Biorthogonal Multiwavelets with Sampling Property and Application in Image Compression

Abstract

This paper discusses biorthogonal multiwavelets with sampling property. In such systems, vector-valued refinable functions act as the sinc function in the Shannon sampling theorem, and their corresponding matrix-valued masks possess a special structure. In particular, for the multiplicity $r=2$, a biorthogonal multifilter bank can be reduced to two scalar-valued filters. Moreover, if the vector-valued scaling functions are interpolating, three different concepts: balancing order, approximation order and analysis-ready order, will be shown to be equivalent. Based on this result, we introduce the transferring armlet order for constructing biorthogonal balanced multiwavelets with sampling property. Also, some balanced biorthogonal multiwavelets will be obtained. Finally, application of biorthogonal interpolating multiwavelets in image compression is discussed. Experiments show that for the same length, the biorthogonal multifilter bank is superior to the orthogonal case. Moreover, certain biorthogonal interpolating multiwavelets are also better than the classical Daubechies wavelets.

Appendices

Appendix 1: Proofs of Theorems

1.1 Proof of Theorem 1

Proof

First, for biorthogonal multifilter banks, perfect reconstruction conditions (1)–(4) are equal to $A\cdot B^{*}=-C\cdot B^{*}=-A\cdot D^{*}=C\cdot D^{*}=2I_2.$ The equation $A\cdot B^{*}=2I_2$ can be written as

$$\begin{aligned} \begin{array}{ll} a(z)a_1(z^{-1})+a(-z)a_1(-z^{-1})=2,\quad a(z)b_1(z^{-1})+a(-z)b_1(-z^{-1})=0,\\ b(z)a_1(z^{-1})+b(-z)a_1(-z^{-1})=0,\quad b(z)b_1(z^{-1})+b(-z)b_1(-z^{-1})=2,\\ \end{array} \end{aligned}$$

which tells us that $b_1(z)=sz^{2m+1}a(-z^{-1}),\ a_1(z)=-sz^{2m+1}b(-z^{-1})$ and $b(z)a(-z)-b(-z)a(z)=2z^{2m+1}s^{-1}$.

By $A\cdot B^{*}=-C\cdot B^{*}=-A\cdot D^{*}=C\cdot D^{*}$, and above equations, it is easy to get $c(z)=-a(z),\ d(z)=-b(z),\ c_1(z)=-a_1(z),\ d_1(z)=-b_1(z)$. Thus, the proof of Theorem 1 is completed. $\square$

1.2 Proof of Theorem 3

Proof

We only need to prove the sufficiency because if one multiwavelet is balanced of order n, it must be an armlet of the same order. So, in the following, we suppose $\varPsi$ and $\widetilde{\varPsi }$ to be armlets of order n.

Let $\{V_{j}\}_{j\in \mathbb {Z}}$ and $\{\widetilde{V}_{j}\}_{j\in \mathbb {Z}}$ be the corresponding MRAs. If $\varPhi$ is interpolating, for any signal f(x) in $V_{N}$, it has the decomposition $f(x)=\sum _{n}c^{T}_{N,n}\varPhi (2^{N}x-n),$ where $c_{N,n}=\left[ c^{1}_{N,n}=f(\frac{n}{2^N}),c^{2}_{N,n}=f(\frac{n}{2^{N}}+\frac{1}{2^{N+1}})\right] ^T$. Proceed to the decomposition:

$$\begin{aligned} c_{N-1,n}=\sum _{k\in \mathbb {Z}}h_{k-2n}c_{N,k},\quad \mathrm{{and}} \quad d_{N-1,n}=\sum _{k\in \mathbb {Z}}g_{k-2n}c_{N,k}. \end{aligned}$$

For the synthesis, we have $c_{N,n}=\sum _{k\in \mathbb {Z}}\left\{ \widetilde{h}^{T}_{n-2k}c_{N-1,k}+\widetilde{g}^{T}_{n-2k}d_{N-1,k}\right\} .$ Then, if we take f(x) as a polynomial of order less than n, by the definition of an armlet, it is easy to obtain that $d_{N-1,k}=0$ in the above equation. And we have

$$\begin{aligned} c_{N,n}=\sum _{k\in \mathbb {Z}}\widetilde{h}^{T}_{n-2k}c_{N-1,k}, \end{aligned}$$

where $c_{N,n}$ and $c_{N-1,k}$ are sampling values of the polynomial f at the level N and $N-1$, respectively. Thus, by the definition of balanced multiwavelets, $\widetilde{\varPsi }$ is a balanced multiwavelet of order n.

By a similar discussion of $\varPsi$, we can show that $\varPsi$ is also a balanced multiwavelet of order n. Hence, the proof of this theorem is completed. $\square$

Appendix 2: The Orthogonal Multifilter Bank $\{H,G\}$ in [10]

$$\begin{aligned} \begin{array}{llll} H=[h_{-3}, h_{-2}, h_{-1}, h_0, h_1, h_2, h_3, h_4]= &{} G=[g_{-3}, g_{-2}, g_{-1}, g_0, g_1, g_2, g_3, g_4]=\\ \begin{array}{lll} h_{-3}\!=\!\left( \begin{array}{ll} 0 &{} 0.0022908\\ 0 &{} 0.0001688 \end{array}\right) &{} h_{-2}\!=\!\left( \begin{array}{ll} 0 &{} 0.0310811\\ 0 &{} 0.0022908 \end{array}\right) \\ h_{-1}\!=\!\left( \begin{array}{ll} 0 &{} 0.2431275\\ 0 &{} 0.0307434 \end{array}\right) &{} h_{0}\ \ \!=\!\left( \begin{array}{ll} 1 &{} 0.9380065\\ 0 &{} 0.2431275 \end{array}\right) \\ h_{1}\ \ \!=\!\left( \begin{array}{ll} 0 &{} -0.243127\\ 1 &{} 0.9380065 \end{array}\right) &{} h_{2}\ \ \!=\!\left( \begin{array}{ll} 0 &{} 0.0307434\\ 0 &{} -0.243127 \end{array}\right) \\ h_{3}\ \ \!=\!\left( \begin{array}{ll} 0 &{} -0.002290\\ 0 &{} 0.0310811 \end{array}\right) &{} h_{4}\ \ \!=\!\left( \begin{array}{ll} 0 &{} 0.0001688\\ 0 &{} -0.002290 \end{array}\right) \\ \end{array} &{} \begin{array}{lll} g_{-3}\!=\!\left( \begin{array}{ll} 0 &{} -0.002290\\ 0 &{} -0.000168 \end{array}\right) &{} g_{-2}\!=\!\left( \begin{array}{ll} 0 &{} -0.031081\\ 0 &{} -0.002290 \end{array}\right) \\ g_{-1}\!=\!\left( \begin{array}{ll} 0 &{} -0.243127\\ 0 &{} -0.030743 \end{array}\right) &{} g_{0}\ \ \!=\!\left( \begin{array}{ll} 1 &{} -0.938006\\ 0 &{} -0.243127 \end{array}\right) \\ g_{1}\ \ \!=\!\left( \begin{array}{ll} 0 &{} 0.2431275\\ 1 &{} -0.938006 \end{array}\right) &{} g_{2}\ \ \!=\!\left( \begin{array}{ll} 0 &{} -0.030743\\ 0 &{} 0.2431275 \end{array}\right) \\ g_{3}\ \ \!=\!\left( \begin{array}{ll} 0 &{} 0.0022908\\ 0 &{} -0.031081 \end{array}\right) &{} g_{4}\ \ \!=\!\left( \begin{array}{ll} 0 &{} -0.000168\\ 0 &{} 0.0022908 \end{array}\right) \\ \end{array} \end{array} \end{aligned}$$