Compactly supported multivariate dual multiframelets with high vanishing moments and high balancing orders

To better explain our motivations, let us recall some basic concepts. Throughout this paper, $\mathsf{M}$ is a $d\times d$ dilation matrix, i.e., $\mathsf{M}\in\mathbb{Z}^{d\times d}$ and its eigenvalues are all greater than one in modulus. For simplicity, let

Denote $(L_{2}(\mathbb{R}^{d}))^{r\times s}$ the linear space of all $r\times s$ matrices of square integrable functions in $L_{2}(\mathbb{R}^{d})$. For simplicity, $(L_{2}(\mathbb{R}^{d}))^{r}:=(L_{2}(\mathbb{R}^{d}))^{r\times 1}$. We introduce the following notion of inner product:

Let $\mathring{\phi},\mathring{\tilde{\phi}}\in(L_{2}(\mathbb{R}^{d}))^{r}$, $\psi,\tilde{\psi}\in(L_{2}(\mathbb{R}^{d}))^{s}$. We say that $\{\mathring{\phi};\psi\}$ is an $\mathsf{M}$-framelet in $L_{2}(\mathbb{R}^{d})$ if there exist positive constants $C_{1}$ and $C_{2}$ such that

where $\psi_{\mathsf{M}^{j};k}:=d_{\mathsf{M}}^{j/2}\psi(\mathsf{M}^{j}\cdot-k)$ and $|\langle f,\psi_{\mathsf{M}^{j};k}\rangle|^{2}:=\|\langle f,\psi_{\mathsf{M}^{j};k}\rangle\|^{2}_{l_{2}}$. $(\{\mathring{\phi};\psi\},\{\mathring{\tilde{\phi}};\tilde{\psi}\})$ is called a dual $\mathsf{M}$-framelet in $L_{2}(\mathbb{R}^{d})$ if both $\{\mathring{\phi};\psi\}$ and $\{\mathring{\tilde{\phi}};\tilde{\psi}\}$ are $\mathsf{M}$-framelets in $L_{2}(\mathbb{R}^{d})$ and satisfy

with the above series converging unconditionally in $L_{2}(\mathbb{R}^{d})$. $(\{\mathring{\phi};\psi\},\{\mathring{\tilde{\phi}};\tilde{\psi}\})$ is called a dual multiframelet if the multiplicity $r>1$, and is called a scalar framelet if $r=1$. Unless specified, we shall use the term framelet to refer both.

For a dual $\mathsf{M}$-framelet $(\{\mathring{\phi};\psi\},\{\tilde{\mathring{\phi}};\tilde{\psi}\})$, the sparseness of the frame expansion (1.1) is closely related to the vanishing moments on the framelet generators $\psi$ and $\tilde{\psi}$. We say that $\psi$ has $m$ vanishing moments if

where $\mathbb{P}_{m-1}$ is the space of all $d$-variate polynomials of degree at most $m-1$. Note that $\psi$ has $m$ vanishing moments if and only if

where $f(\xi)=g(\xi)+\mathcal{O}(\|\xi\|^{m})$ as $\xi\to 0$ means $\partial^{\mu}f(0)=\partial^{\mu}g(0)$ for all $\mu\in\mathbb{N}^{d}_{0;m}$ with $|\mu|:=\mu_{1}+\cdots+\mu_{d}<m$. We define $\operatorname{vm}(\psi):=m$ with $m$ being the largest such integer. It is well known in approximation theory (see e.g. (hanbook, , Proposition 5.5.2)) that if $\operatorname{vm}(\psi)=m$ and $\operatorname{vm}(\tilde{\psi})=\tilde{m}$, then we necessarily have

which plays a crucial role in approximation theory and numerical analysis for the convergence rate of the associated approximation/numerical scheme. Moreover, we have

and

as $\xi\to 0$.

A popular method called oblique extension principle (OEP) has been introduced in the literature, which allows us to construct dual framelets with all generators having sufficiently high vanishing moments from refinable vector functions cpss13 ; cpss15 ; cs08 ; ch01 ; cj00 ; dhacha ; fjs16 ; hjsz18 ; js15 ; kps16 ; lj06 ; sz16 . Denote $(l_{0}(\mathbb{Z}^{d}))^{r\times s}$ the linear space of all $r\times s$ matrix-valued sequences $u=\{u(k)\}_{k\in\mathbb{Z}^{d}}:\mathbb{Z}^{d}\to\mathbb{C}^{r\times s}$ with finitely many non-zero terms. Any element $u\in(l_{0}(\mathbb{Z}^{d}))^{r\times s}$ is said to be a finitely supported (matrix-valued) filter/mask. For $\phi\in(L_{2}(\mathbb{R}^{d}))^{r}$, we say that $\phi$ is an $\mathsf{M}$-refinable vector function with a refinement filter/mask $a\in(l_{0}(\mathbb{Z}^{d}))^{r\times r}$ if the following refinement equation is satisfied:

If $r=1$, then we simply say that $\phi$ is an $\mathsf{M}$-refinable (scalar) function. For $u\in(l_{0}(\mathbb{Z}^{d}))^{r\times s}$, define its Fourier series via $\widehat{u}(\xi):=\sum_{k\in\mathbb{Z}^{d}}u(k)e^{-ik\cdot\xi}$ for $\xi\in\mathbb{R}^{d}$. The Fourier transform is defined via $\widehat{f}(\xi):=\int_{\mathbb{R}^{d}}f(x)e^{-ix\cdot\xi}dx$ for $\xi\in\mathbb{R}^{d}$ for all $f\in L_{1}(\mathbb{R}^{d})$, and can be naturally extended to $L_{2}(\mathbb{R}^{d})$ functions and tempered distributions. The refinement equation (1.3) is equivalent to

where $\widehat{\phi}$ is the $r\times 1$ vector obtained by taking entry-wise Fourier transform on $\phi$. Most known framelets are constructed from refinable vector functions via OEP, and we refer them as OEP-based framelets. There are several versions of OEP which have been introduced in the literature (see chs02 ; dhrs03 ; hanbook ; hl20pp ). Here we recall the following version of OEP for compactly supported multivariate multiframelets:

It is clear that the key step to construct an OEP-based dual framelet is to obtain filters $\theta,\tilde{\theta}\in(l_{0}(\mathbb{Z}^{d}))^{r\times r}$ and $b,\tilde{b}\in(l_{0}(\mathbb{Z}^{d}))^{s\times r}$ such that $(\{a;b\},\{\tilde{a};\tilde{b}\})_{\Theta}$ is a dual framelet filter bank which satisfies (1.7). For any $u\in(l_{0}(\mathbb{Z}^{d}))^{s\times r}$, define

which is an $s\times(rd_{\mathsf{M}})$ matrix of $2\pi\mathbb{Z}^{d}$-periodic $d$-variate trigonometric polynomials. It is obvious that (1.7) is equivalent to

where

For an OEP-based dual $\mathsf{M}$-framelet $(\{\mathring{\phi};\psi\},\{\tilde{\mathring{\phi}};\tilde{\psi}\})$, The orders of vanishing moments of $\psi$ and $\tilde{\psi}$ are closely related to the sum rules of the filters $a$ and $\tilde{a}$ associated to $\phi$ and $\tilde{\phi}$. We say that a filter $a\in(l_{0}(\mathbb{Z}^{d}))^{r\times r}$ has order $m$ sum rules with respect to $\mathsf{M}$ with a matching filter $\upsilon\in(l_{0}(\mathbb{Z}^{d}))^{1\times r}$ if $\widehat{\upsilon}(0)\neq 0$ and

In particular, we define

It can be easily deduced from (1.7) that $\operatorname{vm}(\psi)\leqslant\operatorname{sr}(\tilde{a},\mathsf{M})$ and $\operatorname{vm}(\tilde{\psi})\leqslant\operatorname{sr}(a,\mathsf{M})$ always hold no matter how we choose $\theta$ and $\tilde{\theta}$. Therefore, we are curious about whether or not one can construct filters $\theta,\tilde{\theta}\in(l_{0}(\mathbb{Z}^{d}))^{r\times r}$ in a way such that the matrix $\mathcal{M}_{a,\tilde{a},\Theta}$ admits a factorization as in (1.11) for some $b,\tilde{b}\in(l_{0}(\mathbb{Z}^{d}))^{s\times r}$ such that $\widehat{b}(\xi)\widehat{\phi}(\xi)=\mathcal{O}(\|\xi\|^{\operatorname{sr}(\tilde{a},\mathsf{M})})$ and $\widehat{\tilde{b}}(\xi)\widehat{\tilde{\phi}}(\xi)=\mathcal{O}(\|\xi\|^{\operatorname{sr}(a,\mathsf{M})})$ as $\xi\to 0$.

With OEP, a lot of compactly supported scalar dual framelets with the highest possible vanishing moments have been constructed in the literature, to mention only a few, see cs10 ; ch00 ; chs02 ; dh04 ; dhrs03 ; dhacha ; dh18pp ; han97 ; han09 ; hanbook ; hm03 ; hm05 ; jqt01 ; js15 ; mothesis ; rs97 ; sel01 and many references therein. Though OEP appears perfect for improving the vanishing moments of framelet generators, it has a serious shortcoming. To properly address this issue, we need to briefly recall the discrete framelet transform employing an OEP-based filter bank.

By $(l(\mathbb{Z}^{d}))^{s\times r}$ we denote the linear space of all sequences $v:\mathbb{Z}^{d}\rightarrow\mathbb{C}^{s\times r}$. We call every element $v\in(l(\mathbb{Z}^{d}))^{s\times r}$ a matrix-valued filter. For a filter $a\in(l_{0}(\mathbb{Z}^{d}))^{r\times r}$, we define the filter $a^{\star}$ via $\widehat{a^{\star}}(\xi):=\overline{\widehat{a}(\xi)}^{\mathsf{T}}$, or equivalently, $a^{\star}(k):=\overline{a(-k)}^{\mathsf{T}}$ for all $k\in\mathbb{Z}^{d}$. We define the convolution of two filters via

Let $\mathsf{M}$ be a $d\times d$ dilation matrix, define the upsampling operator $\uparrow\mathsf{M}:(l(\mathbb{Z}^{d}))^{s\times r}\to(l(\mathbb{Z}^{d}))^{s\times r}$ as

We introduce the following operators acting on matrix-valued sequence spaces:

• •

  For $u\in(l_{0}(\mathbb{Z}^{d}))^{r\times t}$, the subdivision operator $\mathcal{S}_{u,\mathsf{M}}$ is defined via

  $$\mathcal{S}_{u,\mathsf{M}}v=|\det(\mathsf{M})|^{\frac{1}{2}}[v\uparrow\mathsf{M}]*u=|\det(\mathsf{M})|^{\frac{1}{2}}\sum_{k\in\mathbb{Z}^{d}}v(k)u(\cdot-\mathsf{M}k),$$

  for all $v\in(l(\mathbb{Z}^{d}))^{s\times r}$.

• •

  For $u\in(l_{0}(\mathbb{Z}^{d}))^{t\times r}$, the transition operator $\mathcal{T}_{u,\mathsf{M}}$ is defined via

  $$\mathcal{T}_{u,\mathsf{M}}v=|\det(\mathsf{M})|^{\frac{1}{2}}[v*u^{\star}]\downarrow\mathsf{M}=|\det(\mathsf{M})|^{\frac{1}{2}}\sum_{k\in\mathbb{Z}^{d}}v(k)\overline{u(k-\mathsf{M}\cdot)}^{\mathsf{T}},$$

  for all $v\in(l(\mathbb{Z}^{d}))^{s\times r}$.

Let $\theta,\tilde{\theta},a,\tilde{a}\in(l_{0}(\mathbb{Z}^{d}))^{r\times r}$ and $b,\tilde{b}\in(l_{0}(\mathbb{Z}^{d}))^{s\times r}$ be finitely supported filters. For any $J\in\mathbb{N}$ and any input data $v_{0}\in(l(\mathbb{Z}^{d}))^{1\times r}$, the $J$-level discrete framelet transform employing the filter bank $(\{a;b\},\{\tilde{a};\tilde{b}\})_{\Theta}$ where $\Theta:=\theta^{\star}*\tilde{\theta}$ is implemented as follows:

1. (S1)

   Decomposition/Analysis: Recursively compute $v_{j},w_{j}$ for $j=1,\dots,s$ via

   $$v_{j}:=\mathcal{T}_{a,\mathsf{M}}v_{j-1},\qquad w_{j}:=\mathcal{T}_{b,\mathsf{M}}v_{j-1}.$$

   (1.14)

2. (S2)

   Reconstruction/Synthesis: Define $\tilde{v}_{J}:=v_{J}*\Theta$. Recursively compute $\tilde{v}_{j-1}$ for $j=J,\dots,1$ via

   $$\tilde{v}_{j-1}:=\mathcal{S}_{\tilde{a},\mathsf{M}}{\mathring{v}}_{j}+\mathcal{S}_{\tilde{b},\mathsf{M}}w_{j}.$$

   (1.15)

3. (S3)

   Deconvolution: Recover $\breve{v}_{0}$ from $\tilde{v}_{0}$ through $\breve{v}_{0}*\Theta=\tilde{v}_{0}.$

We call $\{a;b\}$ the analysis filter bank and $\{\tilde{a};\tilde{b}\}$ the synthesis filter bank. If any input data $v\in(l(\mathbb{Z}^{d}))^{1\times r}$ can be exactly retrieved from the above transform, then we say that the $J$-level discrete framelet transform has the perfect reconstruction property.

Here comes the major shortcoming of OEP. The deconvolution step (S3) is where the trouble arises. If $(\{a;b\},\{\tilde{a};\tilde{b}\})_{\Theta}$ is an OEP-based dual $\mathsf{M}$-framelet filter bank satisfying (1.7), then the original input data $v_{0}$ is guaranteed to be a solution of the deconvolution problem $\mathring{v}_{0}*\Theta=\tilde{v}_{0}$. However, the deconvolution is inefficient and non-stable, that is, there could be multiple solutions to the deconvolution problem. Thus we cannot expect that the input data can be exactly retrieved by implementing the transform. As observed by (hl20pp, , Theorem 2.3), a necessary and sufficient condition for a multi-level discrete framelet transform to have the perfect reconstruction property is that $\Theta$ is a strongly invertible filter.

When $\Theta$ is strongly invertible, the discrete framelet transform is said to be compact, i.e., the transform is implemented by convolution/deconvolution with finitely supported filters only. The strong invertibility of $\Theta$ forces both $\theta$ and $\tilde{\theta}$ to be strongly invertible. In this case, we can define finitely supported filters ${\mathring{a}},{\mathring{\tilde{a}}}\in(l_{0}(\mathbb{Z}^{d}))^{r\times r}$ and ${\mathring{b}},{\mathring{\tilde{b}}}\in(l_{0}(\mathbb{Z}^{d}))^{s\times r}$ via

Moreover, if $(\{\mathring{\phi};\psi\},\{\mathring{\tilde{\phi}};\tilde{\psi}\})$ is a dual $\mathsf{M}$-framelet associated with the OEP-based dual framelet filter bank $(\{a;b\};\{\tilde{a};\tilde{b}\}_{\Theta}$, then the following refinable relations hold:

The underlying discrete framelet transform is now employed with the filter bank $(\{{\mathring{a}};{\mathring{b}}\},\{{\mathring{\tilde{a}}};{\mathring{\tilde{b}}}\})_{I_{r}}$ without the non-stable deconvolution step as as follows:

1. (S1’)

   Decomposition/Analysis: Recursively compute the framelet coefficients ${\mathring{v}}_{j},{\mathring{w}}_{j}$ for $j=1,\dots,s$ via

   $${\mathring{v}}_{j}:=\mathcal{T}_{{\mathring{a}},\mathsf{M}}{\mathring{v}}_{j-1},\qquad{\mathring{w}}_{j}:=\mathcal{T}_{{\mathring{b}},\mathsf{M}}{\mathring{v}}_{j-1},$$

   where ${\mathring{v}}_{0}:=v_{0}$ is an input data.

2. (S2’)

   Reconstruction/Synthesis: Define ${\mathring{\tilde{v}}}_{J}:={\mathring{v}}_{J}$. Recursively compute ${\mathring{\tilde{v}}}_{j-1}$ for $j=J,\dots,1$ via

   $${\mathring{\tilde{v}}}_{j-1}:=\mathcal{S}_{{\mathring{\tilde{a}}},\mathsf{M}}{\mathring{\tilde{v}}}_{j}+\mathcal{S}_{{\mathring{\tilde{b}}},\mathsf{M}}{\mathring{w}}_{j}.$$

For a scalar filter $\Theta$ (i.e., $r=1$), it is strongly invertible if and only if $\widehat{\Theta}$ is a non-zero monomial, i.e., $\widehat{\Theta}(\xi)=ce^{-ik\cdot\xi}$ for some $c\in\mathbb{C}\setminus\{0\}$ and $k\in\mathbb{Z}^{d}$. Thus to have a compact discrete framelet transform in the case $r=1$, $\widehat{\theta}$ and $\widehat{\tilde{\theta}}$ must be both monomials. However, we lose the main advantage of OEP of improving the vanishing moments of framelet generators by choosing such filters $\theta$ and $\tilde{\theta}$.

The previously mentioned shortcoming of OEP motivates us to consider multiframelets, that is, framelets with multiplicity $r>1$. Multiframelets have certain advantages over scalar framelets and have been initially studied in ghm94 ; glt93 and references therein. In sharp contrast to the extensively studied OEP-based scalar framelets, constructing multiframelets through OEP is much more difficult and is much less studied. To our best knowledge, we are only aware of han09 ; hm03 ; hl19pp ; mothesis for studying one-dimensional OEP-based multiframelets, and hl20pp for investigating OEP-based quasi-tight multiframelets in arbitrary dimensions.

Here we briefly explain the difficulties involved in studying multiframelets. We see from Theorem 1.1 that the most important step of constructing OEP-based framelets is choosing the appropriate filters $\theta,\tilde{\theta}\in(l_{0}(\mathbb{Z}^{d}))^{r\times r}$. In many situations, this is not easy. Except for the examples in han09 ; hl19pp , all constructed OEP-based dual framelets with non-trivial $\Theta$ (where $\Theta:=\theta^{\star}*\tilde{\theta}$) do not have a compact underlying discrete framelet transform, i.e., $\Theta$ is not strongly invertible.

On the other hand, the sparsity of a discrete framelet transform is another issue which needs to be worried about when the multiplicity $r>1$. First we look at the scalar case $r=1$. Let $(\{\mathring{\phi};\psi\},\{\mathring{\tilde{\phi}};\tilde{\psi}\})$ be an OEP-based dual $\mathsf{M}$-framelet obtained through Theorem 1.1 with an underlying OEP-based dual $\mathsf{M}$-framelet filter bank $(\{a;b\},\{\tilde{a};\tilde{b}\})_{\Theta}$. Suppose that $\operatorname{vm}(\psi)=m$. Then the framelet representation (1.1) has sparsity in the sense that the polynomial preservation property (1.2) holds. Moreover, item (1) of Theorem 1.1 yields $\widehat{\phi}(0)\neq 0$. Thus it follows from $\widehat{\psi}:=\widehat{b}(\mathsf{M}^{-\mathsf{T}}\cdot)\widehat{\phi}(\mathsf{M}^{-\mathsf{T}}\cdot)$ that $\widehat{b}(\xi)=\mathcal{O}(\|\xi\|^{m})$ as $\xi\to 0$. For any polynomial $\mathsf{p}\in\mathbb{P}_{m-1}$, using Taylor expansion yields $\mathsf{p}(x-k)=\sum_{\alpha\in\mathbb{N}^{d}_{0}}\frac{(-k)^{\alpha}}{\alpha!}\partial^{\alpha}\mathsf{p}(x)$ for all $x,k\in\mathbb{R}^{d}$. Thus for any finitely supported sequence $u\in l_{0}(\mathbb{Z}^{d})$, we have

which is a polynomial whose degree is no bigger than the degree of $\mathsf{p}$, i.e., $\mathsf{p}*u\in\mathbb{P}_{m-1}$. Denote $\mathbb{P}_{m-1}|_{\mathbb{Z}^{d}}$ the linear space of all $d$-variate polynomial sequences of degree at most $m-1$. We now input a polynomial sequence data $\mathsf{p}\in\mathbb{P}_{m-1}|_{\mathbb{Z}^{d}}$ and implement the $J$-level discrete framelet transform with the filter bank $(\{a;b\},\{\tilde{a};\tilde{b}\})_{\Theta}$. Observe that the framelet coefficient $v_{1}$ (see (1.14)) satisfies $v_{1}=\mathcal{T}_{a,\mathsf{M}}\mathsf{p}=|\det(\mathsf{M})|^{\frac{1}{2}}[\mathsf{p}*a^{\star}](\mathsf{M}\cdot)\in\mathbb{P}_{m-1}$, and by induction we conclude that $v_{j}\in\mathbb{P}_{m-1}|_{\mathbb{Z}^{d}}$ for all $j=1,2,\dots,J$. It follows that the framelet coefficients $w_{j}$ (see (1.15)) now satisfy

for all $j=1,\dots,J$, where the last step follows from $\widehat{b}(\xi)=\mathcal{O}(\|\xi\|^{m})$ as $\xi\to 0$ and $v_{j}\in\mathbb{P}_{m-1}|_{\mathbb{Z}^{d}}$. Consequently, all framelet coefficients $w_{j}$ vanish. This means that the sparsity of the framelet expansion (1.1) automatically guarantees the sparsity of the underlying multi-level discrete framelet transform. Unfortunately this is in general not the case when $r>1$, simply due to the fact that $\widehat{\psi}(\xi)=\widehat{b}(\mathsf{M}^{-\mathsf{T}}\xi)\widehat{\phi}(\mathsf{M}^{-\mathsf{T}}\xi)=\mathcal{O}(\|\xi\|^{m})$ does not imply any moment property of $\widehat{b}(\xi)$ at $\xi=0$. This issue is known as the balancing property of a framelet in the literature (cj00 ; cj03 ; han09 ; han10 ; hanbook ; lv98 ; sel00 ). See Section 2 for a brief review of this topic.

From the previous discussion, for OEP-based dual framelets, it seems impossible to achieve high vanishing moments on framelet generators without sacrificing the desired features of the underlying discrete framelet transform. The first breakthrough to this problem is han09 , which proves that for $r\geqslant 2$ and $d=1$, one can always obtain OEP-based dual framelets from arbitrary compactly supported refinable vector functions, such that all framelet generators have the highest possible vanishing moments and the associated discrete framelet transform is compact and balanced. However, the case when $d>1$ is far from being well investigated. We are only aware of han10 which systematically studies the balancing property from the discrete setting for $d>1$ and hl20pp which deals with the problem with the approach of the so-called quasi-tight framelets. In this paper, we will systematically study multivariate OEP-based dual framelets with the three key properties. Our main result is the following theorem.

For $r=1$, we have a similar result which only satisfies item (3), for the following reasons: (1) a filter $\theta\in l_{0}(\mathbb{Z}^{d})$ is strongly invertible if and only if $\theta=c\boldsymbol{\delta}(\cdot-k)$ for some $c\in\mathbb{C}$ and $k\in\mathbb{Z}^{d}$, and using such filters loses the advantage of OEP for increasing vanishing moments on framelet generators; (2) the balancing property does not come in to play when the multiplicity $r=1$. Theorem 1.3 extends the main result of han09 for the case $d=1$ to $d>1$, but is not a simple generalization. Several techniques for the case $d=1$ simply do not work when $d>1$. For instance, a $2\pi$-periodic trigonometric polynomial has $m$ vanishing moments if and only if it is divisible by $(1-e^{-i\xi})^{m}$, which is an important fact for the construction of dual framelets with high vanishing moments when $d=1$. Unfortunately, such factorization is no longer available when $d>1$. A recently developed normal form of a matrix-valued filter (see hl20pp ) plays a crucial role in our study of OEP-based dual framelets with high vanishing moments and high balancing order, and we will provide a short review of this topic in Section 2.

The paper is organized as follows. In Section 2, we briefly review the balancing property of a multi-level discrete transform, as well as a recently developed normal form of a matrix-valued filter. These are what we need to prove our main result. In Section 3, we prove the main result Theorem 1.3. Motivated by the proof of the main theorem, we shall perform structural analysis of compactly supported balanced OEP-based dual multiframelets in Section 4. Finally, a summary of our work and some concluding comments will be given in Section 5.

As mentioned in Section 1, one issue with OEP when the multiplicity $r>1$ is the sparseness of the multilevel discrete framelet transform. In many applications, the original data is scalar valued, that is, an input data $v\in l(\mathbb{Z}^{d})$. Thus to implement a multi-level discrete framelet transform, we need to first vectorize the input data. Let $\mathsf{N}$ be a $d\times d$ integer matrix with $|\det(\mathsf{N})|=r$, and let $\Gamma_{\mathsf{N}}$ be a particular choice of the representatives of the cosets in $\mathbb{Z}^{d}/[\mathsf{N}\mathbb{Z}^{d}]$ given by

We define the standard vectorization operator with respect to $\mathsf{N}$ via

Clearly $E_{\mathsf{N}}$ is a bijection between $l(\mathbb{Z}^{d})$ and $(l(\mathbb{Z}^{d}))^{1\times r}$. The sparsity of a multi-level discrete framelet transform employing an OEP-based dual $\mathsf{M}$-framelet filter bank $(\{a;b\},\{\tilde{a};\tilde{b}\})_{\Theta}$ is measured by the $E_{\mathsf{N}}$-balancing order of the analysis filter bank $\{a;b\}$, denoted by $\operatorname{bo}(\{a;b\},\mathsf{M},\mathsf{N}):=m$ where $m$ is the largest integer such that the following two conditions hold: