## 2 The Theory of Multiwavelets and the Symmetric Lifting Schemes
### Multiwavelet Lifting Scheme  
Lifting scheme [23, 24] is a powerful design and implementation technique for conventional wavelets. It provides a great deal of flexibility and freedom for the constructions of biorthogonal wavelets and it is usually used to construct customized wavelets by the design of prediction operators and update operators. Vanishing moment is an important property of multiwavelet functions, which plays a crucial role in the characterization of local Holder exponent of singularities. In theory, a higher vanishing moment means the scaling functions can represent more complex signals accurately. In this paper, lifting scheme was used to improve the existing multiwavelets such that the obtained multiwavelets had enough vanishing moments to describe the singularity of the measured signals. Figure 2: _Cont._  
**Theorem 2.1**[26] Let the original filter banks of multiwavelets be \(\left\{\mathbf{H}(z),\mathbf{G}(z),\tilde{\mathbf{H}}(z),\tilde{\mathbf{G}}(z)\right\},\) new multiwavelet filter banks \(\left\{\mathbf{H}_{new}(z),\mathbf{G}_{new}(z),\tilde{\mathbf{H}}_{new}(z),\tilde{\mathbf{G}}_{ new}(z)\right\}\) are as follows:  
\[ \begin{split}&\mathbf{H}_{new}(z)=\mathbf{H}(z)\\ &\mathbf{G}_{new}(z)=\mathbf{T}(z^{2})\Big{(}\mathbf{G}(z)+\mathbf{S}(z^{2})\mathbf{ H}(z)\Big{)}\\ &\tilde{\mathbf{H}}_{new}(z)=\tilde{\mathbf{H}}(z)-\mathbf{S}^{*}(z^{2}) \tilde{\mathbf{G}}(z)\\ &\tilde{\mathbf{G}}_{new}(z)=(\mathbf{T}^{*}(z^{2}))^{-1}\tilde{\mathbf{G}}( z)\end{split}  \tag{8}\]  
where the lifting matrices \(\mathbf{S}(z)\) and \(\mathbf{T}(z)\) are finite orders and the determinant of \(\mathbf{T}(z)\) is a monomial. Multiwavelet lifting scheme is similar to scalar lifting scheme except that the filter banks of multiwavelets are matrices, and \(\mathbf{G}_{new}(z)\) contains an extra lifting matrix \(\mathbf{T}(z^{2})\). According to the two-scale relations and Equation (8), the following formula is obtained:  
\[ \begin{split}&\mathbf{\Phi}_{new}(z)=\mathbf{\Phi}(z)\\ &\mathbf{\Psi}_{new}(z)=\mathbf{T}(z^{2})\Big{(}\mathbf{\Psi}(z)+\mathbf{S}(z^{2} )\mathbf{\Phi}(z)\Big{)}\\ &\tilde{\mathbf{\Phi}}_{new}(z)=\tilde{\mathbf{\Phi}}(z)-\mathbf{S}^{*}(z^{2} )\tilde{\mathbf{\Psi}}(z)\\ &\tilde{\mathbf{\Psi}}_{new}(z)=\left(\mathbf{T}^{*}(z^{2})\right)^{-1} \tilde{\mathbf{\Psi}}(z)\end{split}  \tag{9}\]  
Lifting scheme can be used to improve the existing multiwavelets. The obtained multiwavelets are considered ideal since they have a desirable vanishing moment. The key of this method is the design of the lifting matrices \(\mathbf{S}(z)\) and \(\mathbf{T}(z)\)[25]. **Definition 2.2** The \(k\)-th continuous moments of \(\mathbf{\Phi}\), \(\mathbf{\psi}\) are defined as \(M(\mathbf{\Phi},n)=\left\lceil\mathbf{\Phi}(x)x^{*}dx\right.\) and \(M(\mathbf{\Psi},n)=\left\lceil\mathbf{\Psi}(x)x^{*}dx\right.\). Then we have:  
\[ \begin{split}& M(\mathbf{\Phi},n)=\sum_{j=0}^{n}\sum_{k\in Z}\mathbf{H}_{k }\binom{n}{j}k^{j}M(\mathbf{\Phi},n-j)\\ & M(\mathbf{\Psi},n)=\sum_{j=0}^{n}\sum_{k\in Z}\mathbf{G}_{k}\binom{n}{ j}k^{j}M(\mathbf{\Phi},n-j)\end{split}  \tag{10}\]  
Hence, the construction of a wavelet with specified vanishing moments is now a straightforward procedure. An initial wavelet \(\omega_{0}(x)\) (\(\omega_{0}(x)=\psi_{1}\) or \(\psi_{2}\)) is chosen and a set of \(k\) translates of scaling functions and wavelet functions \(\omega_{1}(x),...\omega_{k}(x)\) that will be used to modify the function \(\omega_{0}(x)\) via a lifting step are selected. Then a \"lifting coefficient equation\" is obtained:  
\[\omega_{0}^{new}=\omega_{0}(x)+\sum_{i=1}^{k}c_{i}\omega_{i}(x) \tag{11}\]  
One of the most attractive advantages in multiwavelet lifting is that the functions using to construct new multiwavelets are more than scalar wavelets. When a scalar wavelet is lifted, \(\omega_{i}\) is only \(\mathcal{P}\).