什么是小波|理解小波,第1部分
从系列:理解小波
这篇介绍性视频涵盖了小波是什么以及如何使用它们来探索你的MATLAB中的数据®。视频集中在两个重要的小波变换概念:缩放和转移。的概念可以应用于2 d图像等数据。
你好,每个人。在这篇介绍性的会话,我将介绍一些基本的小波的概念。我将主要使用一个一维的例子,但相同的概念可以应用于图像。首先,让我们来回顾一下什么是小波。现实世界数据或信号经常表现出缓慢变化趋势或振荡伴有瞬变。另一方面,图像平滑区域打断了边缘或突然的变化相反。这些突然的变化往往是最iA =兴趣的部分数据,感知和他们提供的信息。傅里叶变换是一个数据分析的有力工具。然而,这并不代表突然改变效率。原因是傅里叶变换将数据表示为正弦波,不是局部的时间或空间。 These sine waves oscillate forever. Therefore, to accurately analyze signals and images that have abrupt changes, we need to use a new class of functions that are well localized in time and frequency: This brings us to the topic of Wavelets. A wavelet is a rapidly decaying, wave-like oscillation that has zero mean. Unlike sinusoids, which extend to infinity, a wavelet exists for a finite duration. Wavelets come in different sizes and shapes. Here are some of the well-known ones. The availability of a wide range of wavelets is a key strength of wavelet analysis. To choose the right wavelet, you'll need to consider the application you'll use it for. We will discuss this in more detail in a subsequent session. For now, let's focus on two important wavelet transform concepts: scaling and shifting. Let' start with scaling. Say you have a signal PSI(t). Scaling refers to the process of stretching or shrinking the signal in time, which can be expressed using this equation [on screen]. S is the scaling factor, which is a positive value and corresponds to how much a signal is scaled in time. The scale factor is inversely proportional to frequency. For example, scaling a sine wave by 2 results in reducing its original frequency by half or by an octave. For a wavelet, there is a reciprocal relationship between scale and frequency with a constant of proportionality. This constant of proportionality is called the "center frequency" of the wavelet. This is because, unlike the sinewave, the wavelet has a band pass characteristic in the frequency domain. Mathematically, the equivalent frequency is defined using this equation [on screen], where Cf is center frequency of the wavelet, s is the wavelet scale, and delta t is the sampling interval. Therefore when you scale a wavelet by a factor of 2, it results in reducing the equivalent frequency by an octave. For instance, here is how a sym4 wavelet with center frequency 0.71 Hz corresponds to a sine wave of same frequency. A larger scale factor results in a stretched wavelet, which corresponds to a lower frequency. A smaller scale factor results in a shrunken wavelet, which corresponds to a high frequency. A stretched wavelet helps in capturing the slowly varying changes in a signal while a compressed wavelet helps in capturing abrupt changes. You can construct different scales that inversely correspond the equivalent frequencies, as mentioned earlier. Next, we'll discuss shifting. Shifting a wavelet simply means delaying or advancing the onset of the wavelet along the length of the signal. A shifted wavelet represented using this notation [on screen] means that the wavelet is shifted and centered at k. We need to shift the wavelet to align with the feature we are looking for in a signal.The two major transforms in wavelet analysis are Continuous and Discrete Wavelet Transforms. These transforms differ based on how the wavelets are scaled and shifted. More on this in the next session. But for now, you've got the basic concepts behind wavelets.
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