分类：小波变换

Filters method is an efficient way to implement Discrete Wavelet Transform 1. What is a Filter Bank Filter banks play an important role in modern signal and image processing. As its name suggests, a Filter Bank is a bank of filters, meaning a collection of filters, or many filters in simple words, and each of the filter is applied to the original signal. So, one signals can be decomposed into 2 or more signals out. Then what is a Filter bank? Simply speaking, a Filter Bank can be defined as a system (an array ) of bandpass filters that split the input signal (St) into a set of analysis signals (x1t, x2t, …, xnt ) and each one carrying a single frequency sub-band of the original signal. Fig. 1. Structure of a Filter Bank In wavelet analysis, filter banks usually refers to a class of systems that generates scaling and wavelet function. The below figure shows the simple structure of a filter bank can be illustrated by the figure below. In practice, an efficient way to implement Discrete Wavelet Transform (DWT) is to use filters method, which was developed in 1988 by Mallat. This very practical filtering algorithm yields a Fast Discrete Wavelet Transform (FWT). 2. Wavelet filters coefficients Wavelet filters coefficients are the most important information used in Discrete Wavelet Transforms (DWT). In the PyWavelets library of Python, filters coefficients can be obtained via the dec_lo, dec_hi, rec_lo and rec_hi attributes, which correspond to lowpass and highpass decomposition filters and lowpass and highpass reconstruction filters respectively. In a previous article, we have already learned what PyWavelets is, how to install it, and how to display its built-in Wavelets families and their members in PyWavelets, and so on. In the PyWavelets, the filter_bankmethod returns a list for the current wavelet in the following order: [dec_lo, dec_hi, rec_lo, rec_hi]. dec_lo: Decomposition low-pass filterdec_hi: Decomposition high-pass filterrec_lo: Reconstruction low-pass filterrec_hi: Reconstruction high-pass filter 3. Print Filter Bank of a Discrete Wavelet First, let’s import the required packages: one is PyWavelets and another is matplotlib for visualization. # import required packages import pywt import matplotlib.pyplot as plt Next, we define a discrete wavelt, for example, “db5” , and then display its filter bank as follows: wavelet = pywt.Wavelet('db5') wavelet.filter_bank The output is as follows: ([0.0033357252854737712, -0.012580751999081999, -0.006241490212798274, 0.07757149384004572, -0.032244869584638375, -0.24229488706638203, 0.13842814590132074, 0.7243085284377729, 0.6038292697971896, 0.16010239797419293], [-0.16010239797419293, 0.6038292697971896, -0.7243085284377729, 0.13842814590132074, 0.24229488706638203, -0.032244869584638375, -0.07757149384004572, -0.006241490212798274, 0.012580751999081999, 0.0033357252854737712], [0.16010239797419293, 0.6038292697971896, 0.7243085284377729, 0.13842814590132074, -0.24229488706638203, -0.032244869584638375, 0.07757149384004572, -0.006241490212798274, -0.012580751999081999, 0.0033357252854737712], [0.0033357252854737712, 0.012580751999081999, -0.006241490212798274, -0.07757149384004572, -0.032244869584638375, 0.24229488706638203, 0.13842814590132074, -0.7243085284377729, 0.6038292697971896, -0.16010239797419293]) We can just print one filter, for example, the decomposition low-pass filter using the following code: wavelet.dec_lo We get the following results: [0.0033357252854737712, -0.012580751999081999, -0.006241490212798274, 0.07757149384004572, -0.032244869584638375, -0.24229488706638203, 0.13842814590132074, 0.7243085284377729, 0.6038292697971896, 0.16010239797419293] 5. Visualization of Wavelet Bank First, let’s obtain the list of the four filters using the following methods: [dec_lo, dec_hi, rec_lo, rec_hi] = wavelet.filter_bank Then, we can use matplotlib to make a stem plot for ech filter. fig, ((ax1, ax2), (ax3, ax4)) = plt.subplots(2, 2, figsize=(15,7)) ax1.stem(dec_lo) ax1.set_title('Decomposition low-pass filter') ax2.stem(dec_hi) ax2.set_title('Decomposition high-pass filter') ax3.stem(rec_lo) ax3.set_title('Reconstruction low-pass filter') ax4.stem(rec_hi) ax4.set_title('Reconstruction high-pass filter') plt.savefig('./results/filter_bank.png') # save it in any path, here I save it into the results folder in the working directory plt.show() The output looks as follows: Fig. 2. Plots of Filter Banks of db5 6. Online Course If you are interested in learning Practical Wavelet Transforms from very beginning, welcome to the courses via the following linage: Practical Python Wavelet Transforms (I): Fundamentals Practical Python Wavelet Transforms (II): 1D DWT

分类：小波变换

使用Python绘制离散小波滤波器组的图像

Python中绘制小波和尺度函数的图像

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