial domain. We refer to the converted subbands in spatial domain are *x*_{k}(*n*_{1}, *n*_{2}), *k* = 0, 1, ···, *M*−1. According to the description on the PPFT in Section II, each subband *x*_{k}(*n*_{1}, *n*_{2}) has a wedge-shaped frequency support. The similar procedure can be done to *X*′_{BH}(*k*_{1}, *k*_{2}), producing the same number of directional subbands. Therefore, splitting [−π, π)^{2} into 2*M* 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}(*k*_{1}, *k*_{2}) with a 5-channel FB, and the remaining 5 subbands are achieved from *X*′_{BH}(*k*_{1}, *k*_{2}).

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.