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3.1.离散子波变换在实际应用中,象信号和图象处理,我们都会碰到离散信号。为了用子波变换分析这类信号,我们需要采用连续-时间子波变换的离散形式。方程(9)式给出的形式不适合这一用途(定义中我们仍采用了连续函数)。对于离散信号,也不可能确定进行多分辨率分析。我们不能找到一个离散的基函数,该函数经缩放和平移变化的形式构成子空间l~2(R)的一组基函数,这里,l~2(R)是平方求和无限长度序列的空间。从多分辨率分析,可以推出离散-时间子波级数的公式。基本前提是,从前一缩放因子j-1就能够以迭代方式计算系数c_(j,k)和d_(j,k),无需采用函数φ(x)和ψ(x)的显式。显然,我们有
3.1. Discrete Wavelet Transform In practical applications, like signal and image processing, we all encounter discrete signals. In order to analyze such signals with wavelet transform, we need to use the discrete form of the continuous-time wavelet transform. The form given by Equation (9) is not suitable for this purpose (we still use continuous functions in the definition). For discrete signals, it is also impossible to determine whether to perform multiresolution analysis. We can not find a discrete basis function which forms a set of basis functions of subspaces l ~ 2 (R) in the form of scaling and translational changes, where l ~ 2 (R) is the space of the square summation infinite length sequence . From multi-resolution analysis, a formula for discrete-time wavelet series can be derived. The basic premise is that the coefficients c_ (j, k) and d_ (j, k) can be iteratively calculated from the previous scaling factor j-1 without the explicit use of the functions φ (x) and ψ (x). Obviously, we have