Tight Wavelet Filter Banks with Prescribed Directions

A filter $h:\mathbb{Z}^{n}\to\mathbb{R}$ is a function defined on $n$-dimensional integer grids. Although some of the results in this paper, including our main construction theorem (i.e., Theorem 3.1), still hold for general dilation matrices, their interpretation in connection with directional filters could be unclear. Hence, we consider in this paper an integer dilation (factor) $\lambda\geq 2$ only. We say a filter is a lowpass or refinement filter if $\sum_{k\in\mathbb{Z}^{n}}h(k)=\lambda^{n/2}$, and a highpass or wavelet filter if $\sum_{k\in\mathbb{Z}^{n}}h(k)=0$.

A trigonometric polynomial $\tau:\mathbb{T}^{n}\to\mathbb{C}$ is called a mask (associated with the filter $h$) if

In this paper, any property associated with the filter will be related to the mask, and vice versa, without explicit mention. For example, a lowpass mask $\tau$ is a mask with $\tau(0)=\lambda^{n/2}$.

For a dilation factor $\lambda$, let $\Lambda$ be a complete set of representatives of the distinct cosets of the quotient group $\mathbb{Z}^{n}/\lambda\mathbb{Z}^{n}$ containing $0$, and let $\Gamma$ be a complete set of representatives of the distinct cosets of $2\pi((\lambda^{-1}\mathbb{Z}^{n})/\mathbb{Z}^{n})$ containing $0$.

For a nonnegative integer $m$, $\tau$ has accuracy number $m$ if the minimum order of the roots that $\tau$ has at the point in $\Gamma\setminus\{0\}$ is $m$. The flatness number of $\tau$ is the order of the root that $\tau-\lambda^{n/2}$ has at $0$. In particular, a lowpass mask $\tau$ always has positive flatness, and it has positive accuracy if and only if $\tau(\gamma)=0$, for every $\gamma\in\Gamma\setminus\{0\}$. The order of the root that $\tau$ has at $0$ is called its number of vanishing moments. Hence any wavelet mask has at least one vanishing moment.

Let $\tau$ be a lowpass mask with dilation $\lambda$. We recall that $\phi\in L_{2}(\mathbb{R}^{n})$ is a refinable function (associated with $\tau$) if it satisfies $\widehat{\phi}(\lambda\omega)=\lambda^{-n/2}\tau(\omega)\widehat{\phi}(\omega)$, $\omega\in\mathbb{R}^{n}$. Here and below, $\widehat{f}$ is the Fourier transform of $f\in L_{2}(\mathbb{R}^{n})$, defined as, for $f\in L_{1}(\mathbb{R}^{n})\cap L_{2}(\mathbb{R}^{n})$, $\widehat{f}(\omega)=\int_{\mathbb{R}^{n}}f(x)e^{-ix\cdot\omega}\mathrm{d}x,$ $\omega\in\mathbb{R}^{n}$.

Given the wavelet masks $q_{i}$, $1\leq i\leq r,$ $\psi_{i}\in L_{2}(\mathbb{R}^{n})$ is the mother wavelet if it satisfies $\widehat{\psi_{i}}(\lambda\omega)=\lambda^{-n/2}q_{i}(\omega)\widehat{\phi}(\omega),\,\omega\in\mathbb{R}^{n}$, and

is the wavelet system generated by $\psi_{i}$, $1\leq i\leq r$. The wavelet system is called an (MRA-based) tight wavelet frame if it forms a tight frame for $L_{2}(\mathbb{R}^{n})$.

Tight wavelet frames can be obtained by using a method called the unitary extension principle (UEP) [9, 25]. The following statement is taken from Ref. \refciteHanOrtho but adapted to the setting of our paper.

We shall call the set $\{\tau,q_{1},\dots,q_{r}\}$ in the above theorem a tight wavelet filter bank. In particular, the masks $q_{i},\,1\leq i\leq r$ in the tight wavelet filter bank $\{\tau,q_{1},\dots,q_{r}\}$ are wavelet masks.

The polynomial sum of squares representation has been extensively studied in the literature [1, 24] but recently has been connected with tight wavelet filter bank construction [20, 3, 4, 17, 18, 19]. In these approaches, given a lowpass mask $\tau$, one tries to find a sum of hermitian squares representation of a trigonometric polynomial derived from $\tau$.

For a given nonnegative trigonometric polynomial $f$, $f$ is said to have a sum of hermitian squares (SOS) representation if there exist trigonometric polynomials $g_{1},\ldots,g_{N}$ such that

A key observation in the sos-based tight wavelet filter bank construction is that, for a lowpass mask $\tau$ with dilation $\lambda$, finding an sos of the trigonometric polynomial

is sufficient to obtain wavelet masks $q_{1},\dots,q_{r}$ such that $\{\tau,q_{1},\dots,q_{r}\}$ is a tight wavelet filter bank. The precise statement is given below, which is taken from Ref. \refciteLaiStock but adapted to the notation of this paper.

When tight wavelet filter banks are constructed using the method in the above theorem, the number of vanishing moments of wavelet masks can be written in terms of the properties of the trigonometric polynomials $\tau$ and $g_{l}$, $l=1,\ldots,N$.

We mention in passing a related result, which is about the minimum number of vanishing moments of all wavelet masks. This number is precisely $\min\{a,c/2\}$, with $a$ the same as above and $c$ the order of the root that $|\tau|^{2}-\lambda^{n}$ has at $0$ [11].

One of the main difficulties in the sos-based construction methods such as Theorem 2.2 is that, except in the 1-D case, finding an sos representation of a given nonnegative trigonometric polynomial is nontrivial. In the 1-D case, there is no such difficulty thanks to the following Fejér-Riesz Lemma [5].

Fix a positive integer $N\in\mathbb{N}$, the number of directions. Let $\zeta_{1},\ldots,\zeta_{N}\in\mathbb{Z}^{n}$ be the initial points, and let $\eta_{1},\ldots,\eta_{N}\in\mathbb{Z}^{n}$ be the terminal points of the $N$ directions. For each $l=1,\dots,N$, choose a positive integer $m_{l}\in\mathbb{N}$, the number of vanishing moments along the $\xi_{l}:=\eta_{l}-\zeta_{l}$ direction. Then we will show that the directional wavelet mask of our tight wavelet filter bank has the factor (cf. Remark 2 after Theorem 3.1 and (7))

up to scale. Before presenting the main result of this paper, we set some terminology. We refer to $[\xi_{1},\dots,\xi_{N}]\in\mathbb{Z}^{n\times N}$ as the direction matrix and $(m_{1},\dots,m_{N})$ as the vanishing moment vector.

Since any MRA-based wavelet system has fast algorithms, our tight wavelet frame associated with TWFPD has fast algorithms. Because we have a tight wavelet frame, the same filters are used both for the analysis algorithm and the synthesis algorithm. In addition to these standard fast algorithms, because of the specific way that our TWFPD is constructed, an alternative synthesis algorithm is available for our TWFPD. More details about these algorithms are given in Section 3.2.

The TWFPD construction in Theorem 3.1 is a particular case of the sos-based construction of tight wavelet filter banks given in Theorem 2.2. The following corollary is an immediate consequence of Theorem 2.3 when applied to our TWFPD wavelet masks.

From Theorem 3.1 and Corollary 3.3, we see that the two types of wavelet masks in TWFPD, the directional ones and the complementary ones, play quite a different role.

Each directional wavelet mask $q_{D,l}$ has $g_{l}(\lambda\cdot)$ as a factor, entirely determined from the input: the directionality vector $\xi_{l}$, the vanishing moment number $m_{l}$ along the $\xi_{l}$ direction, and the number $\lambda$ with $N\leq\lambda^{n}$, where $N$ is the number of directions. The number of vanishing moments of $q_{D,l}$ is given exactly as $m_{l}$.

On the other hand, the primary role of the complementary wavelet masks in TWFPD is to complement the directional wavelet masks so that when combined, they form a tight wavelet filter bank.

Additionally, these complementary wavelet masks allow TWFPD to have an alternative synthesis algorithm. More precisely, the standard fast analysis algorithm of tight wavelet filter banks using the wavelet masks $q_{2,\nu}$, $\nu\in\Lambda$ in Theorem 2.2 generates the detail coefficients in the analysis part of Laplacian pyramid (LP) algorithms [2]. This connection is easy to observe and can be found, for example, in Ref. \refciteHLO. It is well known that the LP algorithms have a simple reverse process of the analysis algorithm as a synthesis algorithm [7, 16]. Because our complementary wavelet masks $q_{C,\mu}$ correspond to the wavelet masks $q_{2,\nu}$, $\nu\in\Lambda$ in Theorem 2.2, they provide an alternative synthesis algorithm for TWFPD, as shown in detail below.

For a dilation $\lambda\geq 2$, we recall that the downsampling operator _↓ and the upsampling operator _↑ are defined as

Let $\{\tau,q_{D,1},\ldots,q_{D,N},q_{C,1},\ldots,q_{C,\lambda^{n}}\}$ be the TWFPD constructed in Theorem 3.1. We use $x_{j}$ to denote the TWFPD coarse coefficients at level $j$ and use $d_{j,D,l}$, $1\leq l\leq N$ and $d_{j,C,\mu}$, $1\leq\mu\leq\lambda^{n}$ to denote the TWFPD detail coefficients at level $j$. The standard TWFPD analysis algorithm computes the coefficients $x_{j}$, $d_{j,D,l}$, and $d_{j,C,\mu}$ from the coarse coefficients $x_{j+1}$ by convolving with TWFPD filters followed by downsampling. The standard TWFPD synthesis algorithm gets back the coarse coefficients $x_{j+1}$ by upsampling the coefficients $x_{j}$, $d_{j,D,l}$, and $d_{j,C,\mu}$, then by convolving with TWFPD filters, and finally by summing them up. These algorithms are the standard fast algorithms for a tight wavelet filter bank but applied to our TWFPD filters.

input $x_{J+1}:\mathbb{Z}^{n}\to\mathbb{R}$

Standard TWFPD Analysis: computing $x_{j}$, $d_{j,D,l}$, and $d_{j,C,\mu}$ from $x_{j+1}$
for $j=J,J-1,\ldots,0$
   $x_{j}=(\widetilde{h}\ast x_{j+1})_{\downarrow}$                          (i)
   for $l=1,2,\dots,N$
      $d_{j,D,l}=\widetilde{h}_{l}\ast x_{j}$                        (ii)
   end
   for $\mu=1,2,\dots,\lambda^{n}$
      $d_{j,C,\mu}=(x_{j+1}-h\ast(x_{j}{{}_{\uparrow}}))(\lambda\cdot-\nu_{\mu})$       (iii)
   end
end

Standard TWFPD Synthesis: computing $x_{j+1}$ from $x_{j}$, $d_{j,D,l}$, and $d_{j,C,\mu}$
for $j=0,1,\ldots,J$
   $x_{j+1}=h\ast(x_{j}{{}_{\uparrow}})+\sum_{l=1}^{N}((h_{l}{{}_{\uparrow}})\ast h)\ast(d_{j,D,l}{{}_{\uparrow}})$
       $+\sum_{\mu=1}^{\lambda^{n}}(\widetilde{\delta}_{\mu}-\sum_{m\in\mathbb{Z}^{n}}h(-\lambda m-\nu_{\mu})h(\cdot-\lambda m))\ast(d_{j,C,\mu}{{}_{\uparrow}})$
end

The standard TWFPD synthesis algorithm given above is immediate once we observe that the highpass filters corresponding to the TWFPD wavelet masks $q_{D,l}$ and $q_{C,\mu}$ are $(h_{l}{{}_{\uparrow}})\ast h$ and $\widetilde{\delta}_{\mu}-\sum_{m\in\mathbb{Z}^{n}}h(-\lambda m-\nu_{\mu})h(\cdot-\lambda m)$, respectively. In the standard TWFPD analysis algorithm, the detail coefficients are computed easily because of the specific form that our TWFPD wavelet filters take. The coefficients in step (ii) are obtained by observing

and the coefficients $d_{j,C,\mu}$ in step (iii) are from $(\delta_{\mu}\ast x_{j+1})_{\downarrow}=x_{j+1}(\lambda\cdot-\nu_{\mu})$ and

LP-based TWFPD Synthesis: computing $x_{j+1}$ from $x_{j}$ and $d_{j,C,\mu}$
for $j=0,1,\ldots,J$

   for $\mu=1,2,\dots,\lambda^{n}$
      $x_{j+1}(\lambda\cdot-\nu_{\mu})=(h\ast(x_{j}{{}_{\uparrow}}))(\lambda\cdot-\nu_{l})+d_{j,C,\mu}$     (iv)
   end
end

We see that step (iv) is simply the reverse process of step (iii) and that, unlike the standard TWFPD synthesis algorithm we saw earlier, the detail coefficients $d_{j,D,\mu}$ are not used for computing $x_{j+1}$ in this LP-based TWFPD synthesis algorithm.

Depending on what one would like to have for synthesis algorithms, one can choose which synthesis algorithm to use. If one wants a faster synthesis algorithm, then the LP-based TWFPD synthesis algorithm can be used because it is much faster, as we will quantify soon. Suppose one wants to remove noises in the coefficients after the standard TWFPD analysis algorithm. In that case, the standard TWFPD synthesis algorithm is the one to use since it is much more effective in removing such noises [8].

Next, we discuss the complexity of TWFPD algorithms. We measure the complexity by counting the number of multiplicative operations needed in a complete cycle of $1$-level-down analysis and $1$-level-up synthesis, meaning the number of operations required to obtain $x_{j}$, $d_{j,D,l}$, and $d_{j,C,\mu}$ from $x_{j+1}$ and get back $x_{j+1}$. We consider the two fast algorithms with the same analysis algorithm, the one with the standard synthesis algorithm and another with the LP-based synthesis algorithm.

Let $\alpha$ and $\beta_{l}$ be the number of nonzero entries in the lowpass filter $h$ and the highpass filter $h_{l}$. Given $x_{j+1}$ with $L$ data points, the number of multiplicative operations needed in a complete cycle of $1$-level-down analysis and $1$-level-up synthesis with standard TWFPD synthesis algorithm is the sum of the following numbers:

• •

  $\alpha L$              [for step (i) in Standard TWFPD Analysis]

• •

  $(\sum_{l=1}^{N}\beta_{l}/\lambda^{n})L$         [for step (ii) in Standard TWFPD Analysis]

• •

  $\alpha L$            [for step (iii) in Standard TWFPD Analysis]

• •

  $(\alpha+\sum_{l=1}^{N}(\lambda\beta_{l}+\alpha)+2\alpha)L$ [for Standard TWFPD Synthesis]

Thus the complexity in this case is given as $(\sum_{l=1}^{N}\beta_{l}/\lambda^{n}+(N+5)\alpha+\lambda\sum_{l=1}^{N}\beta_{l})L,$ and by considering the average $\beta^{\ast}:=(\sum_{l=1}^{N}\beta_{l})/N$ and using $N\leq\lambda^{n}$, we see that the complexity is bounded by

Therefore, the standard fast TWFPD algorithms have linear complexity with complexity constant $(N+5)\alpha+(\lambda N+1)\beta^{\ast}$. The complexity constant in this case increases as the number of directions $N$ increases.