Wavelets and radial basis functions: a unifying perspective
Wavelets and radial basis functions: a unifying perspective
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小波和径向基函数：统一的视角
DOI：
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发表时间：
2000
期刊：
影响因子：
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通讯作者：
T. Blu
中科院分区：
文献类型：
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作者：
M. Unser;T. Blu
小波和径向基函数是用移位基函数表示信号的两种截然不同的方法。RBF的一个重要方面，使该方法适用于非均匀网格，是基函数，不像小波，是非局部-此外，它们不涉及任何缩放。尽管存在这些根本差异，但我们表明这两种类型的表示是密切相关的。我们用线性样条作为激励的例子。这些可以通过使用单侧斜坡函数的平移来构造，或者，更传统地，通过使用线性b样条的移位来构造。后一个函数是缩放函数的典型例子，可以通过使用有限差分对单侧斜坡函数进行局部化得到。然后，我们推广了这一概念，并确定了整类自相似径向基函数，这些函数可以局部化以产生常规的多分辨率小波基。相反，我们证明了对于任何紧支持的尺度函数，存在一个跨相同多分辨率子空间的单侧中心基函数。其核心特性是多分辨率碱基是通过简单的平移而产生的，没有任何膨胀。
Wavelets and radial basis functions (RBF) are two rather distinct ways of representing signals in terms of shifted basis functions. An essential aspect of RBF, which makes the method applicable to non-uniform grids, is that the basis functions, unlike wavelets, are non-local-in addition, they do not involve any scaling at all. Despite these fundamental differences, we show that the two types of representation are closely connected. We use the linear splines as motivating example. These can be constructed by using translates of the one-side ramp function, or, more conventionally, by using the shifts of a linear B-spline. This latter function, which is the prototypical example of a scaling function, can be obtained by localizing the one-side ramp function using finite differences. We then generalize the concept and identify the whole class of self-similar radial basis functions that can be localized to yield conventional multiresolution wavelet bases. Conversely, we prove that, for any compactly supported scaling function, there exist a one-sided central basis function that spans the same multiresolution subspaces. The central property is that the multiresolution bases are generated by simple translation without any dilation.