Factoring wavelet transforms into lifting steps
Factoring wavelet transforms into lifting steps
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DOI：
10.1007/bf02476026
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发表时间：
1998-01-01
影响因子：
1.2
通讯作者：
Sweldens, W
中科院分区：
文献类型：
--
作者：
Daubechies, I;Sweldens, W
这篇文章本质上是教程。我们展示了如何任何离散小波变换或两个频带的子带滤波与有限滤波器可以分解成一个有限序列的简单的滤波步骤，我们称之为提升步骤，但也被称为梯形结构。这种分解对应于小波或子带滤波器的多相矩阵到基本矩阵的因式分解。这种分解是可能的，代数学家们都知道，并用公式SL（n; R[z，z（-1）]）= E（n; R[z，z（-1）]））表示;它也用于电气工程界的线性系统理论。我们在这里提出了一个自包含的推导，从基本原则，如欧几里德算法的分解，重点是将其应用到小波滤波。这种因式分解为格因式分解提供了一种替代方案，其优点是它也可以用于双正交，即，非单一案例与格分解类似，这里提出的分解将变换的计算复杂度渐近地降低了两倍。红外线还有其他的应用，例如定义一个小波变换的可能性，将整数映射到整数。
This article is essentially tutorial in nature. We show how any discrete wavelet transform or two band subband filtering with finite filters can be decomposed into a finite sequence of simple filtering steps, which we call lifting steps but that are also known as ladder structures. This decomposition corresponds to a factorization of the polyphase matrix of the wavelet or subband filters into elementary matrices. That such a factorization is possible is well-known to algebraists land expressed by the formula SL(n; R[z, z(-1)]) = E(n; R[z, z(-1)])); it is also used in linear systems theory in the electrical engineering community. We present here a self-contained derivation, building the decomposition from basic principles such as the Euclidean algorithm, with a focus on applying it to wavelet filtering. This factorization provides an alternative for the lattice factorization, with the advantage that it can also be used in the biorthogonal, i.e., non-unitary case. Like the lattice factorization, the decomposition presented here asymptotically reduces the computational complexity of the transform by a factor two. Ir has other applications, such as the possibility of defining a wavelet-like transform that maps integers to integers.