Analysis of Framelet Transforms on a Simplex

TL;DR

This paper develops framelet transforms on a 2D simplex, providing exact reconstruction for tight framelets and efficient computation comparable to FFT, advancing analysis tools on simplexes.

Contribution

It introduces a new construction of framelets on the simplex with fast transform algorithms and theoretical guarantees for tight frames.

Findings

Framelet transforms on the simplex are computationally efficient.

Exact reconstruction is achieved with tight framelets.

An explicit example of framelet construction is provided.

Abstract

In this paper, we construct framelets associated with a sequence of quadrature rules on the simplex $T^{2}$ in $\mathbb{R}^{2}$. We give the framelet transforms -- decomposition and reconstruction of the coefficients for framelets of a function on $T^{2}$. We prove that the reconstruction is exact when the framelets are tight. We give an example of construction of framelets and show that the framelet transforms can be computed as fast as FFT.

Equations

$$T^2:=\{\mathbf{x}:=(x_1,x_2)|x_1\ge 0,x_2\ge 0,x_1+x_2\le 1\}.$$

$$\Delta :=\sum _{i=1}^2{x}_i(1-x_i)\frac{{\partial }^2}{\partial x_i^2}-2\sum _{1\le i\le j\le 2}{x}_i{x}_j\frac{{\partial }^2}{\partial x_i\partial x_j}+\sum _{i=1}^2(1-3x_i)\frac{\partial }{\partial x_i},$$

$$-\Delta P_{\ell }={\lambda }_{\ell }^2{P}_{\ell },\ell \in {\mathbb{N}}_0,$$

$$\hat{\alpha }(2\xi )=\hat{a}(\xi )\hat{\alpha }(\xi ),\widehat{{\beta }^n}(2\xi )=\widehat{b_n}(\xi )\hat{\alpha }(\xi ),n=1,\dots ,r,\xi \in \mathbb{R},$$

$$\begin{array}[]{l}\displaystyle\bm{\varphi}_{j,\bm{y}}(\bm{x}):=\sum_{\ell=0}^{\infty}\widehat{\alpha}\left(\frac{\lambda_{\ell}}{2^{j}}\right)\overline{P_{\ell}(\bm{y})}P_{\ell}(\bm{x}),\\ \displaystyle\bm{\psi}_{j,\bm{y}}^{n}(\bm{x}):=\sum_{\ell=0}^{\infty}\widehat{\beta^{n}}\left(\frac{\lambda_{\ell}}{2^{j}}\right)\overline{P_{\ell}(\bm{y})}P_{\ell}(\bm{x}),\quad n=1,\ldots,r.\end{array}$$

$$Q_N[f]:=\sum _{k=1}^N{w}_{j}f({\mathbf{x}}_j)$$

$${\mathbf{\phi }}_{j,k}(\mathbf{x}):=\sqrt{{\omega }_{j,k}}{\mathbf{\phi }}_{j,{\mathbf{x}}_{j,k}}(\mathbf{x})=\sqrt{{\omega }_{j,k}}\sum _{\ell =0}^{\infty }\hat{\alpha }\left(\frac{{\lambda }_{\ell }}{2^j}\right)\overline{P_{\ell }({\mathbf{x}}_{j,k})}{P}_{\ell }(\mathbf{x}),$$

$${\mathbf{\psi }}_{j,k^{\prime }}^n(\mathbf{x}):=\sqrt{{\omega }_{j+1,k^{\prime }}}{\mathbf{\psi }}_{j,{\mathbf{x}}_{j+1,k^{\prime }}}^n(\mathbf{x})=\sqrt{{\omega }_{j+1,k^{\prime }}}\sum _{\ell =0}^{\infty }\widehat{{\beta }^n}\left(\frac{{\lambda }_{\ell }}{2^j}\right)\overline{P_{\ell }({\mathbf{x}}_{j+1,k^{\prime }})}{P}_{\ell }(\mathbf{x}).$$

$$(\mathrm{v}{)}_k=\sqrt{{\omega }_{j,k}}\sum _{\ell =0}^{\infty }{\mathrm{u}}_{\ell }{P}_{\ell }({\mathbf{x}}_{j,k}),k=1,\dots ,N_j.$$

$$(\mathrm{v}{\ast }_{j}h{)}_k:=\sum _{\ell =0}^{\infty }{\hat{\mathrm{v}}}_{\ell }\hat{h}\left(\frac{{\lambda }_{\ell }}{2^j}\right)\sqrt{{\omega }_{j,k}}{P}_{\ell }({\mathbf{x}}_{j,k}),k=1,\dots ,N_j.$$

$$(\mathrm{v}{\downarrow }_j{)}_k:=\sum _{{\lambda }_{\ell }\le 2^{j-1}}{\hat{\mathrm{v}}}_{\ell }\sqrt{{\omega }_{j,k}}{P}_{\ell }({\mathbf{x}}_{j,k}),k=1,\dots ,N_{j-1}.$$

$$({\mathrm{v}}_j{)}_k:=⟨f,{\mathbf{\phi }}_{j,k}⟩,({\mathrm{w}}_j^n{)}_{k^{\prime }}:=⟨f,{\mathbf{\psi }}_{j,k^{\prime }}^n⟩.$$

$${\mathrm{v}}_{j-1}=({\mathrm{v}}_j{\ast }_j{a}^{\star }){\downarrow }_j,{\mathrm{w}}_{j-1}^n={\mathrm{v}}_j{\ast }_j{b}_n^{\star },n=1,\dots ,r.$$

$$\begin{array}[]{ll}(\mathrm{v}_{j-1})_{k}&=\displaystyle\sqrt{\omega_{j-1,k}}\sum_{\ell=0}^{\infty}\overline{\widehat{\alpha}\left(\frac{\lambda_{\ell}}{2^{j-1}}\right)}\widehat{f}_{\ell}\>P_{\ell}(\bm{x}_{j-1,k}),\quad k=1,\dots,N_{j-1},\\[14.22636pt] (\mathrm{w}^{n}_{j-1})_{k^{\prime}}&=\displaystyle\sqrt{\omega_{j,k^{\prime}}}\sum_{\ell=0}^{\infty}\overline{\widehat{\beta^{n}}\left(\frac{\lambda_{\ell}}{2^{j-1}}\right)}\widehat{f}_{\ell}\>P_{\ell}(\bm{x}_{j,k^{\prime}}),\quad k^{\prime}=1,\dots,N_{j},\;n=1,\dots,r.\end{array}$$

$$({\mathrm{v}}_{j-1}{)}_k$$

$$=\sqrt{{\omega }_{j-1,k}}\sum _{{\lambda }_{\ell }\le 2^{j-1}}{\hat{f}}_{\ell }\overline{\hat{\alpha }\left(\frac{{\lambda }_{\ell }}{2^j}\right)}\overline{\hat{a}\left(\frac{{\lambda }_{\ell }}{2^j}\right)}{P}_{\ell }({\mathbf{x}}_{j-1,k})$$

$$=\sqrt{{\omega }_{j-1,k}}\sum _{{\lambda }_{\ell }\le 2^{j-1}}{\widehat{({\mathrm{v}}_j)}}_{\ell }\overline{\hat{a}\left(\frac{{\lambda }_{\ell }}{2^j}\right)}{P}_{\ell }({\mathbf{x}}_{j-1,k})$$

$$\displaystyle=\bigl{(}(\mathrm{v}_{j}\ast_{j}a^{\star})\downarrow_{j}\bigr{)}_{k}.$$

$$({\mathrm{w}}_{j-1}^n{)}_{k^{\prime }}$$

$$=\sqrt{{\omega }_{j,k^{\prime }}}\sum _{{\lambda }_{\ell }\le 2^{j-1}}{\hat{f}}_{\ell }\overline{\widehat{{\beta }^n}\left(\frac{{\lambda }_{\ell }}{2^j}\right)}\overline{\widehat{b_n}\left(\frac{{\lambda }_{\ell }}{2^j}\right)}{P}_{\ell }({\mathbf{x}}_{j,k^{\prime }})$$

$$=\sqrt{{\omega }_{j,k^{\prime }}}\sum _{{\lambda }_{\ell }\le 2^{j-1}}{\widehat{({\mathrm{v}}_j)}}_{\ell }\overline{\widehat{b_n}\left(\frac{{\lambda }_{\ell }}{2^j}\right)}{P}_{\ell }({\mathbf{x}}_{j,k^{\prime }})$$

$$=({\mathrm{v}}_j{\ast }_j{b}_n^{\star }{)}_{k^{\prime }}.$$

$$f=\sum _{k=1}^{N_0}⟨f,{\mathbf{\phi }}_{j,k}⟩{\mathbf{\phi }}_{j,k}+\sum _{j=0}^{\infty }\sum _{k^{\prime }=1}^{N_{j+1}}\sum _{n=1}^r⟨f,{\mathbf{\psi }}_{j,k^{\prime }}^n⟩{\mathbf{\psi }}_{j,k^{\prime }}^n\text{mbox}forallf\in L_2(T^2),$$

$$\|f\|_{L_{2}(T^{2})}^{2}=\sum_{k=1}^{N_{0}}\bigl{|}\langle{f,\bm{\varphi}_{j,k}}\rangle\bigr{|}^{2}+\sum_{j=0}^{\infty}\sum_{k^{\prime}=1}^{N_{j+1}}\sum_{n=1}^{r}\bigl{|}\langle{f,\bm{\psi}_{j,k^{\prime}}^{n}}\rangle\bigr{|}^{2}\quad\mbox{for all}~{}f\in L_{2}(T^{2}).$$

$$\displaystyle\lim_{j\to\infty}\sum_{k=1}^{N_{j}}\bigl{|}\langle{f,\bm{\varphi}_{j,k}}\rangle\bigr{|}^{2}=\|f\|_{L_{2}(T^{2})}^{2},$$

$$\displaystyle\sum_{k=1}^{N_{j+1}}\bigl{|}\langle{f,\bm{\varphi}_{j+1,k}}\rangle\bigr{|}^{2}=\sum_{k=1}^{N_{j}}\bigl{|}\langle{f,\bm{\varphi}_{j,k}}\rangle\bigr{|}^{2}+\sum_{k=1}^{N_{j+1}}\sum_{n=1}^{r}\bigl{|}\langle{f,\bm{\psi}_{j,k}^{n}}\rangle\bigr{|}^{2},\quad j\in\mathbb{N}_{0}.$$

$${\int }_{T^2}{p}_{{\ell }^{\prime }}(\mathbf{x})\mathrm{d}\mu (\mathbf{x})=\sum _{j=1}^N{w}_j{p}_{{\ell }^{\prime }}({\mathbf{x}}_j)\text{mbox}forall{p}_{{\ell }^{\prime }}\in {\mathcal{V}}_{{\ell }^{\prime }}.$$

$$\underset{j\to \infty }{\lim }\hat{a}\left(\frac{{\lambda }_{\ell }}{2^j}\right)=1,{\left|\hat{a}\left(\frac{{\lambda }_{\ell }}{2^j}\right)\right|}^2+\sum _{n=1}^r{\left|\widehat{b_n}\left(\frac{{\lambda }_{\ell }}{2^j}\right)\right|}^2=1\text{mbox}forj,\ell \in {\mathbb{N}}_0,$$

$$(\mathrm{v}{\uparrow }_j{)}_k:=\sum _{{\lambda }_{\ell }\le 2^{j-2}}{\hat{\mathrm{v}}}_{\ell }\sqrt{{\omega }_{j,k}}{P}_{\ell }({\mathbf{x}}_{j,k}),k=1,\dots ,N_j,$$

$${\mathrm{v}}_j=({\mathrm{v}}_{j-1}{\uparrow }_j){\ast }_{j}a+\sum _{n=1}^r{\mathrm{w}}_{j-1}^n{\ast }_j{b}_n.$$

$$(({\mathrm{v}}_{j-1}{\uparrow }_j){\ast }_{j}a{)}_k=\sqrt{{\omega }_{j,k}}\sum _{{\lambda }_{\ell }\le 2^{j-1}}{\widehat{({\mathrm{v}}_j)}}_{\ell }{\left|\hat{a}\left(\frac{{\lambda }_{\ell }}{2^j}\right)\right|}^2{P}_{\ell }({\mathbf{x}}_{j,k})$$

$$({\mathrm{w}}_{j-1}^n{\ast }_j{b}_n{)}_k=\sqrt{{\omega }_{j,k}}\sum _{{\lambda }_{\ell }\le 2^{j-1}}{\widehat{({\mathrm{v}}_j)}}_{\ell }{\left|{\hat{b}}_n\left(\frac{{\lambda }_{\ell }}{2^j}\right)\right|}^2{P}_{\ell }({\mathbf{x}}_{j,k}),n=1,\dots ,r.$$

$$\displaystyle\Bigl{(}(\mathrm{v}_{j-1}\uparrow_{j})\ast_{j}a+\sum_{n=1}^{r}\mathrm{w}^{n}_{j-1}\ast_{j}b_{n}\Bigr{)}_{k}$$

$$=\sqrt{{\omega }_{j,k}}\sum _{{\lambda }_{\ell }\le 2^{j-1}}{\widehat{({\mathrm{v}}_j)}}_{\ell }\left({\left|\hat{a}\left(\frac{{\lambda }_{\ell }}{2^j}\right)\right|}^2+\sum _{n=1}^r{\left|{\hat{b}}_n\left(\frac{{\lambda }_{\ell }}{2^j}\right)\right|}^2\right)P_{\ell }({\mathbf{x}}_{j,k})$$

Full text

Dedicated to Ian H. Sloan on the occasion of his 80th birthday with

our gratitude for his constant supervision, support and encouragement.

1 Introduction

Multiresolution analysis on a simplex $T^{2}$ in $\mathbb{R}^{2}$ has many applications such as in numerical solution of PDEs and computer graphics Dyn1992 ; deFeDeTo2016 ; GrCoMaDe2017 . In this paper, we construct framelets (or a framelet system) on $T^{2}$, following the framework of WaZh2017 , and give the transforms of coefficients for framelets.

Framelets are localised functions associated with quadrature rules of $T^{2}$. Each framelet is scaled at a level $j$, $j=0,1,\dots$ and translated at a node of a quadrature rule of level $j$. The framelet coefficients for a square-integrable function $f$ on the simplex are the inner products of the framelets with $f$ on $T^{2}$. We give the framelet transforms which include the decomposition and reconstruction of the coefficients for framelets. Since the framelets are well-localised, see e.g. MaMh2008 , the decomposition gives all approximate and detailed information of the function $f$. This plays an important role in signal processing on the simplex.

For levels $j$ and $j+1$, the decomposition estimates the framelet coefficients of level $j+1$ by the coefficients of level $j$. The reconstruction is the inverse, which estimates the coefficients of level $j$ by the level $j+1$. Such framelet transforms are significant as by decompositions or reconstructions, we are able to estimate high-level framelet coefficients from the bottom level [math], or the inverse.

We show that when the quadrature rules and masks have good properties, the reconstruction is exact and invertible with the decomposition, see Section 4. We also show that the framelet transforms can be computed as fast as the FFTs, see Section 6.

We construct framelets using tensor-product form of Jacobi polynomials and triangular Kronecker lattices BaOw2015 with equal weights, see Section 5.

2 Framelets on Simplex

In the paper, we consider the simplex (or the triangle)

[TABLE]

Let $L_{2}(T^{2})$ be the space of complex-valued square integrable functions on $T^{2}$ with respect to the normalized Lebesgue area measure $\mu$ on $\mathbb{R}^{2}$ (i.e. $\int_{T^{2}}\mathrm{d}{\mu(\bm{x})}=1$), provided with inner product $\langle f,g\rangle:=\langle f,g\rangle_{L_{2}(T^{2})}:=\int_{T^{2}}f(\bm{x})\mathrm{d}{\mu(\bm{x})}$ , where $\overline{g}$ is the complex conjugate of $g$, and endowed with the induced $L_{2}$-norm $\|f\|_{L_{2}(T^{2})}:=\sqrt{\langle{f,f}\rangle}$ for $f\in L_{2}(T^{2})$.

For $\ell\geq 0$, let $\mathcal{V}_{\ell}:=\mathcal{V}_{\ell}(T^{2})$ be the space of orthogonal polynomials of degree $\ell$ with respect to the inner product $\langle{\cdot,\cdot}\rangle_{L_{2}(T^{2})}$. The dimension of $\mathcal{V}_{\ell}$ is $\ell+1$, see DuXu2014 . The elements of $\mathcal{V}_{\ell}$ are said to be the polynomials of degree $\ell$ on $T^{2}$. The union of all polynomial spaces $\cup_{\ell=0}^{\infty}\mathcal{V}_{\ell}$ is dense in $L_{2}(T^{2})$.

As a compact Riemannian manifold, the simplex $T^{2}$ has the Laplace-Beltrami operator

[TABLE]

with polynomials $P_{\ell}$ in $\mathcal{V}_{\ell}$ as the eigenfunctions and with (square-rooted) eigenvalues $\lambda_{\ell}:=\sqrt{\ell(\ell+2)}$:

[TABLE]

where $\mathbb{N}_{0}:=\{0,1,2,\dots\}$.

Let $L_{1}(\mathbb{R})$ be the space of absolutely integrable functions on $\mathbb{R}$ with respect to the Lebesgue measure and let $l_{1}(\mathbb{Z})$ be the set of $l_{1}$ summable sequences on $\mathbb{Z}$. For $r\geq 1$, let $\Psi:=\{\alpha;\beta^{1},\ldots,\beta^{r}\}$ be a set of $(r+1)$ functions in $L_{1}(\mathbb{R})$, which are associated with a filter bank ${\bm{\eta}}:=\{a;b_{1},\ldots,b_{r}\}\subset l_{1}(\mathbb{Z})$ satisfying

[TABLE]

where $\widehat{g}(\xi):=\int_{\mathbb{R}}g(x)e^{-2\pi ix\xi}\>\mathrm{d}{x}$, $\xi\in\mathbb{R}$ is the Fourier transform for $g\in L_{1}(\mathbb{R})$ and $\widehat{h}(\xi):=\sum_{k\in\mathbb{Z}}h_{k}e^{-2\pi{\mathrm{i}}\xi}$ is the Fourier series of a sequence $h:=(h_{k})_{k\in\mathbb{Z}}$ in $l_{1}(\mathbb{Z})$. Here, the sequences $a$ and $b_{n}$ are said to be low-pass (mask) and high-pass (mask) respectively.

We introduce the continuous and semi-discrete framelets on the simplex following the construction and notation of WaZh2017 ; Dong2015 . The continuous framelets on the simplex $T^{2}$ are, for $j\in\mathbb{N}_{0}$,

[TABLE]

The continuous framelets in (2) are analogues of continuous wavelets in $\mathbb{R}$. The level “$j$” indicates the “dilation” scale and “$\bm{y}$” is the point at which the framelet is “translated”.

Let ${Q}_{N}:=\{(w_{j},\bm{x}_{j})\}_{j=1}^{N}$, which is a set of $N$ pairs of weights $w_{j}$ in $\mathbb{R}\backslash\{0\}$ and points $\bm{x}_{j}$ on $T^{2}$, define the quadrature rule

[TABLE]

for continuous functions $f$ on $T^{2}$. Let ${Q}_{N_{j}}:=\{(\omega_{j,k},\bm{x}_{j,k})\}_{k=1}^{N_{j}}$, $j\in\mathbb{N}_{0}$, be a sequence of such quadrature rules. For $j=0,1,\dots$, the semi-discrete framelets $\bm{\varphi}_{j,k}$ and $\bm{\psi}_{j,k^{\prime}}^{n}$ associated with quadrature rules ${Q}_{N_{j}}$ are defined as the continuous framelets $\bm{\varphi}_{j,\bm{y}}$ and $\bm{\psi}_{j,\bm{y}}^{n}$ translated at $\bm{x}_{j,k}$ and $\bm{x}_{j+1,k^{\prime}}$ respectively. That is, for $k=1,\dots,N_{j}$,

[TABLE]

and for $k^{\prime}=1,\dots,N_{j+1}$ and $n=1,\ldots,r$,

[TABLE]

We say $\bm{\varphi}_{j,k}$ and $\bm{\psi}_{j,k^{\prime}}^{n}$ are low-pass framelet and high-pass framelet respectively.

Note that here we use the ${Q}_{N_{j+1}}$ for high-passes due to the scale of $\bm{\psi}_{j,k^{\prime}}^{n}$ is at $j+1$. This will be clear in Section 5.

We also use the notation $\bm{\psi}_{j,k}^{n}$ for $\bm{\psi}_{j,k^{\prime}}^{n}$ if no confusion arises.

The framelets $\bm{\varphi}_{j,k}$ and $\bm{\psi}_{j,k}^{n}$ corresponding to the low-pass $a$ and high-pass $b_{n}$ carry the information of approximations and details in framelet transforms, as we will show below.

3 Decomposition for Framelets

In practice, we need to estimate the framelet coefficients of high levels from low-level coefficients. This can be achieved by the decomposition of framelets.

The decomposition for framelets can be realized by the operations of convolution and downsampling as we introduce now.

Let $h\in l_{1}(\mathbb{Z})$ be a mask satisfying that the support of the Fourier series $\widehat{h}$ of $h$ is a subset of $[0,1/2]$. Let $l(N)$ be the set of complex-valued sequences with supports in $[0,N]$. Let ${Q}_{N_{j}}:=\{(\omega_{j,k},\bm{x}_{j,k})\}_{k=1}^{N_{j}}$, $j\in\mathbb{N}_{0}$, be the quadrature rules for framelets. Let $l({Q}_{N_{j}})$ be the set of sequences $\mathrm{v}$ in $l(N_{j})$ satisfying that there exists a sequence $\mathrm{u}$ in $\l_{1}(\mathbb{Z})$ such that

[TABLE]

We let $\widehat{\mathrm{v}}_{\ell}:=\mathrm{u}_{\ell}$ (with abuse of notation) be the (generalized) Fourier coefficients of $\mathrm{v}$ for the orthonormal basis $P_{\ell}$ and the quadrature rule ${Q}_{N_{j}}$ on $T^{2}$.

The (discrete) convolution $\mathrm{v}\ast_{j}h$ of a sequence $\mathrm{v}$ with the mask $h$ is a sequence in $l({Q}_{N_{j}})$ given by

[TABLE]

Then, the Fourier coefficients of $\mathrm{v}\ast_{j}h$ are $(\widehat{\mathrm{v}\ast_{j}h})_{\ell}=\widehat{\mathrm{v}}_{\ell}\>{\widehat{h}}\left(\frac{\lambda_{\ell}}{2^{j}}\right)$, $\ell\in\mathbb{N}_{0}$.

The downsampling $\mathrm{v}\downarrow_{j}$, $j\geq 1$, of a sequence $\mathrm{v}\in l({Q}_{N_{j}})$ is a sequence in $l({Q}_{N_{j-1}})$ given by

[TABLE]

For semi-discrete framelets in (3) and (4), the inner products $\langle{f,\bm{\varphi}_{j,k}}\rangle$ and $\langle{f,\bm{\psi}_{j,k^{\prime}}^{n}}\rangle$, $n=1,\ldots,r$, $j\in\mathbb{N}_{0}$, $k=1,\dots,N_{j}$ and $k^{\prime}=1,\dots,N_{j+1}$, are said to be framelet coefficients for $f$. For convenience, we let $\mathrm{v}_{j}$ and $\mathrm{w}^{n}_{j}$ denote the framelet coefficients for $f$:

[TABLE]

The Fourier coefficients of a function $f\in L_{2}(T^{2})$ are $\widehat{f}_{\ell}:=\langle{f,P_{\ell}}\rangle$, $\ell\in\mathbb{N}_{0}$. Let $h$ be a mask in $l_{1}(\mathbb{Z})$ and $h^{\star}$ be the mask whose Fourier series is conjugate to the Fourier series of $h$.

The following proposition shows the decomposition for framelet coefficients between adjacent levels.

Proposition 1

Let the framelet coefficients for semi-discrete framelets in (3) and (4) be given by (7), where the supports of $\widehat{\alpha}$ and $\widehat{\beta^{n}}$ are subsets of $[0,1/2]$. For $j=1,2,\dots$, the decomposition from level $j$ into level $j-1$ is

[TABLE]

Proof

For $f\in L_{2}(T^{2})$, by the orthonormality of $P_{\ell}$ and (7),

[TABLE]

For low-pass, by (1), (5) and (6), for $k=1,\ldots,N_{j-1}$,

[TABLE]

For high-passes, for $k^{\prime}=1,\ldots,N_{j}$ and $n=1,\dots,r$,

[TABLE]

These give (8).

4 Reconstruction for Tight Framelets

We say the set of framelets $\{\bm{\varphi}_{j,k},\bm{\psi}_{j,k^{\prime}}^{n}|n=1,\dots,r,\;k=1,\dots,N_{j},\;k^{\prime}=1,\dots,N_{j+1},\;j\in\mathbb{N}_{0}\}$ a tight frame for $L_{2}(T^{2})$ if the framelets are all in $L_{2}(T^{2})$, and in the $L_{2}$ sense,

[TABLE]

or equivalently,

[TABLE]

The framelets are then said to be (semi-discrete) tight framelets.

If the framelets are tight on the simplex, a function in $L_{2}(T^{2})$ can be represented using the framelet coefficients. The following property as a consequence of (WaZh2017, , Theorem 2.4) shows that the tightness of framelets is equivalent to a multiscale representation of framelets of a level by lower levels.

Proposition 2 (WaZh2017 )

The semi-discrete framelets in (3) and (4) are tight if and only if for all $f\in L_{2}(T^{2})$, the following identities hold:

[TABLE]

The condition in (9) implies that high-level framelet coefficients can be estimated by low levels. This then gives the reconstruction for framelets.

The reconstruction depends on the property of the quadrature rules ${Q}_{N_{j}}$ for framelets. A quadrature rule $Q_{N}:=\{(w_{j},\bm{x}_{j})\}_{j=1}^{N}$ is said to be exact for polynomials up to degree $\ell$ if for $\ell^{\prime}=0,\dots,\ell$,

[TABLE]

When the quadrature rule ${Q}_{N_{j}}$, $j\in\mathbb{N}_{0}$, for framelets $\bm{\varphi}_{j,k}$ and $\bm{\psi}_{j,k}^{n}$ is exact for polynomials up to degree $2^{j}$, the tightness of the framelets is equivalent to the following condition on masks:

[TABLE]

see (WaZh2017, , Theorem 2.1 and Corollary 2.6).

The upsampling $\mathrm{v}\uparrow_{j}$, $j\geq 1$, of a sequence $\mathrm{v}\in l({Q}_{N_{j-1}})$ is a sequence in $l({Q}_{N_{j}})$ given by

[TABLE]

where $\widehat{\mathrm{v}}_{\ell}$ are the Fourier coefficients of $\mathrm{v}$ for basis $P_{\ell}$ and quadrature rule ${Q}_{N_{j-1}}$ on $T^{2}$.

The reconstruction involving the operations of convolution and upsampling is given by the following proposition.

Proposition 3

Let the framelet coefficients for semi-discrete framelets in (3) and (4) be given by (7), where the supports of $\widehat{\alpha}$ and $\widehat{\beta^{n}}$ are subsets of $[0,1/2]$, and (10) holds. Then, for $j\geq 1$, the reconstruction from level $j-1$ to level $j$ is

[TABLE]

Proof

By Proposition 1, for $k=1,\dots,N_{j}$,

[TABLE]

and

[TABLE]

These give

[TABLE]

thus proving (11).

Remark 1

(WaZh2017, , Theorem 3.1) proves (11) for general Riemannian manifolds when the quadrature rule is exact for polynomials up to degree $2^{j}$ and under condition (10). Here we do not require that the quadrature rules of the framelets satisfy the polynomial exactness.

Repeatedly using the decomposition and reconstruction in Propositions 1 and  3 gives multi-level framelet transforms. Figure 1 illustrates the decomposition and reconstruction for levels $0,\dots,j$.

5 Constructive Examples

From the above analysis, the construction of semi-discrete framelets needs an orthonormal basis for $L_{2}(T^{2})$ and appropriate masks and quadrature rules.

Orthonormal bases. One orthonormal basis can be constructed by the tensor product of Jacobi polynomials, see (DuXu2014, , Proposition 2.4.1). For $\tau,\gamma>-1$ and $\ell\geq 0$, let $P^{(\tau,\gamma)}_{\ell}(t)$ be the Jacobi polynomial of degree $\ell$ with respect to the weight $(1-t)^{\tau}(1+t)^{\gamma}$ on [-1,1]. For $\bm{x}:=(x_{1},x_{2})\in T^{2}$ and $\ell\in\mathbb{N}_{0}$ and $m=0,\dots,\ell$, let

[TABLE]

Then $\{P_{\ell,m}|\hskip 0.56905ptm=0,\dots,\ell\}$ is an orthonormal basis of $\mathcal{V}_{\ell}$ and $\{P_{\ell,m}|\hskip 0.56905ptm=0,\dots,\ell,\ell\geq 0\}$ forms an orthonormal basis of $L_{2}(T^{2})$.

Sun Sun2003 constructs another orthonormal basis for $L_{2}(T^{2})$, which is useful in discrete Fourier analysis on $T^{2}$, see LiSuXu2008 ; LiXu2010 .

Masks. We give an example of masks with two high-passes. Let

[TABLE]

By (Daubechies1992, , Chapter 4), the masks $a,b_{1}$ and $b_{2}$ can be defined by their Fourier series as

[TABLE]

which satisfy (10).

The corresponding scaling functions are

[TABLE]

Here, $\mathrm{supp}\>\widehat{\alpha}\subseteq[0,1/2]$ and $\mathrm{supp}\>\widehat{\beta^{n}}\subseteq[1/4,1]$, $n=1,2$. This means that the scaling of the framelet $\bm{\varphi}_{j,k}$ in (3) with the low-pass scaling function in (27) is half of the scaling of the framelets $\bm{\psi}_{j,k}^{1}$ and $\bm{\psi}_{j,k}^{2}$ in (4) with high-pass scaling functions in (31) and (35). The high-pass framelets thus need to use a quadrature rule at the level $j+1$, one level higher than $\bm{\varphi}_{j,k}$.

Figure 2 shows the Fourier series of masks $a$, $b_{1}$ and $b_{2}$ in (16), (20) and (23).

Quadrature rules. We use triangular Kronecker lattices of Basu and Owen BaOw2015 with equal weights as the quadrature rules for framelets, which are shifted lattice points intersecting with the simplex. For the quadrature rule ${Q}_{N_{j}}$ of framelets, we use the triangular Kronecker lattice with at least $2^{2j}$ nodes, which are the translation points of the low-pass framelets $\bm{\varphi}_{j,k}$ at level $j$ and those of high-pass framelets $\bm{\psi}_{j-1,k^{\prime}}^{n}$ at level $j-1$. Figure 3 shows the triangular Kronecker lattice with $N=65$ nodes on $T^{2}$ used for framelets at levels $2$ and $3$.

Framelets. Using the orthonormal basis in (12), scaling functions in (27)–(35) and triangular Kronecker lattices with equal weights, the framelets are, for $j\in\mathbb{N}_{0}$,

[TABLE]

and for $n=1,2$,

[TABLE]

Figure 4 shows the framelets $\bm{\varphi}_{j,k}$, $\bm{\psi}_{j,k^{\prime}}^{1}$ and $\bm{\psi}_{j,k^{\prime}}^{2}$ at level $j=5$ with $k=512$ and $k^{\prime}=2048$, using orthonormal basis (12) and scaling functions (27), (31) and (35), translated at the triangular Kronecker lattice points $\bm{x}_{5,512}$, $\bm{x}_{6,2048}$ and $\bm{x}_{6,2048}$. The total number of low-pass framelets $\bm{\varphi}_{j,k}$ at level $j=5$ is $N_{5}=1025$ and the total number of high-pass framelets $\bm{\psi}_{j,k^{\prime}}^{n}$, $n=1$ or $2$, at level $j=5$ is $N_{6}=4097$. The pictures show that the framelets $\bm{\varphi}_{5,512}$, $\bm{\psi}_{5,2048}^{1}$ and $\bm{\psi}_{5,2048}^{2}$ are radial functions on $T^{2}$ with centers at the translation points $\bm{x}_{5,512}$, $\bm{x}_{6,2048}$ and $\bm{x}_{6,2048}$ respectively.

We observe that the high-pass framelets $\bm{\psi}_{5,2048}^{1}$ and $\bm{\psi}_{5,2048}^{2}$ are highly concentrated at the translation point $\bm{x}_{6,2048}$, and are more localised than the low-pass framelet at the same level. This illustrates that the high-pass framelets can be used to depict details of a function on $T^{2}$ in multiresolution analysis.

6 Fast Computing

The framelet transforms on $T^{2}$ can be represented by discrete Fourier transforms on the simplex. This implies a fast computational strategy of the decomposition and reconstruction for framelets.

We use the notation of Sections 3 and 4. Let $j\in\mathbb{N}_{0}$ and let $\Lambda_{j}$ be the largest integer $\ell$ such that $\lambda_{\ell}\leq 2^{j-1}$. The discrete Fourier transform (DFT) for a sequence $\mathrm{u}\in l(\Lambda_{j})$ is the sequence $\mathbf{F}_{j}\mathrm{u}$ in $l(N_{j})$ such that

[TABLE]

The adjoint discrete Fourier transform (adjoint DFT) $\mathbf{F}_{j}^{*}$ of a sequence $\mathrm{v}\in l(N_{j})$ is the sequence $\mathbf{F}_{j}^{*}\mathrm{v}$ in $l(\Lambda_{j})$ such that

[TABLE]

The DFTs on the simplex in (38) and (39) using the orthonormal basis $P_{\ell}$ are the analogues of DFTs for square-integrable periodic functions on $\mathbb{R}$ which use the orthogonal basis $e^{2\pi{\mathrm{i}}\ell^{\prime}x}$, $\ell^{\prime}\in\mathbb{Z}$.

By (38) and (39), we can rewrite the decomposition in (8) and reconstruction in (11) as

[TABLE]

and

[TABLE]

This means that the decomposition from level $j$ to level $j-1$ is the DFTs of convolutions of the level-$j$ framelet coefficients with masks, and that the reconstruction from level $j-1$ to level $j$ is the sum of convolutions of the adjoint DFTs of level-$(j-1)$ coefficients with masks. Since the convolution is the sum of point-wise multiplications, the computational steps of the framelet transforms are in proportion to those of DFTs on the simplex.

The FFT on $T^{2}$ uses, up to log factors, $\mathcal{O}(N)$ operations for an input sequence of size $N$. If for $j\geq 1$, the ratio $N_{j}/N_{j-1}$ of the numbers of the nodes of the quadrature rules ${Q}_{N_{j}}$ and ${Q}_{N_{j-1}}$ is equivalent to a constant $C$, $C>1$, the computational steps of the framelet transforms (both the decomposition and reconstruction) between levels $0,1,\dots,{J}$, ${J}\geq 1$, are $\mathcal{O}((r+1)N_{J})$ for the sequence $\mathrm{v}_{J}$ of the framelet coefficients of size $N_{{J}}$, and the redundancy rate of the framelet transforms is also $\mathcal{O}((r+1)N_{J})$. The framelets with the quadrature rules using triangular Kronecker lattices, as shown in Section 5, satisfy that $N_{j}/N_{j-1}\sim 4$. Thus, the framelet transforms between levels [math] to $J$ have computational steps in proportion to $2^{2J}$.

7 Discussion

In the paper, we only consider the framelet transforms for one framelet system with starting level [math]. The results can be generalized to a sequence of framelet systems as WaZh2017 ; Han2010 , which will allow one more flexibility in constructing framelets.

The decomposition holds for framelets with any quadrature rules on the simplex. In order to achieve the tightness of the framelets and thus exact reconstruction for functions on the simplex by framelets, the quadrature rules are required to be exact for polynomials, see Sections 3 and 4. However, polynomial-exact rules are generally difficult to construct on the simplex, see (DuXu2014, , Chapter 3).

Triangular Kronecker lattices with equal weights used in Section 5 are low-discrepancy BaOw2015 , but not exact for polynomials. In this case, the reconstruction will incur errors. To overcome this, the masks and quadrature rules shall be constructed to satisfy the condition

[TABLE]

for $j\geq 1$ and for $\ell,\ell^{\prime}\in{\mathbb{N}}_{0}$ satisfying $\overline{\widehat{\alpha}\left(\frac{\lambda_{\ell}}{2^{j}}\right)}{\widehat{\alpha}}\left(\frac{\lambda_{\ell^{\prime}}}{2^{j}}\right)\neq 0$, where $\mathcal{U}_{\ell,\ell^{\prime}}({Q}_{N_{j}}):=\sum_{k=0}^{N_{j}}\omega_{j,k}P_{\ell}(\bm{x}_{j,k})\overline{P_{\ell^{\prime}}(\bm{x}_{j,k})}$ is the numerical integration of $P_{\ell}\overline{P_{\ell^{\prime}}}$ over $T^{2}$ by quadrature rule ${Q}_{N_{j}}$, see (WaZh2017, , Theorem 2.4). This condition requires that the quadrature rules for framelets have good properties for numerical integration over the simplex. Besides the triangular Kronecker lattices used in the paper, one may consider other quadrature rules with low discrepancy on the simplex, for example, the analogues to quasi-Monte Carlo (QMC) points in the cube and spheres, see BrSaSlWo2014 ; DiKuSl2013 ; SlJo1994 .

To implement the fast algorithms for the DFTs in (38) and (39), we need fast transforms for the bases $P_{\ell}$. For example, we can represent the bases $P_{\ell,m}$ in (12) by trigonometric polynomials and apply the FFT on $\mathbb{R}$ to achieve fast algorithms for the DFTs on $T^{2}$.

Acknowledgements.

The authors thank the anonymous referees for their valuable comments. We are grateful to Kinjal Basu, Yuan Xu and Xiaosheng Zhuang for their helpful discussions.