小波变换与傅里叶变换
Wavelet transforms versus Fourier transforms
AI总结：
本文基础介绍小波变换，从分片常数基出发，构造高阶小波，并与傅里叶变换在信号处理（如视频压缩）中比较，指出小波在指纹识别中领先。
摘要：
本文是对小波的一个非常基础的介绍。文章从一组由伸缩和平移构造的分片常数函数的正交基开始。"小波变换"将每个$f(x)$映射到其相对于该基的系数。其数学简单，变换速度快（比快速傅里叶变换更快，我们对此进行了简要解释），但用分片常数逼近的效果较差。为了改进这第一个小波，我们引入了伸缩方程及其不寻常的解。我们构造了高阶小波，并且令人惊讶的是，用它们进行计算非常快速——总是间接且递归地进行。我们非正式地讨论了这些变换在信号处理中的竞争，特别是在视频和图像压缩（包括高清电视）中的应用。到目前为止，傅里叶变换——或其8乘8分块版本，即离散余弦变换——常常被选用。但小波已经具有竞争力，并且在指纹识别方面处于领先地位。我们展示了这一发展理论的一个样本。
英文摘要：
This note is a very basic introduction to wavelets. It starts with an orthogonal basis of piecewise constant functions, constructed by dilation and translation. The ``wavelet transform'' maps each $f(x)$ to its coefficients with respect to this basis. The mathematics is simple and the transform is fast (faster than the Fast Fourier Transform, which we briefly explain), but approximation by piecewise constants is poor. To improve this first wavelet, we are led to dilation equations and their unusual solutions. Higher-order wavelets are constructed, and it is surprisingly quick to compute with them --- always indirectly and recursively. We comment informally on the contest between these transforms in signal processing, especially for video and image compression (including high-definition television). So far the Fourier Transform --- or its 8 by 8 windowed version, the Discrete Cosine Transform --- is often chosen. But wavelets are already competitive, and they are ahead for fingerprints. We present a sample of this developing theory.
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